Structural dynamics and acoustics

Lecture notes

D. Clouteau, R. Cottereau, P.E. Gautier

September 9, 2008

Contents

I Structural Dynamics 30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.2 What should be known to start . . . . . . . . . . . . . . . . . 50.3 Balance of momentum in a moving frame . . . . . . . . . . . 5

0.3.1 Reference and current configurations . . . . . . . . . . 60.3.2 Small displacement field in a moving frame . . . . . . 60.3.3 Velocity and acceleration in the moving frame . . . . . 70.3.4 Strain and strain rate tensors in the moving frame . . 80.3.5 Balance of momentum . . . . . . . . . . . . . . . . . . 9

0.4 Linearized equations in the moving reference configuration . . 120.4.1 Constitutive behavior . . . . . . . . . . . . . . . . . . 120.4.2 Balance of momentum . . . . . . . . . . . . . . . . . . 130.4.3 Boundary and initial conditions . . . . . . . . . . . . . 14

0.5 Free vibrations of small amplitude with geometric stiffness :some examples . . . . . . . . . . . . . . . . . . . . . . . . . . 140.5.1 The pendulum . . . . . . . . . . . . . . . . . . . . . . 150.5.2 The boat . . . . . . . . . . . . . . . . . . . . . . . . . 16

0.6 Important formulae . . . . . . . . . . . . . . . . . . . . . . . . 180.7 Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

0.7.1 Free vibrations of a string . . . . . . . . . . . . . . . . 180.7.2 Drums . . . . . . . . . . . . . . . . . . . . . . . . . . . 190.7.3 Pressured pipes . . . . . . . . . . . . . . . . . . . . . . 200.7.4 Free vibrating modes of the a self graviting earth . . . 21

1 Virtual Power Principle and energy balance 241.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.2 Virtual power principle . . . . . . . . . . . . . . . . . . . . . . 24

1.2.1 Weak formulation and Virtual Power Principle . . . . 251.2.2 Power balance . . . . . . . . . . . . . . . . . . . . . . 261.2.3 Free energy without initial stresses . . . . . . . . . . . 261.2.4 Mathematical framework . . . . . . . . . . . . . . . . 28

1.3 Free vibrating modes of non pre-loaded structures . . . . . . 281.3.1 The Rayleigh quotient . . . . . . . . . . . . . . . . . . 291.3.2 Eigenmode expansion . . . . . . . . . . . . . . . . . . 30

1

2

1.3.3 general solution for an undamped single oscillator . . 311.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 31

1.4 Free vibrating modes of loaded structures in moving frames . 321.4.1 Energy balance . . . . . . . . . . . . . . . . . . . . . . 341.4.2 Stability in a galilean frame . . . . . . . . . . . . . . . 341.4.3 Eigenmodes in a galilean frame . . . . . . . . . . . . . 351.4.4 Eigenmodes in a moving frame . . . . . . . . . . . . . 35

1.5 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.5.1 VPP with viscous damping . . . . . . . . . . . . . . . 361.5.2 Proportional viscous damping . . . . . . . . . . . . . . 36

1.6 Important formulae . . . . . . . . . . . . . . . . . . . . . . . . 371.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.7.1 Rayleigh Quotient . . . . . . . . . . . . . . . . . . . . 381.7.2 The string with a moving load . . . . . . . . . . . . . 381.7.3 Rigid structure . . . . . . . . . . . . . . . . . . . . . . 381.7.4 Response of an undamped SDOF . . . . . . . . . . . . 381.7.5 Approximate fundamental frequency of the self grav-

iting earth . . . . . . . . . . . . . . . . . . . . . . . . . 38

Part I

Structural Dynamics

3

4

Figure 1: Small transformation in a Galilean frame

0.1 Introduction

Vibrations of mechanical structures play an important role in many fields ofmechanics, ranging from safety concerns (aircraft and rocket stability, wavewind and earthquake loaded civil engineering structures...), aging issues (fa-tigue failure of turbines, settlement of railway tracks,...), serviceability andcomfort (equipment vibrations, acoustic comfort of vehicles...), or non de-structive testing.

