Structural Sound and Vibrations
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Vocabulary:
discrete system: mass is concentrated in isolated points
flexible solid: mass is distributed continuously
structure: solid satisfies certain geometrical assumptions (e.g. thin-walled). Beams, plates, shells.
Structural Sound and Vibrations
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 1
plates, shells.
structural vibrations: free and forced vibrations of (bounded) structures.
structural sound: waves propagating through (unbounded) structures.
A bar is vibrating in its axial direction. The axial displacements are denoted U(x,t).Consider an arbitrary section of length dx
m = dx
x,UE,A,
A=FdA dx = + F
dx
1. Longitudinal Waves in Bars
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 2
m = dx
dxUAUmm....
==F
A=
F A dxdx
+
= +
F
= E , dxdu
= 2
2
2
2
t
UEx
U
=
22
2
2
t
UEx
U
=
-i( , ) ( ) e tU x t u x =
2 ( ) ( ) 0u x u x
E
+ =
( ) i -i1 2L Lk x k xu x C e C e= + ( ) ( )i -i1 2( , ) L Lk x t k x tU x t C e C e += +
The infinite bar admits longitudinal waves, traveling right and left, with the
Speed of Sound and Wave Length
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 3
The infinite bar admits longitudinal waves, traveling right and left, with thespeed of sound,
corresponding to the longitudinal wave number . Waves in an infinite bar can propagate at all frequencies.
LE
c
=
LL
kc
=
2LL
c Ef
pi
= =
Longitudinal Modes
0)()0( == luu
l
If the bar is of finite length, the displacement must satisfy boundary conditions, e.g. (0) = (l) = 0 for the free bar.
, 0,1, 2,n E npi = =
Free vibrations in finite bars are possible only at certain discrete eigenfrequencies. The corresponding shapes of vibration are called eigenmodes.
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 4
, 0,1, 2,nn E
nlpi
= =
( ) cosnn
u x xlpi
=
2. Bending Waves in Beams
Replace in beam equation EIw(x)=q(x) the static load by the inertia force Aw
x
E,I,
( ),W x t
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 5
4 2
4 2( , ) ( , ) 0W x t W x tEI Ax t
+ =
( ) ( ), i tW x t w x e =4
24 0
d w Aw
dx EI
+ = ( )i t kxW e
4B
AkEI
=
Speed of Sound and Wave Length
412B
EIA
pi
=
Remark: the speed of sound depends on the driving frequency. Such waves arecalled dispersive.
4BBk EI
cA
= =
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 6
nlpi
Bending Modes (Standing Waves)Beams of finite length admit free vibrations only at certain frequencies, depending on the beam length and the boundary conditions at the ends. Free vibrations can be interpreted also as standing waves, which result from the interference of traveling waves and their reflections.
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 7
pi
2pi3pi4pi
AEI
nn 2=
Nodes of eigenmode
The eigenfrequencies and modes can be analytically computed for various boundary conditions; see table in the appendix.
hx
y,z W
( )3
20,
12 1EhD W hW D
+ = =
i t=
3. Bending Waves in Plates
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 8
( )( , , ) ( , ) i tW x y t w x y e =
4 0D W W =4 2 h
D
=
4BD
ch
=
Bending waves can propagate through infinite plates at arbitrary frequencies. The speed of sound and the wavelength are
412B
Dh
pi
=
Plates with finite dimensions can vibrate freely at their eigenfrequencies. Each eigenfrequency corresponds to a standing wave (bending mode).
a
h
b
Standing Waves (Bending Modes)
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 9
The eigenfrequencies and modes are known analytically only for some specialcases. For instance, the eigenfrequencies of a rectangular plate with simplysupported edges are
2 22
mn
m n Da b h
pi
= +
cf. Gross, Hauger, Schnell, TM4 5. Auflage S. 247
, 1, 2,m n =
where m,n indicate the number of sine-half-waves in the mode shape.
The FEM is a discretisation method. FE models of structural vibrations are linear matrix-vector equations, similar to the vibrations of discrete systems.
