Comparison between linear and nonlinear vibration...

Willem Coudron

absorbers for seismic activityComparison between linear and nonlinear vibration

Academic year 2015-2016Faculty of Engineering and ArchitectureChair: Prof. dr. ir. Jan MelkebeekDepartment of Electrical Energy, Systems and Automation

Master of Science in Civil EngineeringMaster's dissertation submitted in order to obtain the academic degree of

Counsellor: Kevin DekemeleSupervisor: Prof. dr. ir. Mia Loccufier

Permission for use of content

The author gives permission to make this master dissertation available for consultation and

to copy parts of this master dissertation for personal use. In the case of any other use, the

copyright terms have to be respected, in particular with regard to the obligation to state

expressly the source when quoting results from this master dissertation.

June 15, 2016

Acknowledgements

I would like to thank my promotor prof. dr. ir. Mia Loccufier for the assistance during the

academic year and the provision of great research material for this dissertation. I also want to

thank assistants Kevin Dekemele and Bram Vervisch for their tips and help in tackling many

problems on the way.

Abstract

Comparison between linear and nonlinear vibration absorbers

for seismic activity

Willem Coudron

Supervisor: Prof. dr. ir. Mia Loccufier

Counsellor: Kevin Dekemele

Master's dissertation submitted in order to obtain the academic degree of

Master of Science in Civil Engineering

Department of Electrical Energy, Systems and Automation

Chair: Prof. dr. ir. Jan Melkebeek

Faculty of Engineering and Architecture

Academic year 2015-2016

Summary

This dissertation considers the implementation of a linear and nonlinear vibration absorber

on a building model to reduce the mechanical vibrations under dynamic loading. A multi-story

building model was designed in aluminum and placed on a horizontal shaker. This shaker is

able to perform predefined displacement signals to load the building model via base

excitation. Two excitation types are considered: harmonic excitation and impulse excitation.

The harmonic excitation was applied by moving the shaker with a multisine displacement

signal and the impulse excitation was applied by giving the shaker a step displacement. Both

a linear and nonlinear vibration absorber were designed and tuned to these excitations to

reduce the vibrations of the building model. The performance of both absorbers is evaluated

by comparing the response of the modified building model to the response of the original

building model.

Keywords

Multi-story building model, vibration absorber, linear, nonlinear, NES, harmonic excitation,

impulse excitation

Comparison between linear and nonlinear vibration absorbers for seismic activity

Willem Coudron

Supervisor: prof. dr. ir. Mia Loccufier

Abstract - This dissertation considers the design, implementation and performance assessment of a linear and nonlinear vibration absorber. A three-story building model was designed and placed on an existing shaker to serve as main system for the absorber implementation. The shaker was used to load the building model with a harmonic and impulse excitation. Both absorber types were tuned to these excitations and installed to reduce the vibrations of the building model. The performance of both absorbers was evaluated by looking at the reduction of the interstory drifts and floor accelerations in case of harmonic excitation, and the reduction of the settling time and maximum displacement in case of impulse excitation.

Keywords - Multi-story building model, vibration absorber, linear, nonlinear, NES, harmonic excitation, impulse excitation

I. INTRODUCTION

Mechanical vibrations are a common disturbance occurring in buildings and civil structures. They are excited by dynamic loads (e.g. earthquakes, wind, etc.) and range from some minor oscillations to very large and destructive vibrations depending on the load intensity. Earthquakes in particular are well known for their destructive power. Buildings in seismic zones are therefore constructed according to specific design rules and often special devices are installed to reduce the mechanical vibrations. A vibration absorber is a very effective device to control mechanical vibrations.

Both a linear and nonlinear absorber are considered in this dissertation. Their difference is situated in the resilient member. The spring of a linear absorber obeys Hooke's law and shows a linear relation between the force in the spring and the displacement of the absorber mass. The spring of a nonlinear absorber, on the other hand, results in a nonlinear relation between displacement and force. This leads to the disappearance of a preferential frequency. Unlike the linear absorber, the nonlinear absorber does not oscillate at one natural frequency, but it is able to oscillate at a range of frequencies depending on the energy level in the nonlinear absorber. Another, less convenient, consequence is the existence of an energy threshold. Efficient vibration reduction of the nonlinear absorber is only possible when the input energy of the excitation exceeds the energy threshold of the absorber.

Both absorber types are designed and they are implemented for harmonic and impulse excitation on a three-story building model. The excitation is introduced at the base via horizontal movements of the shaker on

W. Coudron. E-mail: [email protected] .

which the building model is mounted. The mass of the building is concentrated on the floor elements, which will attract inertial forces under base excitation. The concerning vibrations are thus horizontal vibrations of the floor elements. These are reduced due to the action of the absorbers. The performance of both absorbers is evaluated by comparing the response of the modified building to the original one with several criteria. The interstory drifts and floor accelerations are considered for the harmonic excitation while the 10 % settling time and maximum floor displacements are checked for the impulse excitation.

II. VIBRATION ABSORBER DESIGN

A. Linear vibration absorber

The linear absorber is designed as a translational spring-mass-damper system that is attached to the desired floor element of the building (figure 1). A four wheel cart is used for the oscillating mass that is connected to the building with tension springs at both sides. There is some unwanted damping due to the slippage of the wheels, but the main damping is installed as magnetic damping. The latter is based on the energy dissipation of motional eddy currents [1]. An array of magnets is attached to the cart and a hollow aluminum tube is installed on the building so that the magnet array moves back and forth within the tube. The movement of the magnets cause eddy currents in the tube, which produce a repulsive force to the magnets proportional to their velocity. The magnetic damping is thus a viscous type of damping, which is convenient to model.

Figure 1: Schematic image of the linear absorber

The mass of the absorber is adjusted with additional steel plates and washers on the cart. It can be measured accurately with a scale. The stiffness is adjusted with the number of tension springs and the amount of damping is altered by the number and/or strength of the magnets. Both the stiffness and damping are measured experimentally by considering the free vibration of the linear absorber after an initial displacement. The stiffness is obtained from the damped

m

c

k k

natural frequency, which is equal to the inverse time period between two consecutive peaks. The damping factor is determined with the logarithmic decrement method.

B. Nonlinear vibration absorber

The nonlinear vibration absorber is also designed as a translational spring-mass-damper system. The mass is realized with a slider mounted on a rail which is attached to the concerning floor element (figure 2). The stiffness is provided with a piano-wire concept [2]. A steel wire is installed in the direction perpendicular to the rail and the slider is attached to the center of the wire. The displacement of the slider deflects the wire transversely and the elastic elongation of the wire results in a restoring force F according to equation 1. This is the nonlinear spring characteristic of a so-called NES (nonlinear energy sink).

(1)

E, A and L are respectively the young's modulus, the

cross-section and the length of the wire. The nonlinear spring constant can thus be altered by adjusting the diameter and length of the steel wire.

Figure 2: Schematic top view of the nonlinear absorber installed on a floor element of the building

The mass and damping are again determined with a scale and the logarithmic decrement method respectively. The damping originates from the friction between the slider and the rail and an additional amount of magnetic damping. The rail friction is assumed to be viscous by applying sewing machine oil on the rail. An amount of magnetic damping is added to increase the dissipative capacity of the NES, which was found to be beneficial for the performance of the nonlinear absorber. The nonlinear spring constant of the wire construction is measured with a static loading test. The results are close to the theoretical prediction of equation 1, which can thus be used for the estimation of knl.

III. DYNAMIC LOADS

A. Harmonic excitation

In a first phase, the building is loaded with a harmonic base excitation. This is applied by moving the shaker with a multisine displacement signal. This multisine is the sum of ten sines with frequencies ranging from 3 to 5 Hz to excite the first mode (f1= 4,2Hz). The amplitude is limited to 1, 2, 3 or 4 mm to consider the absorber implementation for different load intensities. The response of the building to an excitation of 2 mm is

shown in figure 3; the first mode is clearly excited as the three floors of the building move in phase constantly.

B. Impulse excitation

A second loading type is an impulse excitation. This is also applied at the base by giving the shaker a step displacement. Different intensities are again considered by varying the step displacement (5, 10, 15 or 20 mm). The response of the building to an impulse of 15 mm is shown in figure 4; each floor gets a large, initial displacement immediately after the impulse, which is followed by a free vibration. This vibration lasts relatively long due to the low damping factors of the aluminum building model. The impulse mainly excites the first mode, but second and third mode vibrations are also present to a lesser extent during the first oscillations.

Figure 3: Response of the building to the multisine excitation

Figure 4: Response of the building to the impulse excitation

IV. ABSORBER IMPLEMENTATION

A. Tuning to the harmonic excitation

The linear absorber is tuned to the first mode and it is placed on the third floor, which is the antinode, for an efficient vibration reduction. The tuning procedure of Rana & Soong [3] for harmonic base excitation is applied. This approach considers the MDOF building model as a SDOF system and applies the formulas of Den Hartog by using the modal mass ratio μ1 of the first mode. The formulas to determine the optimal frequency and damping factor of the absorber are given in equations 2 and 3; fopt represents the optimal ratio of the absorber's frequency to the first eigenfrequency of the building, while ζopt is the optimal damping factor of the absorber.

μ

μ

(2)

(3)

Based on the spring choice, the parameters of the

absorber were selected to satisfy these equations. The response of the modified building with a linear absorber

L

to the multisine excitation of 2 mm is shown in figure 5. Compared to the original case, the floor vibrations are much more reduced while the linear absorber oscillates heavily to dissipate the vibration energy.

Figure 5: Response of the modified building with a linear

absorber to the multisine excitation

The nonlinear absorber is also placed on the third floor and tuned to the first mode according to a tuning method optimized for NES in case of harmonic excitation [4]. This method optimizes the nonlinear spring constant based on the chosen absorber mass and damping, the main system's properties and the input energy of the harmonic excitation. The latter is an important parameter to optimize the absorber with respect to its energy threshold; the tuning procedure actually comes down to the optimal placement of the energy threshold so that the action of the NES is triggered for the concerning excitation. The nonlinear spring constant knl should be larger than or equal to the optimal value knl,opt according to equation 4.

ζ

(4)

Parameters ζ and

- depend on the absorber's

damping. Parameter X takes the harmonic excitation frequency into account, which was taken equal to the first eigenfrequency, and 0 depends on the energy introduced by the multisine excitation. The optimal spring constant is thus different for each load intensity. The modified response to a multisine excitation of 2 mm is shown in figure 6. The vibrations of the building floors are again reduced while the nonlinear absorber undergoes wide oscillations. The action of the nonlinear absorber is, however, smaller than the linear absorber.

Figure 6: Response of the modified building with a nonlinear

absorber to the multisine excitation

B. Tuning to the impulse excitation

Both the linear and nonlinear absorber are again tuned to the first mode and placed on the third floor because the impulse excitation mainly triggers the first mode. The

linear absorber is also tuned with the mentioned procedure of Rana and Soong; this is an approximate approach as this method is optimized for harmonic base excitation. The response of the modified building with a linear absorber to an impulse of 15 mm is shown in figure 7. The decay of the free vibration happens much faster and the duration of the transient vibrations is seriously reduced. Just after the impulse the linear absorber reacts heavily and dissipates a lot of vibration energy while the first mode vibrations of the building floors decays rapidly. The second and third mode vibrations go on relatively undisturbed and are mainly damped by the building's inherent damping.

Figure 7: Response of the modified building with a linear

absorber to the impulse excitation

The nonlinear absorber is tuned to the impulse

excitation with an optimization method designed for NES

in impulse load conditions [4]. The nonlinear spring

constant should satisfy equation 5 to obtain an optimal

placement of the energy threshold.

ζ

(5)

The energy of the impulse is taken into account with the parameter 00; this depends on the start velocities of the building floors imposed by the impulse. The response of the modified building with a nonlinear absorber to an impulse of 15 mm is shown in figure 8. The first mode vibrations are again heavily reduced while the nonlinear absorber dissipates a large amount of energy through some wide oscillations. The vibrations of the second and third mode are also left unaltered by the nonlinear absorber. The frequency-energy dependence of the NES is visible in figure 8: the frequency decreases with decreasing energy in the NES.

Figure 8: Response of the modified building with a nonlinear absorber to the impulse excitation

V. PERFORMANCE

A. Performance for harmonic excitation

The performance of both absorbers is quantified based on two criteria: the interstory drifts and the floor accelerations. The interstory drift is a measure that is related to the occurring stresses in the bearing elements of the floors (walls, columns) and to the damage in non-structural elements like partition walls or glazing units. Hence it is beneficial to reduce the interstory drifts. The floor accelerations should be reduced as well to enhance the comfort and safety of building users. Both the maximum and root mean square value of these quantities during the multisine excitation were selected for comparison.

First of all, it is noticed that both absorbers work well under harmonic load conditions. Both the interstory drifts and floor accelerations are reduced with ± 40 to 70 %. Besides there is a trend visible in function of the load intensity: the relative reduction increases for a larger intensity of the multisine excitation. When comparing both absorbers, the interstory drift criterion reveals that the nonlinear absorber performs slightly better while both absorbers perform equally well with respect to the reduction of the floor accelerations. The action of the nonlinear absorber is lower than that of its linear counterpart; this is based on the absorber's displacement relative to the building and the acceleration of the absorber. Hence it can be concluded that the nonlinear absorber performs better for the multisine excitation and is especially suited when the action of the absorber should be limited.

B. Performance for impulse excitation

The performance criteria for impulse excitation are the 10% settling time and maximum value of the floor displacements. The 10% settling time is the time after which the vibration is smaller than 10% of its initial value. This quantity is related to the duration of the transient vibrations after the impulse.

The 10% settling time of each building floor is reduced heavily with values of ± 80 to 90 % for both absorbers so that the duration of free vibration after the impulse is significantly reduced. The nonlinear absorber performs better than the linear absorber with ± 6 %. The addition of a second linear absorber, tuned to the second mode, was also considered to tackle the second mode vibrations occurring after the impulse. This proved to be beneficial for the nonlinear absorber but detrimental for the linear absorber. The 10% settling time is reduced slightly more when the second mode linear absorber is added to the configuration with a first mode nonlinear absorber. In the case of a first mode linear absorber, however, the addition of a second mode linear absorber has a negative effect; the operation of the first mode absorber is affected by the second mode absorber.

The maximum displacement, which occurs just after the impulse is not altered by the linear absorber. The nonlinear absorber, on the other hand, is able to reduce this quantity moderately, especially for the second floor. The nonlinear absorber reduces the maximum displacement of the first, second and third floor with ± 10 %, ± 30 % and ± 20 % respectively.

Based on these results the nonlinear absorber is also better suited for the impulse excitation.

VI. CONCLUSION

The response of the modified building with both a linear and nonlinear absorber indicated that the nonlinear absorber achieves the best performances for both the harmonic and impulse excitation. Moreover, the nonlinear absorber undergoes smaller oscillations during its operation so that this absorber type is preferred when the minimization of the absorber action is an objective. The tuning and implementation of the nonlinear absorber, however, requires more effort than the linear absorber. The nonlinear spring constant needs to be optimized separately for the different intensities of the excitations and the energy threshold was only estimated roughly with the mentioned formulas. An experimental finetuning was necessary to approximate the optimal value of the nonlinear spring constant as good as possible.