In most of these cases, the vibrations are not desired, and occur in ad-dition to some nominal mechanical behavior, which they may significantlyinfluence. For example, the natural frequency of a string depends on itstension. Similarly, the vibration and stability of a turbine depend on itsangular velocity.

However, design requirements limit these vibrations to small amplitudes,so that a linearized theory can be used to describe the dynamic behavior ofthe vibrating structures. The aim of the chapter is to derive such a theoryin the framework of continuum mechanics, and to describe some examplesof free vibration modes.

Section 0.2 recalls the continuum mechanics model for small vibrationsof non pre-stressed structures in a fixed frame, as given in section 1 of [1],and following the same notations. Section 0.3 concentrates on the additionalterms introduced by the consideration of a moving frame as the reference,rather than a fixed one, and section 0.4 presents the geometrical stiffnessterms induced by the presence of nominal pre-stresses and the so-calledfollower forces. Finally, section 0.5 introduces the notion of free vibratingmodes, with some examples, and insists on the role of the geometrical stiff-ness terms.

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0.2 What should be known to start

Small vibrations of non pre-stressed structures in a fixed frame

The balance of momentum in a fixed frame of reference reads at any pointx of a structure occupying, at time t, a volume Vt (see fig. 0.1):

Divxσ + fv = ργ (1)

where fv = ρf is the vector of body force density, γ(x, t) denotes theacceleration of the particle, σ(x, t) is the symmetric Cauchy’s stress tensorand Divxσ =

∑3i=1

∂σ∂xi

ei is the divergence of a tensor. Besides, prescribedboundary conditions must be satisfied on the boundary of the domain:

σ n = text(x, t) on Γσ ⊂ ∂Vt (2)u = uext(x, t) on Γu =⊂ ∂Vt \ Γσ (3)

where u(x, t) is the displacement field and and n is the unit outward normalvector on the boundary. Finally, initial conditions for u and the particlevelocity v have to be satisfied inside Vt at t = 0.

Under the hypothesis of small transformations (small displacements andsmall strains), the dynamic behavior of the structure is fully characterized,at time t, by the displacement field u(x, t) with respect to the referenceconfiguration, at time t = 0, and the structure remains close to the referenceconfiguration, so that x, the position at time t, and p, the position in thereference configuration, are equivalent. Moreover, in this case, the subscriptt on the boundary conditions (2) and (3) can be removed.

Assuming a linear isotropic elastic behaviour with respect to a non pre-stressed configuration, this Cauchy’s stress tensor is a linear function of thestrain tensor also known as the Hooke’s law:

σ = λtr(ε)Id + 2µε ε = (Dx(u) + Dx(u)T )/2, (4)

where λ and µ are the two Lame’s coefficients, and the second order tensorDx(u) =

∑3i=1

∂u∂xi

⊗ ei is the gradient of the vector field u. Under thesehypotheses, the acceleration is the double partial derivative of the displace-ment field with respect to time γ = ∂2u

∂t2= u.

0.3 Balance of momentum in a moving frame

When considering the vibration of rotating machines, robots, vehicles, thehypothesis of small transformations with respect to a given Galilean frame(O, i1, i2, i3) is not valid anymore. However, a non Galilean moving frame(C(t), e1(t), e2(t), e3(t)) can often be defined, such that the transformationsremain small within that frame. This moving frame is characterized by

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xC(t), the position of point C in the reference frame, and R(t), the rotationtensor given by:

R(t) =3∑

k=1

ek(t)⊗ ik RT R = Id

0.3.1 Reference and current configurations

Given a particle, characterized by its position vector p in the referenceconfiguration and with respect to a given Galilean frame, its position vectorat time t in the same Galilean frame is denoted x(p, t).

The set of points p ∈ Vo defines the so-called reference configuration inthe reference Galilean frame whereas the set of points x ∈ Vt defines thecurrent configuration in the same Galilean frame.

