Optimal Semi-Active Damping Of

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Chapter 2

Optimal Semi-Active Damping of

Cables

2.1 Introduction

A closed-form solution exists for the optimal damping of cable vibrations

using a linear viscous damper representing the prototype linear passivedevice [5]. An analytical solution also exists for the addition of a positive

spring element in parallel to the linear viscous damper [6].

Exact analytical solutions for other damping devices that are either

non-linear passive, or are passive and have time dependent properties, do

not exist. In general, the exact solution of a non-linear differential equation

is unknown, and only approximate solutions can be obtained [7]. These

kinds of damper are classed as, or emulated by, semi-active devices which

are the subject of this investigation.

In the literature there are several different approaches to obtaining

approximate solutions for the optimal operation of semi-active devices.

From one direction, damping strategies are developed from the linear pas-

sive damper theory based either on energy equivalence [8, 9] or mean forces

[10]. From the other direction, active control strategies obtained as numer-

ical solutions [11] are modified to exclude active forces, of which clipped

LQR/LQG control is a typical example [12]. There are also studies whichconsider the optimal tuning of combinations of elements, i.e., a Maxwell


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2.2. Problem Formulation 8

element [13].

In all of the above approaches to finding the best semi-active damping

strategy, a predefined behavior of the damping device is assumed. For

example, the optimal tuning of a pure friction device involves varying thesingle parameter which defines the dimensions of its rectangular trajectory

on the force displacement map [14], not modifying the shape of the rect-

angle itself. In contrast to these approaches this paper attempts to find an

optimal semi-active damping strategy with no constraints or limitations

other than those imposed by the definition of a semi-active device.

For this problem an evolutionary algorithm (EA) is employed [15],

which takes its inspiration from the evolutionary mechanism of naturalselection proposed by Darwin. An EA has not yet been applied to the

problem of optimal damping as far as the author is aware, however, its

interdisciplinary character means that it has already been applied in such

diverse fields as genetics [16], road traffic control [17], and structural de-

sign [18] to name a few. An EA lends itself well to the problem of optimal

damping, in that it is able to search the massive fitness landscape involved

very efficiently. Due to the relatively expensive fitness evaluation associ-

ated with a numerical cable simulation, this is a major advantage of theEA over other optimization techniques.

2.2 Problem Formulation

2.2.1 The System

The system under consideration is a cable with a transverse damper near

to one end. If the cable is modeled as a taut string (Appendix A) the cable

damper system can be represented in the following mathematical form

T2v(x, t)x2

m2v(x, t)t2

= f(t)(x a) (2.1)where T is the cable force, m is the cable mass per unit length, v

(x, t

)is the transverse cable displacement, and f(t) is the force applied to thecable. Furthermore, a is the location of the damper from the left support,


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9 Chapter 2 Optimal Semi-Active Damping of Cables

x is the distance along the cable from the same support, and is a Dirac

function.

TT

c

aL

mx

Figure 2.1: Schematic of cable damper system

In this formulation the semi-active device considered can only remove

energy from the cable, i.e., it is limited to the development of dissipative

forces only. Such a device can be described mathematically as

f(t) = c(t)v(a, t)t

(2.2)

where c

(t

)is the damper viscosity which is an exogenously controllable

function of time t and must by definition satisfy the equality c(t) 0. Aschematic of the entire system is displayed in Fig. 2.1. For the purposesof this investigation, the internal damping of the cable is neglected.

The semi-active device is restricted to purely negative powers P(t) suchthat P(t) 0, where

P

(t

)= f

(t

)v

(a, t

)t

= c

(t

) v

(a, t

)t

2

. (2.3)

Following this convention, the force displacement trajectory and force

velocity trajectory of a spring with positive stiffness and linear viscous

damper, respectively, show the behaviors depicted in Fig. 2.2.

