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AA242B: MECHANICAL VIBRATIONS 1 / 29

AA242B: MECHANICAL VIBRATIONSApproximation of Continuous Systems by Displacement Methods

These slides are partially based on the recommended textbook: M. Geradin and D. Rixen,“Mechanical Vibrations: Theory and Applications to Structural Dynamics,” Second Edition,

Wiley, John & Sons, Incorporated, ISBN-13:9780471975465

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Outline

1 The Rayleigh-Ritz Method

2 The Finite Element Method

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The Rayleigh-Ritz Method

Choice of Approximation Functions

Rayleigh-Ritz approximation

ui (x1, x2, x3, t) =n∑

j=1

nij(x1, x2, x3)qj(t), i = 1, 2, 3

qj , j = 1, · · · , n are the generalized coordinatesvector form of “assumed modes”

nj(x1, x2, x3) =

n1j(x1, x2, x3)n2j(x1, x2, x3)n3j(x1, x2, x3)

u(x1, x2, x3, t) =n∑

j=1

nj(x1, x2, x3)qj(t)

admissibility conditions: C 0 continuity and satisfaction of theessential boundary conditions

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Discretization of the Displacement Variational Principle

Rayleigh-Ritz approximation in matrix form

u(x1, x2, x3, t) = N(x1, x2, x3)q(t)

N ∈ R3×n is the displacement interpolation matrix

[N(x1, x2, x3)]ij = nij(x1, x2, x3), i = 1, 2, 3, j = 1, · · · , n

q(t) =[q1 · · · qn

]Tis the vector of generalized coordinates

Strain componentsrecall the spatial differentiation operator

DT =

∂

∂x10 0 ∂

∂x20 ∂

∂x3

0 ∂∂x2

0 ∂∂x1

∂∂x3

0

0 0 ∂∂x3

0 ∂∂x2

∂∂x1

continuous strain vector

ε(x, t) = Du = DN(x1, x2, x3)q(t) = B(x1, x2, x3)q(t)

where (B(x1, x2, x3) = DN(x1, x2, x3)) ∈ R6×n is the straininterpolation matrix

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Displacement variational principle

kinetic energy

u = Nq⇒ T =1

2

∫V

ρ (Nq)T NqdV =1

2qTMq

where M =

∫V

ρNTNdV is the symmetric positive definite mass

matrix of the discrete systemstrain energy

Vint =1

2

∫V

εTHεdV =1

2

∫V

(Bq)T HBqdV =1

2qTKq

where H is the Hooke matrix of elastic coefficients, and

K =

∫V

BTHBdV is the symmetric positive semi-definite stiffness

matrix of the discrete systemthe rigid body modes u are the solutions of Ku = 0 — they produceno strain energy

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Displacement variational principle (continue)external potential energy

Vext = −∫Sσ

(Nq)T tdS −∫V

(Nq)T XdV = −qTg

where g(t) =

∫Sσ

NT t(t)dS +

∫V

NT X(t)dV is called the external

load factordiscretized displacement variational principle

δ

∫ t2

t1

{1

2qTMq−

(1

2qTKq− qTg

)}dt = 0

=⇒∫ t2

t1

{δqTMq−

(δqTKq− δqTg

)}dt = 0

=⇒[δqTMq

]t2t1−∫ t2

t1

δqT {Mq + Kq− g} dt = 0

recall that δq(t1) = δq(t2) = 0

=⇒ Mq + Kq = g(t)

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Computation of Eigensolutions by the Rayleigh-Ritz Method

Discretized eigenvalue problem

assume free vibrationsassume harmonic motion

Mq + Kq = 0 ⇒ Kqa = ω2Mqa

Theorem: Each eigenvalue ω2i resulting from the discretization of

the displacement variational principle by the Rayleigh-Ritz method isan upper bound to the corresponding exact eigenvalue ω2

i,e

ω2i,e ≤ ω2

i , i = 1, · · · , n

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Computation of Eigensolutions by the Rayleigh-Ritz Method

Eigenmodesonce the eigenvalues ω2

i are determined, the associated eigenmodesqai are obtained from the solution of

Kqai− ω2

i Mqai= 0

the corresponding approximate eigenmodes u(i)1 are given by

u(i) = N(x1, x2, x3)qai

orthogonality conditions

M and K symmetric⇒

{qTaiMqaj

= δij

qTaiKqaj

= δijω2j

assuming the eigenvectors are normalized with respect to Msubstitute the expressions for M and K in the above orthogonalityconditions

=⇒∫VρuT

(i)u(j)dV = δij∫V

(Du(i)

)TH(Du(j)

)dV = δijω

2i

1In this chapter, the subscript (i) is used to denote the i-th mode of u instead ofthe subscript i to avoid confusion with the i-th direction of this vector

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Computation of the Response to External Loading by the Rayleigh-Ritz Method