ACKNOWLEDGEMENTS

The author would like to acknowledge the support of prof. dr. ir. Mia Loccufier, Kevin Dekemele and Bram Vervisch.

REFERENCES

[1] B. Ebrahimi, B. Khamesee and F. Golnaraghi. A novel eddy current damper: theory and experiment. Journal of physics D: applied physics, 2009.

[2] M. McFarland, A. Vakakis, L. Bergman and T. Copeland. Characterization of a strongly nonlinear laboratory benchmark system. Dynamics of civil structures volume 4, 2010.

[3] R. Rana and T. Soong. Parametric study and simplified design of tuned mass dampers. Engineering structures, 1998.

[4] B. Vaurigaud, A. T. Savadkoohi and C.-H. Lamarque. Targeted energy transfer with parallel nonlinear energy sinks. Part I: theory and numerical results. Nonlinear Dynamics, 2011.

Table of contents

1 Introduction ........................................................................................................... 1

1.1 Context of vibration absorbers .................................................................................... 1

1.2 Outline of this dissertation ........................................................................................... 2

2 Design of a test model ......................................................................................... 3

2.1 Conceptual design ...................................................................................................... 3

2.2 Dimensioning of the components ................................................................................ 4

2.3 Experimental setup ..................................................................................................... 7

2.4 Identification of the modal parameters......................................................................... 7

3 Design of a linear vibration absorber ............................................................... 11

3.1 Practical realization ................................................................................................... 11

3.1.1 Stiffness .............................................................................................................. 12

3.1.2 Magnetic damping ............................................................................................... 13

3.2 Design parameters .................................................................................................... 16

3.3 Experimental determination of the design parameters ............................................... 17

4 Design of a nonlinear vibration absorber ......................................................... 20

4.1 Features of a nonlinear vibration absorber ................................................................ 20

4.1.1 Nonlinear spring characteristic ............................................................................ 20

4.1.2 Frequency-energy dependence ........................................................................... 21

4.1.3 Energy threshold & targeted energy transfer ....................................................... 23

4.2 Practical realization ................................................................................................... 24

4.2.1 Piano wire spring concept ................................................................................... 25

4.3 Design parameters .................................................................................................... 27

4.4 Experimental determination of the nonlinear spring constant .................................... 28

5 Performance under harmonic excitation .......................................................... 31

5.1 Multisine excitation .................................................................................................... 31

5.2 Linear vibration absorber........................................................................................... 34

5.2.1 Tuning ................................................................................................................. 34

5.2.2 Frequency response function .............................................................................. 35

5.2.3 Performance ....................................................................................................... 36

5.3 Nonlinear vibration absorber ..................................................................................... 37

5.3.1 Tuning ................................................................................................................. 37

5.3.2 Performance ....................................................................................................... 39

5.3.3 Frequency response function .............................................................................. 41

5.4 Comparison of the performance ................................................................................ 42

6 Performance under impulse excitation ............................................................. 46

6.1 Base step excitation .................................................................................................. 46

6.2 Linear vibration absorber........................................................................................... 49

6.2.1 Tuning ................................................................................................................. 49

6.2.2 Performance ....................................................................................................... 49

6.3 Nonlinear vibration absorber ..................................................................................... 50

6.3.1 Tuning ................................................................................................................. 50

6.3.2 Performance ....................................................................................................... 52

6.3.3 Energy threshold ................................................................................................. 53

6.4 Combined implementations ....................................................................................... 54

6.4.1 Linear absorber (mode 1) and linear absorber (mode 2) ..................................... 55

6.3.3 Nonlinear absorber (mode 1) and linear absorber (mode 2) ................................ 56

6.5 Comparison of the performance ................................................................................ 57

7 Concluding remarks ........................................................................................... 60

References ............................................................................................................... 62

Appendix A ............................................................................................................... 63

Symbols & abbreviations

A Cross-section of the wire m²

A Amplitude m

Ai,max Maximum acceleration of the ith floor m/s²

Ai,rms Root mean square acceleration of the ith floor m/s²

Aa,max Maximum acceleration of the absorber m/s²

Aa,rms Root mean square acceleration of the absorber m/s²

C Damping matrix Ns/m

C* Modal damping matrix Ns/m

c Damping constant Ns/m

ca Damping constant of the absorber Ns/m

d Depth of the wall elements m

d Diameter of the wire m

Di-j,max Maximum interstory drift between story i and story j m

Di-j,rms Root mean square interstory drift between story i and story j m

Di-a,max Maximum displacement between the absorber and story i m

Di-a,rms Root mean square displacement between the absorber and story i

m

DOF Degree of freedom

E Young's modulus N/m²

E Mode shape matrix

E1 Mode shape of the first eigenmode

E1n Normalized mode shape of the first eigenmode

F Force N

f Frequency Hz

fn Natural frequency of the absorber Hz

fd Damped natural frequency of the absorber Hz

fopt Optimum ratio of the linear absorber's frequency to the frequency of the main system

fi Eigenfrequency of the ith mode Hz

FRF Frequency response function

g Gravity constant 9,81 m/s²

h Height of the wall elements m

I Moment of inertia m4

K Stiffness matrix N/m

K* Modal stiffness matrix N/m

ki Stiffness of the ith floor N/m

kl Linear spring constant N/m

knl Nonlinear spring constant N/m³

knl,opt Optimum value of the nonlinear spring constant N/m³

L Length of the conductor tube in the free fall test m

L Length of the wire between the nonlinear absorber's mass and clamp

m

∆L Elongation of the wire m

M Mass matrix kg

M* Modal mass matrix kg

M1 First modal mass kg

ma Mass of the absorber kg

mi Mass of the ith floor kg

μ Ratio of the absorber's mass to the total structure's mass

μ 1 First modal mass ratio

MDOF Multi-degree-of-freedom

NES Nonlinear energy sink

ω Angular frequency rad/s

ωi Angular eigenfrequency of the ith mode rad/s

ωn Natural angular frequency of the absorber rad/s

ω 1* Modal angular eigenfrequency of the first mode rad/s

φ0 Energy parameter in the harmonic tuning procedure for the nonlinear absorber

m/s²

φ00 Energy parameter in the impulse tuning procedure for the nonlinear absorber

m/s

q Modal coordinate

SDOF Single-degree-of-freedom

t Thickness of the wall elements m

T Time of the magnet's free fall s

T Tension force in the wire N

T0 Pretension in the wire N

TET Targeted energy transfer

θ Angle between the original and deformed wire °

ustatio Stationary displacement of the building m

u Displacement of the center of mass m

v Velocity m/s

vi(0) Start velocity of the ith floor m/s

x Displacement m

X Parameter for the excitation frequency in the tuning procedure of the nonlinear absorber

ζ Damping parameter in the tuning procedure of the nonlinear absorber

ζi Damping factor of the ith mode

ζopt Optimum damping factor for the linear absorber

1

Chapter 1

Introduction

1.1. Context of vibration absorbers

Mechanical vibrations are a common disturbance occurring in buildings and civil structures.

They are excited by a variety of dynamic loads such as earthquakes, wind loads, traffic,

impact loads, etc. Depending on the intensity of these loads, vibrations range from some

minor oscillations which produce annoying noise and compromise the comfort of building

users, to very large vibrations which lead to failure of structural components or total collapse

of structures. Earthquakes in particular are well known for their destructive power. They are

responsible for loss of lives and a huge amount of material damage. To reduce the extent of

such disasters, mankind has taken measures to arm structures for earthquakes. Buildings in

seismic zones are constructed according to specific design rules and often special devices

are added to reduce mechanical vibrations to an acceptable level. A vibration absorber is

one of the oldest, simplest, and most effective devices to control vibrations and it has already

been successfully applied worldwide.

The concept of a vibration absorber was invented in 1909 by Hermann Frahm. It is a local

addition of a spring-mass-damper system to the main structural system. Due to the relative

movement of the absorber with respect to the main system, a large part of the vibration

energy is absorbed and hence the vibrations of the main system are reduced. The mass,

stiffness and damping of the absorber are design parameters that can be chosen specifically

to control the vibrations of the main system in a particular loading case. To optimize these

parameters in a variety of situations, a lot of updates and optimization methods followed

Frahm's initial invention of the vibration absorber.

This dissertation considers the vibration absorber both in its linear and nonlinear form. From

a morphological point of view, the difference between a linear and nonlinear vibration

absorber is situated in the resilient member of the absorber. The spring of a linear absorber

obeys Hooke's law, which results in a linear relation between the force in the spring and the

displacement of the mass. The spring of a nonlinear absorber, on the other hand, does not

obey Hooke's law and yields a nonlinear relation between force and displacement. A strongly

nonlinear spring with a cubic force-displacement relation was implemented in the nonlinear

absorber to clearly see the special behavior of a nonlinear absorber. From a behavioral point

of view, the nonlinear absorber is more versatile than its linear counterpart. A linear absorber

oscillates at one specific frequency so that it can be used to reduce vibrations at one target

frequency. A nonlinear absorber does not have a preferential frequency and can oscillate at

a range of frequencies depending on the input energy. The downside of the nonlinear

2

absorber is the existence of an energy threshold; efficient vibration reduction is only possible

above a minimal amount of input energy into the main system. The ability to capture a band

of frequencies is, however, very interesting and makes the nonlinear absorber more robust to

variations in the excitation frequency and modifications of the dynamic properties of the main

system.

The main system in this dissertation is a multi-story building which is subject to base

excitation. The mass of buildings can be modeled to be lumped in the floor slabs because

the heavy, concrete floor slabs form the major portion of the building's mass. These floor

slabs will attract large inertial forces under base excitation so that the vibrations of the

building are horizontal oscillations of the floor slabs. Hence, the resulting multi-degree-of-

freedom system is a stack of floor masses interconnected by springs and dashpots, which

represent the stiffness and damping of the supporting walls and columns. The attachment of

one or multiple vibration absorbers to the desired floor masses increases the number of

degrees of freedom (DOF) and will alter the building's response to dynamic loads. If the

absorbers are tuned properly, the harmful resonances will be flattened out in case of

nonlinear absorbers or replaced by two smaller, neighboring resonances in case of linear

absorbers.

1.2. Outline of this dissertation

Chapter 2 introduces the multi-story building, which will act as the main system on which

vibration absorbers will be implemented. The composition of the model will be discussed and

the important aspects of the design are explained. An overview of the experimental setup for

excitation and measurement of the building model is given and finally the identification of the

dynamic properties of the building model is elaborated.

The design and practical realization of a linear and nonlinear vibration absorber is dealt with

in chapters 3 and 4 respectively. The composition of both the absorbers is explained to give

a clear insight into the action of the vibration absorbers and the way they are attached to the

building model. Then the design parameters (i.e. mass, stiffness and damping) are discussed

and the way to alter and measure these parameters is elaborated.

The implementation of the vibration absorbers for two types of dynamic excitation is

discussed in the following two chapters. Both excitation types are introduced at the base of

the structure with the horizontal shaker. In chapter 5, the building model is loaded with a

harmonic excitation by moving the shaker with a multisine displacement signal. Both

vibration absorbers are tuned to this excitation and implemented to reduce the vibrations of

the building. The performance of both absorbers is evaluated and compared to each other.

Chapter 6 considers the transient vibrations of the building after an impulse. This impulse is

introduced at the base by giving the shaker a step displacement. Both absorbers are also

tuned to this excitation and their performance is evaluated. Finally chapter 7 closes this

master's dissertation with some concluding remarks.

3

Chapter 2

Design of a test model

2.1. Conceptual design

A model of a multi-story building was designed to serve as the main system on which

vibration absorbers can be implemented and tested. The number of stories chosen for this

research is three so that a 3-DOF model is obtained. The building model was designed in

aluminum because of its excellent workability, relatively low density and low damping

characteristics. The density of aluminum (2700 kg/m³) is low compared to other metals so the

gravity load on the horizontal shaker is kept to a minimum. Furthermore, the use of aluminum

leads to low damping factors because of its low inherent material damping. This ensures that

the vibrations are well visible during a sufficient amount of time for observations of the

building's response.

The building model is a composition of four components: fixations, wall elements, floor

elements, and additional mass plates. Figure 2.1 shows these components and their position

within the model. Two fixations connect the model with the horizontal shaker at the base of

the building. These L-shaped profiles are screwed in the movable plate of the shaker and

have longitudinal gaps for the connection to the wall elements. The wall elements are thin

plates that can be pushed into the gaps of the fixations and the end-blocks of the floor

Figure 2.1: Composition of the building model

Fixations

Wall element

Floor element

Additional mass plate

4

elements. The specific geometry of these plates allows movement of the floor elements in

only one direction and hence prevents unwanted vibrations in other directions. The floor

elements can only vibrate in the out-of-plane direction because the in-plane stiffness of the

wall elements is much larger than their out-of-plane stiffness. The floor elements have

specific end-blocks with longitudinal gaps at the top and bottom for the connection to the wall

elements of the upper and lower story. The flat part of the floor elements contains four screw-

threaded holes for the attachment of additional mass plates or the installation of a vibration

absorber.

The connections between the wall elements on

the one hand and the floor elements or fixations

on the other, are designed to be clamped.

Figure 2.2 shows a detailed view of a

connection between a floor and wall element.

The width of the longitudinal gaps in the end-

blocks is equal to the thickness of the wall

elements so that the wall elements can be

pushed into the end-blocks. The latter contain

screw-threaded holes on the side to fixate the

wall elements with screws for a firm connection.

These holes are also useful for the installation

of vibration absorbers.

The combination of the four mentioned components develops a modular building model. The

way of connecting the different elements allows a relatively quick assembly or modification. It

is easy to add or remove a story so that the number of DOF can vary as a function of the

research. Furthermore it is possible to adapt the dynamic properties of the building with

additional mass plates. The increase of one or more floor masses will lower the

eigenfrequencies and damping factors, and will alter the shape of the eigenmodes.

2.2. Dimensioning of the components

The frequency bandwidth of the horizontal shaker, used to excite the building at the base,

extends from 0 to 20 Hz. The dimensions of the floor and wall elements are determined in

order to keep the eigenfrequencies of the basic building (i.e. without additional mass plates)

within this range. This ensures that each of the three eigenmodes can be excited for

research purposes. The dimensioning was elaborated with the aid of a MATLAB simulation,

in which the building was modeled as a 3 DOF system. After the mass and stiffness matrices

were constructed, the eigenfrequencies were calculated by solving the eigenvalue problem.

The use of a MATLAB script allows a quick and iterative adaptation of the dimensions until

the eigenfrequencies are within the desired range.

Figure 2.2: Clamped connection detail

End-block

Wall element

Floor element

5

The mass of the building is assumed to be concentrated in the floors. The thin wall elements

are neglected because their mass is much lower than the mass of the floor elements. Hence

a diagonal mass matrix M is obtained in which mi is the mass of the ith floor. The stiffness

matrix K has a symmetrical shape and ki represents the equivalent spring constant of the ith

story.