All quantities (mass density, strains, stresses, forces, velocities...) de-pending on p are material quantities also called Lagrangian quantities. Theywill often be written in the text using a o subscript. On the contrary allquantities depending on the space coordinate x are spatial quantities alsocalled Eulerian quantities. They will often be written in the text using a t

subscript.

0.3.2 Small displacement field in a moving frame

Since we are interested in small vibrations with respect to a nominal movingstate, it is hereafter assumed that the position vector x(p, t) of particle p attime t can be expressed as the combination of a small displacement u(p, t)with respect to the reference configuration (‖Dp(u)‖ << 1) and an arbitraryrigid body motion xo following the moving frame (see fig. 0.3.2):

x(p, t) = xo(p, t) + R(t)u(p, t) (5)xo(p, t) = xC(t) + R(t)(p− pC) (6)

where pC is the position vector in the reference Galilean frame of the par-ticle associated with point C in the reference configuration (for convenienceand, without any loss of generality, we will assume in the following thatpC = 0). In fact, the same function u(xo = p, t) can also be seen as thecoordinate vector of the displacement field in the moving frame1, where x

1Indeed, the position vector components xm of the same particle observed in the movingframe is defined by:

xm = (x− xC) · em

Noticing that em = Rim leads to:

x− xC =Xm

xmem = R(Xm

xmim) = Rx.

7

Figure 2: Small transformation combined with large rotation in a Galileanframe

is the coordinate vector in the moving frame defined as a function of itscoordinate vector x in the reference Galilean frame x by :

xdef= RT (x− xC)

0.3.3 Velocity and acceleration in the moving frame

When observed in the moving frame, the particle velocity v and the particleacceleration γ are not anymore simply given as the partial time derivativesof the displacement u as in a Galilean frame.

This section shows that, in the moving frame, the particle velocity v(xo, t)and the particle acceleration γ(xo, t) read:

v = vo + ˙u + Ωu (7)

γ = γo + ¨u + 2Ω ˙u + Ω2u, (8)

where Ω(t) = RT R, and the particle velocity vo(xo, t) and acceleration

As a consequence, xo = p−pC = p and thus, defining u(xo, t) as the displacement in themoving frame gives :

u(xo, t) = x− xo = RT (x− xo) = u(p, t)

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γo(xo, t) are defined, in this very frame, by:

vo = vC + Ωxo

γo = γC + ( ˜Ω + Ω2)xo

Apart from the classical inertial terms γo and ¨u, two additional terms areto be accounted for when working in a moving frame: the Coriolis term2Ω ˙u, and an additional inertial term Ω2u. Note that the latter, dependinglinearly on the displacements, can be seen as an inertial stiffness, while theformer should not be seen as a damping term, although it depends linearlyon the velocity, because it does not dissipate energy ( ˙uT Ω ˙u = 0).

Proof: Indeed, the particle velocity v in the reference frame reads:

v = xC + Ω(xo − xC)︸ ︷︷ ︸vo

+R ˙u + ΩRu,

with Ω(t) = RRT the skew-symmetric rotation velocity tensor. Defining Ω =RT ΩR = RT R the same tensor observed in the moving frame, this particle velocityv, observed in the moving frame, reads:

v = RT v = vo + ˙u + Ωu vo = RT xC︸ ︷︷ ︸vC

+Ω(p− pC).

Taking once more the partial derivative with respect to time gives the particleacceleration in the reference frame:

γ =dv

dt= xC + (Ω + Ω2)(xo−xC)︸ ︷︷ ︸

γo

+R¨u + 2ΩR ˙u + Ω2Ru,

leading to the equivalent expression in the moving frame:

γ = RT γ = γo + ¨u + 2Ω ˙u + Ω2u,

with:γo = γC + ( ˜Ω + Ω

2)xo γC = RT γC

˜Ω = RT ΩR = ˙Ω

0.3.4 Strain and strain rate tensors in the moving frame

Since we are interested in large transformations, the small strains tensor εdefined in equation (4) as the symmetrical part of the gradient is not a goodmeasure of the strain in the continuum. The proper measure of this strainstate is the Green-Lagrange strain tensor E(p, t) defined (see [1, section 1])in the reference Galilean frame as:

E =F T F − Id

2, F = Dp(x).

