Post on 01-Feb-2021
Guideline development for offshore structure vibration analysis
Andrii Pishchanskyi
Master Thesis
presented in partial fulfillment of the requirements for the double degree:
“Advanced Master in Naval Architecture” conferred by University of Liege "Master of Sciences in Applied Mechanics, specialization in Hydrodynamics,
Energetics and Propulsion” conferred by Ecole Centrale de Nantes
developed at ICAM in the framework of the
“EMSHIP” Erasmus Mundus Master Course
in “Integrated Advanced Ship Design”
Ref. 159652-1-2009-1-BE-ERA MUNDUS-EMMC
Supervisor:
Prof. Hervé Le Sourne, ICAM
Reviewer: Prof. Maciej Taczala, West Pomeranian University of
Technology
Nantes, February 2015
Guideline development for offshore structure vibration analysis 3
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
CONTENTS
ABSTRACT ............................................................................................................................... 5
1. INTRODUCTION .................................................................................................................. 7
2. ANALYSIS OF BASIC SIMPLE STRUCTURES ............................................................. 10
2.1. Frequency Range of Mode Extraction .......................................................................... 10
2.2. Infinitely Long Cylindrical Shell .................................................................................. 10
2.2.1. Analytical Solution ................................................................................................. 10
2.2.2. Numerical results ................................................................................................... 12
2.3. Beam .............................................................................................................................. 13
2.3.1. Analytical Solution ................................................................................................. 13
2.3.2. Numerical Results .................................................................................................. 15
2.4. Plate ............................................................................................................................... 19
2.4.1. Analytical Solution ................................................................................................. 19
2.4.2. Numerical Results .................................................................................................. 20
2.5. Finite Length Cylindrical Shell ..................................................................................... 22
2.5.1. Analytical Solution ................................................................................................. 22
2.5.2. Numerical results ................................................................................................... 24
2.6. Stiffened Plate ............................................................................................................... 26
2.6.1. Analytical Solution ................................................................................................. 26
2.6.2. Numerical results ................................................................................................... 27
3. SENSITIVITY STUDIES .................................................................................................... 31
3.1. Material sensitivity ........................................................................................................ 31
3.1.1. Analytical solution .................................................................................................. 31
3.1.2. Numerical results ................................................................................................... 32
3.2. Utilization of Solid Elements ........................................................................................ 34
3.2.1. Shear locking .......................................................................................................... 34
3.2.2. Hourglassing .......................................................................................................... 35
3.2.3. Plate example ......................................................................................................... 36
3.3. Effective Modal Mass ................................................................................................... 38
3.4. Added Mass ................................................................................................................... 41
4. SIMPLE REAL CASE ......................................................................................................... 44
4.1. Model Description ......................................................................................................... 44
4.2. Modal Analysis ............................................................................................................. 45
4.2.1. Frequency Range of Mode Extraction ................................................................... 46
4.2.2. Mesh Convergence ................................................................................................. 46
4.2.3. Result Comparison of Different Finite Elements Models....................................... 47
P 4 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
4.2.4. Numerical Results of Modal Analysis .................................................................... 48
4.2.5. Modal Effective Mass ............................................................................................. 50
4.3. Frequency Response Analysis ....................................................................................... 50
4.3.1. Frequency Response Analysis Procedure .............................................................. 51
4.3.2. Numerical Results of Modal Frequency Response ................................................. 54
4.4. Vibration Isolation ......................................................................................................... 57
4.4.1. Mounting System .................................................................................................... 59
4.4.2. FEA Modelling ....................................................................................................... 63
4.4.3. Numerical Results .................................................................................................. 64
5. COMPLEX REAL CASE .................................................................................................... 67
5.1. Structure Description ..................................................................................................... 67
5.1.1. Modification of the Static FE Model for the Frequency Response Analysis .......... 68
5.1.2. Result extraction ..................................................................................................... 73
5.2.1. Frequency Range for Mode Extraction .................................................................. 74
5.2.2. Modal effective mass .............................................................................................. 74
5.3. Frequency Response Analysis ....................................................................................... 77
CONCLUSION AND FUTURE PROSPECTS ....................................................................... 82
REFERENCES ......................................................................................................................... 85
Guideline development for offshore structure vibration analysis 5
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
ABSTRACT
Vibration is the part of dynamics that deals with repetitive motion around the state of
equilibrium. There are two main aspects that differ structural dynamic problem from its static
analogy. The first one is that load, by definition, is the time-varying for the dynamic problem,
which causes that response also varies with time. The second principal difference is that if time-
varying load �(�) is applied, the resulting displacements depend not only on this load but also on inertial forces which resist the acceleration causing them. Structural response to any dynamic
load is expressed in terms of the displacement time-history of the structure. It is obtained by
solving the equation of motion of the structure that defines the dynamic displacement.
The objective of the master thesis is to develop a guideline for the vibration analysis of offshore
structures which will allow SOFRESID ENGINEERING in addition to the static analysis to
perform also vibration one.
Typical designs of the company (pipelaying vessel, field development ship, self-propeller
dynamically positioned vessel etc.) distinguish themselves with high structural complexity that
increases the risk of resonance and requires big computational effort for vibration analysis.
Therefore, modal analysis is introduced as an effective means to evaluate the response of the
structures and to identify beforehand if resonance occurs. A coordinate transformation in modal
analysis allows to calculate free vibration response with the mathematics of symmetric
eigenvalue problem. The orthogonality of the eigenvectors lets to uncouple the equations of
motion. Moreover, it is assumed that the rate of change of excitation frequency is negligibly
low. Thus, the vibration response is a steady state and transient response is not considered.
Finally, no stress analysis after vibration analysis is required. Therefore, small model features
that do not significantly influence neither mass nor stiffness can be excluded.
The results of the master thesis include an optimal finite element size for the typical structures
and constrains; a number of sensitivity studies of the key structural and numerical parameters;
flowcharts of vibration analysis and the design of vibration isolation system; the velocities of
vibration responses for the simple and complex cases as a function of excitation frequency.
Finite elements solver NASTRAN is used to solve the equations of motion, whereas pre- and
post-processor PATRAN is used to create input files for the solver and post-process results.
P 6 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
DECLARATION OF AUTHORSHIP I declare that this thesis and the work presented in it are my own and have been generated by me as the result of my own original research. Where I have consulted the published work of others, this is always clearly attributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. This thesis contains no material that has been submitted previously, in whole or in part, for the award of any other academic degree or diploma. I cede copyright of the thesis in favour of ICAM.
Date: January 15, 2015 Signature: Andrii Pishchanskyi
Guideline development for offshore structure vibration analysis 7
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
1. INTRODUCTION
Problem Traditional vibration analysis is based on rules, defined by classification societies. According
to ISO classification, vibration phenomena is related to a service limit state. Therefore, it is not
explicitly defined by class rules. This may cause that a vessel, even though is well-designed
from strength point of view, has excessive vibration. This, in its turn, leads to malfunction of
onboard equipment and fatigue problem.
Therefore, SOFRESID ENGINEERING in collaboration with ICAM decided to develop
guideline for vibration analysis of its typical structures to ensure on a design stage that vibration
response stays below required level.
Objective This master thesis includes a part of the work performed by ICAM for SOFRESID
ENGINEERING within the framework of SOFRESID STRUCTURE VIBRATION
ANALYSIS project.
The objective of the master thesis is to develop a guideline for the vibration analysis of offshore
structures. The structure of ongoing project of Field Development Ship (FDS) is used to verify
developed guideline, identify and solve possible implementation difficulties. Among them a
definition of the optimal finite element size in order to ensure that the numerical solution is
convergent and at the same time does not requires extra computational effort. Another aspect
is to provide the insight of numerical vibration analysis using NASTRAN finite elements
solver.
The propulsion system of FDS consists of four azimuth and two tunnel thrusters. Its components
such as propellers and rotating machines are among the main sources of dynamic load. In fact,
without a dynamic study, the implementation of such sophisticated propulsion system may
cause excessive vibration of the structure near the azimuth thrusters and malfunction of the
thrusters themselves. In this regard, the master thesis is limited to periodic load from the
electrical motor and the propeller of an azimuth thruster. Figure 1-1 shows that periodic
loadings demonstrate iterative alteration in time for a number of cycles. The simplest periodic
loading can be represented as a sine function (Figure 1-1a), which is called simple harmonic.
Rotating machines as electrical motors transmit a sinusoidally varying force to surrounding
structure. Other types of periodic loadings, e.g., those induced by hydrodynamic pressure
P 8 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
generated by a propeller, are more complex (Figure 1-1b). Nonetheless, the Fourier theorem
indicates that any periodic load can be expressed as the sum of simple harmonic components.
Thus, the analysis of response to any periodic load uses the same common procedure [1].