Example: E,A,, l
le le
4. Modal analysis with FEM
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 10
=
eAl
e
EAc
l=
e
EAc
l=
2 0 = K M u
2eAl 2eAl
2 2
2 2 0U Ux E t
=
4 2
4 2( , ) ( , ) 0W x t W x tEI Ax t
+ =
0D W h W + =
Bar:
Beam:
Plate:
Analytical and Computational (FEM) Models of Structural Vibrations
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 11
0D W h W + =
Work of stresses on strains (strain energy) =
potential energyWork of inertia forces on displacements =
kinetic energy
FE models of free structural vibrations
2 0 = K M u
Stiffness matrix Mass matrix
Plate:
Example: simply supported beam
FEM
Exact
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 12
Clamped/ clamped
Example ctd.
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 13
Longitudinal stiffness of beams >> bending stiffness large eigenfrequencies
Example ctd. (Longitudinal vibrations)
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 14
, 0,1,2,3,...nn E
nlpi
= =
In principle, any construction from flexible material represents a distributed-mass system that will vibrate under dynamic loads. The loads can be:- harmonic or periodic, - transient, - stochasticOnly harmonic loads are considered in this course.
5. Forced vibrations (Frequency Response Analysis)
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 15
x( ),F x t
( )4 2
4 2( , ) ( , )
,
W x t W x tEI A F x tx t
+ =
1. Separation ansatz:
2. Decomposition of load
( , ) ( ) , ( , ) ( )i t i tW x t w x e F x t f x e = =
( ) ( )f x f w x=
5.1. Analytical Solution
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 16
into the eigenfunctions
3. Decomposition of unknown displacement ( ) ( )1
k kk
w x y w x
=
=
( ) ( )1
k kk
f x f w x=
=
4. Solution of modal equations ( ) 2 21 ik
kk
fyA
=
+
x( ) ( )0, cosF x t x t =
0x x=
1. Fourier decomposition of force:
( ) ( ) 00
2sin
l
k kxf f x w x dx k
l lpi
= =
( ) 2 sin2
sin
k kw x xlxk
l l
pi
=
=
Normed by:
l
Modal Basis
Example: Local Stiffness (1/4)
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 17
( ) ( )1
k kk
w x y w x
=
=
0 l l
2. Solution of modal equations: ( )20
1l
kw x dx =
( )( )
0
42
2sin
1 ik
xkl ly
EI kAA l
pi
pi
=
+
3. Modal Superposition:
4. Local Stiffness: 0x x=
( ) ( )( )
0
42
sin sin2
1 ik k
x xk kl ly w x
Al EI kA l
pi pi
pi
=
+
Local Stiffness (2/4)
3. Modal Superposition:
where
( ) ( )1
k kk
w x y w x
=
=
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 18
( ) ( ) ( )( )
2 0
0 0 41 1 2
sin2,
1 ik k
k k
xklw x y w x
m EI kA l
pi
pi
= =
= =
+
0, 0 = =5. Verification: Static Stiffness:
( ) ( ) ( )
1
4 4 42 2 2
1 1 12 3 51 i 1 i 1 i
m
EI EI EIA l A l A l
pi pi pi
+ + + + + +
0, 0 = = 1
41 1 1Al EI pi
+ + +
Local Stiffness (3/4)x
0 2x l=
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 19
0, 0 = =
x=1:2:1000;0.5*pi^4/sum(1./x.^4)
48.0000
44 4 4 41 1 1
2 1 3 5Al EI
Al
pi
+ + +
Mechanical Eng. Table:
100
101
102
103Local Stiffness of Beam @ le/2, material damping 0.02
1/ ( )w
Local Stiffness (4/4)
EigenMode 7
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 20
0 10 20 30 40 5010-3
10-2
10-1
10
( ) 1f
EigenMode 1
EigenMode 3
EigenMode 5Static stiffness
Continuous Structure Discrete System
FEM
5.2. Frequency response with FEM
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 21
- direct or
- modal reduction
( ) ( )2i + K C M u = fComputational model: Solution:
Example: Plate Vibrations, Direct solution
PARAMETERS FOR SPARSE DECOMPOSITION OF DATA BLOCK SCRATCH ( TYPE=RDP ) FOLLOWMATRIX SIZE = 36966 ROWS NUMBER OF NONZEROES = 386249 TERMS
Size of system matrix = # of unsupported DOF ( = 6*(# of free Nodes) for shell models)
( ) ( )1
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 22
12:40:24 Analysis started.12:40:24 Geometry access/verification to CAD part initiated (if needed).12:40:24 Geometry access/verification to CAD part successfully completed (if needed).12:40:24 Finite element model generation started.12:40:25 Finite element model generated 12000 degrees of freedom.12:40:25 Finite element model generation successfully completed.12:40:25 Application of Loads and Boundary Conditions to the finite element model started.12:40:25 Application of Loads and Boundary Conditions successfully completed.12:40:25 Solution of the system equations for frequency response started.12:40:25 Solution of the system equations for frequency response successfully completed.12:40:25 Frequency response analysis completed.