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We herein show that, given the decomposition (5), this tensor and the smallstrains tensor ε(xo) in the moving frame coincide:

ε(xo) =Dxo(u) + Dxo(u)T

2= E(xo, t).

but, due to large rotations, they differ from the symmetrical part of thegradient computed in the reference Galilean frame.

The strain rate D(x, t) is defined as the symmetric part of the gradientof the velocity v with respect to the space variable x. It is shown to beequal to the partial time derivative of the small strain tensor computed inthe moving frame :

D(x) ≈ ε( ˙u) (9)

Hence, as far as strains are of interest, the moving frame appears to be themost simple choice, since it allows the use of the small strains approximationwhile using a fixed frame requires a full Green-Lagrange approach.

Proof: Indeed, since xo = p:

F = R (Id + Dp(u)) ⇒ E(p, t) ≈ Dp(u) + Dp(u)T

2.

On the contrary the symmetrical part of the gradient in the reference frame reads :

ε =Dp(x) + Dp(x)T − 2Id

2=

RDp(u) + Dp(u)T RT + R + RT − 2Id

2

and it is small and equivalent to E only for R ≈ Id. As far as the strain rate isconcerned one has:

D =12(Dx(v) + Dx(v)T ) = F−T dE

dtF−1 ≈ R

dE

dtRT

and thus :D = RT DR ≈ dε

dt= ε( ˙u)

0.3.5 Balance of momentum

Let us consider a continuous medium occupying, at time t, the volume Vt

with boundary ∂Vt (see fig. 0.3.5). The Cauchy’s hypothesis (see [1]) statesthat the traction vector t, defined on any surface inside a continuum, de-pends linearly on the outer normal vector n. This leads to the following def-inition of the Cauchy’s total stress tensor σt(x, t), in the reference Galileanframe, or σt(x), in the moving frame:

t(x, t,n) = σt(x, t)n, σt(x) = RT σt(xC + Rx)R.

10

Figure 3: Initial and current configuration for small transformations in amoving frame ( s have been omited).

This Cauchy’s stress tensor σt(x, t) satisfies the balance of momentum (1)in the reference Galilean frame. Simply multiplying the equation by RT

one can easily show that Cauchy’s stress tensor σt(x, t) satisfies the sameequation in the moving frame:

Divxσt(x, t) + ρf(x, t) = ργ(x, t), ∀x ∈ Vt, (10)

where ρf is the applied force density vector, at point x and time t, inthe moving frame, ρ(x, t) is the mass density and γ is given by (8). TheCauchy’s stress tensor also satisfies the boundary conditions (2) in the ref-erence Galilean frame and which reads in the moving frame :

σtn = text ∀x ∈ Γσ ⊂ ∂Vt. (11)

As far as vibrations of structures are concerned, the main drawback ofthese equations is that they are written in terms of so-called Eulerian quan-tities (functions of the space variables x or x) whereas Lagrangian quantities(functions of the material variables p or xo) would be more convenient todefine a constitutive law.

Choosing the moving reference configuration with its associated movingframe, this change of variables leads to 2:

Divxo(F S) + ρofo = ρoγ (12)2Indeed, for any material volume Ωo ∈ Vo in the reference configuration, and occupying

volume Ω(t) at time t:ZΩ(t)

DivxσtdV =

Z∂Ω(t)

σt(n)dS =

ZΩ(t)

ρ(d

dtv − f)dV

F being the gradient of transform from p to x, any elementary volume dV (t) = n · ddSis the transform of dVo = no · (F−1d)dSo = (F−T no) · ddSo = dV (t)/ det(F ), leading to:

ndS = det(F )F−T nodSo

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where F is the gradient of the transform from xo to x, S is the symmetricsecond Piola stress tensor to satisfy a constitutive law, and fo is the massforce density in the moving configuration. These quantities are given by:

F (xo, t) = Id + Dxo(u) (13)

S(xo, t) = det(F )F−1

σ(x)F−T

(14)fo(xo, t) = f(x(xo, t), t) (15)

These equations are valid in any frame, moving or Galilean, and for bothsmall and large transformations. In particular, one can show that the secondPiola tensor S is frame-invariant, as well as the Green-Lagrange strain tensorE. More precisely:

S(p) = S(xo), E(p) = E(xo).