(2.1)

(2.2)

The equivalent spring constant of each story depends on the out-of-plane stiffness of the wall

elements. The latter is a consequence of the assumed bending shape of the walls. Due to

the clamped nature of the connections, the wall elements have no rotation at the

connections. Hence, the wall elements can be modeled as illustrated in the left part of Figure

2.3 and the equivalent spring constant can be calculated based on the relation between the

indicated force F and displacement x. To facilitate the derivation of this relation, the deformed

wall element can be considered to be a combination of two vertical cantilevers as shown in

the right part of Figure 2.3 (Loccufier, 2015). Each of these vertical cantilevers is half of the

height of a wall element and takes half of the horizontal displacement x. The relation

between the force and displacement is calculated for one cantilever with the principle of

virtual work and is given in equation 2.3.

Figure 2.3: Modeling of the wall elements

(2.3)

The moment of inertia depends on the geometry of the wall elements and can be calculated

F

F

F

F

h

x

h/2

h/2

x/2

x/2

6

with the formula for a rectangular cross-section as in equation 2.4. By inserting the moment

of inertia in equation 2.3, the expression for the equivalent spring constant of a wall element

is obtained. The equivalent spring constant of one story is twice this value as each story has

two supporting wall elements. This spring constant is the same for each of the three stories

in the building model and depends on the Young's modulus of aluminum (69 GPa), the depth

d, the thickness t and the height h of the wall elements.

(2.4)

(2.5)

After a series of trial simulations, a practical solution was chosen that yields floor and wall

elements and the related eigenfrequencies as listed in table 2.1. It can be noted that the third

eigenfrequency is somewhat higher than 20 Hz, which could cause problems to excite the

third eigenmode. The used simulation is, however, an estimation and not an exact

representation of the building model in reality. Some reasonable assumptions were made to

simplify the modeling process; this will introduce deviations of the reality. Firstly, the mass of

the wall elements was neglected for a simple construction of the mass matrix; the wall

elements add a certain mass to the building though. Secondly, the spring constant of a

building story was calculated based on fully clamped connections. It seems, however,

unlikely that each manufactured connection will perfectly yield zero rotation of the wall

elements at the connection. Based on these considerations, the real eigenfrequencies are

expected to be lower than 20 Hz. Nonetheless, if this would not be the case, the additional

mass plates are still available to lower the eigenfrequencies.

Floor elements mi 1,031 kg

Wall elements

h 250 mm

d 200 mm

t 1,5 mm

Eigenfrequencies

f1 5,4 Hz

f2 15,1 Hz

f3 21,8 Hz

Table 2.1: Chosen dimensions and related eigenfrequencies

Other dimensions of the building model were determined by practical considerations and in

agreement with the dimensions from table 2.1. Detailed design drawings with dimensions of

each component are added in appendix A.

7

2.3. Experimental setup

After construction by a skilled manufacturer, the building model was placed in an

experimental setup for the application of dynamic loads and the measurement of the

building's response. A picture of the experimental setup with indication of the main

components is given in Figure 2.4. The building model is mounted on a horizontal shaker,

which is connected to a computer. The computer has specific software to drive the shaker

with a predefined motion in one horizontal direction (i.e. the out-of-plane direction of the wall

elements). Accelerometers are mounted on each floor element to measure the response of

the building. These accelerometers are plugged in a data acquisition module, which transfers

the acceleration signals to the computer. The computer, on its turn, records the signals with

data-logging software for further analysis. An impact hammer is also connected to the data

acquisition module so that the building can be excited with a known force signal for

frequency response function (FRF) measurements.

Figure 2.4: Main components of the experimental setup

2.4. Identification of the modal parameters

The experimental setup was firstly used for the identification of the main system. The modal

parameters of the building model (eigenfrequencies, eigenmodes and damping factors) were

estimated with the use of the ABRAVIBE toolbox (Brandt, 2016). That is a free toolbox for

MATLAB, which includes a series of very interesting algorithms and program routines for

vibration analysis.

One row of the FRF-matrix was measured for the extraction of the modal parameters. More

specifically the third row was measured which means that the acceleration of the third floor

Horizontal shaker

DAQ module

Computer

Building

model

8

was measured for an impact force on each of the three floors. Due to the measurement of

the response in accelerations, the measured FRFs are of the accelerance type which has

units of (m/s²)/N or g/N. Each force signal was a series of 10 impacts with a time period of 20

seconds in between. This allows the application of time domain averaging, which improves

the signal-to-noise ratio of the measured signals and hence the quality of the FRFs. The

magnitude of the measured FRFs is shown in the first row of Figure 2.5, together with the

related force spectrum and coherence in the rows underneath. The coherence is a quality

measure of the estimated FRFs and drops below unity if there is contaminating noise in the

force and/or acceleration signal. It can be seen that the quality of the FRFs is very good for

the larger part of the frequency range of interest except for some minor drops.

Figure 2.5: Frequency response functions of the third row of the FRF-matrix (31: impact on the first floor, 32: impact on the second floor, 33: impact on the third floor). For each FRF measurement the magnitude of the FRF (dB scale), the force spectrum and the coherence are plotted in the first, second and third row respectively.

With the knowledge of one row of the FRF-matrix, the poles and eigenmodes of the building

model can be estimated with specific routines from the ABRAVIBE toolbox. The poles are

complex numbers that contain information about both the eigenfrequencies and damping

factors; they are estimated by using the least-squares complex exponential method. The

9

eigenmodes are estimated with another routine that uses curve-fitting of the FRFs. The

resulting pole vector P and mode shape matrix E are given in equations 2.6 and 2.7.

(2.6)

(2.7)

According to the definition of a pole in equation 2.8, both the real and imaginary part of a

pole depend on the angular eigenfrequency ω and the damping factor ζ. The latter two can

be calculated with suitable transformations of the pole definition.

ζ ω ω ζ

(2.8)

ω (2.9)

ζ

ω

(2.10)

The eigenfrequencies and damping factors are determined with these equations and the

vectors with their results are given in equations 2.11 and 2.12. It can be noticed that the

eigenfrequencies are indeed lower than the values predicted by the MATLAB simulation and

they are within the desired range of 0 to 20 Hz. The dimensioning of the floor and wall

elements was thus done successfully. The damping factors are very low and hence a lightly

damped system is obtained. This is a consequence of the aluminum use and is beneficial for

a clear observation of the vibrations.

(2.11)

(2.12)

The mode shape matrix in equation 2.7 contains complex eigenmodes, which suggests non-

proportional damping. However, when looking at the phase angles of the mode shape matrix

in equation 2.14, it can be noticed that the phase differences are small or almost equal to π

radians. With a reasonable sense of approximation, this means that the different floors are

vibrating either in phase or out of phase. The building model can thus be considered as a

proportional damped system. The mode shapes of the three eigenmodes are visualized in

Figure 2.6; these are indeed the expected mode shapes for a 3 DOF system.

10

(2.13)

(2.14)

Figure 2.6: Visual representation of the three mode shapes

Mode 1 Mode 3 Mode 2

0,52

1,03

0,90

1,17

0,39

-1 0,78

-1,38

1

11

Chapter 3

Design of a linear vibration absorber

3.1. Practical realization

A linear vibration absorber is essentially a spring-mass-damper system, added to a certain

degree of freedom of the main system. In this dissertation a translational absorber was

constructed of which a picture is shown in figure 3.1. The oscillating mass is a four wheel

cart, attached to the building with two springs and two small aluminum plates. The cart is

simply placed on the flat part of a floor element and does not need a guiding system to move

back and forth. Hence the installation of the vibration absorber does not add a lot of

additional mass to the original building, which is desirable for the reduction of the structure's

total weight and the similarity in dynamic properties of the building before and after

installation of the absorber. The movement of the cart involves a low amount of unwanted

damping coming from the rolling resistance of the wheels. The plastic wheels and the

aluminum surface of the floor element do not deform during the rolling motion so that the

rolling resistance is only the result of slippage, which is rather small. This low amount of

unwanted damping is beneficial to the control of the absorber's damping, which is an

important design parameter. The added damping is implemented as magnetic damping. An

array of magnets is attached to the oscillating mass in a way that it moves back and forth

through a conductor tube of aluminum. The latter is attached to the building via one of the

small aluminum plates, which was also used for the attachment of the springs. The relative

movement of the magnets and the conductor tube involves energy dissipation and hence

damps the motion of the absorber.

Figure 3.1: Linear vibration absorber installed on the third floor

12

3.1.1. Stiffness

The stiffness of the linear absorber is implemented with tension springs. This type of spring

shows a low amount of inherent damping and is provided with hooks at the ends, which is

convenient for the connection to the oscillating mass. An equal number, dependent on the

desired stiffness, of tension springs is attached to both sides of the cart. The other ends of

the springs are attached to the building via small aluminum plates, which are fastened to the

sides of the floor element by using the screw threaded holes in the end-blocks. The plates

are provided with holes with a diameter somewhat larger than the spring diameter. The

springs are pulled through these holes to apply pretension in the springs, and they are fixed

by placing a very small plate between two windings on the outer side of the aluminum plate.

The pretension is essential to ensure that the springs work in tension and are not

compressed into a wavy shape. The equivalent spring constant of this spring configuration is

derived with the aid of figure 3.2. The first part of this figure illustrates the cart with two

tension springs in their unstretched length L0 and a spring constant k. There are no forces

acting on the cart because the springs are not deformed from their initial shape in this case.

Figure 3.2: Derivation scheme for the equivalent spring constant

In the second part of figure 3.2, the springs are pretensioned and attached to the aluminum

plates at the sides. The length of the springs is now L1 and both springs are equally and

sufficiently pretensioned to prevent compression of the springs during the oscillation of the

cart. Both springs exert an equal force on the cart, given in equation 3.1, so that the cart is in

equilibrium and does not undergo any displacement.

(3.1)

If the cart does undergo a displacement x, as illustrated in the third part of figure 3.2, the

length of the springs and hence the force in the springs becomes dissimilar. This results in a

net force that opposes the displacement of the cart, which is given in equation 3.2.

(3.2)

k, L0 k, L0

k, L1 k, L1

F1,L F1,R

F2,R F2,L

k, L1 + x k, L1 - x

13

From the last equality in equation 3.2, it can be noticed that the equivalent spring constant of

the applied spring configuration is equal to twice the spring constant of the tension spring

used at both sides of the cart. In case multiple parallel springs are used at both sides of the

cart, the spring constant at each side is the sum of the individual spring constants. The use

of two or three springs at each side will thus create an equivalent spring constant of 4k or 6k

respectively.

To get an idea of the magnitude of the implemented stiffness, the spring constant of the used

tension spring is determined experimentally with a vertical loading test. For this purpose, the

tension spring is suspended vertically and loaded with increasing weights. The vertical

displacement of the spring is measured for each weight and the resulting measurement pairs

are plotted on a force-displacement graph (figure 3.3). The measurement is executed twice

to exclude errors in the measurement procedure. It can be seen that both measurements

show similar results and a linear relation, which is expected from Hooke's law. A linear trend

line is drawn through the measurement points to obtain the spring constant; the equations of

the trend lines are also plotted on the graph. The coefficient of the linear term is the spring

constant and the average of both measurements yields a spring constant of 29,2 N/m. When

one or two tension springs are installed at both sides of the cart, the equivalent spring

constant of the absorber should thus be 58,4 N/m or 116,8 N/m based on this measurement

and the derivation of the equivalent spring constant.

3.1.2. Magnetic damping

The damping of the linear absorber is implemented as magnetic damping; a concept in which

eddy currents are responsible for the energy dissipation of the oscillating mass (Ebrahimi et

al., 2009). An array of magnets is therefore attached to a horizontal rod, which is mounted on

the cart so that the magnets move along with the oscillating cart. A hollow, aluminum tube is

y = 28.055x + 0.1116

y = 30.337x + 0.0998

0

0.5

1

1.5

2

2.5

3

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Forc

e [N

]

Displacement [m]

Figure 3.3: Measured force-displacement relation of the used tension spring

14

also installed on one of the aluminum plates at the side of the floor element. The axis of the

tube is aligned with the array of magnets so that the magnets move horizontally within the

tube. The configuration of these elements is schematically shown in figure 3.4.

Figure 3.4: Schematic representation of the linear vibration absorber

The movement of the magnets relative to the conductor tube causes motional eddy currents

in the conductor. These eddy currents produce a repulsive force to the magnets that is

proportional to the velocity of the moving magnets. Hence the magnetic damping behaves as

viscous damping which is very convenient to model and results in linear differential

equations. The amount of added damping is also well adjustable by varying the amount of

magnets in the array or changing the strength of the used magnets. It should, however, be

considered that the magnets are attached to the cart and are part of the oscillating mass.

The mass, which is an important design parameter of the absorber, should thus be kept even

when adapting the array of magnets. This is done by adding or removing additional masses

to the cart, like washers or small steel plates.

The used magnets are made of neodymium and their specifications are listed in table 3.1.

The aluminum conductor tube has an internal diameter of 18 mm and a thickness of 2 mm.

To have an indication of each magnet's amount of damping, an estimation of the damping

constant was executed with a free fall test. In this test, the magnet is dropped in a vertical

conductor tube and the fall time is measured. The conductor tube for this test has the same

cross-section as the one in the absorber, but it has a higher length to result in a fall time that

is measurable in practice. The vertical equilibrium of the magnet is considered during the free

fall and is given in equation 3.3. It is assumed that the velocity of the falling magnet is

constant over the length of the tube, which is reasonable when observing the magnet during

a test. The equilibrium equation is then integrated over time and the upper boundary T

represents the time period the magnet needs to fall from top to bottom. Further elaboration of

the integrals results in expression 3.6 for the damping constant; it depends on the mass of

the magnet m, the gravity constant g, the time period of the free fall T and the length of the

conductor tube L.

(3.3)

Conductor tube Array of magnets

15

(3.4)

(3.5)

(3.6)

Magnet type Shape Diameter

[mm]

Height

[mm]

Adhesive

force [kg]

Damping

constant [Ns/m]

S-12-02-N Disk 12 2 1,6 0,025

S-12-03-N Disk 12 3 2,5 0,058

S-12-06-N Disk 12 6 3,9 0,174

S-12-10-N Disk 12 10 5,3 0,461

R-10-04-05-N Ring 10 5 2,2 0,035

ITN-13

Disk with internal

screw thread 13 4.5 3 0,029

Table 3.1: Specifications of the neodymium magnets used for magnetic damping

The last two columns of table 3.1 indicate that the damping constant increases with

increasing adhesive force of the magnet, which is expected. Only magnet type ITN-13 does

not comply to this trend. This is, however, the result of a problem in the free fall test of this

magnet type. When this magnet is dropped in the tube, it spins around its horizontal axis and

constantly hits the conductor tube during the fall. This is probably due to the internal screw

thread that causes an uneven mass distribution in the magnet. The measured damping

constant of this magnet type is thus not that reliable.