Response to external loading

modal superposition

q =n∑

s=1

ηs(t)qas

substitute in dynamic equations of equilibrium, premultiply by qTar ,

and exploit orthogonality conditions of the discrete eigenmodes

=⇒ ηr + ω2r ηr = qT

ar g = φr , r = 1, · · · , n

reconstruct the continuous solution as follows

u =n∑

s=1

ηs(t)uas

recall that g =∫SNT tdS +

∫VNT XdV

=⇒ φr = qTar g =

∫S

(Nqar )T tdS +

∫V

(Nqar )T XdV =

∫S

uTar tdS +

∫V

uTar XdV

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The Case of Prestressed Structures

Structure subjected to an initial stress field σ0ij

recall the additional term in the strain energy of the system

Vg =

∫V

σ0ijε

(2)ij dV

where ε(2)ij =

1

2

∂um∂xi

∂um∂xj

=1

2

∂nmk

∂xi

∂nml

∂xjqkql

=⇒ Vg =1

2qTKgq

where

[Kg ]kl =

∫V

σ0ij∂nmk

∂xi

∂nml

∂xjdV

dynamic equations of equilibrium

Mq + (K + Kg ) q = g(t)

associated eigenvalue problem

(K + Kg ) qa = ω2Mqa

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Structure subjected to an initial stress field σ0ij (continue)

geometric stiffness driven by a parameter λ

σ0ij = λσ0?

ij ⇒ Kg = λK?g

parameterized (λ) eigenvalue problem(K + λK?

g

)q = ω2Mq

static case (ω = 0) (K + λiK

?g

)qi = 0

=⇒ eigenproblem where the eigenvalues λi correspond to criticalloadsthe eigenvalue λ1 with the smallest module yields the prestress statein which the system buckles

=⇒ σ0ijcr = λ1σ

0?ij

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Application of Rayleigh-Ritz method to a continuous systemThe clamped-free uniform bar

u(x) =x

lq1 +

x2

l2q2

=⇒ q =

q1

q2

M = ml

13

14

14

15

K =EA

l

1 1

1 43

=⇒

ω21 = 2.49 EA

ml2> ω2

1,e = 2.46 EAml2

ω22 = 32.18 EA

ml2> ω2

2,e = 22.21 EAml2

u(x) =x

lq1 +

x2

l2q2 +

x3

l3q3

=⇒

ω2

1 = 2.47 EAml2

> ω21,e = 2.46 EA

ml2

ω22 = 23.4 EA

ml2> ω2

2,e = 22.21 EAml2

ω23 = 109.2 EA

ml2> ω2

3,e = 61.68 EAml2

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The Finite Element Method

Finite Element Approximation

Finite Element approximation

particular application of the Rayleigh-Ritz methodthe structural domain is subdivided into a finite number of“elements” defined by the connectivity of “nodes”the assumed modes are local “shape” functions“local” refers to an element of the structural domain

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Interpolation of the displacement field

u(x1, x2, x3, t) =n∑

j=1

nj(x1, x2, x3)qj(t)

let (x(l)1 , x

(l)2 , x

(l)3 ), l = 1, · · · , n denote the coordinates of the nodes

of the elementsthe generalized coordinates ql are the “nodal” values of thedisplacement field

ql(t) = ui(x

(l)1 , x

(l)2 , x

(l)3 , t

)=⇒ nij(x

(l)1 , x

(l)2 , x

(l)3 ) = δjl , i = 1, 2, 3

indeed

ui (x(l)1 , x

(l)2 , x

(l)3 , t) =

n∑j=1

nij(x(l)1 , x

(l)2 , x

(l)3 )qj(t) =

n∑j=1

δjlqj(t) = ql(t)

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u(x1, x2, x3, t) =n∑

j=1

nj(x1, x2, x3)qj(t)

nj are continuous functions — usually, polynomials

global viewpoint: nj is attached to node j and has support in theelements attached to this node — it vanishes elsewhere

local viewpoint: inside each element e, the displacement field isapproximated by a linear combination of a small number of shapefunctions ϕe

i equal to the number of nodes i attached to element e

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Examplesfinite element approximation of a scalar field in one dimension

nj(x (l))

= δjl

⇒ u(x (l), t

)=

n∑j=1

nj(x (l))qj(t) = ql(t)

finite element approximation of a vector field in one dimension

n1j

(x (l))

= δj(3l−2), n2j(x(l)) = δj(3l−1), n3j(x

(l)) = δj(3l)

⇒

u1

(x (l), t

)=

n∑j=1

n1j

(x (l))qj(t) = q3l−2(t)

u2

(x (l), t

)=

n∑j=1

n2j

(x (l))qj(t) = q3l−1(t)

u3

(x (l), t

)=

n∑j=1

n3j

(x (l))qj(t) = q3l(t)