Figure 1-1. Characteristics and sources of periodic loads: (a) simple; (b) complex
Content description The first part of the master thesis is dedicated to the definition of an optimal finite element size
for the typical structures and constrains of SOFRESID ENGINEERING. For this purpose a
bibliographic research on available analytical solutions and recommendations for vibration
analysis is performed. Later they are verified by using NASTRAN solver on the simple
elements of structure. In addition, possible numerical problems, their solution and helpful
features of the solver are indicated. Finally, a number of sensitivity studies of the key
parameters are performed to define their influence.
Next part deals with the global response of the simple structure of electrical motor assembly.
The ways to simplify the vibration analysis (model simplification, the use of 1D finite elements)
are analysed. The suitability of the analytical solutions for definition of the finite element size
described in the previous section are checked for the specific structure of the electrical motor
assembly. All steps of the vibration analysis are highlighted in details and the analysis itself is
implemented. Finally, the design of a vibration isolation system is investigated in order to
reduce vibration response in case if resonance is unavoidable.
Final part describes the local response of FDS’s structure due to dynamic loading from the
thruster. Finite elements model used by SOFRESID ENGINEERING for static analysis is
adapted for the purpose of vibration analysis. The structural elements that can be modified by
SOFRESID ENGINEERING in order to change vibration response are identified. Moreover,
Guideline development for offshore structure vibration analysis 9
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
dynamic loading from the thruster is investigated. Lastly, vibration analysis shows that
resonance occurs within the frequency range of dynamic loading.
P 10 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
2. ANALYSIS OF BASIC SIMPLE STRUCTURES
The definition of the optimal finite elements size is among difficulties of the implementation of
the vibration analysis. For a complex mesh, an automatic refinement is impossible and a manual
refinement is not feasible due to tremendous amount of man-hour prone to mistakes. Thus, the
knowledge of the optimal finite elements size is extremely desirable. This section deals with its
definition for a set of typical structural elements.
2.1. Frequency Range of Mode Extraction
The classical frequency range of ship vibration is 1-80 Hz, ISO (2000). However, the higher
the frequency, the finer must be the mesh and more difficult it is to obtain realistic numerical
results using finite elements method (FEM). It is generally recognized that good results can be
obtained in the frequency range 1-30 Hz. When the modal superposition method is used for
forced vibration analysis, the frequency range of mode extraction of free vibration should be at
least 1.4 times wider that the excitation frequency range. This criterion assures the calculation
of the correct vibration amplitude [2].
In this section of the master thesis which presents the analysis of basic simple structures the
frequency range of mode extraction equals �0 − 150�� in order to assess the discrepancy between analytical and numerical solution in higher frequency range.
2.2. Infinitely Long Cylindrical Shell
2.2.1. Analytical Solution
The cylindrical coordinate system used to describe a cylinder is shown in Figure 2-1.
Figure 2-1. Coordinate system for cylinders [3]
Guideline development for offshore structure vibration analysis 11
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
The cylindrical shell coordinates are measured at the mid-surface. Here, u, v, w correspond to
the axial, circumferential and radial displacements respectively. To characterize the various
vibration modes of a cylindrical shell, i and j wave numbers are introduced. For a certain
vibration mode, they provide the number of circumferential waves and axial half-waves
respectively.
The equations describing the free vibration of cylindrical shell admit solutions, which are
independent of the axial coordinate (x). These solutions apply to infinitely long cylinders in the
sense that the distance between axial nodes (~L/j) is much greater than the cylinder radius, so
that the boundary conditions at the end of the cylinder do not influence the solution and the
solution does not vary along the length of the cylinder [3].
The i=0 modes correspond to breathing behaviour, where the cross section at any axial location
remains circular and no circumferential bending of the involved shell occurs. The i=1 mode
corresponds to the beam-type bending vibration mode where the cross section remains
undeformed. These bending modes will be more significant for longer, more slender cylinders,
which more closely approximate a beam of column [4]. As infinitely long cylindrical shell is
independent of boundary conditions, beam-type bending mode (i=1) is omitted for its study.
Hence, symmetry can be utilized while modelling, as no anti-symmetric mode shape is present.
Moreover, for axisymmetric structures, numerical results give repetitive modes with the plane
of vibration rotated by 90°. They have identical mode shape and minor natural frequency
discrepancy. This is due to discretization error that makes the stiffness of finite element model
not perfectly axisymmetric [5]. Natural frequency for radial-circumferential modes of infinitely
long cylindrical shell is given by:
� = ��2�� � ��(1 − ��)��/�
(2-1)
where �� – dimensionless parameter which specifies the circular natural frequency (�) of the cylindrical shell for different modes (see Reference [3] for more details), R – cylinder radius to
mid-surface, E – modulus of elasticity, � – density of shell material, � – Poisson’s ratio. Figure 2-2 shows the finite element mesh developed to model an infinite cylindrical shell, where
shell thickness � = 20��, radius � = 1000��, height ℎ = 100��. Cylindrical coordinate system and symmetry boundary conditions are used in order to reproduce infinitely long
cylindrical shell behaviour.
P 12 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
Figure 2-2. Modelling of the infinitely long cylinder
Table 2-1 presents the analytical frequencies obtained by using above formula. It demonstrates
that only 2-7 radial-circumferential flexure modes should be considered in the frequency range
of mode extraction. Natural frequencies of extension and axial modes are given to show that
even the first corresponding natural frequencies lies out of the frequency range of mode
extraction when considering typical maritime structures.
Table 2-1. Analytical natural frequencies of the infinitely long cylinder
Mode number
Natural frequency [Hz]
Radial-circumferential modes Extension mode Axial mode
0 844.83
1 499.81
2 6.54
3 18.51
4 35.49
5 57.39
6 84.20
7 115.89
2.2.2. Numerical results
A modal analysis is run using the finite element code NASTRAN and the obtained natural
frequencies are presented in Table 2-2 as a function of mesh density. 4-node shell elements are
used. Then, analytical and numerical results are compared. Figure 2-3 shows the discrepancy
as a function of the number of elements used to model the infinitely long cylindrical shell.
Guideline development for offshore structure vibration analysis 13
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
Table 2-2. Numerical natural frequencies of infinitely long cylindrical shell
Mode
Natural frequency [Hz]
Analytical solution
Number of elements [/] (size of the elements [mm])
10 (313x100)
14 (224x100)
18 (174x100)
21 (150x100)
25 (126x100)
30 (105x100)
35 (90x100)
2 6.54 6.66 6.60 6.58 6.57 6.56 6.56 6.55 3 18.51 18.84 18.68 18.61 18.59 18.56 18.55 18.54 4 35.49 36.06 35.81 35.69 35.63 35.59 35.56 35.54 5 57.39 58.01 57.85 57.69 57.61 57.55 57.50 57.46 6 84.2 84.06 84.68 84.56 84.48 84.40 84.33 84.28 7 115.89 112.67 116.05 116.23 116.19 116.11 116.03 115.98
Figure 2-3. Natural frequency convergence study for the infinitely long cylindrical shell
The numerical solution converges and asymptotically approaches a constant value. The
discrepancy between convergent numerical and analytical results is less than 1 %. Therefore,
analytical solution can readily be used to estimate the number of mode shapes to consider. The
size of elements for converged solution is 150x100mm. Nonetheless, as it was mentioned
before, the solution for infinitely long cylindrical shell is independent of axial coordinate. Thus,
the axial size of element (100mm) is irrelevant and can be any value. Its influence is eliminated
by applying appropriate multiple-point constrains.
2.3. Beam
2.3.1. Analytical Solution
A straight elastic beam possesses both the mass and stiffness to resist bending. During
transverse vibration, the beam flexes perpendicular to its own axis to alternate stored potential
energy in the elastic bending of the beam and then release into the kinetic energy of transverse
motion [3]. Transverse vibration occurs in two perpendicular planes.
-4
-3
-2
-1
0
1
2
3
10 15 20 25 30 35
Dis
crep
ancy
[%]
Number of elements [/]2 waves 3 waves 4 waves 5 waves 6 waves 7 waves
P 14 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
The mode shape and natural frequency of a slender beam are function of an integer index (i).
The number of flexural half-waves in the mode shape might be associated with this index. This
means that each i corresponds to a certain natural frequency and mode shape. The natural
frequency in general can be expressed in the form:
� = ���2��� �� ���/�
(2-2)
where �� – dimensionless parameter which is a function of the boundary conditions applied to the beam (see Reference [3] for more details), � – length of the beam, m – is the mass per unit length of the beam, I – area moment of inertia, E – modulus of elasticity.
Longitudinal vibration arises from stretching and contracting of the beam along its own
longitudinal axis. The longitudinal vibration is assumed to be uniform over the cross section
[3]. The natural frequency in general can be expressed in the form:
� = ��2�� �����/�
(2-3)
where �� – dimensionless parameter which is a function of the boundary conditions applied to the beam (see Reference [3] for more details), � – length of the beam, ρ – mass density of the beam material, E – modulus of elasticity.