( ) ( )12i + u = K C M fExtract from solver protocol:
Numerical Results
Forced Vibration @200Hz
LF:
MF:
Forced Vibration @74Hz
Eigenmode @ 74Hz
Forced Vibration @500Hz
Eigenmode @ 50Hz
Harmonic force
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 23
HF:
Forced Vibration @5kHz
Forced Vibration @1kHz
Forced Vibration @500Hz
2000 1200 2mm,=0.1 (0.2 HF)
MAT1 1 2.1+11 .3 8000.
Example ctd./ Modal Reduction (S111)
( ) ( )2i K C M u = fFind: Solution of in a frequency band:0 min max
1. Modal Analysis: Solve eigenvalue problem 2 0i i K M x =
1
1.5
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 24
3. Solution and Back-Transform:
1max1.5n
2. Modal Superposition: 1 1 2 2 n ny y y= + + =u x x x Xy
2iT T + = X K C M Xy X f
K f1
,
= =y K f u Xy
Modal Reduction (Details)
Recall:
2iT T + = X K C M Xy X f
2, ,
T Tn n= =X KX X M X I
1
1
n
=
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 25
:TX CX
diagonal: modal damping, Rayleigh damping, material damping fully populated: local damping (G on MAT1, PBUSH, )
Example: Plate Vibrations
1,
= =y K f u Xy
2 0i i K M x =
( ) ( )2i + K C M u = f
2iT T = X K C M Xy X f
K f
1
2
3
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 26
09:58:52 Finite element model generation started.09:58:52 Finite element model generated 7349 degrees of freedom.09:58:52 Finite element model generation successfully completed.09:58:52 Application of Loads and Boundary Conditions started.09:58:52 Application of Loads and Boundary Conditions successfully completed.09:58:53 Solution of the system equations for normal modes started.09:58:57 Solution of the system equations for normal modes successfully completed.09:58:57 Solution of the system equations for frequency response started.09:58:57 Solution of the system equations for frequency response successfully completed.09:58:57 Frequency response analysis completed.
Extract from solver protocol:
12
3
Appendix: Longituninal vibrations of bars
Ec =
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 27
Quelle: Inman S. 322
Ec
=
Bending vibrations of beams (1/2)
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 28
Quelle: Inman S. 3352
n n
EIA
=n s. Folgeseite
Bending vibrations of beams (2/2)
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 29
The table entries are obtained from the general ansatz
xAxAxAxAxw coshsinhcossin)( 4321 +++=and the boundary conditions of the particular load case.
nL4.7300
nL1.875
AEI
nn 2=
Eigenmodes of beams
clamped/ clamped clamped (cantilever)
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 30
4.7300
7.8532
10.996
14.1372
1.875
4.6941
7.8547
10.995
pi2
12:5 + nn pi
212
:5 nn
The modal effective mass (MEM) of a vibration mode with respect to a rigid body mode is defined as
It can be seen from the definition that the MEM are non-negative and their unit is kg (assumingthat the modes are non-dimensional). Essentially the MEM represent mass-weighted scalarproducts of vibration modes and rigid body modes,
ix
, 1, , , 1, ,6.L i N j= = =x Mr
jr
( )2i jij
i i
m =x Mrx Mx
Appendix: Modal Effective Mass
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 31
where N is the number of vibration modes. The rigid body modes consist of uniform unittranslations/ rotations in the direction of/ around each coordinate axis. The scalar product is takenonly over the free nodes of the FE model.
, 1, , , 1, ,6.ij i jL i N j= = =x Mr
Example: Vertical rigid-body mode of simply supported beam.
The scalar product indicates how close a vibrational mode is to a rigid body mode. If the scalar product is one the modes are exactly aligned, if it is zero they are orthogonal.
