Guideline development for offshore structure vibration analysis · 2018. 12. 12. · The objective of the master thesis is to develop a guideline for the vibration analysis of offshore

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


2.3.2. Numerical Results .................................................................................................. 15

2.4. Plate ............................................................................................................................... 19



2.5. Finite Length Cylindrical Shell ..................................................................................... 22



2.6. Stiffened Plate ............................................................................................................... 26



3. SENSITIVITY STUDIES .................................................................................................... 31

3.1. Material sensitivity ........................................................................................................ 31

3.1.1. Analytical solution .................................................................................................. 31


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


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



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.



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



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



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,



dynamic loading from the thruster is investigated. Lastly, vibration analysis shows that

resonance occurs within the frequency range of dynamic loading.



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]



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.



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.



Table 2-2. Numerical natural frequencies of infinitely long cylindrical shell

Mode


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


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



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



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



Table 2-5. Convergence study of natural frequency discrepancy for beam with simply supported BC

Mode shape


Analytical solution


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


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



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

[%]


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



Table 2-7. Convergence study of natural frequency discrepancy for beam with simply supported BC

Mode shape


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



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


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



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


An

alyt

ical

so

lutio

n


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



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



2.5. Finite Length Cylindrical Shell


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



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).



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


Order Number of axial half-waves, j

Number of circumferential waves, i


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


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)


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



(i·j)


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



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


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.



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


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.



Table 2-14. Natural frequency numerical results of stiffened plate

Order


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

[%]


1x1

2x1

3x1

4x1

1x2

2x2

3x2

4x2

5x1

5x2

1x3



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;



• The size of elements for converged solution is 250x250mm for the plate and 250x50 for

stiffeners.



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



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.


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)



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



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



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



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



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



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:



�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:

Γ� = ��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



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



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



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.



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.



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



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




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,



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



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

Documents

Guideline development for offshore structure vibration analysis · 2018. 12. 12. · The objective of the master thesis is to develop a guideline for the vibration analysis of offshore