Torsional vibration is the result of local twisting of the beam or shaft about its axis. Exact
closed form results for torsional vibrations can generally be obtained only for the shafts or pipes
with circular cross sections [3]. Moreover, in torsional mode, centre of gravity does not translate
significantly, thus effective modal mass is negligible. Generally, sections of beams used in
naval industry are rather I- or T-shape and further investigation have to be performed to get
analytical solutions.
For comparison between finite element and analytical solutions, an I-beam is chosen and its
section dimensions are given in Figure 2-4 (dimensions are in mm). Beam length is 6000mm.
Figure 2-4. Beam cross section
Guideline development for offshore structure vibration analysis 15
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
Simply supported and clamped boundary conditions are successively applied to the beam
extremities and obtained result are compared. Tables 2-3 and 2-4 present natural frequencies
obtained analytically for simply supported and clamped boundary conditions respectively.
Natural frequency of longitudinal vibration is given to show that even the first longitudinal
mode exceeds frequency range of mode extraction.
Table 2-3. Beam natural frequencies for simply supported boundary conditions
Table 2-4. Beam natural frequencies for clamped boundary conditions
Mode shape
Transverse vibration in x-z plane
Transverse vibration in x-y plane
Longitudinal vibration
1 26.90 12.39 635.33 2 74.16 34.16 3 145.39 66.96 4 110.68
2.3.2. Numerical Results
Two models implemented by using either 1D FE or 2D FE are analysed in order to evaluate the
discrepancy for these model techniques.
Beam modelled with 1D FE
NASTRAN simulations are then carried out using decreasing finite element size. Tables 2-5
and 2-6 present the resulting beam natural frequencies with simply supported and clamped
boundary conditions respectively. Their number of half-waves (HW) identifies the
corresponding mode shapes. Finite element results are then compared to analytical ones.
Figures 2-5 and 2-6 show the discrepancy as a function of the number of elements used to model
the beam for the simply supported and clamped boundary condition respectively.
Mode shape
Transverse vibration in x-z plane
Transverse vibration in x-y plane
Longitudinal vibration
1 11.87 5.47 421.98 2 47.47 21.86 3 106.81 49.19 4 87.45 5 136.65
P 16 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
Table 2-5. Convergence study of natural frequency discrepancy for beam with simply supported BC
Mode shape
Natural frequency [Hz]
Analytical solution
Number of elements [/] (size of the elements [mm])
5 (1200)
7 (857)
9 (667)
12 (500)
15 (400)
18 (333)
1 HW in x-y 5.47 5.46 5.46 5.46 5.46 5.46 5.46 1 HW in x-z 11.87 11.83 11.83 11.83 11.83 11.83 11.83 2 HW in x-y 21.86 21.77 21.82 21.82 21.83 21.83 21.83 2 HW in x-z 47.47 46.70 46.83 46.87 46.88 46.89 46.89 3 HW in x-y 49.19 48.06 48.83 48.95 49.00 49.01 49.01 4 HW in x-y 87.45 78.79 85.51 86.46 86.77 86.84 86.87 3 HW in x-z 106.81 101.22 103.18 103.60 103.79 103.86 103.90 5 HW in x-y 136.65 161.51 128.32 133.31 134.73 135.06 135.18
Figure 2-5. Convergence study of NF discrepancy for beam with simply supported BC
Table 2-6. Convergence study of natural frequency discrepancy for beam with clamped BC
Mode shape
Natural frequency [Hz]
Analytical solution
Number of elements [/] (size of the elements [mm]) 5
(1200) 7
(857) 9
(667) 12
(500) 15
(400) 18
(333) 1 HW in x-y 12.39 12.35 12.36 12.37 12.37 12.37 12.37 1 HW in x-z 26.90 26.46 26.48 26.49 26.49 26.50 26.50 2 HW in x-y 34.16 33.59 33.93 33.98 34.00 34.00 34.01 2 HW in x-z 66.96 62.10 65.70 66.23 66.39 66.43 66.45 3 HW in x-y 74.16 70.65 71.46 71.62 71.70 71.72 71.73 4 HW in x-y 110.68 86.40 105.08 108.20 109.10 109.30 109.37 3 HW in x-z 145.39 127.51 135.15 136.56 137.12 137.31 137.40
-15-10
-50
51015
20
4 6 8 10 12 14 16 18 20
Dis
crep
ancy
[%]
Number of elements [/]
1 HW in x-y
1 HW in x-z2 HW in x-y2 HW in x-z3 HW in x-y4 HW in x-y3 HW in x-z5 HW in x-y
Guideline development for offshore structure vibration analysis 17
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
Figure 2-6. Convergence study of natural frequency discrepancy for beam with clamped BC
Conclusions:
• Numerical results correlate well enough with used analytical solution but discrepancy
tends to increase with frequency. Nevertheless, analytical solution can readily be used to
estimate number of half-waves to consider. The size of elements for converged solution
is 500mm for both cases of boundary conditions;
• Applying clamped boundary conditions for a beam makes it stiffer in comparison to
simply supported ones. Therefore, bracketing assumption can be utilized to estimate
natural frequency of the structure, if the model does not closely approximate real
structure. For example, if boundary conditions are intermediate between simply supported
and clamped, it is useful to bracket the natural frequency by using both approximations
in analysis. For further elements of structure the same effect of switching from simply
supported to clamped BC is taken for granted;
• Numerical results understate analytical results.
Beam modelled with 2D finite elements
Providing beam has connections to multiple structural elements, 1D simplification may not be
sufficient to estimate behaviour of real structure. In this case, beam geometry should be
modelled explicitly using 2D shell elements. Thus, a basic case of I-beam modelled with 2D 4-
node shell elements is investigated. Beam scantling is the same as in the case with 1D FE and
simply supported boundary conditions are considered. Table 2-7 presents the resulting beam
natural frequencies. Their number of HW identifies the corresponding mode shapes. Finite
element results are then compared to analytical ones. Figure 2-7 shows the evolution of the
discrepancy when the total number of shell elements used to model the beam increases. The
size of elements for converged solution is 333x62 mm.
-25
-20
-15
-10
-5
0
4 9 14 19
Dis
crep
ancy
[%]
Number of elements [/]
1 HW in x-y
1 HW in x-z
2 HW in x-y
2 HW in x-z
3 HW in x-y
4 HW in x-y
3 HW in x-z
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Master Thesis developed at ICAM, Nantes
Table 2-7. Convergence study of natural frequency discrepancy for beam with simply supported BC
Mode shape
Natural frequency [Hz]
Analytical solution
Total number of elements [/] (size of the elements [mm])
54 (667x62)
72 (500x62)
90 (400x62)
108 (333x62)
210 (286x62)
1 HW in x-y 5.47 5.35 5.33 5.32 5.32 5.31
1 HW in x-z 11.87 11.94 11.91 11.90 11.89 11.89 2 HW in x-y 21.86 21.71 21.48 21.37 21.32 21.29
2 HW in x-z 47.47 48.07 47.60 47.39 47.28 47.20 3 HW in x-y 49.19 50.24 48.99 48.45 48.17 48.04 4 HW in x-y 87.45 93.49 88.90 87.08 86.16 85.76 3 HW in x-z 106.81 110.73 106.89 105.84 105.29 104.95 5 HW in x-y 136.65 161.87 148.85 144.94 135.75 134.71
Figure 2-7. Convergence study of natural frequency discrepancy for beam with simply supported BC
In Table 2-8, the converged solution discrepancies of 1D beam and 2D shell elements are
compared.
Table 2-8. Comparison of natural frequency discrepancy obtained from 1D beam and 2D shell element meshes
Mode shape 1D beam elements Comparison 2D shell elements
1 HW in x-y -0,04 > -2,79 1 HW in x-z -0,31 < 0,15 2 HW in x-y -0,16 > -2,63 2 HW in x-z -1,22 < -0,57 3 HW in x-y -0,37 > -2,34 4 HW in x-y -0,67 > -1,94 3 HW in x-z -2,73 < -1,74 5 HW in x-y -1,08 > -1,42
The table above demonstrates that solutions obtained by using beam elements and shell
elements converge to different values. The difference is due to the difficulty in modelling beam
using 2D shell elements. Indeed, when 2D shell elements are used, the intersection of web and
flange is counted two times in stiffness and mass matrices (Figure 2-8).
-5
0
5
10
15
20
50 100 150 200
Dis
crep
ancy
[%]
Axial number of elements [/]
1 HW in x-y
1 HW in x-z
2 HW in x-y
2 HW in x-z
3 HW in x-y
4 HW in x-y
3 HW in x-z
5 HW in x-y
Guideline development for offshore structure vibration analysis 19
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
Figure 2-8. Web and flange intersection by using 2D shell elements in modelling
This explains the difference pattern between results (all mode shapes in x-y plane correspond
to higher frequency for 1D beam elements and all mode shapes in x-z plane correspond to higher
frequency for 2D shell elements). While bending occurs about z-axis, intersection does not
influence bending stiffness significantly, as it is located close to the axis. Therefore, combined
effect of mass and stiffness magnification decreases the natural frequency. On the other hand,
while bending occurs about y-axis, the intersection of web and flange is remote from the axis
that changes bending stiffness more significant. Combined effect of mass and stiffness
magnification causes the natural frequency to increase.