Based on some trial tests, it became clear that the damping constant of a magnet array is not

simply the sum of the individual damping constants. The graph in figure 3.5 shows the

damping constant in function of the number of magnets in the magnet array for two different

magnet types. A linear relation would allow the simple addition of the individual damping

constants. However, the relation is nonlinear; the gain in damping constant per added

magnet becomes smaller as the number of magnets increases. The underlying cause of this

nonlinear relation can probably be found by studying the magnetic field properties of a

magnet array. This is not of particular interest to this research though. Hence the damping

constant of each different magnet array should be determined experimentally with a free fall

test.

16

3.2. Design parameters

The dynamic behavior of the linear absorber is determined by three important design

parameters: the mass m, the spring constant k and the damping constant c. These

parameters determine the natural frequency fn and the damping factor ζ of the absorber

according to equations 3.7 and 3.8.

π

(3.7)

ζ

(3.8)

The mass of the absorber is the composite mass of the four wheel cart and all the parts

attached to it (magnet array, washers, steel plates, accelerometer). This can be measured

quite precisely with a scale (accuracy 0,001 kg). The mass is also well adjustable by adding

or removing washers and steel plates of different sizes. Hence the control of the mass

parameter is very good. The stiffness parameter is less controllable because springs are only

available in discrete spring constants. The added stiffness is also known rather

approximately because the spring constant of the tension springs is estimated with a static

load test while the springs in the absorber setup will be loaded dynamically. The damping

parameter is also low controllable because of the unavoidable presence of unwanted

damping. Although the damping constant of the magnetic damping can be estimated with the

explained free fall test, the rolling resistance of the wheels will add an unknown amount of

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5 6 7 8 9 10

Dam

pin

g c

onsta

nt

[Ns/m

]

Number of magnets

Figure 3.5: Relation between the damping constant and number of magnets

Magnet type S-12-06-N

Magnet type S-12-03-N

17

damping. Due to these uncertainties in the magnitudes of the spring constant and damping

constant, a better method is required to quantify the stiffness and damping of the absorber.

This is necessary because the accurate knowledge of these parameters is of paramount

importance in the tuning procedure of the absorber.

3.3. Experimental determination of the design parameters

The design parameters are determined experimentally by attaching the absorber to an

isolated floor element as in the working conditions and fixing this floor element onto a firm

table. The obtained setup is a SDOF system in which the absorber can move back and forth

on the floor element. The absorber is brought to an initial displacement of a few centimeters

by hand and released suddenly so that a decaying oscillation of the absorber is obtained.

The acceleration signal of the free vibration is measured with an accelerometer and recorded

for further analysis in MATLAB. Figure 3.6 shows the acceleration signal after the application

of a low pass butterworth filter to remove disturbing noise. The natural frequency and

damping factor are extracted from this acceleration signal; these two parameters

characterize the dynamic behavior of the absorber completely. It is also possible to calculate

the exact values of the spring constant and damping constant with equations 3.7 and 3.8

because the mass of the absorber is known accurately with the use of a scale.

The damped natural frequency fd is equal to the inverse of the time period between two

consecutive peaks in the acceleration signal. The natural frequency fn is derived from the

damped natural frequency according to equation 3.9.

ζ

(3.9)

Figure 3.6: Acceleration signal of the decaying oscillation

exp(-ζωnt)

18

The damping factor is calculated with the logarithmic decrement method. This method is

developed for a viscous damped system and considers the natural logarithm of the ratio of

two consecutive amplitudes. The amplitudes can be taken from the displacement, velocity or

acceleration signal. The latter is used here and the damping factor is calculated by applying

a suitable transformations of equation 3.10.

πζ

ζ

(3.10)

It can be noticed that there is a series of peaks between which the natural frequency and

damping factor can be calculated. It is chosen to calculate an average value over a range of

peaks in the middle section, where the signal has gone through some cycles already but is

not too damped yet. The peaks used for the calculation are indicated with red dots in figure

3.6. After the determination of the natural frequency and damping factor, an exponential

decay with the calculated values is plotted through the first peak of the acceleration signal to

check if the determined parameters fit the oscillating behavior. This exponential decay is

plotted in green on figure 3.6 and matches the acceleration signal quite well in most cases.

The extent of the fit depends on the damping of the absorber: the more magnetic damping is

added, the better the exponential fit. This is logical because an exponential decay is inherent

to viscous damping. The addition of magnetic damping (viscous) enlarges the portion of

viscous damping and makes the unwanted damping of the rolling friction (non-viscous)

negligible. This is well visible in the two opposing cases shown in figures 3.7 and 3.8. In the

first case no magnetic damping is added so that the decay is caused completely by the

slippage of the wheels. This results in a bad fit of the exponential decay; the peaks of the

oscillating motion show a linear decay. This is a typical feature of dry-friction damping or so-

called Coulomb damping. In the second case a large amount of magnetic damping is added

so that the damping is mostly of the viscous type. The exponential decay fits very well in this

case, which is expected from the theory of viscous damping.

Figure 3.7: Decaying oscillation in case of no added magnetic damping (Coulomb damping)

Figure 3.8 Decaying oscillation in case of a large amount of added magnetic damping

(viscous damping)

fn = 4,1 Hz

ζ = 0,023

fn = 3,7 Hz

ζ = 0,149

exp(-ζωnt) exp(-ζωnt)

19

From the case where no magnetic damping was added, the equivalent damping constant of

the rolling resistance can be calculated as if it were a viscous damping type. Even though

this case shows Coulomb damping, the equivalent damping constant of this part is useful to

know the added damping constant of the magnets in other cases. The mass in this case was

0,115 kg and the natural frequency and damping factor were experimentally found to be 4,1

Hz and 2,3 %. Based on equations 3.7 and 3.8 this leads to an equivalent damping constant

of 0,14 Ns/m.

The second case with a large amount of added damping is now considered to check the

accuracy of the estimated spring constant and damping constant. The specifications of the

absorber are listed in table 3.2. By the use of one tension spring at each side of the cart, a

spring constant of 58,4 N/m is expected based on the vertical load test and derivation of the

equivalent spring constant. The exact spring constant obtained from the free vibration

analysis, is 80,3 N/m. Hence the estimation of the spring constant gives a relative error of

27 % and underestimates the exact spring constant. Two reasons come to mind to explain

this discrepancy. Firstly, the oscillations of the absorber load the springs in a dynamic way,

while the spring constant was estimated in a static loading test. Secondly, not the full length

of the spring is active in the absorber setup; a part of the tension spring is located outside the

aluminum plates to provide sufficient pretension (see figure 3.1). The damping constant of

the considered magnet array is estimated at 0,97 Ns/m with the free fall test. The exact

damping constant of the magnets is 0,89 Ns/m. This value is obtained by subtracting the

equivalent damping constant of the rolling resistance from the total damping constant which

is determined experimentally. The amount of magnetic damping is thus overestimated with a

relative error of 9 %. The discrepancy can partly be attributed to the time measurement error

in the free fall test because this relies on the human reaction capacity. There is also a

difference in the motion of the magnet array; the free fall test considers a steady downward

motion of the magnets while the magnet array in the absorber will move back and forth in a

faster way. It can be concluded that the estimations of spring and damping constant give a

good indication of the exact values, but the experimental determination is necessary to

obtain reliable parameters for the tuning procedure.

Absorber specifications

ma 0,150 kg

Stiffness 2 x 1 tension spring

Magnet array 1 x ITN-13 + 3 x S-12-06-N

Estimated constants

k 58,4 N/m

c 0,97 Ns/m

Experimentally obtained

parameters (exact)

fn 3,7 Hz

ζ 0,149

k 80,3 N/m

ctotal 1,03 Ns/m

cwheels 0,14 Ns/m

cmagnets 0,89 Ns/m

Table 3.2: Absorber parameters for a case with a large amount of magnetic damping

20

Chapter 4

Design of a nonlinear vibration absorber

4.1. Features of a nonlinear vibration absorber

4.1.1. Nonlinear spring characteristic

The difference between a linear and nonlinear vibration absorber is situated in the resilient

member. The spring of a linear absorber has a linear spring characteristic and obeys Hooke's

law. The spring of a nonlinear absorber, on the other hand, has a spring characteristic that is

partly or completely nonlinear. Equation 4.1 considers a general form of a spring

characteristic with kl being the linear spring constant [N/m] and knl being the nonlinear spring

constant [N/mn]. The values of these constants and the power n determine the relation

between the force in the spring F and the displacement of the spring x. Figure 4.1 shows the

different possibilities dependent on the sign of the spring constants.

with n > 0 (4.1)

kl knl Spring characteristic

+ + Nonlinear (hardening)

+ 0 Linear

+ - Nonlinear (softening)

0 + Pure nonlinear (hardening)

The power n affects the strength of the nonlinearity in the spring characteristic. In this

dissertation a power of three will be considered to obtain a strongly nonlinear absorber that

clearly brings the nonlinear phenomena forward. Two common nonlinear oscillators with this

power are the Duffing oscillator and the nonlinear energy sink (NES); their spring

characteristics are given in equations 4.2 and 4.3. It can be seen that the NES is purely

nonlinear, while the Duffing oscillator has an additional linear term.

(Duffing oscillator) (4.2)

(NES)

(4.3)

Forc

e

Displacement

Figure 4.1: Graphical representation of different spring characteristics

21

4.1.2. Frequency-energy dependence

A beneficial consequence of the nonlinear spring characteristic is the disappearance of a

single natural frequency of the absorber. The nonlinear absorber is able to oscillate at a

range of frequencies instead and the governing frequency depends on the energy level in the

absorber. This energy dependence can be illustrated by considering the free vibration of a

Duffing oscillator and applying the harmonic balancing method (Marijns & Geeroms, 2012).

The equation of motion for the free vibration of an undamped Duffing oscillator is given in

equation 4.4.

(4.4)

If the harmonic balancing method is applied for the fundamental harmonic (HB1 method), the

solution of the displacement can be written in the form of expression 4.5.

(4.5)

This solution is put in the equation of motion to find an expression for the frequency of the

Duffing oscillator.

(4.6)

(4.7)

The transition to equation 4.7 makes use of a goniometric formula that simplifies cos³ into the

sum of two cosines. The cosine term with frequency 3ω may be omitted because the solution

of the HB1 method only considers the fundamental frequency ω.

ω

ω (4.8)

ω

(4.9)

ω

(4.10)

Elaboration of the equation of motion finally results into expression 4.10, which shows that

the frequency of the undamped Duffiing oscillator is dependent on the amplitude A. The

amplitude of an undamped free vibration depends on the initial displacement and the initial

velocity according to equation 4.11.

(4.11)

22

Fre

quency

Amplitude

From equations 4.10 and 4.11, it can be seen that the frequency of the oscillator depends on

the input energy due to the nonlinear term in the spring characteristic. If the initial

displacement and/or initial velocity of the oscillator increases, the amplitude of the free

vibration will increase and consequently the frequency of the vibration. This relation is

illustrated in figure 4.2 for both the Duffing oscillator and the NES. The expression for the

frequency-energy dependence of the NES can also be found by applying the HB1 method or

by simply omitting the linear spring constant in expression 4.10. This results in a linear

frequency-amplitude relation for the NES, that starts in the origin. Hence the NES can

theoretically oscillate at any frequency. The Duffing oscillator, on the other hand, has a lower

limit to its frequency range, which is the natural frequency resulting from its linear spring

constant (i.e. ωn=√kl/m).

In reality damping will always be present and introduce a decay in the free vibration. It was

neglected here for the sake of simplicity in the derivation of expression 4.10. The explained

frequency-energy dependence remains present when damping is added. This is visible when

looking at the free vibration of the designed nonlinear absorber (figure 4.3). The time period

between consecutive peaks increases as the peaks decay with time. Or, in other words, the

frequency decreases along with the energy dissipation in the absorber due to damping.

Figure 4.2: Frequency-energy dependence of two nonlinear oscillators

NES

Duffing oscillator

ωn

Figure 4.3: Frequency decrease in the free vibration of a nonlinear absorber

f1

f1 = 6,0 Hz

f2 = 5,1 Hz

f3 = 4,0 Hz

f4 = 3,5 Hz

f2

f3 f4

23

4.1.3. Energy threshold & targeted energy transfer

A drawback caused by the nonlinear spring characteristic is the existence of an energy

threshold; the efficiency of a nonlinear vibration absorber depends on the energy present in

the main system. If the input energy is lower than the threshold, the nonlinear absorber

reacts weakly and does not reduce the vibrations of the main system significantly. Beyond

the energy threshold, however, a heavy reaction of the nonlinear absorber occurs which

results in a so-called targeted energy transfer (TET) (Petit, 2012). TET is the passive and

irreversible energy transfer from the main system to the nonlinear absorber, which is very

desirable for vibration reduction purposes. The existence of an energy threshold is well

visible during the application of a nonlinear absorber for impulse excitations. Figure 4.4

shows the response of the building model with a nonlinear absorber attached to the third

floor, for both a soft and strong impulse excitation. In the first case the input energy is lower

than the energy threshold; hence the absorber reacts weakly and does not really alter the

free vibration of the building model. For a larger impulse load intensity as in the second case,

the input energy is higher than the threshold and the absorber reacts strongly. This results in

an energy transfer from the building to the absorber so that the free vibration of the building

dies out much faster.

Figure 4.4: Response of the building model and nonlinear vibration absorber for two impulse excitations with different intensity (input energy below the threshold (above) and input energy above the threshold (below))

24

4.2. Practical realization

The design of the nonlinear vibration absorber depends mostly on the chosen concept to

realize the nonlinear spring characteristic. A spring with a cubic nonlinearity is constructed to

obtain a strongly nonlinear absorber; this is achieved with a piano wire spring concept

(McFarland, 2010). In this concept, the absorber mass is attached to a tensed wire and it

pulls the wire transversely in its center. The stretching of the elastic steel wire results in a

restoring force so that the absorber oscillates in the direction perpendicular to the wire.

The piano wire construction of a previous master dissertation is reused and adapted for the

installation on the building model (Marijns & Geeroms, 2012). Figure 4.5 shows the

construction of the nonlinear absorber installed on the third floor of the building model. The

absorber mass is an aluminum slider installed on a guiding rail so that it moves back and

forth in the desired direction. The slider is composed of two movable blocks between which

the steel wire is clamped to provide the stiffness. It is possible to attach additional steel

plates on the slider to increase the mass of the absorber.

An aluminum profile with a hollow, square cross-section is attached to the bottom of the floor

element; this profile is provided with movable clamps to set up the wire. The clamps consist

of a block and a plate that is fastened to the block with screws; this allows to fix the wire

between the block and the plate. The profile has ten holes on each side of the floor element

to modify the position of the clamps. This allows to vary the length of the wire, which is an

important parameter for the magnitude of the nonlinear spring constant. At both ends of the

profile a guitar tuner knob is installed with the aid of small aluminum plates. These tuner

knobs are convenient to fix the ends of the wire and allow to tension or loosen the wire.

Moreover they prevent unwanted loosening of the wire during the oscillating motion of the

absorber, which is essential for a consistent nonlinear spring characteristic.

In contrast to the linear absorber, the installation of the nonlinear absorber adds a

considerable amount of non-moving mass to the building model, which is an inherent

disadvantage of this absorber construction. Although aluminum was used to reduce the

additional mass, the absorber adds 1,131 kg to the concerning floor element.