Origin of Method, References
Modal effective masses are frequently calculated in spacecraft design. Spacecraft structures undergo large accelerated motions during launch and landing, and hence it is of critical interest to establish which deformational modes are likely to be excited by rigid body motions.
In practice, the MEM are also used to determine whether a mode is global or local.
The following simple tests illustrate the method. The results will show that a small MEM does not necessarily indicate local modes.
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 32
Tom Irvine, EFFECTIVE MODAL MASS & MODAL PARTICIPATION FACTORS, http://www.vibrationdata.com/tutorials2/ModalMass.pdf (28.01.2010)
Wijker, Jacob Job, Spacecraft structures, Springer Verlag Berlin, 2008 (Chapter 16: Modal effective mass)
A. Paolozzi and I. Peroni, A PROCEDURE FOR THE DETERMINATION OF EFFECTIVE MASS SENSITIVITIES IN A GENERAL TRIDIMENSIONAL STRUCTURE, Computers & Structures Vol. 62, No. 6. pp. 1013-1024. 1997
RK, Using Modal Effective Mass to Determine Modes for Frequency Response Analysis, http://feadomain.com/e107_plugins/content/content.php?content.76 (28.01.2010)
FE model: four elements, rectangular cross-section, both ends simply supported. Mass:
30.8m, =8000kgml =x
y
z
0.02m 0.04m
800*20*40*8 9 5.12 3tm e e= =
Test 1: Simply Supported beam
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 33
Result: Nine modes. Computation with lumped masses rotational dof are condensed out, 3 trans dof per node.
1,2 (y and z)3,4 (y and z)
5,6 (y and z)7-9: Longitudianal and
torsional modes.
EFFECTIVE MASS MATRIX *** ***
* 3.840000E-03 7.595561E-12 -3.844592E-12 0.000000E+00 -6.286070E-09 -2.332034E-09 ** 7.595561E-12 3.840000E-03 1.766097E-15 0.000000E+00 -8.835683E-13 1.536000E+00 ** -3.844592E-12 1.766097E-15 3.840000E-03 0.000000E+00 -1.536000E+00 -8.579544E-13 ** 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ** -6.286070E-09 -8.835683E-13 -1.536000E+00 0.000000E+00 7.168000E+02 4.088132E-10 ** -2.332034E-09 1.536000E+00 -8.579544E-13 0.000000E+00 4.088132E-10 7.168000E+02 **** ***
A-SET RIGID BODY MASS MATRIX*** ***
3 3.84 3t4
m e=
Rigid body mass matrix
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 34
*** ***
* 3.840000E-03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ** 0.000000E+00 3.840000E-03 0.000000E+00 0.000000E+00 0.000000E+00 1.536000E+00 ** 0.000000E+00 0.000000E+00 3.840000E-03 0.000000E+00 -1.536000E+00 0.000000E+00 ** 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ** 0.000000E+00 0.000000E+00 -1.536000E+00 0.000000E+00 7.168000E+02 0.000000E+00 ** 0.000000E+00 1.536000E+00 0.000000E+00 0.000000E+00 0.000000E+00 7.168000E+02 **** ***
Observation:
rigid body translations in y, z direction are coupled with rotations.
effective mass identical with rigid body mass
four elements each interior node gets of total mass, free boundary nodes get 1/8.
Since boundary nodes are fixed only of total mass is included in rigid body mass matrix.
MODAL EFFECTIVE MASS FRACTION(FOR TRANSLATIONAL DEGREES OF FREEDOM)
MODE FREQUENCY T1 T2 T3NO. FRACTION SUM FRACTION SUM FRACTION SUM
1 7.077026E+01 9.314766E-31 9.314766E-31 9.714045E-01 9.714045E-01 2.673639E-32 2.673639E-322 1.411843E+02 1.945421E-32 9.509307E-31 1.114075E-34 9.714045E-01 9.714045E-01 9.714045E-013 2.802581E+02 1.935277E-25 1.935287E-25 8.024709E-29 9.714045E-01 6.461121E-29 9.714045E-014 5.541207E+02 1.947883E-22 1.949818E-22 3.381378E-25 9.714045E-01 2.764477E-25 9.714045E-015 5.913062E+02 1.512678E-22 3.462496E-22 2.859548E-02 1.000000E+00 5.836389E-26 9.714045E-016 1.148360E+03 8.641317E-22 1.210381E-21 8.764064E-24 1.000000E+00 2.859548E-02 1.000000E+007 3.045298E+03 9.714045E-01 9.714045E-01 5.222461E-18 1.000000E+00 1.346433E-18 1.000000E+008 5.626977E+03 3.037006E-15 9.714045E-01 1.207543E-17 1.000000E+00 3.084168E-18 1.000000E+009 7.352000E+03 2.859547E-02 1.000000E+00 2.592362E-18 1.000000E+00 6.609832E-19 1.000000E+00
Effective modal mass fraction
Observation:
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 35
Observation:
MEM is zero for symmetric mode shapes 3,4.