The moving frame has been favored here because, in our case, the Green-Lagrange strain tensor and the small strains tensor ε in this very framecoincide. This will allow us to linearize the equations obtained.

In the proposed formulation, the boundary conditions can be writtenon the boundary of the reference configuration. Indeed, equation (11) nowreads:

Sno = F−1

text(x(xo, t),n) dSdSo

∀xo ∈ Γσo ⊂ ∂Vo, (16)

Since the conservation of mass gives ρ(x) det F = ρo(p), one has:ZΩo

Divp(FS)dVo =

Z∂Ωo

FS(no)dSo =

ZΩo

ρo(d

dtv − fo)dVo

with S(p, t) = det(F )F−1σt(x(p, t), t)F−T and fo = f(x(p, t), t). Since this is true forall Ωo :

Divp(FS) + ρofo = ρoγ

Once multiplied by RT , this equation is nothing but the balance of momentum writtenon the moving reference configuration, the second Piola Tensor being identical in bothreference configurations. Indeed, the tangent linear deformation in the moving frame isdefined as:

F (xo, t) = Dxo(x) = Id + Dxo(u) = Id + Dp(u) = RT F

and the second Piola’s tensor is given by:

S(xo) = det(F )F−1

σ(x(xo, t))F−T

leading to equation (12), since the change of coordinate from the moving to the fixedframe gives :

Rσ(x)RT = σ(x),

and accounting for F = RF , one can show that the second Piola tensor is invariant:

S(p) = det(R)| z 1

det(F )F−1

RT σ(x(p, t))R| z σ(x)

F−T

= S(xo)

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where dS = det F ‖F−Tno‖dSo. This equation is quite difficult to handle

when text is given as a function of the Eulerian variable x, which happensfor example in fluid-structure interaction poblems. It only simplifies whentext vanishes. When text(x) = −pext(x)n, we have :

Sno = −pext(x) detF (Id + 2E)−1no ∀xo ∈ Γσo ⊂ ∂Vo (17)

0.4 Linearized equations in the moving referenceconfiguration

Since we have shown that the proposed formulations can equivalently bewritten in a Galilean or in a moving frame, we will ignore in the followingthe ˜ notation, hence implicitly writing all equations in a moving frame.

In this section, we are interested in small perturbations with respect toa steady-state, in a frame moving at a constant angular frequency Ωref = 0and a constant velocity vref . This state is characterized by a steady-statestress tensor σref(xo) satisfying :

Divxo(σref) = ρo(Ω2refxo − f ref) in Vo (18)

σref(no) = tref on Γσo ⊂ ∂Vo (19)

with f ref the stationary part of fo and tref the stationary boundary condi-tions on Γσo :

ρofo(xo) = ρof ref(x(xo, t)) + fd(xo, t) (20)text(x(xo, t), t) = tref(x(xo, t)) + td(xo, t) (21)

In the following fd and td will be first order quantities with respect to f ref

and tref .

0.4.1 Constitutive behavior

Assuming a linear elastic behavior with respect to perturbations around thisstress state, and assuming ‖F − Id‖ = ‖Dxo(u)‖ 1, we have that :

σdef= S − σref = Cε(u) (22)

with σ at first order quantity with respect to σref and C the fourth orderelastic tensor. Due to the pre-stressed state this elastic behavior is oftennon-isotropic. When isotropy can still be assumed the relationship betweenthe dynamic stress σ and the small strain tensor ε reduces to the classicalHooke’s law :

σ = λtr(ε)Id + 2µε (23)

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Figure 4: Tangent elastic behavior around the pres-stressed state

0.4.2 Balance of momentum

Using a first order expansion the first term on the left hand side of equation(12) now reads :