Figure 4.5: Nonlinear vibration absorber installed on the third floor

Profile

Slider

Rail

Clamp

Guitar

tuner knob

Wire

25

4.2.1. Piano wire spring concept

The transverse deflection of the wire results in nonlinear spring behavior. The nonlinear

spring characteristic is derived by considering the restoring force due to the tension in the

wire for a certain displacement. If the slider undergoes a displacement x, the wire elongates

by ∆L at both sides of the slider. The elongation is calculated with Pythagoras' theorem

according to expression 4.12.

(4.12)

This elastic elongation causes tension in both parts of the wire with a magnitude according to

equation 4.13. The tension force T depends on the cross-section of the wire A, the Young's

modulus E, the elongation ∆L and the initial length of the wire part L.

(4.13)

The tension forces in both parts of the wire cancel out each other in the direction of the

profile, but yield a net force in the direction of the rail. If there is an initial tension T0 present in

the wire, the net force is given in expression 4.14.

(4.14)

Angle θ is indicated in figure 4.6; it is the angle formed by the deformed wire with its initial

position. The sine of this angle is equal to expression 4.15.

θ

(4.15)

Assembly of the former equations results in relation 4.16 between the restoring force F and

the displacement x. This relation is simplified into a practical expression 4.17 for convenience

Figure 4.6: Schematic top view of the nonlinear vibration absorber in deformed position

L

x θ

T T

F

26

by expanding F(x) in a Taylor series about x = 0 (it is assumed that the displacement x is

small compared to the length L).

(4.16)

(4.17)

It is sufficient to retain the linear and cubic term because the higher order terms are

negligibly small; this results in the general expression 4.18 for the nonlinear spring

characteristic of the piano wire spring. This is the spring characteristic of a Duffing oscillator.

(4.18)

With the construction of the nonlinear absorber it possible to add some pretension in the wire

by winding the wire around the axes of the guitar tuner knobs. The added pretension is,

however, limited and during experiments it was found to make the wire more prone to

rupture. Moreover it is difficult to measure the added pretension. Hence it is attempted to

eliminate all pretension by installing the wire in a loose but straight manner. This results in

the loss of the linear term in equation 4.18 and consequently a pure nonlinear spring

characteristic is obtained. The designed nonlinear absorber is thus of the NES type.

(4.19)

Based on equation 4.19, the nonlinear spring constant depends on the Young's modulus, the

cross-section and the length of the wire. Steel guitar strings with diameters 0,2 mm, 0,25 mm

and 0,51 mm are used for the wire. It is assumed that this steel has a Young's modulus of

210 GPa. The wire can be set at ten different lengths depending on the position of the

clamps. Hence there are thirty design options for the magnitude of the nonlinear spring

constant. The theoretical values according to equation 4.19 are listed in table 4.1.

27

knl [N/m³] d = 0,2 mm d = 0,25 mm d = 0,51 mm

A = 0,031 mm² A = 0,049 mm² A = 0,204 mm²

Hole 1 L = 0,329 m 186107 290792 1210162

Hole 2 L = 0,306 m 230253 359770 1497219

Hole 3 L = 0,277 m 312093 487645 2029383

Hole 4 L = 0,254 m 404982 632784 2633393

Hole 5 L = 0,229 m 549367 858386 3572260

Hole 6 L = 0,204 m 782845 1223196 5090451

Hole 7 L = 0,180 m 1131232 1767550 7355836

Hole 8 L = 0,153 m 1842022 2878160 11977750

Hole 9 L = 0,129 m 3073265 4801977 19983907

Hole 10 L = 0,101 m 6499366 10155259 42262126

Table 4.1: Design options for the nonlinear spring constant

4.3. Design parameters

Like the linear absorber, there are three important parameters that influence the dynamic

behavior of the nonlinear absorber: the mass, stiffness and damping. The mass is again the

best controllable parameter; it can be measured accurately with a scale and adjusted with

additional steel plates or washers.

The stiffness is somewhat less controllable because of the similar reasons mentioned earlier.

The nonlinear spring constant is available in discrete magnitudes depending on the position

of the clamps and the diameter of the wire. There are, however, more possibilities than in the

linear case. The magnitude of the nonlinear spring constant can theoretically be estimated

with equation 4.19 and experimentally be measured with a static load test. It is possible that

the latter differs from the real spring constant in the absorber setup, when loaded

dynamically. But this cannot be verified with a free vibration analysis because the frequency

of the nonlinear absorber varies according to the energy level in the absorber. Hence the

nonlinear spring constant will be determined with a static load test, explained in section 4.4.

In contrast to the linear vibration absorber, the damping is not a design parameter in the

used tuning procedures for the nonlinear absorber. The damping present in the nonlinear

absorber is due to the friction between the slider and the rail and an additional amount of

magnetic damping (c = 1,03 Ns/m). The magnetic damping was obtained with a magnet

array of one ITN-13 magnet and two S-12-06-N magnets; this additional damping was

installed to increase the energy dissipation, which improves the performance of the nonlinear

absorber. The rail is treated with sewing machine oil to reduce the amount of dry-friction and

create so-called viscous friction between the slider and rail. Hence the rail damping is

considered to be viscous and the magnitude is determined with the method explained in

section 3.3. The rail and slider are therefore installed with linear springs on a fixed floor

element and a free vibration analysis is performed. The damping constant of the rail friction is

28

0

5

10

15

20

25

30

0 0.01 0.02 0.03 0.04

Fo

rce [

N]

Displacement [m]

experimentally found to be 1,0385 Ns/m. This results in a total damping constant ca = 2,0685

Ns/m for the nonlinear absorber.

4.4. Experimental determination of the nonlinear spring constant

To know the added stiffness of the piano wire spring, the nonlinear spring characteristic is

determined experimentally with a static loading test. The absorber mass is therefore

removed from the setup and the wire is loaded transversely in its center with a dynamometer.

The necessary force to increase the displacement with half a centimeter is read from the

dynamometer until a displacement of 0,04 m is reached. The measurement pairs are

consequently placed in a graph and a polynomial trend line is drawn through them with the

use of Excel to estimate the nonlinear spring constant. The upper displacement of 0,04 m is

already considerable and will probably not be reached in operation conditions. When loading

the wire further, the wire even breaks after several tests.

Figure 4.7 shows the course of the nonlinear spring characteristic in case the clamps are set

at the second hole (L=0,306 m) and a wire diameter of 0,25 mm is used. The wire is installed

in a loose manner to avoid pretension and consequently a linear term in the nonlinear spring

characteristic. Both the theoretical values calculated with equation 4.19 and the experimental

values determined with the static loading test are plotted in the figure. The difference

between both is small and equation 4.19 thus gives a good approximation of the nonlinear

spring constant in practice. Consequently it is reasonable to use this equation to quantify the

nonlinear spring constant during the tuning procedure, to avoid the static loading test of each

of the thirty spring configurations.

Figure 4.7: Nonlinear spring characteristic for a wire diameter 0,25 mm and wire length 0,306 m

Experimental

knl = 385859 N/m³

Theoretically

knl = 359770 N/m³

29

The equations of the polynomial trend line through the measured and theoretical points are

given in equations 4.20 and 4.21 respectively. The latter one is by definition purely cubic.

Although the trend line constructed by Excel adds terms of lower order, these can be

neglected because of the very small coefficients with respect to the coefficient of the cubic

term. The trend line of the experimental values is also mainly cubic; the lower order terms

have negligible coefficients so that these may be omitted. Hence the experimental value of

the nonlinear spring constant is 385859 N/m³. The theoretical nonlinear spring constant

underestimates the real value in this case with a relative error of 6,7 %.

Experimental: (4.20)

Theoretical: (4.21)

The effect of pretension on the nonlinear spring characteristic is also briefly considered. The

wire in the same setup (d=0,25 mm; L=0,306 m) is pretensioned by turning the guitar tuner

knobs up. Because the pretension cannot be measured in this setup, the added pretension is

quantified in terms of knob revolutions. The loose wire was tensioned by turning up the

knobs at both sides with increments of two revolutions. The spring characteristic of each

case is determined to consider the effect of pretension. Figure 4.8 shows the spring

characteristic in case of loose and tensioned conditions. The pretension ranges from 2 knob

revolutions till 6 knob revolutions; during the static loading test of pretension due to 8 knob

revolutions, the wire broke in a brittle manner.

A first observation is that the force for a certain displacement increases when pretension is

added. A second observation is established when considering the coefficients of the

polynomial trend lines given in equations 4.22 till 4.25. The coefficient of the cubic term

decreases while the coefficient of the linear term increases. This observation is in agreement

with equation 4.18 where it can analytically be seen that an increasing value of the

pretension enlarges the linear term and reduces the cubic term in the spring characteristic.

The cubic term remains, however, dominant because the added pretension with the guitar

tuner knobs is relatively low. A high amount of pretension is not desirable though, because it

leads to failure of the wire in this absorber setup. The failure in the wire occurs right next to

the clamps because there is a kink over there. The kink changes in orientation during the

oscillations leading to a higher load intensity at that point, which results in failure ultimately.

30

0

5

10

15

20

25

30

35

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

Fo

rce [

N]

Displacement [m]

Loose Turn knobs 2x Turn knobs 4x Turn knobs 6x

Loose: (4.22)

2 knob revolutions: (4.23)



Figure 4.8: Effect of pretension on the nonlinear spring characteristic

31

Chapter 5

Performance under harmonic excitation

5.1. Multisine excitation

A first excitation type for which both vibration absorbers are implemented and compared, is a

harmonic base excitation. This excitation is applied by moving the horizontal shaker with a

multisine displacement signal. The concerning multisine is the sum of ten sines with

frequencies ranging from 3 to 5 Hz to excite the first eigenmode of the building model. The

signal has a duration of 60 seconds and the amplitude is kept within predefined limits (1 mm,

2 mm, 3 mm & 4 mm). This variation in amplitude is used to implement the vibration

absorbers for different load intensities. The time signal and the related frequency spectrum of

the multisine excitation with an amplitude limit of 2 mm, are shown in figure 5.1.

Figure 5.1: Multisine excitation with an amplitude limit of 2 mm (complete time signal

(above), zoomed view of the time signal (middle) and frequency content (below))

32

A part of the response of the building model to this multisine excitation is shown in figure 5.2;

the first eigenmode is clearly excited as the three floors move in phase constantly.

For the purpose of comparing responses of different configurations and assessing the

performance of the vibration absorbers, the response is quantified by looking at the interstory

drifts and the floor accelerations during the multisine excitation. The interstory drift is the

relative displacement between two neighboring floors; its magnitude is related to the stresses

occurring in the vertical bearing elements of a building (columns, walls) and the damage of

non-structural elements (partition walls, glazing units, equipment). Based on these

considerations it is of great interest to reduce the interstory drifts as much as possible. The

floor accelerations in a building should also be reduced, to enhance the comfort and safety of

building users. Both the maximum value and the root mean square value are considered for

the interstory drifts and floor accelerations.

For the response of the original building model (i.e. without vibration absorber), a distinction

is made between the building model before the implementation of the linear absorber (m3 =

1,031 kg) and the building model before the implementation of the nonlinear absorber (m3 =

2,162 kg). The second case has a higher mass in the third floor due to the installation of the

rail and the profile necessary for the functioning of the nonlinear absorber. This difference in

mass results in a different response; hence both cases are considered separately to make a

correct assessment of the performance of both vibration absorbers. The response of the

original building model for both types of absorbers is quantified in tables 5.1 and 5.2. It can

be noticed that the response is somewhat smaller for the second case.

For both cases it is clear that the interstory drift is the highest between the base and the first

floor and the lowest between the second and the third floor. This stems from the mode shape

of the first eigenmode, visible in figure 2.6. The accelerations are larger for the higher

positioned floors, which is also a consequence of the mode shape. Higher positioned floors

are subject to larger displacements within the same time interval; hence they need higher

Figure 5.2: Response of the original building model to the multisine excitation of 2 mm

33

accelerations to complete the oscillations. Both the interstory drifts and floor accelerations

logically increase with increasing amplitude of the multisine excitation.

Multisine excitation 1 mm 2 mm 3 mm 4 mm

Interstory

drift

[mm]

MA X

D0-1,max 3,9 7,2 17,0 21,4

D1-2,max 3,2 6,1 12,7 17,2

D2-3,max 1,9 3,5 7,8 10,0

RM S

D0-1,rms 1,4 2,7 7,0 9,0

D1-2,rms 1,1 2,1 5,7 7,4

D2-3,rms 0,6 1,2 3,1 3,9

Floor

acceleration

[m/s²]

MA X

A1,max 3,5 6,6 19,0 23,5

A2,max 4,6 9,7 22,2 29,0

A3,max 5,5 10,6 27,0 32,3

RM S

A1,rms 1,0 1,9 6,0 7,6

A2,rms 1,5 3,1 8,4 11,0

A3,rms 1,9 3,8 10,7 13,9

Table 5.1: Quantitative response of the original building model to multisine excitations


Interstory

drift

[mm]

MA X

D0-1,max 2,5 6,7 8,4 19,0

D1-2,max 2,1 5,5 7,2 17,6

D2-3,max 1,5 3,2 4,6 11,6

RM S

D0-1,rms 1,0 2,2 3,7 8,2

D1-2,rms 0,8 1,8 3,2 7,1

D2-3,rms 0,6 1,2 2,1 4,9

Floor

acceleration

[m/s²]

MA X

A1,max 1,4 4,1 4,7 10,6

A2,max 2,0 6,7 8,2 18,7

A3,max 2,5 7,2 10,5 23,2

RM S

A1,rms 0,5 1,0 1,8 4,2

A2,rms 0,8 1,8 3,1 7,3

A3,rms 1,0 2,3 4,0 9,3

Table 5.2: Quantitative response of the building model adapted for the installation of the nonlinear vibration absorber to multisine excitations

34

5.2. Linear vibration absorber

5.2.1. Tuning

A linear vibration absorber is installed on the building model to reduce the vibrations under

harmonic excitation. The tuning procedure of Rana & Soong is applied to set the design

parameters of the linear absorber (Rana & Soong, 1998). This method applies the SDOF

formulas of Den Hartog on MDOF systems by using a modal mass ratio in the formulas. In

Den Hartog's derivation of optimal absorber parameters, it is assumed that the main system

is undamped. In the presence of damping in the main system, no closed-form expressions

can be derived for the optimum parameters. The latter have to be determined by numerical

optimization in this case. The main system in this case (i.e. the building model) is lightly

damped, which makes it reasonable to neglect the main system damping and apply the

closed-form formulas with good approximation.

The linear absorber is tuned to the first mode of the building model as the multisine excitation

triggers vibrations of this mode. The mode shape of the first eigenmode is given in equation

5.1. The largest displacement of this mode is situated at the third floor, hence it is most

efficient to place the absorber on the third floor.

(5.1)

Since the absorber is placed on the third floor, the eigenmode is normalized to the third

coordinate. This results in the normalized mode shape of equation 5.2.

(5.2)

The first modal mass is consequently calculated with the normalized first mode shape and

the mass matrix according to equation 5.3.