97%
0%
3%
Same Beam model, four elements, left end clamped.
x
y
z
Result: 12 modes. Computation with lumped masses rotational dof are condensed out, 3 trans dof per node.
Test 2: Cantilever beam
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 36
Result: 12 modes. Computation with lumped masses rotational dof are condensed out, 3 trans dof per node.
1,2 (y and z)
3,4 (y and z)
68% 22%
Mode shapes and modal effective mass fraction..
5,7 (y and z)6,8(y and z)
7% 3%
MODAL EFFECTIVE MASS FRACTION(FOR TRANSLATIONAL DEGREES OF FREEDOM)
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 37
(FOR TRANSLATIONAL DEGREES OF FREEDOM)
MODE FREQUENCY T1 T2 T3NO. FRACTION SUM FRACTION SUM FRACTION SUM
1 2.452855E+01 8.438346E-39 8.438346E-39 6.846376E-01 6.846376E-01 1.080871E-31 1.080871E-312 4.900471E+01 4.317162E-40 8.870062E-39 9.489781E-32 6.846376E-01 6.851764E-01 6.851764E-013 1.438673E+02 8.686789E-41 8.956930E-39 2.171252E-01 9.017628E-01 1.157240E-34 6.851764E-014 2.856255E+02 1.992160E-33 1.992169E-33 1.382351E-31 9.017628E-01 2.179096E-01 9.030859E-015 3.793786E+02 6.398226E-32 6.597443E-32 7.258439E-02 9.743472E-01 6.754913E-29 9.030859E-016 6.570369E+02 8.113327E-30 8.179302E-30 2.565278E-02 1.000000E+00 1.016997E-26 9.030859E-017 7.441080E+02 6.852732E-30 1.503203E-29 7.966597E-28 1.000000E+00 7.224850E-02 9.753344E-018 1.266119E+03 5.962455E-27 5.977486E-27 1.014358E-26 1.000000E+00 2.466557E-02 1.000000E+009 1.552479E+03 9.026479E-01 9.026479E-01 2.163141E-29 1.000000E+00 2.481903E-28 1.000000E+0010 4.421087E+03 7.999389E-02 9.826418E-01 3.634619E-29 1.000000E+00 2.992522E-28 1.000000E+0011 6.616625E+03 1.594510E-02 9.985870E-01 2.315860E-28 1.000000E+00 8.756785E-28 1.000000E+0012 7.804841E+03 1.413076E-03 1.000000E+00 8.412071E-27 1.000000E+00 4.169802E-27 1.000000E+00
A-SET RIGID BODY MASS MATRIX*** ***
* 4.480000E-03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ** 0.000000E+00 4.480000E-03 0.000000E+00 0.000000E+00 0.000000E+00 2.048000E+00 ** 0.000000E+00 0.000000E+00 4.480000E-03 0.000000E+00 -2.048000E+00 0.000000E+00 ** 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ** 0.000000E+00 0.000000E+00 -2.048000E+00 0.000000E+00 1.126400E+03 0.000000E+00 ** 0.000000E+00 2.048000E+00 0.000000E+00 0.000000E+00 0.000000E+00 1.126400E+03 **** ***
78
m
Rigid body mass matrix (clamped beam)
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 38
*** ***
The MEM
indicate the dominant direction of global modes. allow to separate lateral bending modes from axial modes. can be zero for global modes of symmetric models.
Conclusions
HAW/M+P, Ihlenburg, CompA Vibrations: Flexible Bodies 39
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