2.2.2 Optimization Problem

The optimization problem can be formulated as follows: Given a cable withinitial energy E, being statically displaced in a single mode n, assuming


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2.2. Problem Formulation 10

1 0 11

0

1

Displacement (m)

Force(N)

1 0 11

0

1

Velocity (m/s)

Force(N)

Figure 2.2: Force displacement trajectory of spring with positive stiffness (left)and force velocity trajectory of linear viscous damper (right)

a sinusoidal mode shape

v(x,0) = sin(nx/L) (x,0)t

= 0, (2.4)

copt(t) is sought such thatE(copt(t)) E(c(t)) [0, tp] (2.5)

for all c(t) in the search space, where E is the energy remaining in thecable after a time tp, wheretp =

2

n

mL2

T. (2.6)

The time tp defined in (2.6) corresponds to one time period of the mode

under consideration.

The problem is formulated so that the solution is as close as possiblyoptimal for a single mode of vibration. To this end, the initial conditions

dictate that the energy of the cable is completely in one mode n. However,

when applying a nonlinear damper to a cable with energy in a single mode,

spillover of energy to higher modes occurs, resulting in a distribution of en-

ergy over several modes [8]. By limiting the time tp to a single period, the

effects of spillover on the desired conditions of the optimization are min-

imized. Reducing the number of time periods considered also minimizes

the computation time needed to solve the problem when it is reformulatedfor an EA in the next part of the investigation.


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2.3 Methodology

The two main components of the optimization method are an EA and a

numerical cable simulation. The following subsections will describe how toadapt the problem formulated in the previous section for an EA, and detail

how the two components of the method should be implemented thereafter.

2.3.1 Chromosome Encoding

The time dependent viscosity c

(t

), given as the candidate solution of the

optimization in the problem formulation, is not directly applicable in itscontinuous form to the encoding of a chromosome in the EA. However, by

discretizing c(t), a finite array of real numbers can be obtained which isof a suitable form.

[ck] = [ck = c(kTs)k = 0, 1, 2,...,Nc 1] (2.7)Equation (2.7) describes the chromosome as a series ofNc discrete viscous

levels with time steps Ts over the optimization time tp.

2.3.2 Cable Simulation

For the simulation of the cable and damper response, the partial differential

equation is dicretized with the spatial sampling interval x and put into

the standard form

Mv(t) +Cv(t) +Kv(t) = f(t) (2.8)where M is the mass matrix, C is the structural damping matrix, and K

is the stiffness matrix. From this, the standard state space form can be

constructed

z

(t

)=

0 I

M1K M1C

z

(t

)+

0

M1

f

(t

)(2.9)

z(t) = v(t)v(t) (2.10)


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2.3. Methodology 12

where is the shape function which defines the point at which the

damper force f(t) acts on the cable.The integration method of choice to solve (2.9) and (2.10) is a fourth-

order accurate method based on integration by parts [19]. This method isimplemented with the sampling time t.

2.3.3 Evolutionary Algorithm

Although the choice of chromosome has already been discussed, there are

many more design parameters to consider for the EA. An overview of the

EA flow diagram is presented in Fig. 2.3.

Test fitness of

individuals in

population

Select

two individuals

to reproduce

Is Eq. (11) true?

Create two new

variations of

selected individuals

Replace two

individuals in

population

Generate

initial

population

Terminate

algorithm

Figure 2.3:Flow diagram


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Fitness Function

The fitness function is slightly different to the optimization criterion given

in the problem formulation, i.e., the energyE remaining in the cable afterthe time tp. Instead, the fitness function is defined as the total energy

removed by the damper from the cable over the time tp, E - E . This

is done so that the fitness increases towards an optimal solution following

convention.

Initial Population and Selection

The initial population is created as a set of random individuals based on

a Wiener process. An individual of the initial population is given by

[ck0] = [ck0 = c0(kTs)k = 0, 1, 2,...,Nc] (2.11)c0 =W(t) min(W(t)) [0, tp] (2.12)

W

(t

)W

(s

)

t sN

0, 2initial

0 s < t tp (2.13)

W(0) = 0 (2.14)where W is a random variable, and N is a normal distribution with

standard variance initial.

The selection method chosen is rank selection. This means that the

probability that a creature is selected is f

/F, where f is the ranking of the

creature (1 being the least fit) and F the sum of the rankings of the entire

population. This method outperforms other common methods with fitnessbias such as roulette selection. Its advantage over the latter method is its

independence of fitness bias with changing fitness gradients [20].