(5.3)

The modal mass ratio μ1 depends on the mass of the absorber and the first modal mass.

This modal mass ratio is used in the formulas of Den Hartog for harmonic base excitation to

find the optimum parameters for the frequency and damping factor of the absorber.

μ

(5.4)

μ μ

(5.5)

35

ζ μ

μ

μ (5.6)

fopt represents the optimal ratio of the absorber's frequency to the first eigenfrequency of the

building model and ζopt represents the optimal damping factor of the absorber. These

quantities are calculated for a range of practical masses of the absorber. After the choice of

the amount of tension springs, the mass and damping of the absorber are adjusted until the

frequency and damping factor match the optimum values as close as possible. During this

iterative adjustment, the frequency and damping factor of the absorber are determined as

explained in section 3.3.

When using one tension spring at each side of the four wheel cart, the optimum parameters

were best approximated for an absorber mass of 0,147 kg and a magnet array constituted

out of one ITN-13 magnet and two S-12-06-N magnets. The optimum and real parameters of

the linear vibration absorber are listed in table 5.3.

Absorber mass ma 0,147 kg

Mass ratio μ 3,34 %

Optimum parameters

Natural frequency fn,opt 3,88 Hz

Damping factor ζopt 15,98 %

Real parameters

Natural frequency fn 3,89 Hz

Damping factor ζ 15,28 %

Stiffness k 87,90 N/m

Damping constant c 1,03 Ns/m

Table 5.3: Parameters of the linear absorber in case of harmonic excitation

5.2.2. Frequency response function

After the implementation of the linear vibration absorber, the FRF of the modified building

model is measured with the use of an impact hammer excitation and it is compared to the

original FRF. Both FRF's are plotted in figure 5.3; these FRF's are the accelerance when the

acceleration is measured on the third floor for an impact force on the third floor. The original

FRF has three resonances, marking the three eigenmodes, and two antiresonances in

between. The addition of the linear vibration absorber alters the FRF significantly around the

first resonance; this resonance peak is replaced by two neighbouring peaks with smaller

magnitude. The response of the building model's first eigenmode will thus be much more

reduced. The two smaller peaks differ in magnitude, which is normally not the case for the

formulas of Den Hartog. This is, however, a consequence of the damping in the building

model, which was neglected by using the closed-form expressions of Den Hartog. The linear

absorber, tuned to the first mode, has an effect on the other modes as well: the second mode

36

is moderately reduced and the third mode is slightly reduced. These observations agree with

the findings in the paper of Rana & Soong.

5.2.3. Performance

The implementation of the linear vibration absorber improves the response of the building

model during the multisine excitation significantly. Figure 5.4 shows a part of the response of

the modified building model to the multisine excitation with an amplitude limit of 2 mm. It is

visible that the response of the three floors is reduced with respect to the original case, while

the linear absorber reacts heavily to the multisine excitation. The linear absorber thus

alleviates the building model by taking and dissipating a large portion of the vibration energy.

Figure 5.4: Response of the building with linear absorber to the multisine excitation of 2 mm

Figure 5.3: Frequency response function before and after implementation of the linear vibration absorber

37

The response of the modified building model is quantified and compared to the original

building model in table 5.4. Two parameters are added to quantify the action of the linear

absorber: the relative displacement of the absorber with respect to the third floor D3-a and the

acceleration of the absorber Aa. It is visible that the absorber undergoes much higher

displacements and accelerations than the floors of the building model for each intensity of

the multisine excitation. The response of the floors, on the other hand, is significantly

reduced with ± 40 to 60 % for the interstory drifts and with ± 40 to 70 % for the floor

accelerations. There is a trend visible in the performance of the linear absorber as well: the

response is reduced relatively more for a higher excitation intensity.


Interstory

drift

[mm]

MA X

D0-1,max 2,1 (-46,2%) 4,0 (-44,4%) 7,8 (-54,1%) 9,1 (-57,5%)

D1-2,max 1,8 (-43,8%) 3,0 (-50,8%) 6,0 (-52,8%) 6,9 (-59,9%)

D2-3,max 1,1 (-42,1%) 1,9 (-45,7%) 4,0 (-48,7%) 4,2 (-58,0%)

D3-a,max 10,2 20,6 41,9 51,7

RM S

D0-1,rms 0,8 (-42,9%) 1,5 (-44,4%) 2,6 (-62,9%) 3,4 (-62,2%)

D1-2,rms 0,6 (-45,5%) 1,2 (-42,9%) 2,1 (-63,2%) 2,7 (-63,5%)

D2-3,rms 0,3 (-50,0%) 0,7 (-41,7%) 1,2 (-61,3%) 1,6 (-59,0%)

D3-a,rms 3,6 7,9 14,3 19,3

Floor

acceleration

[m/s²]

MA X

A1,max 2,0 (-42,9%) 2,9 (-56,1%) 5,9 (-68,9%) 7,3 (-68,9%)

A2,max 2,3 (-50,0%) 4,4 (-54,6%) 8,1 (-63,5%) 10,1 (-65,2%)

A3,max 2,8 (-49,1%) 4,6 (-56,6%) 9,4 (-65,2%) 11,5 (-64,4%)

Aa,max 6,9 13,9 28,2 35,5

RM S

A1,rms 0,5 (-50,0%) 0,9 (-52,6%) 1,5 (-75,0%) 2,0 (-73,7%)

A2,rms 0,7 (-53,3%) 1,5 (-51,6%) 2,6 (-69,0%) 3,5 (-68,2%)

A3,rms 1,0 (-47,4%) 1,9 (-50,0%) 3,3 (-69,2%) 4,4 (-68,3%)

Aa,rms 2,5 5,4 9,6 12,8

Table 5.4: Quantitative response of the building with a linear absorber to multisine excitations

5.3. Nonlinear vibration absorber

5.3.1. Tuning

In a second approach, a nonlinear vibration absorber is installed on the building model to

reduce the vibrations under the multisine excitation. The nonlinear absorber is also tuned to

the first mode and placed on the third floor. A tuning procedure designed for a NES is applied

to find the optimal stiffness of the nonlinear absorber (Vaurigaud et al., 2011). This method

only optimizes the stiffness of the nonlinear absorber based on the absorber's mass and

damping, the dynamic properties of the building model and the energy introduced by the

excitation. The damping is thus not a design parameter but some damping has to be present

38

to dissipate the vibration energy of the building model and create the so-called targeted

energy transfer (TET). The viscous damping between the slider and the rail is always present

and has an amount of c = 1,0385 Ns/m. Some additional magnetic damping, like in the case

of the linear absorber, is also added to increase the energy dissipation in the NES and hence

improve its efficiency. Experimentally the NES was found to work more efficient when a

magnet array of one ITN-13 magnet and two S-12-06-N magnets is added to the nonlinear

absorber. This magnet array imposes an additional damping constant of c = 1,03 Ns/m so

that the total damping constant of the nonlinear absorber is ca = 2,0685 Ns/m.

The vibration energy in the building model, due to the mulitisine excitation, is a crucial

quantity in the determination of the optimal nonlinear stiffness. The value of the nonlinear

stiffness is adjusted so that the energy threshold of the nonlinear absorber is lower than the

energy level in the building. This creates the right conditions for the TET phenomenon and

the reduction of the building's vibrations. The tuning procedure of the nonlinear absorber thus

comes down to the optimization of the energy threshold. To this end the nonlinear spring

constant knl should be equal to or larger than the optimal value according to equation 5.7.

ω

ζ

(5.7)

ma is the mass of the absorber; this includes the slider and all the extra elements attached to

it (additional steel plates, screws, magnet array, accelerometer). ω1* is the first angular

eigenfrequency calculated with the first modal mass M1* and the first modal stiffness K1*

according to equation 5.8.

ω

(5.8)

The first modal mass and stiffness are extracted from the concerning modal matrices, which

are calculated with the modal shape matrix E according to equations 5.9 and 5.10.

(5.9)

(5.10)

E1(3) is the coordinate of the first mode shape at the location of the nonlinear absorber,

which is the third floor in this case. Parameter X depends on the frequency of the excitation

on the main system. The multisine displacement signal excites the building model at different

frequencies but mainly triggers the first mode so that the first eigenfrequency is chosen as

excitation frequency ω in this tuning procedure. This results in a unit value of parameter X.

ω

ω (5.11)

39

The damping of the nonlinear absorber is included in parameter ζ, which is given in equation

5.12. The parameter - depends on this parameter and is given in equation 5.13.

ζ

ω

(5.12)

ζ

(5.13)

At last there is the parameter 0 to include the energy level of the building model in the

tuning formula. This parameter depends on the first modal eigenfrequency and the stationary

value of the displacement of the building model ustatio.

ω (5.14)

ustatio is calculated by considering the MDOF building model as a SDOF model with the modal

parameters of the first mode and applying a sine excitation at the base; this is modeled with

equation 5.15. The sine has a frequency equal to the first eigenfrequency of the building

model and its amplitude A is varied from 1 to 4 mm. The peak acceleration of this sine is

taken to model the inertia forces induced by the base excitation; this is equal to Aω .

(5.15)

The frequency response function of this SDOF system is calculated and ustatio is set equal to

the response at the first eigenfrequency due to the base sine excitation. This results in the

values listed in table 5.5 for ustatio and the related optimum values for the nonlinear stiffness

constant knl,opt.

Multisine excitation ustatio [m] knl,opt [N/m³]

1 mm 0,0019 17544000

2 mm 0,0037 1220000

3 mm 0,0056 232490

4 mm 0,0074 76248

Table 5.5: Values for ustatio and the optimum stiffness constant knl,opt

of the nonlinear absorber for each intensity of the multisine excitation

5.3.2. Performance

During the implementation of the nonlinear absorber, it is experimentally found that the

optimum value predicted by the tuning procedure differs from the real optimum value. Due to

the various assumptions in the theoretical tuning procedure, the energy threshold and the

related optimum value for the nonlinear spring constant is only estimated approximately.

Hence the obtained values are considered as an indication of the real optimum value after

40

which an experimental finetuning is executed. This is done by increasing and decreasing the

value of the spring constant and observing the change in performance of the absorber. The

best performance was obtained for spring constants higher than the predicted optimum

value; the applied stiffness constants for each intensity of the multisine excitation are listed in

table 5.6 along with the other characteristics of the nonlinear absorber.



Damping constant ca 1,88 Ns/m

Spring constant 1 mm knl,1mm 42262126 N/m³




Table 5.6: Parameters of the nonlinear absorber in case of harmonic excitation

The response of the modified building model to the multisine excitation of 2 mm is shown in

figure 5.5. It is clear that the vibrations of the building model are reduced significantly when

comparing this response to the original one in figure 5.2. A large part of the excitation energy

is dissipated by the action of the NES, which decreases the vibration energy in the building

and hence the oscillations of the building floors. The performance of the nonlinear absorber

is also quantified by looking at the interstory drifts and the accelerations of the floors. Both

performance criteria are listed in table 5.7 for each intensity of the multisine excitation;

relative reductions of ± 40 to 70 % are obtained. When comparing the response of the

building with a nonlinear absorber to the response of the building with a linear absorber

(figure 5.4), it is visible that the oscillations of the nonlinear absorber are smaller than those

of its linear counterpart while the reduction of the building vibrations is more or less the

same. Hence the nonlinear absorber works more efficient when the magnitude of the

absorber's oscillations has to be limited.

Figure 5.5: Response of the building with nonlinear absorber to the multisine excitation of 2mm

41


Interstory

drift

[mm]

MA X

D0-1,max 1,3 (-48,0%) 2,3 (-65,7%) 4,1 (-51,2%) 7,3 (-61,6%)

D1-2,max 1,1 (-47,6%) 2,2 (-60,0%) 3,7 (-56,9%) 5,4 (-69,3%)

D2-3,max 0,9 (-40,0%) 1,7 (-46,9%) 2,8 (-39,1%) 4,1 (-64,7%)

D3-a,max 2,4 7,1 20,7 32,5

RM S

D0-1,rms 0,5 (-50,0%) 0,9 (-59,1%) 1,3 (-64,9%) 2,3 (-72,0%)

D1-2,rms 0,5 (-37,5%) 0,8 (-55,6%) 1,1 (-65,6%) 2,0 (-71,8%)

D2-3,rms 0,3 (-50,0%) 0,6 (-50,0%) 0,8 (-61,9%) 1,5 (-69,4%)

D3-a,rms 0,8 2,7 7,1 10,7

Floor

acceleration

[m/s²]

MA X

A1,max 0,9 (-35,7%) 1,6 (-61,0%) 3,7 (-21,3%) 6,6 (-37,7%)

A2,max 1,2 (-40,0%) 2,2 (-67,2%) 4,5 (-45,1%) 8,1 (-56,7%)

A3,max 1,3 (-48,0%) 2,6 (-63,9%) 4,9 (-53,3%) 8,0 (-65,5%)

Aa,max 3,1 8,4 19,6 29,5

RM S

A1,rms 0,2 (-60,0%) 0,5 (-50,0%) 0,8 (-55,6%) 1,6 (-61,9%)

A2,rms 0,4 (-50,0%) 0,7 (-61,1%) 1,2 (-61,3%) 2,0 (-72,6%)

A3,rms 0,5 (-50,0%) 1,0 (-56,5%) 1,6 (-60,0%) 2,5 (-73,1%)

Aa,rms 0,9 2,1 3,8 5,0

Table 5.7: Quantitative response of the building with a nonlinear absorber

to multisine excitations

5.3.3. Frequency response function

The FRF of the modified building model with a nonlinear absorber is determined by applying

a sweep excitation and measuring the frequency content. An impact hammer test is not

effective in this case as there is hardly some relative movement of the absorber after a knock

of the impact hammer. The FRF is obtained by estimating the cross spectral density of the

Figure 5.6: Frequency response function before and

after implementation of the nonlinear vibration absorber

42

excitation and the response of the third floor of the building. A sweep excitation with an

amplitude of 1 mm is applied so that the nonlinear spring constant knl,1mm is installed for the

modified building. It is visible that the nonlinear absorber affects the resonance peak of the

first mode heavily; the peak is flattened out and the nonlinear absorber is thus effective for

wider frequency band than its linear counterpart. The resonance peaks of the second and

third mode are also reduced considerably.

5.4. Comparison of the performance

In this section, the quantitative performance of both absorbers under the multisine excitation

is discussed and visualized with diagrams. Figures 5.7 and 5.8 present the relative reduction

of the maximum value and RMS value of the interstory drifts for both absorbers. Based on

this performance criterion, it can be seen that both absorbers work well under harmonic

excitation because the values are reduced with 40 to 70 % in all cases. Besides, there is a

trend visible in the performance of both absorbers: the relative reduction of the interstory drift

is higher for a larger intensity of the multisine excitation. When comparing the performance of

the absorbers, it is visible that the nonlinear absorber performs better or equally good than

the linear absorber. There are some exceptions here and there, but the overall conclusion is

that the nonlinear absorber performs slightly better when the minimization of the interstory

drifts is an objective.