2.4. Plate
2.4.1. Analytical Solution
Unlike for beam and infinite cylinder, the mode shapes and natural frequencies of plates are
functions of the two integers indices i and j. The number of flexural half-waves in one of the
two orthogonal plate dimensions might be associated with each of these indices. One natural
frequency and the associated mode shape exist for each unique combination of i and j.
Providing the mode shape of a rectangular plate is approximated by the mode shape of single
beams along x and y axes, the approximate closed form solutions for the natural frequency can
be developed using the Rayleigh’s energy approach. This gives an approximate natural
frequency, in hertz, of the form [3]:
�! = �2 "#�$
%$ + #�$
'$ + 2(�(� + 2�(���� − (�(�)%�'� )�/� " �ℎ*12+(1 − ��)
�/� (4-4)
where the dimensionless parameters G, H and J – functions of the indices i and j and the
boundary conditions imposed to the plate (see more details in Reference [3]). In this formula,
a – plate length, b – plate width, h – plate thickness, E – modulus of elasticity, � – Poisson’s
P 20 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
ratio, + – mass per unit area of the plate. Figure 2-9 shows the finite element modelling of the plate configuration (a=4000mm, b=1000mm, h=10mm) used for its study. Two boundary
condition cases are considered with all edges are either simply supported or clamped.
Figure 2-9. Finite element modelling of the plate
2.4.2. Numerical Results
Table 2-9 presents the plate natural frequency results for different mesh sizes. The discrepancy
between analytical and numerical results is plotted in Figure 2-10 as a function of the total
number of elements. Finally, Table 2-10 shows the discrepancy chart of mode shapes
(discrepancy ascends).
Table 2-9. Convergence study of natural frequency discrepancy for plate with simply supported BC
Mo
de
shap
e
Natural frequency [Hz]
An
alyt
ical
so
lutio
n
Number of elements [/] (size of the elements [mm])
27
x6
(148
x167
)
36
x8
(111
x125
)
45
x10
(89x
100
)
54
x12
(74x
83)
63
x14
(64x
71)
72
x16
(56x
63)
90
x20
(45x
50)
11
8x24
(3
7x42
)
12
6x28
(3
2x36
)
16
2x32
(2
5x28
)
22
5x50
(1
7x20
)
1x1 25.6 25.5 25.5 25.5 25.5 25.5 25.5 25.6 25.6 25.6 25.6 25.6
2x1 30.1 29.7 29.8 29.9 29.9 30.0 30.0 30.0 30.0 30.0 30.0 30.0
3x1 37.6 36.8 37.1 37.2 37.3 37.4 37.4 37.5 37.5 37.5 37.5 37.5
4x1 48.1 46.7 47.2 47.5 47.6 47.7 47.8 47.9 47.9 48.0 48.0 48.0
5x1 61.7 59.6 60.3 60.7 60.9 61.1 61.2 61.3 61.4 61.4 61.5 61.5
6x1 78.2 75.3 76.4 76.9 77.2 77.4 77.6 77.7 77.8 77.9 78.0 78.0
7x1 97.8 94.0 95.3 96.0 96.4 96.7 96.9 97.1 97.3 97.4 97.5 97.5
1x2 97.8 97.4 97.5 97.6 97.6 97.6 97.7 97.7 97.7 97.7 97.7 97.7
2x2 102.3 101.1 101.4 101.7 101.8 101.9 102.0 102.0 102.1 102.1 102.1 102.2
3x2 109.8 107.3 108.0 108.5 108.8 109.0 109.1 109.3 109.4 109.5 109.6 109.6
8x1 120.4 115.6 117.2 118.1 118.6 119.0 119.2 119.5 119.7 119.8 119.9 120.0
4x2 120.4 116.0 117.3 118.1 118.6 119.0 119.2 119.5 119.7 119.8 119.9 120.0
5x2 133.9 127.1 129.2 130.4 131.2 131.8 132.2 132.7 132.9 133.1 133.3 133.4
9x1 145.9 140.1 142.1 143.2 143.8 144.2 144.5 144.9 145.1 145.3 145.4 145.5
6x2 150.4 140.7 143.8 145.6 146.7 147.5 148.0 148.7 149.1 149.4 149.6 149.8
Guideline development for offshore structure vibration analysis 21
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
Figure 2-10. Convergence study of natural frequency discrepancy for plate with simply supported BC
Table 2-10. Discrepancy chart of mode shapes
Order of convergence
Mode shape Order of convergence
Mode shape Order of convergence
Mode shape
1 1x1 6 3x2 11 4x2
2 1x2 7 4x1 12 8x1
3 2x2 8 5x1 13 9x1
4 2x1 9 6x1 14 5x2
5 3x1 10 7x1 15 6x2
Conclusion:
• The discrepancy value correlates more with the product of the number of longitudinal and
transverse half-wave (, ∙ .), rather than with frequency value. For example, mode shapes 2x2 with /�0� = 102.3�� and 3x1 with /*0� = 37.6�� require approximately the same number of elements to converge;
• At contrary, the convergence rate is inversely proportional to longitudinal and transverse
half-waves product (, ∙ .); • The size of elements for converged solution is 36x32mm;
• Numerical results understate analytical.
-7
-6
-5
-4
-3
-2
-1
0
0 2000 4000 6000 8000 10000 12000
Dis
crep
ancy
[%]
Total number of elements [/]
1x12x13x14x15x16x17x11x22x23x28x14x25x29x16x2
P 22 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
2.5. Finite Length Cylindrical Shell
2.5.1. Analytical Solution
The solution of a simply supported finite length cylindrical shell can be based on previously
used solution for infinitely long cylindrical shell. The boundary conditions at both extremities
of the cylinder considered in this study are:
5 = 6 = 0,80 = 90 = 0 :;= = 0, = = � (2-5) It is seen that no axial constrain at shell ends is considered. This kind of boundary condition
can be approximated by using a thin, flat, circular diaphragm attached to the ends of the
cylinder. The term ”shear diaphragm” is usually adopted for this kind of boundary conditions.
Corresponding cylinder natural frequencies may then be calculated as:
�! = ��!2�� � ��(1 − ��)��/�
(2-6)
where ��! – dimensionless parameter (see Reference [3] for more details), i – number of circumferential waves in mode shape, j – number of axial waves in mode shape, R – cylinder
radius to mid-surface, E – modulus of elasticity, � – density of shell material, � – Poisson’s ratio. Therefore, for the defined cylindrical shell configuration, natural frequency is a function
of a number of axial half-waves and circumferential waves. Analytical solutions for extension
mode (i= 0) and bending mode (i= 1) relay on assumption of long cylinder L/(jR)>8. The
analysed configuration does not correspond to this assumption. Therefore, as shown in Figure
2-11, eq. 2-6 gives inaccurate results for i=0 and i=1, especially for growing j.
Figure 2-11. Finite cylindrical shell analytical solution
Guideline development for offshore structure vibration analysis 23
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
For these modes only the results corresponding to j= 1 are close to real. Nonetheless, for the
studied configuration /�>?,!>� = 412��, /�>�,!>� = 312��, which are far above the frequency range of mode extraction. Therefore, these mode shapes with j=0, j=1 are intentionally omitted.
As it was done for infinitely long cylindrical shell, symmetry can be utilized while modelling,
as no anti-symmetrical mode shape is present. Figure 2-12 demonstrates finite elements model
of finite length cylindrical shell with dimensions: � = 2000��, ℎ = 6000��, � = 20��.
Figure 2-12. FE model of finite length cylindrical shell
Figure 2-13 presents the calculated natural frequencies associated with the mode shapes
obtained by varying the number of circumferential waves in range �2 − 12 and number of axial half-waves in a range �1 − 5.
Figure 2-13. Natural frequencies of cylindrical shell
It worth to mention that the lowest frequency does not correspond to mode shape i=1 and j=1.
Table 2-11 shows the analytical frequencies within frequency range of mode extraction (,AB0 =11, .AB0 = 4).