Replacement

The method of choice here is rank replacement. For this method the

probability that a member of the population will be replaced by a childis D/d, which depends inversely on the ranking of the population member


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2.3. Methodology 14

where d is the inverse ranking of a creature and D is the sum of the

inverse rankings of the entire population. This method is chosen in favor of

other methods such as random replacement, roulette replacement, absolute

fitness replacement, and locally elite replacement for the same reason as

in with the selection method [20].

Crossover

When creating two children from two parents, crossover enables the par-

ents genes to be combined to create the childrens genes. This facilitates a

more global search of all data structures accessible to the EA. The methodchosen here is two-point crossover. This means that two random loci are

generated on each of the childrens genes, whereby the first parents genes

are copied to before and after the two loci, and the second parents in

between.

Mutation

Mutation differs as an operator for diversification from crossover in that it

facilitates a more local search of data structures. It does this by making

small random changes to the children created by their parents. The chosen

method of mutation here is probabilistic mutation with rate , where the

rate is usually equal to the reciprocal of the gene length. This means that

a mutation occurs at every point on the gene with a probability . The

type of mutation that occurs is a Gaussian mutation, which means the

addition of a random variable, this variable having a normal distributionwith mean 0 and standard variance .

Population Size and Model of Evolution

The population size is a trade-off between diversity and computing time.

Smaller populations may not contain enough diversity or randomness to

find a good solution to the problem posed to the EA, however, too largea population may lead to the burning of excessive randomness and hence


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much larger computation times. The model of evolution used for this

investigation is a steady-state evolutionary algorithm. For this model every

act of selecting a pair of parents and replacing (or not) two members of

the population with children is counted as one generation.

Termination Condition

Without knowing the solution to the optimization problem in advance,

it is difficult to devise a good termination condition. However, with an

idea of the computation time needed to run one generation of the EA, a

reasonable condition is to say, stop the algorithm when

mating events = 10000 (2.15)

This end condition is reasonably good if the EA is run many times with

different initial conditions. From these multiple runs, the probability that

the final solution is globally optimal will be greatly increased.

2.4 Results

2.4.1 Parameters of Cable Simulation

A cable is considered with the following set of fictitious properties: tension

force T = 300N, mass per unit length m = 2kg

/meter, cable length L =

4 meters, and damper position a = 0.02L (2% of cable length). For the

cable simulation the number of discrete increments of viscosity Nc = 66,the cable is divided into 100 segments whereby X= L/100, the first modeis considered n = 1, and the sampling time T = Ts/2.2.4.2 Parameters of Evolutionary Algorithm

When looking at the evolution of the population over time, the fittest in-

dividual in the population of each generation was seen to converge asymp-totically to a solution. For different parameters of the evolution, this


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2.4. Results 16

convergence was quicker or slower. It was found that a population of 600,

initial = 3000, = 100/n, and = 1/Nc yielded good results. Figure 2.4and Fig. 2.5 show the best fitness evolution and standard deviation evolu-

tion, respectively, for all of the EA runs. The run which performed the best

was selected as the solution and is depicted in both plots with a thicker

line. Figure 2.6 shows a fitness histogram for the selected run, where the

frequency is the number of population individuals within each bin.

0 2000 4000 6000 8000 100000.4

0.5

0.6

0.7

0.8

0.9

1

Mating Events

BestFitness

Other runs

Best run

Figure 2.4: Best fitness

2.4.3 Results of Evolutionary Algorithm for a Single

Cable

The fittest individual in the final generation is shown in Fig. 2.7. The

solution shows two large peaks which represent periods of time in which

the damper effectively tries to hold the cable. It is at the lower viscosity

levels between these peaks where most of the damper displacement and

energy dissipation occur. A consequence of this is that there is still some

randomness at the higher levels of the profile, due to its insignificance

with respect to the damper performance, and a much smoother profileat lower viscosities for the contrary reason. Like many optimal solutions


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0 2000 4000 6000 8000 100000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Mating Events

FitnessStandardDeviation

Other runs

Best run

Figure 2.5: Standard deviation

Figure 2.6: Population fitness histogram evolution with 50 bins


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2.4. Results 18

the damper holds the cable at certain points in time, not because this

optimally removes energy at this time instant, but because it will improve

the potential of the damper to remove energy more efficiently at later time

instants. Figure 2.7 also shows the damper power which illustrates this

point.