Divxo (FS) = Divxo(σ + Dxo(u)σref) + Divxoσref

The second term on the left hand side of equation (12) must be handled withgreat care. When function fo is explicitly known, no further treatment isneeded. However, force densities are often due to external fields such asgravity or electromagnetic fields. In these cases, fo as a function of xo, isgiven by equation (15) as the composition of a known function f and theunknown function x(xo, t). This type of forces is referred to as followerforces. Since we are interested in small perturbations around xo, a firstorder differentiation leads to:

fo(xo, t) ≈ f ref(xo) + Dxo(f ref)(xo)u(xo, t) +fd(xo, t)

ρo

Since all functions of space variables are evaluated at xo and all space deriva-tives are with respect to xo, the latter expression can be simplified removingsubscripts o. Given these notations, the linearized balance of momentumreads, for all x in Vo:

Divx (σ + Dx(u)σref) + ρDx(f ref)u + fd = ρ(Ω2u + 2Ωu + u

)(24)

where fd is the dynamic force vector. It represents the geometric stiffnessdue to rotation and follower forces. Another geometric stiffness is broughtin by the term Divx(Dx(u)σref), but this time the stiffness applies to thegradient of the displacement field rather than to the displacement field itself.It can be very much compared to the elastic stiffness relating σ and ε.

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0.4.3 Boundary and initial conditions

In order to fulfill the linearization, the same first order approximation hasto be applied to boundary conditions (16). This also gives rise to followerforces. The most common loading case3 consists in an imposed pressurefield, as given in (17). When the dynamic applied pressure pd = pext − pref

is a first order quantity compared to pref , the linearized boundary conditionreads on Γσo :

σn = − (pd + preftr(ε) + (gradpref · u))n + 2prefεn, (25)

showing that the equivalent applied traction, on the reference configura-tion, is not anymore along the normal vector n. Moreover, this boundarycondition is not a standard one since it relates the traction vector to thedisplacement and its derivatives. The terms added to the applied dynamictraction vector are also called geometric stiffness terms.

Kinematic boundary conditions are much easier to write since they areconditions on particles located on some part of the boundary. As a conse-quence the current and the reference configuration coincide at these pointsleading to :

u = uext(x, t) on Γuo ⊂ ∂Vo (26)

Initial conditions are also kept unchanged as a given displacement field uand a given velocity field u at time t = 0.

0.5 Free vibrations of small amplitude with geo-metric stiffness : some examples

Small amplitude free vibrating modes, or eigenmodes, are displacement fieldsun(x) such that u(x, t) = un cos(ωnt) satisfies the homogeneous linearizedbalance of momentum (24), boundary conditions (25) (26) and constitutive

3The general case gives, for td = text − tref , and because the applied dynamic tractionvector is a first order quantity compared to tref :

σn = td+tr (ε) tref−“(Dx(u)tref + Dn(text)Dx(u)T n

”+Dx(text)u+εnn(Dn(text)n−tref)

on Γσo with εnn = nT εn. When the current traction vector depends linearly on the currentnormal vector (text = σextn), as for the case of an applied pressure, Dn(tref) = σref andthe last term vanishes. On Γσo, and for all constant vector a, this leads to :

a · (σ + Dx(u)σref)n = a · td + (σrefa) ·“divxu−DT

x (u)”

n + (Dx(σrefa)u) · n

= a · td + rot (u ∧ (σrefa)) + a · (Divx(σref)⊗ u)n

15

Figure 5: Free vibration of a pendulum

equation (22):

Divx(σ+Dx(u)σref)+ ρDx(f ref)u=ργ in V (27)γ = u + 2Ωu + Ω2u

σn = − (pref(Idtr(ε)− 2ε) + gradpref · u) n on Γσ ⊂ V (28)u = 0 on Γu (29)

fn = ωn/2π is the associated natural frequency or eigen frequency and Tn =2π/ωn is the natural period. The following chapters will give a rational studyof the existence of such modes and frequencies, and, in particular, it will beshown that, for a bounded domain , such a set of solutions exists, and thatit is at most countable, and with increasing natural frequencies. The modewith the smallest eigenfrequency is referred to as the foundamental mode.At this stage, it is worth noticing that when the foundamental frequency iszero the structure becomes unstable even in the reference state. It occursfor example for an unclamped structure, the corresponding modes being therigid body modes. This may also happen due to geometric stiffness termsleading to the so-called buckling phenomena (more precisely, the linearizedEuler buckling).