P 24 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
Table 4-11. Natural frequencies and mode shapes of finite length cylindrical shell
Order Number of axial half-waves, j
Number of circumferential waves, i
Natural frequency [Hz]
Order Number of axial half-waves, j
Number of circumferential waves, i
Natural frequency [Hz]
1 1 4 33.21 15 3 6 103.18 2 1 5 36.05 16 3 8 104.94 3 1 3 45.47 17 2 9 106.16 4 1 6 46.78 18 3 9 119.14 5 1 7 61.72 19 3 5 121.72 6 2 6 65.88 20 1 10 123.36 7 2 5 70.01 21 2 10 128.41 8 2 7 73.04 22 4 8 131.93 9 1 8 79.67 23 2 3 133.05 10 1 2 86.93 24 4 7 133.69 11 2 8 87.30 25 3 10 138.78 12 2 4 90.20 26 4 9 139.96 13 3 7 98.58 27 4 6 147.30 14 1 9 100.25 28 1 11 148.93
2.5.2. Numerical results
NASTRAN modal analysis have been run from different meshes, starting by a coarse mesh and
ending by a refined one. As a result, Table 2-12 presents the obtained natural frequencies, while
Figure 2-14 shows the evolution of the discrepancy between numerical and analytical solutions
when the number of finite element in the circumferential and axial direction increases.
Table 4-12. Numerical natural frequencies of finite length cylindrical shell
(i·j)
Natural frequency [Hz]
Analytical solution
Number of elements in circumferential and axial direction of the cylinder[/] (size of the elements [mm])
11x12 (570x500)
17x16 (369x375)
22x20 (285x300)
28x24 (224x250)
34x28 (185x214)
4x1 33.21 31.59 31.42 31.39 31.38 31.37
5x1 36.05 34.58 34.23 34.19 34.18 34.19
3x1 45.47 42.82 42.68 42.64 42.62 42.61
6x1 46.78 45.58 44.93 44.86 44.85 44.86 7x1 61.72 60.74 59.88 59.77 59.75 59.76 6x2 65.88 65.38 63.78 63.66 63.67 63.71
5x2 70.01 69.09 68.12 67.94 67.87 67.84
7x2 73.04 74.10 70.70 70.50 70.55 70.64
8x1 79.67 77.08 76.94 76.90 76.87 76.86
2x1 86.93 78.08 77.80 77.68 77.65 77.66
8x2 87.30 88.56 85.03 84.54 84.57 84.69
4x2 90.20 91.43 87.53 87.26 87.11 87.03
7x3 98.58 95.38 96.14 95.75 95.74 95.84
9x1 100.25 100.90 98.21 98.17 98.16 98.18
6x3 103.18 104.33 101.46 100.99 100.82 100.78
Guideline development for offshore structure vibration analysis 25
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
(i·j)
Natural frequency [Hz]
Analytical solution
Number of elements in circumferential and axial direction of the cylinder[/] (size of the elements [mm])
11x12 (570x500)
17x16 (369x375)
22x20 (285x300)
28x24 (224x250)
34x28 (185x214)
8x3 104.94 108.92 102.04 101.37 101.43 101.65
9x2 106.16 112.20 104.24 103.27 103.20 103.33
9x3 119.14 113.49 116.56 115.05 115.01 115.30
5x3 121.72 114.18 120.09 119.45 119.13 118.98
10x1 123.36 122.80 120.77 121.07 121.14 121.18
10x2 128.41 127.98 126.78 125.44 125.24 125.36
8x4 131.93 134.08 127.04 126.46 126.29 126.20
3x2 133.05 136.36 128.66 127.86 127.87 128.11
7x4 133.69 138.01 131.68 130.91 130.69 130.71
10x3 138.78 140.43 136.17 134.50 134.14 134.43
9x4 139.96 143.02 137.53 134.65 134.70 135.12
6x4 147.30 151.32 145.05 145.24 144.78 144.63
11x1 148.93 156.65 146.20 146.26 146.52 146.61
Figure 2-14. Natural frequency convergence study for finite length cylindrical shell
-12
-10
-8
-6
-4
-2
0
2
4
6
8
0 200 400 600 800 1000
Dis
crep
ancy
[%]
Total number of elements [/]
4x1
5x1
3x1
6x1
7x1
6x2
5x2
7x2
8x1
2x1
8x2
4x2
7x3
9x1
6x3
8x3
9x2
9x3
5x3
10x1
10x2
8x4
3x2
7x4
10x3
9x4
6x4
11x1
P 26 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
Conclusion:
• Striking features of discrepancy study is that certain mode shapes (2x1, 3x1, 4x1, 5x1)
converge faster, but at same time, they correspond to the biggest discrepancy (Δ�,� =10.5%). The reason of it is not defined. As the magnitude of a mode shape displacement is not quantitative, subsequent frequency response analysis should be performed in order
to figure out obtained phenomena;
• The size of elements for converged solution is 285x300mm.
2.6. Stiffened Plate
2.6.1. Analytical Solution
Stiffened plate differs from uniform plate by the fact that its properties are directional; its
bending rigidity about one axis is not necessary the same as the bending rigidity about a
perpendicular axis [3]. Certain conditions are considered in order to simplify vibration analysis.
The way to analyse a stiffened panel is to consider it as an unstiffened orthotropic plate. In order
to use orthotropic plate solution, smearing analysis is performed. An equivalent uniform
orthotropic plate is created by smearing. The properties of the discrete stiffeners are spread over
the plate surface by including coefficients which are defined by combined bending rigidity of
the plate and stiffeners. Approximate solutions for the natural frequency of the rectangular
orthotropic plate are found by:
�! = �2+�/� "#�$E0%$ + #�
$EF'$ + 2����E0F%�'� + 4EG((�(�−����)%�'� )�/�
(2-7)
where i and j – the number of longitudinal and transverse half-waves, E0, EF, E0F, EG–equivalent orthotropic constant for stiffened plate, the dimensionless parameters G, H and J are
functions of the indices i and j and the boundary conditions of the plate, a – plate length, b –
plate width, h – plate thickness, E – modulus of elasticity, � – Poisson’s ratio, + – average mass per unit area. For example, equivalent orthotropic constant for stiffened plate with bar along x-
axis is defined as:
E0 = �ℎ*12(1 − ��) + �B B'� (2-8) where h – plate thickness, B – moment of inertia of stiffener alone with respect to midsurface, �B – elasticity of stiffener, '� – stiffener spacing along y-axis.
Guideline development for offshore structure vibration analysis 27
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
Studies configuration is a plate with flat bar stiffeners in two directions on both sides. The
modelling is shown on Figure 2-15. The configuration of the stiffened plate is as follows:
- Plate length% = 15000��; - Longitudinal spacing between stiffeners %� = 3000��; - Number of transverse stiffeners H = 4; - Plate width' = 10000��; - Transverse spacing between stiffeners '� = 1000��; - Number of longitudinal stiffeners � = 9.
Figure 2-15. Stiffened plate modelling
Table 2-13 presents the natural frequencies calculated analytically within a range �0 − 110��. Table 2-13. Analytical frequencies of stiffened plate
Order / [Hz] Order / [Hz] Order / [Hz] Order / [Hz] Order / [Hz] 1 3.79 11 30.96 21 53.55 31 69.24 41 92.28
2 5.04 12 33.21 22 54.76 32 70.77 42 92.57
3 8.54 13 33.39 23 56.66 33 72.53 43 93.29
4 14.16 14 34.11 24 59.02 34 76.86 44 94.76
5 14.78 15 34.16 25 59.13 35 80.65 45 97.36
6 15.16 16 35.97 26 59.55 36 85.44 46 101.47
7 16.67 17 39.57 27 60.65 37 86.69 47 103.34
8 20.16 18 42 28 62.87 38 91.16 48 104.05
9 21.67 19 44.43 29 64.08 39 91.75 49 104.39
10 26.01 20 45.36 30 66.69 40 92.21 50 107.45
2.6.2. Numerical results
Table 2-14 presents the natural frequencies obtained numerically for the stiffened plate whereas
Figure 2-16 shows the discrepancy between analytical and numerical results as a function of
the number of finite elements used to model the panel.
P 28 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
Table 2-14. Natural frequency numerical results of stiffened plate
Order
Natural frequency [Hz]
Analytical solution
Number of elements [/] (size of the elements for plate & stiffener [mm])
580 (1000x1000 &
1000x100)
1460 (500x500 & 500x100)
5840 (250x250 &
250x50)
23360 (125x125 &
125x50) 1x1 3.79 3.58 3.56 3.55 3.55 2x1 5.04 4.71 4.68 4.67 4.66 3x1 8.54 7.96 7.84 7.80 7.79 4x1 14.16 13.33 12.92 12.78 12.75 1x2 14.78 13.88 13.44 13.29 13.26 2x2 15.16 14.11 13.67 13.52 13.49 3x2 16.67 15.29 14.82 14.64 14.62 4x2 20.16 18.29 17.57 17.30 17.27
5x1 21.67 21.99 20.61 20.10 20.05 5x2 26.01 22.10 20.68 20.18 20.13 1x3 33.21 29.49 26.00 25.04 24.91
Figure 2-16. Natural frequency convergence study for stiffened plate
The above Fig. demonstrates that numerical results converge but discrepancy is significant for
certain modes. This can be due to the facts that:
• Software considers volume of shell element intersection two times in mass calculation
(as in case of I-beam). For current study, the difference in mass is around 1%, therefore,
it is assumed to be negligible;
• Analytical approximation assumes that distance between vibration nodes is significantly
bigger that spacing between stiffeners. For mode shapes with drastic discrepancy
increase, the distance between vibration nodes approaches the spacing between stiffeners.