0 0.1 0.2 0.3 0.4 0.5 0.6

0

1

2

x 104

Time (s)

Viscosity(Ns/m)

0 0.1 0.2 0.3 0.4 0.5 0.6

8

6

4

2

0

Time (s)

Power(J/s)

Figure 2.7: Viscosity profile of solution (above) and damper power (below)

In Fig. 2.8, the solution is plotted as force against displacement at

damper position, where the path of the plot proceeds in an anticlockwise

direction with time due to (2.2). Characteristics emerge from the plot

which are remarkably similar to those of a friction device with a negative

stiffness [4]. A negative stiffness acts contrary to a positive stiffness in that

as deformation occurs an active force is developed which helps the deforma-tion to proceed. The reason that this phenomenon can be observed here,

where the damper is limited to passive forces and is clearly not unstable,

is that the friction device acts in parallel, keeping the summed forces of

the combined elements from becoming active. The negative power seen

in Fig. 8 also confirms the dissipative nature of the simulated semi-active

damper. A gap in the force displacement loop is present due to the differ-

ence between the theoretical time period for a cable without an external

damper used in the simulation, and the actual time period when applyingthe nonlinear device.


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A qualitative explanation can be given for the presence of a negative

stiffness in the EA result. It is known that for energy dissipating elements,

such as viscous dampers and friction devices, the optimal tuning is a com-

promise between the maximum force developed and amplitude of response,

i.e., the maximum dissipation of energy occurs when the enclosed area of

the force displacement trajectory is maximized. A negative stiffness, as

mentioned above, develops a force in the direction of displacement. This

means that as the damper goes towards its maximum displacement, the

negative stiffness can be viewed as acting to increase the amplitude of

response, thereby increasing the total energy dissipated.

0.008 0.006 0.004 0.002 0 0.002 0.004 0.006 0.008

60

40

20

0

20

40

Displacement (m)

Force(N)

Cross indicating step in control viscosity

Figure 2.8: Damper force against displacement

A comparison by numerical simulation is now made with two populardamping strategies. These are optimal linear viscous damping and clipped

LQR control assuming an ideal semi-active device, i.e., no dynamics and

constraints. The weighting of the cost function in the LQR solution is

optimally tuned so as to remove the maximum energy from the cable over

the optimization time tp. In practise, an observer must be used with fewer

states than the continuous cable on which it is implemented. This makes

the LQG controller unreliable, due to the unobserved modes. However,

this is not the case for a semi-active device which is limited to passiveforces [21].


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2.4. Results 20

0 0.1 0.2 0.3 0.4 0.5 0.6

2.6

2.8

3

3.2

3.4

3.6

3.8

Time (s)

Energy

(J)

optimal viscosity found with EA

optimal linear damping

clipped LQR with optimal weighting

Figure 2.9: Comparison of damping strategies

Figure 2.9 shows the simulated change in energy of the cable over time

for the three different strategies. The linear viscous damper, clipped LQR

controller, and time varying viscosity found with the EA remove 12.7%,21.6% and 24.9%, respectively, of the total initial cable energy E over

the time tp. Assuming the energy to be completely in the first mode, the

change in energy can be expressed in terms of the logarithmic decrement

and the damping ratio , as shown in (2.16) and (2.17).

= lnA/A = lnE/E (2.16)= /4

2 + 2 (2.17)

The performances of the respective strategies are equivalent to damping

ratios of 1.08%, 1.94%, and 2.28%.

2.4.4 Results for Single Cable at Higher Modes

An identical procedure to that carried out in the previous section was

applied to higher cable modes. Table 2.1 shows the damping ratios for allcontrol strategies considered for the first three cable modes. Although the


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Table 2.1: Comparison between damping ratios from EA results with thosefrom optimal viscous damping and clipped LQR control for first three cablemodes

Mode Optimal Linear Viscous Clipped LQR EA Result1 1.08% 1.94% 2.28%

2 1.05% 1.83% 1.99%

3 1.00% 1.65% 1.85%

performance of the EA result decreases with increasing mode number, it

still outperforms the clipped LQR strategy which also shows a decrease in

performance with increasing mode number.