Figure 5.7: Relative reduction of the maximum interstory drifts of the building model due to the implementation of the linear and nonlinear vibration absorber

0

10

20

30

40

50

60

70

80

1 2 3 4

Rela

tive r

eduction [

%]

Maximum amplitude multisine excitation [mm]

Maximum interstory drift

D0-1 Linear absorber

D1-2 Linear Absorber


D0-1 Nonlinear Absorber



43

When looking at the relative reduction of the floor accelerations of the building, it can also be

concluded that both absorbers perform well with reductions ranging from 40 to 70 %. The

same trend as for the interstory drifts is visible: the absorber performance improves with

increasing intensity of the multisine excitation. The comparison between both absorbers does

not really yield a better absorber type. Based on the RMS values, both absorbers perform

equally well. The maximum floor accelerations are better reduced in the case of a linear

absorber, especially for the higher excitation intensities.

Figure 5.9: Relative reduction of the maximum floor accelerations of the building model due to the implementation of the linear and nonlinear vibration absorber

0

10

20

30

40

50

60

70

80

1 2 3 4

Rela

tive r

eduction [

%]


Maximum Floor acceleration

A1 Linear absorber

A2 Linear Absorber

A3 Linear Absorber

A1 Nonlinear Absorber



0

10

20

30

40

50

60

70

80

1 2 3 4

Rela

tive r

eduction [

%]


RMS interstory drift

D0-1 Linear absorber






Figure 5.8: Relative reduction of the root mean square interstory drifts of the building model due to the implementation of the linear and nonlinear vibration absorber

44

Figure 5.10: Relative reduction of the root mean square floor accelerations of the building model due to the implementation of the linear and nonlinear vibration absorber

At last the action of both absorbers is considered in figures 5.11 and 5.12. A first logical

observation is that both the relative displacement and the acceleration of the absorbers

increases with an increasing intensity of the multisine excitation. A higher load intensity

simply brings more vibration energy into the building which has to be dissipated. Hence the

action of the absorbers increases to dissipate this larger amount of energy. Secondly it is

clearly visible that the nonlinear absorber undergoes smaller oscillations and accelerations.

When the minimization of the absorber's actions is an objective, the nonlinear absorber is

thus more suited for vibration reduction.

Figure 5.11: Relative displacement of both absorber types in case of multisine excitation

0

10

20

30

40

50

60

70

80

1 2 3 4

Rela

tive r

eduction [

%]


RMS Floor acceleration

A1 Linear absorber

A2 Linear Absorber

A3 Linear Absorber




0

10

20

30

40

50

60

1 2 3 4

Dis

pla

cem

ent [m

m]


Relative displacement absorber

Dmax linear absorber

Drms linear absorber

Dmax nonlinear absorber

Drms nonlinear absorber

45

Figure 5.12: Acceleration of both absorber types in case of multisine excitation

As a final, general conclusion on the use of both vibration absorbers for the multisine

excitation, it can be said that both absorbers perform well but that the nonlinear absorber is

better suited for this excitation type. The latter statement is based on the better performance

for interstory drift reduction and the lower action of the absorber itself, while the performance

for floor acceleration reduction is similar to the linear absorber. When looking at the

frequency content of the harmonic excitation, this observation is somewhat expected. The

multisine signal excites the building with a range of frequencies around the first

eigenfrequency. The nonlinear absorber is able to capture different frequencies while the

linear absorber always oscillates at its natural frequency. This frequency characteristic of the

nonlinear absorber is advantageous in the dissipation of the multisine excitation and leads to

a better performance.

0

5

10

15

20

25

30

35

40

1 2 3 4

Accele

ration [

m/s

²]


Acceleration absorber

Amax linear absorber

Arms linear absorber

Amax nonlinear absorber

Arms nonlinear absorber

46

Chapter 6

Performance under impulse excitation

6.1. Base step excitation

This chapter considers the implementation of vibration absorbers to reduce the transient

vibrations of the building under impulse excitation. The impulse is introduced at the base of

the building via a displacement step of the horizontal shaker. A theoretical step signal would

involve an instantaneous displacement, which results into a vertical line connecting two

horizontal branches in the displacement signal. The shaker, however, needs a finite time to

complete the predefined displacement. This is visible on the zoomed time signal in the right

part of figure 6.1; a step displacement of 15 mm takes ± 0,08 s. After the completion of the

step, some minor oscillations of the shaker occur, but overall the horizontal shaker is able to

produce a good step displacement for the impulse excitation of the building model.

Figure 6.1: Step displacement of the horizontal shaker for impulse excitation

To assess the performance of the absorbers on various load levels, impulse excitations of

different intensities are considered by varying the magnitude of the displacement step (5 mm,

10 mm, 15 mm & 20 mm). The response of the original building model is similar for each

impulse; it is shown for a displacement step of 15 mm in figure 6.2. Immediately after the

step of the shaker, each floor gets a large initial displacement. This is then followed by a free

vibration which lasts relatively long because of the low damping factors of the building model.

The first mode is dominant during this free vibration as the three floors roughly move in

phase. Just after the impulse, vibrations of higher frequency are also visible coming from the

second and third mode. These vibrations decay faster than the first mode vibration because

47

theoretically each mode of a proportional damped system decays exponentially according to

exp(-ωi ζi t). The three damping factors are similar in magnitude but the higher frequency of

the second and third mode results in a faster exponential decay of these modes. The

dominance of the first mode is also visible in the spectrum of the displacement signals of

each floor (figure 6.3).

Figure 6.2: Response of the building model to a step displacement of 15 mm (overall view of the decaying vibration (above) and zoomed view on the first oscillations (below))

Figure 6.3: Spectrum of each floor displacement signal in case of impulse excitation

48

To quantify the performance of the vibration absorbers, the response of the original building

model will be compared to the response of the modified building model. This is done by

looking at the magnitude of the first displacement and the 10% settling time of each floor.

The 10% settling time is the time after which the displacement of the vibration is smaller than

10% of its initial value. Again a distinction is made between the original building model and

the building model with the attached profile and rail for the installation of the nonlinear

absorber. The second case adds a considerable mass to the building model, which alters the

dynamic properties of the building model and hence its response on the impulse excitations.

The performance of the nonlinear absorber should be compared to this original model to

obtain a correct idea of its performance. The quantitative response in both cases is given in

tables 6.1 and 6.2.

Step displacement 5 mm 10 mm 15 mm 20 mm

Maximum

displacement

[mm]

Floor 1 2,8 5,5 8,6 11,7

Floor 2 3,9 8,1 12,7 16,4

Floor 3 5,5 11,4 17,1 22,7

10% Settling

time [s]

Floor 1 23,7 21,7 18,0 15,4

Floor 2 23,4 23,3 21,2 18,3

Floor 3 25,0 21,4 19,4 16,7

Table 6.1: Quantitative response of the original building model


Maximum

displacement

[mm]

Floor 1 2,8 6,5 9,0 12,5

Floor 2 3,8 8,3 12,6 17,1

Floor 3 4,8 10,1 15,1 19,9

10% Settling

time [s]

Floor 1 48,9 27,1 21,8 16,7

Floor 2 46,5 36,6 29,2 22,7

Floor 3 42,7 33,3 28,3 25,3

Table 6.2: Quantitative response of the building model adapted

for the installation of the nonlinear vibration absorber

For both cases, it can be observed that the maximum floor displacement increases with an

increasing impulse intensity, which is logical. The settling time, however, decreases with an

increasing intensity of the impulse excitation. This means that larger vibrations are reduced

relatively faster due to the inherent damping of the building model. A larger initial

displacement results in a longer free vibration though. It can also be observed that the

adapted building model, which is heavier, gets a somewhat smaller initial displacement and a

longer settling time. The latter is a result of the larger mass, which lowers the damping

factors as can be seen in the definition of the damping factor (equation 3.8). Due to the lower

damping factors, it will take longer to damp the vibrations under 10 % of its initial value.

49

6.2. Linear vibration absorber

6.2.1. Tuning

The tuning procedure of Rana & Soong as explained in section 5.2.1 is again used to tune

the parameters of the linear absorber. This tuning procedure is actually optimized for

harmonic base excitation but is used here for impulse base excitation as an approximate

method. The linear absorber is tuned to the first mode and placed on the third floor because

the first mode is dominant in the free vibration caused by the impulse excitation. The

parameters of the linear absorber are given in table 6.3.



Optimum parameters



Real parameters





Table 6.3: Parameters of the linear absorber in case of impulse excitation

6.2.2. Performance

The installation of a linear vibration absorber has a serious influence on the duration of the

free vibration of the building model under impulse excitation. Figure 6.4 shows the response

of the modified building model to an impulse with a step displacement of 15 mm. After the

step displacement of the horizontal shaker, the absorber reacts heavily while the vibrations of

the building floors decrease rapidly. The absorber keeps oscillating with respect to the third

Figure 6.4: Response of the building with a linear absorber to a step displacement of 15 mm

50

floor until the first mode is died out almost completely. The second and third mode vibrations

last longer because these are not really reduced by the linear absorber; they are mainly

damped by the inherent damping of the building model.

The performance of the linear absorber on impulse excitations is quantified in table 6.4. Note

that there are no results for a base step displacement of 20 mm; the intensity of this impulse

excitation was too high for a proper functioning of the damping in the linear absorber (the

magnet array constantly bumped against the interior or the aluminum conductor tube). The

results show that the maximum, initial displacement of the floors is hardly modified; at some

cases it is slightly lower while at other cases it is slightly higher. Hence it can be concluded

that the linear absorber does not alter the maximum floor displacements under impulse

excitation. The 10% settling time, on the other hand, is modified greatly: it is reduced with ±

80 to 90 % for each floor at each intensity of the impulse excitation. The linear absorber thus

has very good effect on the duration of the transient vibrations of the building model.

Step displacement 5 mm 10 mm 15 mm

Maximum

displacement

[mm]

Floor 1 2,6 (-7,1%) 5,6 (+1,8%) 8,6 (0%)

Floor 2 3,6 (-7,7%) 7,9 (-2,5%) 11,8 (-7,1%)

Floor 3 5,2 (-5,5%) 11,0 (-3,5%) 17,2 (+0,6%)

Absorber 10,1 21,3 33,5

10% Settling

time [s]

Floor 1 4,0 (-83,1%) 3,8 (-82,5%) 3,6 (-80,0%)

Floor 2 4,0 (-82,9%) 2,8 (-88,0%) 2,8 (-86,8%)

Floor 3 2,9 (-88,4%) 2,5 (-88,3%) 2,2 (-88,9%)

Table 6.4: Quantitative response of the building with a linear absorber for impulse excitations

6.3. Nonlinear vibration absorber

6.3.1. Tuning

In this section, a nonlinear vibration absorber is installed on the building model to reduce the

transient vibrations under impulse excitation. The nonlinear absorber is also tuned to the first

mode and placed on the third floor. A similar tuning procedure to the one in section 5.3.1 is

applied to find the optimal stiffness of the nonlinear absorber (Vaurigaud et al., 2011). This

tuning method comes down to the optimal placement of the energy threshold based on the

energy introduced by the impulse excitation. To obtain targeted energy transfer in a stable

manner, the stiffness of the nonlinear absorber should be equal to or larger than the optimal

value knl,opt given in equation 6.1.

ω

ζ

(6.1)

51

ma is the mass of the absorber; this is the slider and all the extra elements attached to it

(additional steel plates, screws, magnet array, accelerometer). ω1* is the first eigenfrequency

calculated with the first modal mass M1* and the first modal stiffness K1* according to

equation 6.2.

ω

(6.2)

E1(3) is the coordinate of the first eigenmode on the third floor (location of the nonlinear

absorber). The damping of the absorber due to the viscous friction between the slider and

the rail, and the added magnetic damping is quantified by the damping constant ca; this is

included in the parameter ζ according to equation 6.3. The same amount of magnetic

damping as in the case of harmonic excitation (ca=2,0685 Ns/m), is added to obtain sufficient

energy dissipation in the NES.

ζ

ω (6.3)

- depends on the parameter ζ according to equation 6.4.

(6.4)

Finally there is the parameter 00, which brings the energy in the building model due to the

impulse excitation into account. This parameter can be approximated with equation 6.5, in

which u is the displacement of the center of mass of the total system.

ω (6.5)

The impulse gives the building model a start velocity but no start displacement so that the

second term in equation 6.5 disappears. Parameter 00 can consequently be written as

equation 6.6 by applying the definition of the displacement of the center of mass u.

(6.6)

The second term in equation 6.6 can also be omitted because the absorber undergoes no

initial displacement. q1 is the first modal coordinate so that 00 finally can be expressed as

equation 6.7; it depends on the modal matrix E and the start velocity of the three floors in the

building model.

(6.7)

52

The start velocities of the floors are experimentally determined by measuring the response of

the original building model (adapted for the installation of the nonlinear absorber) to the step

displacement of the horizontal shaker. The obtained acceleration signals are integrated into

the velocity signals and the first peak velocity is selected as the start velocity for each floor.

The start velocities of the floors and the resulting optimum spring constant for each impulse

intensity are listed in table 6.5.


v1(0) [m/s²] 0,1003 0,2150 0,3178 0,4238

v2(0) [m/s²] 0,1095 0,2268 0,3343 0,4457

v3(0) [m/s²] 0,1293 0,2761 0,4117 0,5461

knl,opt [N/m³] 1777700 395530 179330 101520

Table 6.5: Start velocities of the floors and optimum

spring constant of the NES for each impulse excitation

6.3.2. Performance

Like in the case for harmonic excitation, the tuning procedure predicts the position of the

energy threshold only approximately. Some experimental trials were executed to find the

best possible stiffness of the nonlinear absorber for each step displacement. It was again

found that the real optimum value of the spring constant is higher than the predicted one.

The characteristics of the nonlinear absorber are listed in table 6.6. Note that the mass of the

absorber is chosen fairly large; this is done to keep the optimum values of the spring

constants within the practical range imposed by the available wire diameters and possible

wire lengths in the nonlinear absorber design.



Damping constant ca 1,88 Ns/m





Table 6.6: Parameters of the nonlinear absorber in case of impulse excitation

The implementation of the nonlinear absorber has a very good effect on the duration of the

transient vibrations of the building. Figure 6.5 shows the response of the modified building to

a step displacement of 15 mm. Immediately after the impulse, the absorber reacts heavily

and dissipates a large amount of energy through a number of wide oscillations. At the same

time the first mode vibrations of the building decrease rapidly. The second and third mode

vibrations, however, last longer and fade out due to the building's inherent damping which is

53

rather low. The nonlinear absorber does not affect these vibrations and its action stops

because the energy in the building is too low to trigger its motion. It is visible that the

frequency of the nonlinear absorber decreases during the energy dissipation; this is the

consequence of the mentioned frequency-energy dependence in section 4.1.2.

The performance of the nonlinear absorber for each intensity of the impulse is quantified in

table 6.7. The 10% settling time of the building floors is again heavily reduced with relative

reductions of ± 90%. The nonlinear absorber thus greatly shortens the vibration time; a still

situation of the building model is reached in just a few seconds. The maximum displacement

of each floor is also reduced, but to a smaller extent. A relative reduction of ± 10 to 30 % is

obtained and especially the maximum displacement of the second floor is reduced.