-28
-23
-18
-13
-8
-3
2
0 5000 10000 15000 20000 25000
Dis
crep
ancy
[%]
Number of elements [/]
1x1
2x1
3x1
4x1
1x2
2x2
3x2
4x2
5x1
5x2
1x3
Guideline development for offshore structure vibration analysis 29
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
This causes that stiffened panel begins to vibrate locally and assumption of orthotropic
homogeneous plate is no more valid.
Further comparison of natural frequency is cumbersome as there is a number of mode shapes
with prevailing local displacement. For example, between mode 5x2 and 1x3 instead of one
mode obtained by analytical approximation, numerical results contain 13-14 modes, depending
on mesh density. Figure 2-17 shows an example of mode shape with prevailing local
deformation. Figure 2-18 demonstrates that even global mode 1x3 is influenced by local
displacement, which is the source of discrepancy.
Figure 2-17. Mode shape with prevailing local deformation
Figure 2-18. Local displacement in global mode
Conclusion:
• Analytical approximation based on homogenization does not account for existence of
local modes. On the other hand, if the adopted mesh is sufficiently fine, finite elements
model is able to capture correctly local mode shapes, when the plate deforms between
two stiffeners. Finally, as shown in Figure 2-16, using a mesh of 5000 shell elements
seems sufficient to converge to correct natural frequency values;
P 30 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
• The size of elements for converged solution is 250x250mm for the plate and 250x50 for
stiffeners.
Guideline development for offshore structure vibration analysis 31
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
3. SENSITIVITY STUDIES
This section consists of the set of separate sensitivity studies. The influence of material is
studies in order to determine if it possible to modify the results of vibration analysis only by
changing material. Another study is dedicated to the numerical difficulties of using 3D finite
elements for vibration analysis. Next, the advantages and insight of the utilization of modal
effective mass as a means to determine the significant vibration modes for analysis is
investigated. Finally, in order to consider that structure vibrates in fluid, virtual mass approach
is tested and verified by analytical formulation.
3.1. Material sensitivity
3.1.1. Analytical solution
Free vibration analysis results depend on global mass and stiffness matrices which occur in the
equation of motion of the form:
(−�8�� + �J)KLM = 0 (3-1) Changing a material does not influence the geometric characteristics of the model and they stay
the same. Therefore, the equation of motion for undamped system can be written as:
N−��O:PL�Q�� + �R� �S:Q
,S,QH�TU KLM = 0 (3-2) In eq. 3-2 only density and modulus of elasticity are influenced by material change. A
coordinate transformation lets to calculate free vibration response with the mathematics of
symmetric eigenvalue problem. The orthogonality of the eigenvectors allows to uncouple the
equations of motion. It causes that a certain mode of structure vibration may be interpreted as
a single dof oscillator. The corresponding natural frequency is found by:
� = VW� = V�R� �S:Q
,S,QH�T��O:PL�Q = V�� ∙ S:HT�%H� (3-3)
Therefore, natural frequencies of two geometrically identical structures but using different
material can be compared as:
���� = V���� ���� (3-4) In order to capture material influence on modal analysis results, 1D beam models with steel and
aluminium material are analysed. Two non-realistic cases of only density or modulus of
P 32 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
elasticity change are implemented to see separate effect of both. As the solution of modal
analysis is linear, later they can be superimposed to obtain real case solution. The influence of
material change from steel to aluminium is defined as:
�XY�BZ = V�XY�BZ �BZ�XY = V200 ∙ 10[
69 ∙ 10[ 27007800 = 1 (3-5) where �XY , �BZ – modulus of elasticity for steel and aluminium respectively, �XY , �BZ – mass density of steel and aluminium respectively. Considering specified case (�BZ = 2700 G]A^, �XY =7800 G]A^), material change should not influence natural frequencies of the structure.
3.1.2. Numerical results
Table 3-1 presents obtained numerical results and shows that numerical results correlate well
with analytical ones.
Table 3-1. Material characteristics influence on natural frequency of the beam
Mode �?(�XY , �XY) ��(�XY , �BZ) ��(�BZ, �XY) �*(�BZ, �BZ)
[Hz] [Hz] ��/�? [Hz] ��/�? [Hz] �*/�? 1 5.46 9.29 1.70 3.21 0.59 5.45 1.00 2 11.83 20.12 1.70 6.95 0.59 11.82 1.00 3 21.83 37.10 1.70 12.82 0.59 21.79 1.00 4 46.88 79.85 1.70 27.59 0.59 46.90 1.00 5 49.00 83.30 1.70 28.79 0.59 48.93 1.00 6 86.77 147.56 1.70 50.99 0.59 86.67 1.00 7 103.79 177.22 1.71 61.24 0.59 104.09 1.00 8 134.73 229.21 1.70 79.21 0.59 134.63 1.00
In usual modal analysis procedure, while transformation from physical to modal coordinate
system, all the mode shapes are normalized. By default, the mass is used for normalization.
Each mode shape KL�M with its corresponding natural frequency �� is normalized with respect to the mass matrix �8. Scaling factor _� for each mode shape is determined by: (_�KLM�)`�8(_�KLM�) = 1 (3-6)
_� = 1/aKLM�̀ �8KLM� (3-7) In this case, the corresponding generalized mass matrix is of the form:
��b = �L`�8�L = � (3-8)
Guideline development for offshore structure vibration analysis 33
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
where � – unit matrix. Even if the obtained natural frequencies are the same when using different material, eigenvectors are normalized by different mass matrices. Therefore, resulting
mode shapes are different. Figure 3-1 and Table 3-2 present some comparisons of mass
normalized mode shapes of steel and aluminium beams for mode shape 1.
Figure 3-1. Comparison of mass normalized mode shapes of steel and aluminium beams
Table 3-2. Comparison of mass normalized mode shapes of steel and aluminium beams
Node Eigenvector (T2)
Ratio Steel Aluminium
1 0,00 0,00 0,0
2 0,68 1,15 1,7
3 1,31 2,22 1,7
4 1,85 3,15 1,7
5 2,27 3,85 1,7
6 2,53 4,30 1,7
7 2,62 4,45 1,7
8 2,53 4,30 1,7
9 2,27 3,85 1,7
10 1,85 3,15 1,7
11 1,31 2,22 1,7
12 0,68 1,15 1,7
13 0,00 0,00 0,0
The last column shows the ratio between nodal coordinates which is equal to inverse of density
ratio:
;%�,: = c2BZc2XY = 1/ N�BZ�XYU = 1.7 (3-9) On the other hand, if the eigenvector are normalized by the maximum displacement of the
eigenvector, both cases gives the same mode shape. As shown on Figure 3-2 which compares
the translation along y-axis for mode shape 1, it is seen that both mode shapes fully coincide.
Subsequent frequency response analysis may to be carried out in order to prove that
displacement is the same for steel and aluminium beams.
0
1
2
3
4
5
1 2 3 4 5 6 7 8 9 10 11 12 13
Mod
al d
ispl
acem
ent [
/]
Node
SteelAluminium
P 34 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
Figure 3-2. Comparison of mode shapes of steel and aluminium beams normalized by maximum displacement
Conclusion:
• The influence of material change for the same structure can be evaluated by calculation
of the square root of corresponding moduli of elasticity and densities;
• Eigenvector normalization option should be taken into account while comparing mode
shapes.
3.2. Utilization of Solid Elements
The numerical problems of shear locking and hourglassing should be addressed before using
3D elements to simulate problems with dominant bending deformation as it is possible to obtain
untrue results in certain situations.
3.2.1. Shear locking
Fully integrated linear 8-node elements tend to be overly stiff in modal analysis where bending
deformation is dominant. This numerical phenomena is known as shear locking. In an ideal
situation, a block of material under a pure bending moment experiences a curve shape shown
in Figure 5-3. In order to clarify the presentation, straight dotted lines are drawn on the surface
of the block. When the block is submitted to a bending moment, horizontal dotted lines and
edges bend to curves, while vertical dotted lines remain straight. The angle A remains at 90
degrees after bending, just as predicted by classical beam theory [7].
Figure 3-3. Shape change of the material block under the moment in the ideal situation [6]
0.00
0.30
0.60
0.90
1.20
1 2 3 4 5 6 7 8 9 10 11 12 13
u [/
]
Node
SteelAluminium
Guideline development for offshore structure vibration analysis 35
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
Therefore, an element should be capable to deform to curve shape, in order to resemble the
ideal shape change. But the edges of the fully integrated first order element are not able to bend
to curves. When submitted to a pure bending moment, the linear element will have shape as
shown in Figure 3-4.