In few ideal cases, these eigenmodes and eigenfrequencies have a closed-form solution, enlightening the influence of the geometric stiffness. Some ofthis cases will be studied in section 0.7, and more general methods will begiven in subsequent chapters.

0.5.1 The pendulum

Let us consider a pendulum with a mass M , attached to a string withnegligible mass of length L and section S. We consider the string as a

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continuous medium. At rest, the initial stress state is :

σref =Mg

Sez ⊗ ez,

where g is the gravity acceleration, and ez is a constant unit vector, orientedupwards. We now consider a small displacement of the string:

u(x, t) = α(t)zex

Taking the dynamical stress σ = 0, assuming grad g = 0 and ρ = 0 leadsto :

Div(σ + Dx(u)σref) =αMg

SDivx(ex ⊗ ez) = 0

The dynamic force applied on the mass is given by (z = −L) :

−∫

S(σ + Dx(u)σref)ndS = αMgex

and the equilibrium of the mass gives :

αMg + MLα = 0

with natural frequency ωo =√

gL

We could have proceeded differently considering the equilibrium of massM on which a constant negative pressure p = −Mg/S is applied on theinterface between the mass and the string. The displacement of the mass asa rigid body reads :

u = θ(t)ey ∧ (−zez) = −zθ(t)ex

and thus Dx(u) = −θex ⊗ ez Integrating the balance equation over themasse leads to : ∫

S(0 + Dx(u)σrefn)dS = LMθex

since n = ez, σrefn = Mg/Sez and thus :

−θexMg = LMθex

and again : ωo =√

gL

0.5.2 The boat

We are interested in the vertical oscillation of a boat on a calm sea. Vi isthe volume of the boat under the water, M is the mass of the boat and ρf

is the mass density of the water. The equilibrium position is given by:

ρfVi = M

17

Figure 6: Pumping of a rigid boat

The pressure field in the water is linearly increasing with depth (z is orientedpositive upward and cancels on the free surface of the water):

pref = −ρfgz

Integrating the balance of momentum (27) on the entire rigid boat V sub-jected to only vertical displacements u = uez leads to (Dx(u) = 0):∫

∂Vσn = Muez

Accounting for the boundary condition (28) with ε = 0 and grad pref =−ρfgez leads to:

ρfgu

∫Sw

ndS = Muez

where Sw is the wet surface of the boat. As So = ∂Vi \Sw is the upper partof the immersed part of the boat, one obtains:

ωo =

√ρfgSo

M

since : ∫∂Vi

ndS = 0

As a consequence a submarine does not suffer from such vibrations.

18

0.6 Important formulae

In a moving frame :

γ = γo + Ω2u + 2Ωu + u

v = vo + Ωu + u

The linearized balance of momentum in a moving frame with follower forcesand applied pressure :

Divx (σ + Dx(u)σref)− ρDx(f ref)u + fd = ρ(γ − γref)

The boundary condition with follower pressure on Γσ :

σn = − (pd + preftr(ε) + (gradpref · u))n + 2prefεn

The linear isotropic elastic constitutive behavior or the Hooke’s law for smallstrains :

σ = λtr(ε)Id + 2µε , ε =Dx(u) + DT

x(u)2

0.7 Exercices

0.7.1 Free vibrations of a string

Let V be a cylindrical string along axis ez with a length L and a circularcross section of radius a. This string is pre-loaded with a stress field σref =σoez⊗ez. The string is clamped at the two ends. The string has an isotropiclinear elastic behavior around this state with a shear modulus µ σo. Weare looking for a free vibrating solution of small amplitude with respect tothis state having the following simple form :

u = u(z) cos(ωt)ey

Dx(u) = u′(z) cos(ωt)ey ⊗ ez ε = u′(z) cos(ωt)ey ⊗s ez

σ = 2µu′(z) cos(ωt)ey ⊗s ez

Divx(σ + Dx(u)σref) ≈ u′′(z)σo cos(ωt)ey = ρu = −ω2ρu(z) cos(ωt)ey

u = sin(nπ

z

L

)cos

(nπ

√σo/ρ

Lt

)ey

19

Figure 7: The Bessel function

0.7.2 Drums

The circular membrane of a drum of radius a and thickness e is pre-loadedwith the following uniform stress field :