2.4.5 Results of Evolutionary Algorithm for a General

Cable

The procedure for finding the optimal time dependent viscosity profile

was applied to cables with different properties. All of the cable properties

were varied by plus and minus 50% from the bench mark cable properties,

including the tension force T, length L, mass per unit lengthm and damper

position a. Mode 2 is also considered with the benchmark cable properties.

In Fig. 2.10, the results from the EA for each variation of cable properties

are plotted as force against displacement. It is clear from this plot that

the characteristics of the solutions plotted as force against displacement

are very similar to those of Fig. 2.8, i.e., a friction element with a negativestiffness.

It is now interesting to see how the EA results described as a relation-

ship between force and displacement for cables with different properties

depend on the properties of the cable itself. A qualitative analysis of the

results in Fig. 2.10 leads to the redefinition of the axes of the plot to a nor-

malized force f

(t

) against a normalized displacement v

(a, t

) at damper

position.

f(t) = f(t)LTn

(2.18)


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2.4. Results 22

0.015 0.01 0.005 0 0.005 0.01 0.015

100

80

60

40

20

0

20

40

60

80

100

Damper Displacement (m)

DamperForce(N)

m m/2

a a/2

T T/2

m + m/2

T + T/2

a + a/2

L L/2

L + L/2

n + 1

Figure 2.10: Solutions plotted as force against displacement at damper position

for cables with different properties and modes before normalization


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v(a, t) = v(a, t)Lan

(2.19)

Figure 2.11 shows plots of the normalized damper force against nor-

malized damper displacement for cables with the aforementioned varyingparameters. It is clear from the close convergence of the different trajecto-

ries to a single trajectory that the solutions are strongly dependent on the

mode number n, and the cable properties length L, tension force T, and

damper position a used in (2.18) and (2.19), and weakly dependent on the

remaining parameter mass per unit length m.

0.004 0.002 0 0.002 0.0040.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

Normalised Damper Displacement (m)

NormalisedD

amperForce(m)

m m/2

L L/2

a a/2

T T/2

m + m/2

L + L/2

a + a/2

T + T/2

n + 1

Figure 2.11: Solutions plotted as normalized force against normalized displace-ment at damper position for cables with different properties and modes

2.4.6 Characterization of Solution

As mentioned earlier, the results clearly suggest that the optimal behaviorof the semi-active damper pertains to that of a friction device in parallel


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2.4. Results 24

with a spring element with a negative stiffness, whereby a control law can

be defined as

f

(t

)FN = kEAsign

(x

(t

))X

(t

)+ kEAx

(t

)(2.20)

where X(t) is the displacement envelope of the vibration at the damperposition and kEA is defined as

kEA = T/a (2.21)where is the single parameter required for the control law and is inde-

pendent of the cable properties.

To ensure that the desired force is always dissipative when there are

errors in the amplitude envelope estimation, the following condition is

added

f(t) = { f(t)FN iff(t)x(a, t) 00 iff(t)x(t) > 0. (2.22)

2.4.7 Fitting of Single Parameter to results and Es-

timation of Displacement Envelope

The displacement envelope X(t), used in the control law of (2.20), is notestimated in real-time within the simulations described subsequently. In-

stead, an iterative method is used to find the amplitude peaks at the

damper position, where the continuous envelope X(t) is derived by thefitting of a smooth function to these peaks. Amplitude estimation in real-

time based on a model predictive approach is the topic of future work.

The single parameter can now be fitted to the mean of the normalizedEA results for cables with different properties as depicted in Fig. 2.11. The

value of which results in the closest match between the force from the

control law and the force given by the EA result is = 0.88. In Fig. 2.12,

the control law and EA result are compared by plotting the damper force

against displacement for both. Figure 2.13 shows the amplitude estimation

X

(t

)and the damper displacement over the simulation time tp. As can be

seen in Fig. 2.14, there is almost no difference between the performance of

the control law and the EA solution in terms of the energy removed overthe simulation time tp.