Figure 3-4. Shape change of the fully integrated first order element under bending moment [6]
Despite of what was mentioned above, the angle A changes under pure moment. It causes an
artificial shear stress to the element and corresponding strain energy developing in the element
generates shear deformation instead of bending deformation. Consequently, the linear fully
integrated element gets overly stiff or locked under the bending moment.
Unlike the first order element, the edges of the fully integrated second order element are capable
to bend as expected. Therefore, the element deformation shape will correctly model the material
block behaviour shown in Figure 3-5.
Figure 3-5. Shape change of the fully integrated second order element under the moment [6]
3.2.2. Hourglassing
In order to tackle the shear locking and decrease computation effort, the use of reduced
integration elements is sometimes suggested. For 8-node CHEXA element, instead of eight
integration points, a single point is used. Nevertheless, this solution of shear locking prone to
its own numerical difficulty called hourglassing. The reduced integration first order element
tends to be excessively flexible. Figure 3-6 depicts the element deformation subjected to a
bending moment. The striking feature is that vertical and horizontal dotted lines and the angle
A remain unchanged. This means that normal stresses and shear stresses are zero at the
integration point and that no strain energy is generated by the deformation. This zero energy
P 36 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
mode is a purely numerical nonphysical response, which may propagate when a somewhat
coarse mesh is used. The propagation of such a mode may therefore produce meaningless
results. The results often indicate that the structure is excessively flexible [6].
Figure 3-6. Shape change of the reduced integration element under the moment [6]
The reduced integration second order solid element is prone to hourglassing, if only one layer
of elements is used. Practically problem can be solved by the use of two or three layer of
elements.
3.2.3. Plate example
Previously studied model of simply supported plate is further considered to capture numerical
problems and suggest way to avoid them. Obtained results are normalized to the previously
obtained converged solution using shell elements d;QTLP� = efghijefklhhm. This way it is easy to see the discrepancy between obtained results and converged solution. Figure 3-7 shows the part of
the plate modelled with 8-node finite elements.
Figure 3-7. Part of the plate modelled with 8-node finite elements
Guideline development for offshore structure vibration analysis 37
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
Table 3-3 presents the normalized first 3 modes obtained by using fully integrated 8-node
CHEXA elements. This table demonstrates that the results are overly stiff. Moreover, the mesh
refinement does not influence results. To conclude, this type of element should be avoided when
the modelled structure is expected to bend.
Table 3-3. Normalized natural frequency from fully integrated 8-node CHEXA elements
Table 3-4 shows the normalized first 3 modes obtained by using this time fully integrated 20-
node CHEXA elements. Even when using a very coarse mesh, use of this element type leads to
some results close to the converged solution.
Table 3-4. Normalized natural frequency from fully integrated 20-node CHEXA elements
Mode Mesh size (height x length x width)
1x36x8 4x36x8 8x36x8
1 1,03 1,02 1,00
2 1,03 1,03 1,01
3 1,04 1,03 1,02
Following Table 3-5 shows the normalized first 3 modes calculated by using reduced 8-node
CHEXA elements.
Table 3-5. Normalized natural frequency from reduced integration 8-node CHEXA elements
Mode Mesh size (height x length x width)
1x36x8 4x36x8 8x36x8
1 1,03 1,06 1,06
2 1,03 1,06 1,06
3 1,03 1,05 1,05
Despite of expected problem of hourglassing, obtained results are close to the converged
solution even when using a coarse mesh. From the NASTRAN manual, it is seen that a special
technique “bubble function” is employed while using reduced integration scheme. This is
recommended because it minimizes shear locking and Poisson’s ratio locking and does not
cause hourglass deformation modes associated with no strain energy [10]. Moreover, for
CHEXA elements with no mid-side nodes, reduced shear integration with bubble function is
the default. This way, the human errors of hourglassing are reduced.
Attempt to perform calculation of 20-node CHEXA elements with reduced integration is not
possible, as bubble function is not allowed for elements with mid-side node. Nevertheless,
Mode Mesh size (height x length x width) 1x36x8 4x36x8 8x36x8
1 7,36 7,35 7,35
2 7,05 7,04 7,04
3 6,65 6,64 6,63
P 38 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
NASTRAN internally switches off bubble function and calculation gives no fatal error. Table
3-6 shows obtained results for the plate with reduced integration scheme and no bubble function
when using 20-node CHEXA elements. Results are accurate, even when modelling the plate
with only one layer of elements.
Table 3-6. Normalized natural frequency from reduced integration 20-node CHEXA elements
Mode Mesh size (height x length x width) 1x36x8 4x36x8 8x36x8
1 1,03 1,02 1,01
2 1,03 1,02 1,02
3 1,04 1,03 1,03
Conclusion:
• The fully integrated first order solid elements (8-node CHEXA elements) should be
avoided for problems with dominant bending deformation. If fully integrated solid
elements are used, they should be 20-node CHEXA;
• Reduced integration elements are computationally inexpensive and tolerant to element
distortion.
3.3. Effective Modal Mass
Numerical models of complex structures usually possess a significant number of dof. It
corresponds to the number of computable vibration modes of the structure. It is common
practice to perform vibration mode truncation in order to obtain accurate result and increase
computational efficiency. It is possible to use the effective modal mass as a means to determine
the number of modes that should be considered when performing a global vibration response
analysis in order to obtain realistic results after modal superposition.
Modes with relatively high effective masses can be easily excited by external load. On the
contrary, modes with low effective masses cannot be easily excited in such a way. Main
contribution of the use of effective modal mass are:
- The effective modal mass can be utilized as a means to evaluate the “significance” of
vibration mode;
- As a rule of thumb, the number of vibration modes to consider in a typical global
vibration response analysis is determined so that the total effective mass of the model is
at least 90% of the actual mass [8].
The governing matrix equation for an undamped dynamic system is:
Guideline development for offshore structure vibration analysis 39
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
�8K=n M + �JK=M = KoM (3-10) where �8 – mass matrix, �J – stiffness matrix, K=n M – acceleration vector, K=M – displacement vector, KoM – excitation force vector. A solution of the homogeneous form of eq. 3-10 can be expressed in terms of eigenvalues � and eigenvectors KpM. The eigenvectors represent the mode shapes. Let the matrix composed of eigenvector: �5 = �Kp�M, Kp�M, … , KprM. The system’s generalized mass matrix is given by:
��b = �5`�8�5 (3-11) Let K;M be an influence vector, which represents the displacement of the masses resulting from static application of a unit ground displacement. The influence vector induces a rigid body mode
in all modes.
Define the coefficient vector as:
K�M = �5`�8K;M (3-12) The modal participation factor for mode i is:
� = ���b �� (3-13) The effective modal mass for mode i is:
muvv,� = ����b �� (3-14) It worth to mention that the off-diagonal normalized modal mass matrix terms w�b �! , , ≠ .y are zero regardless of the normalization and even if the physical mass matrix �8 has distributed mass. This is due to the orthogonality property of the eigenvectors. The off-diagonal modal
mass terms do not appear in eq. 3-14 [8].
NASTRAN makes it possible to derive modal mass participation. NASTRAN version 2001 and
beyond has the capability to extract modal mass participation directly from normal modes
analysis [9]. The card used to extract the modal mass is MEFFMASS (see Reference [10] for
more details).
Previously analysed case of clamped I-beam is considered to show the use of effective modal
mass. For the simple case of I-beam, mesh density may limit the number of modes to consider.
Let consider the example of I-beam meshed using 6 nodes and 5 elements. Clamped boundary
conditions are imposed at the beam ends, so only 4 intermediate nodes are free to move. The
maximum number of modes to analyse is limited to 9Az{uX = 9{zv ∙ 9rz{uX = 6 ∙ 4 = 24. Nevertheless, the maximum number of modes in NASTRAN output is only 12 (it is not clear
why rotation dof are not considered). For this number of modes, the percentage of the
P 40 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
considered effective modal masses is less than 90%. Table 3-7 presents the ratio between the
effective modal mass for the translations along x-, y-, z-axes (denoted by T1, T2, T3) over the
total beam mass.
Table 3-7. Total effective modal mass as a function of the mesh density and number of modes to consider.
In the first and second configurations listed in Table 3-7, the elements mesh density limits the
number of modes to consider. The third case (15 elements) corresponds to a mesh sufficiently
fine to consider at least 90% of actual mass for the translation in three orthogonal directions.
Figure 3-8 depicts the evolution of the effective modal mass sum as a function of the vibration
mode number, for the I-beam with 15 elements.
Figure 3-8. Evolution of effective modal mass with mode number
Above given Fig. demonstrates that it is reasonable to perform truncation of vibration mode to
consider and it is done in the Table 3-7 in the last row. Moreover, from Figure 3-8, one can
distinguish the modes with relatively big effective modal mass that will significantly contribute
to inertia force (1, 4, 8, 11). Figure 3-9 demonstrate the first four mode shapes for the translation
along y-axis with relatively big effective modal mass are shown.