Maximum

displacement

[mm]

Floor 1 2,5 (-10,7%) 5,3 (-18,5%) 8,3 (-7,8%) 11,0 (-12,0%)

Floor 2 2,8 (-26,3%) 5,9 (-28,9%) 8,8 (-30,2%) 12,5 (-26,9%)

Floor 3 3,9 (-18,8%) 8,1 (-19,8%) 12,4 (-17,9%) 16,5 (-17,1%)

Absorber 4,7 10,5 16,5 26,2

10% Settling

time [s]

Floor 1 3,8 (-92,2%) 2,8 (-89,7%) 3,1 (-85,8%) 3,1 (-81,4%)

Floor 2 3,3 (-92,9%) 2,9 (-92,1%) 3,0 (-89,7%) 2,5 (-89,0%)

Floor 3 2,3 (-94,6%) 2,0 (-94,0%) 2,0 (-92,9%) 2,1 (-91,7%)

Table 6.7: Quantitative response of the building with a nonlinear absorber for impulse

excitations

6.3.3. Energy threshold

During the experimental search for the optimum spring constant, the existence of the energy

threshold for nonlinear absorbers was observed many times. Figure 6.6 and 6.7 show the

response of the modified building to a step displacement of 10 mm in case of a good and bad

Figure 6.5: Response of the building with a nonlinear absorber to a step displacement of 15mm

54

placement of the energy threshold. In the first case, the stiffness is sufficient (knl = 1131232

N/m³) to trigger a large reaction of the nonlinear absorber necessary for an efficient vibration

reduction. As a result the first mode vibration of the building decreases quickly. The energy

threshold is thus well positioned for the input energy caused by the step displacement of 10

mm. In the second case, the stiffness is too low (knl = 404982 N/m³) and the nonlinear

absorber only shows a moderate reaction. Hence there is less energy dissipation in the

absorber and the first mode vibration decreases slower and in a more gradual manner.

6.4. Combined implementations

Both the linear and nonlinear vibration absorber reduce the first mode vibration greatly after

the impulse excitation but the second and third mode vibrations go on relatively undisturbed.

In this section a linear absorber tuned to the second mode is added to the previous

configurations to tackle the second mode vibration. The antinode of the second mode is

Figure 6.6: Response of the building with a nonlinear absorber to a step displacement of 10mm in case of a good placement of the energy threshold (knl = 1131232 N/m³)

Figure 6.7 Response of the building with a nonlinear absorber to a step displacement of 10mm in case of a bad placement of the energy threshold (knl = 404982 N/m³)

55

situated at the first floor as can be seen in figure 2.6. Hence the linear absorber tuned to the

second mode is placed on the first floor to obtain an efficient vibration reduction. The tuning

procedure of Rana and Soong as explained in section 5.2.1 is applied on the second mode;

this results in an absorber with the parameters listed in table 6.8.



Optimum parameters



Real parameters





Table 6.8: Parameters of the linear absorber tuned to the second mode

6.4.1. Linear absorber (mode 1) and linear absorber (mode 2)

The addition of the second mode absorber to the configuration with a first mode linear

absorber has two effects on the response of the building. Figure 6.8 shows the response of

the modified building to a step displacement of 15 mm. Just after the impulse, the second

mode absorber is active for a short period and damps the second mode vibration quickly. But

the addition of the second mode absorber also affects the performance of the first mode

absorber in a detrimental way. The presence of the second mode absorber makes the

damping of the first mode vibration to occur slower, which results in a worse performance

related to the reduction of the 10 % settling time (table 6.9).

Figure 6.8: Response of the modified building with a first mode linear absorber and a second mode linear absorber to a step displacement of 15 mm

56

Step displacement 5 mm 10 mm 15 mm

Maximum

displacement

[mm]

Floor 1 2,9 (+3,6%) 5,5 (0%) 8,5 (-1,2%)

Floor 2 3,7 (-5,1%) 7,5 (-7,4%) 11,3 (-11,0%)

Floor 3 5,3 (-3,6%) 10,6 (-7,0%) 16,2 (+5,3%)

Absorber mode 1 9,6 19,8 31,3

Absorber mode 2 4,1 8,5 12,9

10% Settling

time [s]

Floor 1 5,9 (-75,1%) 5,4 (-75,1%) 3,7 (-79,4%)

Floor 2 6,5 (-72,2%) 4,8 (-79,4%) 5,1 (-75,9%)

Floor 3 5,6 (-77,6%) 4,3 (-79,9%) 4,0 (-79,4%)

Table 6.9: Quantitative response of the building with two linear absorbers

tuned to mode 1 & 2 for impulse excitations

6.4.2. Nonlinear absorber (mode 1) and linear absorber (mode 2)

The addition of the second mode absorber to the configuration with a first mode nonlinear

absorber has a better effect on the overall performance. The second mode is again damped

quickly after the impulse due to the action of the second mode absorber, and the

performance of the first mode absorber is not affected. The first mode vibration is also

damped quickly so that this configuration leads to a very good performance (table 6.10).

Figure 6.9: Response of the modified building with a first mode nonlinear absorber and a second mode linear absorber to a step displacement of 15 mm

57

0

10

20

30

40

50

60

70

80

90

100

5 10 15

Rela

tive r

eduction [

%]

Step displacement shaker [mm]

10% settling time

Linear

Nonlinear

Linear + Linear

Nonlinear + Linear


Maximum

displacement

[mm]

Floor 1 2,6 (-7,1%) 5,2 (-20,0%) 8,2 (-8,9%) 10,8 (-13,6%)

Floor 2 2,8 (-26,3%) 5,5 (-33,7%) 8,7 (-31,0%) 11,5 (-32,7%)

Floor 3 4,1 (-14,6%) 8,3 (-17,8%) 12,8 (-15,2%) 16,8 (-15,6%)

Absorber mode 1 4,9 11,3 17,3 22,9

Absorber mode 2 4,0 8,2 12,7 17,1

10% Settling

time [s]

Floor 1 3,6 (-92,6%) 2,0 (-92,6%) 2,5 (-88,5%) 1,9 (-88,6%)

Floor 2 2,5 (-94,6%) 2,4 (-93,4%) 5,0 (-82,9%) 3,3 (-85,5%)

Floor 3 2,1 (-95,1%) 1,4 (-95,8%) 3,2 (-88,7%) 3,7 (-85,4%)

Table 6.10: Quantitative response of the building with a nonlinear absorber tuned to the first

mode and a linear absorber tuned to the second mode for impulse excitations

6.5. Comparison of the performance

In this final section, the quantitative performance of the different absorber configurations

under impulse excitation is discussed and visualized with diagrams. A first performance

criterion that is related to the duration of the transient vibrations is the 10% settling time of

the building floors. The 10% settling time of the three floors is very similar for each impulse

intensity and absorber configuration; the average value of the three floors is taken here to

compare the absorbers. A first observation is that each absorber configuration has a very

good effect on the duration of the transient vibrations; the reduction of the average settling

time ranges from 75 to 94 % and it is similar for different intensities of the impulse. The

nonlinear absorber performs better than the linear absorber with ± 6 %. The installation of a

second linear absorber to reduce the second mode vibrations has a good effect on the first

mode nonlinear absorber but makes the performance of the first mode linear absorber worse.

Figure 6.10: Relative reduction of the 10% settling time for each absorber configuration

58

The maximum displacement of the building's floors, occurring just after the impulse, is

considered as a second criterion. This is reduced much less than the 10 % settling time. The

linear absorber is not really able to reduce the maximum floor displacements consistently;

sometimes it is reduced with ± 5 % while in other cases it is kept equal or even increased

(negative reduction in diagram). The nonlinear absorber, on the other hand, reduces the

maximum floor displacements with ± 10 to 30 %. Especially the second floor is altered

moderately. The addition of a second mode absorber does not really affect the performance

of this criterion.

Figure 6.11: Relative reduction of the maximum floor displacements for each absorber configuration

When looking at the action of the first mode absorber, it is visible that the absorber

undergoes higher oscillations for a higher intensity of the impulse. Like in the case of

harmonic excitation, the nonlinear absorber takes less action than the linear absorber under

Figure 6.12: Maximum displacement of the first mode absorber for each absorber configuration

-5

0

5

10

15

20

25

30

35

40

5 10 15 5 10 15 5 10 15

Rela

tive r

eduction [

%]


Maximum floor displacement

Linear

Nonlinear

Linear + Linear

Nonlinear + Linear

0

5

10

15

20

25

30

35

40

5 10 15

Dis

pla

cem

ent [m

m]


Maximum displacement absorber

Linear

Nonlinear

Linear + Linear

Nonlinear + Linear

Floor 1 Floor 3

Floor 2

59

impulse excitation. The addition of the second mode absorber has no influence on the action

of the first mode absorber.

As a general conclusion for the impulse excitation on the building model, it can be stated that

the use of a nonlinear vibration absorber is superior to the use of a linear vibration absorber.

Although the linear absorber is able to reduce the duration of the transient vibrations

considerably, the nonlinear absorber reduces the settling time even better. It also reduces

the maximum displacements to a modest extent and shows a lower action compared to the

linear absorber. The addition of a second mode linear absorber has positive effect on the

performance of the nonlinear absorber, while it affects the performance of the linear absorber

in a negative way.

60

Chapter 7

Concluding remarks

In this master dissertation the design, implementation and performance assessment of a

linear and nonlinear vibration absorber was executed. A new multi-story building model of

aluminum was designed to serve as main system for the absorber implementation. This

building model proved to be very convenient for the installation of various absorber

configurations. It was installed on an existing shaker table to apply dynamic excitations. The

building model showed to be strong enough to withstand various dynamic loads and at the

same time it is sufficiently flexible and lightly damped to clearly show the modal vibrations for

research purposes.

Both the linear and nonlinear absorber were designed as translational masses attached to

the building with springs and provided with magnetic damping. A four wheel cart serves as

absorber mass for the linear absorber; this cart is attached to the building with linear tension

springs. The mass of the nonlinear absorber is a slider mounted on a rail that is fastened to

the concerning building floor. The nonlinear spring was realized with a piano-wire concept.

The non-moving mass of both absorbers was minimized as much as possible to preserve the

dynamic properties of the original building model. This worked out well for the linear

absorber, but was not really feasible for the nonlinear absorber. The piano-wire concept

simply needs more material to realize the wide fixation of the wire. This is a disadvantage of

the nonlinear absorber design and results in two different building models to evaluate the

performance of both absorbers on. Apart from this, both absorber designs worked out well

during their implementation. The action of the absorbers was not affected by unwanted

effects and the stiffness and damping were controllable and adjustable in function of the

research. Some caution was, however, necessary for the slackness of the wire in the

nonlinear absorber. After the installation of the wire, some preparatory cycles of excitation

and retensioning are necessary to avoid a loose wire during the recording of the response to

a certain excitation.

The tuning procedure of the linear absorber proved to be more straightforward than the one

of the nonlinear absorber. The existence of an energy threshold introduces an energy

parameter in the tuning formula, which requires some theoretical assumptions and the

determination of additional parameters. Moreover it was only possible to estimate the

position of the energy threshold roughly and an experimental finetuning was necessary to

approximate the optimal nonlinear spring constant as close as possible.

Based on tests with two types of excitation, the nonlinear absorber showed a better

performance than its linear counterpart. In the case of harmonic base excitation, the

61

nonlinear absorber was able to reduce the interstory drifts more and achieve an equal

performance on the reduction of the floor accelerations. For the impulse excitation, the

nonlinear absorber performed better on both considered criteria; the 10 % settling time and

the maximum displacement of the floors was reduced more. To achieve this good

performance, the nonlinear absorber had to be tuned separately for each different intensity of

the excitation, while the same setup of the linear absorber functioned well for each intensity.

The action of the nonlinear absorber was clearly less for both excitations; the displacements

with respect to the building model and the absorber's accelerations during the dynamic

loading was considerably smaller than for the linear absorber. This makes the nonlinear

absorber more suited when a minimal action of the absorber is desired.

The general conclusion is that a better performance can be achieved with the nonlinear

vibration absorber, but this requires more effort in the tuning procedure and the adjustment of

the design parameters. The input energy from the dynamic excitation also needs to be

considered and can change the optimal stiffness of the nonlinear absorber considerably.

62

References

Loccufier, M. (2015). Structural dynamics. Ghent, Belgium.

Brandt, A. (2016). ABRAVIBE toolbox for MATLAB. Retrieved from http://www.abravibe.com/

Brandt, A. (2011). Noise and vibration analysis. Odense, Denmark: Wiley.

Ebrahimi, B., Khamesee, B., & Golnaraghi, F. (2009). A novel eddy current damper: theory

and experiment. Journal of physics D: applied physics, 42 (7), 075001 (6pp)

Marijns, L., & Geeroms, M. (2012). Ontwerp van een niet-linear trillingsabsorptie element

(Master dissertation, University of Ghent, Belgium). Retrieved from http://lib.ugent.be/

Petit, F. (2012). Exploring the limitations of linear and nonlinear vibration absorbers (Doctoral

thesis, University of Ghent, Belgium). Retrieved from https://biblio.ugent.be/

McFarland, M., Vakakis, A., Bergman, L., & Copeland, T. (2010). Characterization of a

strongly nonlinear laboratory benchmark system. Dynamics of civil structures volume 4, 1-6

Rana, R., & Soong, T. (1998). Parametric study and simplified design of tuned mass

dampers. Engineering structures, 20 (3), 193-204

Vaurigaud, B., Savadkoohi, A.T., & Lamarque, C-H. (2011). Targeted energy transfer with

parallel nonlinear energy sinks. Part I: Design theory and numerical results. Nonlinear

dynamics, 66, 763-780

Savadkoohi, A.T., Vaurigaud, B., Lamarque, C-H. & Pernot, S. (2012). Targeted energy

transfer with parallel nonlinear energy sinks. Part II: Theory and experiments. Nonlinear

dynamics, 67, 37-46

Appendix A: Design drawings building model

Fixations to the shaker Installation of wall elements

Additional mass platesClamped connections between floor and

wall elements

Fixation

Top view [mm] Cross-section [mm]

25 50 50 50 25

40606040

200

10 19,5

4

1,5

4

4

5

5

14

4 holes Ø 5 mm

For the attachment to the shaker

3 screw-threaded holes Ø 4 mm

To fixate the connection with the wall element

Wall element

Cross-section [mm]

Side view [mm]

200

270270

1,5

Floor element

Top view [mm]

Cross-section end-block [mm]

5

4

1,5

4

10

10

5

15

5

40

60

60

40

200

320

9,5 301

9,5

50

150

50

60,5 180

60,5

12 screw-threaded holes Ø 4 mm at the sides of the end-blocks

for the fixation of the wall elements in the connection

4 screw-threaded holes Ø 5 mm

For the attachment of additional mass plates

Additional mass plate

Top view [mm]

Cross-section [mm]

4 holes Ø 5 mm

for the attachment to the floor elements

3

20 180 20

220

20

120

20

20 200 20

140

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