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0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4

0.5

0

0.5

1

Normalised Damper Displacement (m)

NormalisedForce(m)

control law


Figure 2.12: Comparison of damper force against displacement for control lawand EA result over simulation time

0 0.1 0.2 0.3 0.4 0.5 0.6

0.01

0.005

0

0.005

Time (s)

Displacement(m)

X(t)

v(a,t)

Figure 2.13: Displacement envelope X(t) found by iteration (not in real-time)


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2.4. Results 26

0.1 0.2 0.3 0.4 0.5 0.6

2.8

3

3.2

3.4

3.6

3.8

4

Time (s)

Energy

(J)

control law


Figure 2.14: Comparison of cable energy with control law and EA result oversimulation time

2.4.8 Performance of Simplified Solution with Fitted

A simple control law has been defined in (2.20). It has been shown to

be almost identical in behavior and performance to the results of the EA

over a single time period tp. Now the control law will be applied to a

more realistic vibration situation, a free vibration decay test. In this test,

the cable is excited to a steady state condition by an excitation force

with a distribution and frequency aimed at the first cable mode. The

excitation is then stopped, and the cable is allowed to decay freely. From

the decay of the cable, the damping ratio of the system in the given mode of

vibration can be measured. This test differs from the problem formulation

of the EA in that the cable energy is not restricted to only the target

mode of excitation. Due to the nonlinearity of the damper multiple modes

of vibration are excited, and it is of interest to assess the control law

performance in this more realistic situation.

Figure 2.15 shows the displacement at damper position and amplitude

estimated by the iterative method mentioned earlier for the decay test.

The damping ratios for each successive period of vibration, once the ex-

citation force is stopped, are plotted against time in Fig. 2.16. It can

be seen that at the start of decay, the damping ratio is equal to 1.9%,this is 17 percent less efficient than the performance predicted by the EA,


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which is an extrapolation of the energy decay found over one period. As

the decay proceeds with time the damping ratio drops further to around

1.6%, equivalent to a 30% drop in efficiency. The reason for this loss in

performance is the presence of multiple modes of vibration, which become

more prevalent once the excitation force is stopped. The same effect can

be observed in Fig. 2.17 which shows the energy stored in the cable over

the decay time as a percentage of the maximum energy over the whole

decay test. In this figure, the clipped LQR controller is included for com-

parison. In contrast to the control law derived from the EA, the clipped

LQR controller performs better when multiple modes of vibration occur,

approximately equalling the performance of the EA control law for the first

period of decay and then outperforming it as the decay proceeds since the

EA solution is optimized for single mode vibrations.

2 4 6 8 10 120.0015

0.0005

0

0.0005

0.001

Time (s)

Displacement(m)

X(t)

v(a,t)

0.001

Figure 2.15: Damper displacement and displacement envelope for excitationtest


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2.4. Results 28

6 7 8 9 10 11 12

1.5

1.6

1.7

1.8

1.9

Time (s)

DampingRatio(%)

Figure 2.16: Damping ratio against time during decay

6 7 8 9 10 11 12 130

20

40

60

80

Time (s)

EnergyPercentage(%)

Control law Eq. (15)

EA single mode performance

Clipped LQR

100

Figure 2.17: Percentage of energy at excitation turn off over decay time


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2.4.9 Performance of Control Law for Multiple Mode

Vibrations

Very often it is observed that multiple modes of vibration occur in cables.Therefore, it is of interest to test the proposed control law for multiple

mode vibrations. As before, a decay test is considered for the assessment of

the control law performance. However, instead of a single mode excitation,

an excitation consisting of modes 1 and 3 will be used where the maxi-

mum excitation force for each mode is identical. Figure 2.18 and Fig. 2.19

show the simulation results from this test where it can be observed that

the presence of mode 3 has no detrimental effect on the damper perfor-

mance compared to the performance of the control strategy for single modeexcitations, in fact there is even a slight improvement.

0 2 4 6 8 10 12

0.0015

0.001

0.0005

0

0.0005

0.001

0.0015

Time (s)

Displacement(m)

X(t)

v(a,t)

Figure 2.18: Damper displacement and displacement envelope for mixed modeexcitation test


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2.4. Results 30

6 7 8 9 10 11 12

1.6

1.7

1.8

1.9

2

Time (s)

DampingRatio(%)

Figure 2.19: Damping ratio against time during mixed mode decay

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