0.0
0.2
0.4
0.6
0.8
1.0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Sum
of e
ffec
tive
mod
al m
ass
frac
tion
[/]
Mode numberT1 T2 T3
Number of elements
Number of modes
Modal effective mass fraction
T1 T2 T3
5 12 0,800 0,800 0,800
9 24 0,889 0,889 0,889
15 40 0,933 0,933 0,933
15 31 0,929 0,933 0,933
Guideline development for offshore structure vibration analysis 41
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
Figure 3-9. Mode shapes with relatively big effective modal mass
As the mode number increases, the mode shape becomes increasing more complex and
displacement is distributed more uniformly over the model. It is possible to state that
corresponding kinetic energy decreases and strain energy increases with mode number.
Therefore, first modes are of more importance, if displacement is goal of study. It cannot be
proven by performing only a modal analysis because results are normalized and are not
comparable between different modes. To prove that lower modes have higher displacement
amplitude, a subsequent frequency response analysis should be performed.
Conclusion:
• Figure 3-8 presents that effective modal mass between the modes with its significant
value, gradually decreases with mode number. From this figure, it is possible to determine
the mode shapes that can be readily excited by excitation;
• Figure 3-9 demonstrates that all the modes with significant effective modal mass are
symmetric and their centre of gravity experience significant translation during excitation.
3.4. Added Mass
Considering a case of surrounding fluid to the structure, vibration is coupled to fluid motions.
Coupled fluid-structure analysis must be perform to find natural frequency of a structure in a
fluid. For small displacement during vibration, no flow detachment occurs. In addition, fluid is
considered to be incompressible. Therefore, potential flow theory can be utilized. For a two-
dimension section with two perpendicular axes of symmetry, fluid effect is specified by the
P 42 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
added mass associated with the acceleration along each of the axes of symmetry and the added
mass moment of inertia for rotation about these axes. Added mass often specified as an added
mass coefficient which is defined as the added mass per unit length. In case of an I-beam for a
specified dimension ratio: a/t=2.6, b/t=3.6 (Figure 3-10) the added mass coefficient for
displacement along z-axis is defined as [3]:
8B = 2.11��%� (3-15)
Figure 3-10. I-beam scantling [3]
The influence on the added mass on the modal frequencies is given by:
|uY{}F = 1(1 + 8B/8)�/� (3-16) where |uY – submerged frequency, {}F – dry frequency, 8B – added mass of fluid and 8 – structural mass.
NASTRAN has an option of virtual mass (VM) approach to consider fluid. In this case, fluid
should not be explicitly modelled. Nevertheless, it has inherent limitations, for example, shell
elements (CQUAD4 or CQUAD3) are only allowed (see Reference [10] for more information
regarding virtual mass). This makes virtual mass approach not applicable for certain finite
elements configurations. Previously studied case of I-beam modelled with 180 2D shell
elements is considered to investigate virtual mass approach. Figure 3-11 presents a comparison
of natural frequencies obtained analytically and numerically.
Guideline development for offshore structure vibration analysis 43
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
Figure 3-11. Natural frequency of wetted slender beam vibration in x-z plane with simply supported
BC
There is a good agreement between obtained results for simple I-beam model. Therefore, virtual
mass approach should be considered as a mean to take into account for fluid presence in case if
it is applicable for studied model.
P 44 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
4. SIMPLE REAL CASE
This section considers the global response of the simple structure of an electrical motor
assembly due to a dynamic unbalance. The electrical motor is typical source of the dynamic
loading for naval and offshore structure and identified as one of the potential cause of excessive
vibration response. The objective is to perform a vibration analysis and to analyse obtained
results.
General Assumption:
• No stress analysis after vibration analysis is required;
• The rate of change of excitation frequency is negligibly low. Thus, the vibration response is
a steady state response and transient response is not considered.
4.1. Model Description
Figure 4-1 shows that structural elements of a foundation are difficult interconnected between
each other. This fact significantly changes the boundary conditions and consequently the modal
behaviour. Therefore, the analytical solutions for the definition of finite element size described
in the section of simple basic structures are not applicable for the study of electrical motor (E-
motor) and its foundation. Moreover, it is a local study, hence the size of finite elements is not
critical and it is chosen in order to represent correctly the mode shapes that occur within the
frequency range of interest in the dynamic response calculation.
Figure 4-1. E-motor foundation
Guideline development for offshore structure vibration analysis 45
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
As it was assumed, no subsequent stress analysis is required. Therefore, small model features
that do not significantly influence neither mass nor stiffness can be excluded. Table 4-1 presents
the bill of materials for the E-motor assembly (electrical motor and its foundation).
Table 4-1. The bill of materials for the E-motor assembly
Number Quantity Description Material Dimensions Remarks Total mass [kg]
Mass fraction
[/]
1 1 Electrical motor
6000 0,612
2 2 HEM 280 S235JR L = 3630 Longitudinal
beam 1372 0,140
3 2 HEM 280 S235JR L = 2081.5 Transverse
beam 787 0,080
4 1 Steel sheet S275JR 3190x2200x25 Foundation
plate 1215 0,124
5 4 Steel sheet S275JR 100x80x30 7 0,001
6 4 Steel sheet S275JR 345x210x100 Mounting
plate 226 0,023
7 4 Steel sheet S275JR 250x162x30 38 0,004
8 2 Steel sheet S275JR 450x400x25 70 0,007
9 2 Steel sheet S275JR 490x430x25 82 0,008
Sum 9798 1,000
Above Table demonstrates that first four items represent 96% of the total mass. Moreover, these
elements except of the E-motor are structural elements that stiffen the foundation. Therefore,
for finite element model only foundation plate, longitudinal and transverse beams are explicitly
modelled. Furthermore, modal behaviour of the electrical motor itself is of minor interest,
hence, it is assumed perfectly rigid and modelled as a lumped mass in its COG connected by
rigid elements to the foundation. It worth to mention that exact vertical location of the COG is
unknown. For the master thesis COG is located at � = 3187�� above foundation plate. This is a potential source of the discrepancy between obtained numerical and measurement campaign
results [11].
Two models were implemented by using either 1D finite elements or 2D finite elements for
beam structural elements in order to check if 1D simplification of beams influences results.
4.2. Modal Analysis
Prior to the frequency response analysis, a modal analysis is performed in order to investigate
the natural frequencies of E-motor assembly. It allows to identify beforehand if resonance
occurs. In addition, mesh convergence can be verified and the number of vibration modes to
consider can be defined. Considering that modal analysis uses modal decomposition,
P 46 Andrii Pishchanskyi
Master Thesis developed at ICAM, Nantes
performing first the modal analysis increases overall computational efficiency of the frequency
response.
4.2.1. Frequency Range of Mode Extraction
The nominal speed rate of the electrical motor is 900RPM, which corresponds to maximum
excitation frequency u0~ = 15��. According to recommendation from “18th International ship and offshore structure congress” – ISSC (2012) [2], frequency range of vibration mode
extraction should be increased by the factor W = 1.4. Later the suitability of this empirical guide for the study of E-motor assembly is checked.
u0Y = W ∙ u0~ = 15 ∙ 1.4 = 21�� (4-1)
4.2.2. Mesh Convergence
Mesh convergence study of the model using solely 2D finite elements is performed in order to
validate that the solution is convergent. Figure 4-2 demonstrates finite elements model of the
E-motor assembly using 2D finite elements for both plate and beams structural elements. In
order to provide congruence between the nodes of plate and beams, they are modelled in the
same plane. This simplification of numerical model eases modelling and is supposed to have
minor effect on results.
Figure 4-2. Finite elements model of E-motor assembly
Guideline development for offshore structure vibration analysis 47
“EMSHIP” Erasmus Mundus Master Course, period of study September 2013 – February 2015
Table 4-2 demonstrates that the refinement of finite element mesh by a factor of 4 causes only
one percent difference of results.
Table 4-2. Convergence study of the finite elements model
Mode
Natural frequency [Hz] Discrepancy [/]
Δ = ,H,�,%P − ;Q,HQ;Q,HQ Initial mesh (CQUAD=3930, CTRIA3=8)
Refined mesh (CQUAD4=15692, CTRIA3=16)
1 9.10 9.08 0.00
2 13.39 13.20 0.01
3 40.83 40.33 0.01
Therefore, initial mesh with the edge length of 2D finite element varying within the range 30-
75mm is considered to be sufficient for subsequent frequency response analysis. Frequency of
the third mode greatly exceeds both the range of excitation frequency and the frequency range
of mode extraction. Therefore, the inclusion of the first two modes only in modal superposition
gives accurate results in modal frequency response analysis.
4.2.3. Result Comparison of Different Finite Elements Models
Once the mesh density for convergent solution is defined, 2D finite elements used to model
beams are replaced by 1D beam elements. Figure 4-3 shows the finite element model of E-
motor assembly using 2D finite elements to model foundation plate and 1D elements for
longitudinal and transverse beams