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Partition of unity

u(x1, x2, x3, t) =n∑

j=1

nj(x1, x2, x3)qj(t)

the chosen shape functions nj must satisfy

n∑j=1

nij(x1, x2, x3) = 1, ∀i = 1, 2, 3

in order to preserve a constant displacement field (rigid body mode)

for example, in one dimension

qj (t) = u(t), ∀j = 1, · · · , n

⇒ u(x , t) =n∑

j=1

nj (x)qj (t) = u(t)n∑

j=1

nj (x) = u(t)

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Generation of P1 Elements

P1 family of elements

in one dimension

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Restriction of a global function to an element

consider the domain V as a union of elements

V =N⋃

e=1

Ve

restriction of the global function nj(x1, x2, x3) to an element e

ϕej (x1, x2, x3) = ϕj(x1, x2, x3) = nj(x1, x2, x3)|Ve

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P1 family of elementsshape functions in one dimension

Ne(x) =[ϕ1(x) ϕ2(x)

]=[

1− x

h

x

h

]

displacement field

u(x , t) = ϕ1(x)u1(t) + ϕ2(x)u2(t) = Ne(x)qe(t)

with qe(t) =[q1(t) q2(t)

]T20 / 29



Element-Level Matrices

Element energies

element-level kinetic energy

Te =1

2

∫Ve

ρe uT udV

where Ve is the volume of element eelement-level strain energy

Vint,e =1

2

∫Ve

(Du)T H (Du) dV

element-level potential energy

Vext,e = −∫Sσ,e

uT tdS −∫Ve

uT XdV

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Element-level mass and stiffness matrices

starting from u = Neqe

consider the element-level kinetic energy

Te =1

2

∫Ve

ρe (Ne qe)T Ne qedV =1

2qTe Me qe

where the element-level mass matrix is defined as

Me =

∫Ve

ρeNTe NedV

consider the element-level strain energy

Vint,e =1

2

∫Ve

(DNeqe)T H (DNeqe) dV =1

2qTe Keqe

where the element-level stiffness matrix is defined as

Ke =

∫Ve

(DNe)T H (DNe) dV

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Generalized loadconsider the element-level potential energy

Vext,e = −∫Sσ,e

qTe N

Te tdS −

∫Ve

qTe N

Te XdV = −qT

e ge

where

ge =

∫Sσ,e

NTe tdS +

∫Ve

NTe XdV

is the generalized load

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Global Matrices

Global energies

global kinetic energy and mass matrix

T =N∑

e=1

Te =N∑

e=1

1

2

∫Ve

ρe uT udV =

N∑e=1

1

2qTe Me qe

global strain energy and stiffness matrix

Vint =N∑

e=1

Vint,e =N∑

e=1

1

2

∫Ve

(Du)T H (Du) dV =N∑

e=1

1

2qTe Keqe

global potential energy and generalized force vector

Vext =N∑

e=1

Vext,e = −N∑

e=1

∫Sσ,e

uT tdS −∫Ve

uT XdV = −N∑

e=1

qTe ge

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Global Matrices

Localization operator

consider the global vector of generalized displacements

q =[q1 q2 q3 · · · qn

]Tconsider an element e and its generalized displacements qe

qe = Leq

where Le is a localization operatorfor example, if e = (1, 2)

Le =

[1 0 0 · · · 00 1 0 · · · 0

]⇒ qe =

[q1 q2

]T

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Global Matrices

Global mass matrix

N∑e=1

1

2qTe Me qe =

1

2qT(

N∑e=1

LTe MeLe

)q =

1

2qTMq =⇒ M =

N∑e=1

LTe MeLe

Global stiffness matrix

N∑e=1

1

2qTe Keqe =

1

2qT(

N∑e=1

LTe KeLe

)q =

1

2qTKq =⇒ K =

N∑e=1

LTe KeLe

Global generalized force vector

N∑e=1

qeT ge = qT

(N∑

e=1

LTe ge

)= qT g =⇒ g =

N∑e=1

LTe ge

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Assembly Process

Scatter and gather

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Assembly Process

Assembly of the mass matrix

loop over all the elements e = 1, · · · ,Nadd the local contribution LT

e MeLe to the mass matrix of theassembled system

M =N∑

e=1

LTe MeLe

Assembly of the stiffness matrix

similarly

K =N∑

e=1

LTe KeLe

Assembly of a load vector

similarly

g =N∑

e=1

LTe ge

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Lumped Mass Matrix

Lumping of the mass matrix

consistent mass M = [mij ]M is usually sparsemass lumping

m11 m12 . . . m1n

m21

. . ....

.... . .

...mn1 . . . . . . mnn

︸︷︷︸

M

→

n∑i=1

m1i 0 . . . 0

0n∑

i=1

m2i

...

.... . . 0

0 . . . 0n∑

i=1

mni

︸︷︷︸

M(L)

M(L) is the lumped mass matrix

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