σref = T (Id − ez ⊗ ez) T ≥ 0

With respect to this state, the membrane has an isotropic elastic behaviorwith a shear modulus µ T . The gravity forces are neglected and thehorizontal displacements vanish at r = R.

Look for the axisymmetrical natural modes of the drum having the fol-lowing form u = u(r)ez. (The non singular solution of the Bessel equa-tion f ′′ + f ′/r + f = 0 is the Bessel function Jo whose first roots arexo = 2.40, x1 = 5.52, x2 = 8.65, ... (see figure 7).

The balance of momentum gives :

Divx(σ + (Dx(u)T ) = −ρω2u

Since divu = 0, the constitutive equation gives σ ≈ 0 and, finally, sinceDx(u) = u′ez ⊗ er :

u′′ + u′/r =ρ

Tω2u u = Jo(ω

√ρ

Tr)

The boundary condition at r = R giving :

ωn = xn

√T

ρR2

20

0.7.3 Pressured pipes

A axisymetrical cylindrical pipe with inner radius R and thickness e Rand length L >> R is under an internal pressure pref . Free surface boundaryconditions are applied on the other part of its boundary.

Show that the initial stress state in the pipe can be approximated as :

σ = Teθ ⊗ eθ

Assuming a isotropic linear elastic behavior of the pipe around this stressstate find the first axisymmetrical modes and frequencies of the pipe.

Due to symmetry and to the free surface condition at the two ends thestress field in the pipe is looked for as :

σ = σrr(r)er × er + σθθ(r)eθ ⊗ eθ

The balance of momentum gives :

σ′rr +σrr − σθθ

r= 0 ⇒ 〈σθθ〉 = pref

R

e

where 〈σθθ〉 is the mean circonferential stress. Since σrr is of the order ofpref it is negligible compared to 〈σθθ〉 when e/R 1.

The balance of momentum with respect to this stress state and for anaxisymetrical displacement field u = u(r) cos(ωt)er with Dx(u) = u′er ⊗er + u/reθ ⊗ eθ reads :

Divx(λ(u′+u/r)Id +2µ(u′er⊗er +u/reθ⊗eθ)+Tu/r)eθ⊗eθ) = ρω2uer

Divx

(((λ + 2µ)u′ + λu/r)er ⊗ er + (λu′ + (λ + 2µ + T )u/reθ ⊗ eθ

)= ρω2uer

(λ + 2µ)(u′′ + u′/r)− (λ + 2µ + T )u/r2 = −ρω2u

r2u′′ + ru′(k2r2 − (1 + α))u = 0

with k = ω√

ρλ+2µ , α = T

λ+2µ . The solution of this equation is a combinationof two Bessel functions :

u = aJ1+α(kr) + bY1+α(kr)

When ke 0Boundary conditions at r = R reads :

(λ + pref)(u′ + u/R) + 2(µ− pref)u′ = 0

R(λ + 2µ− pref)u′ + (λ + pref)u = 0

whereas at r = R + e :

(R + e)(λ + 2µ)u′ + (λ)u = 0

When ke 0Ru′ = u

21

0.7.4 Free vibrating modes of the a self graviting earth

We are looking for the free vibrating modes of a sphere filled with an ho-mogeneous isotropic elastic material Ωo of radius Ro, mass density ρo andLamE’s coefficients λo and µo.

1. The sphere is assumed free of initial stress state and small transfor-mation solutions are looked for.

(a) We look for a stationary solution with spherical symmetry u =uR(R)eR cos(ωt). Show that uR satisfies :

(λ + 2µ)∆u = −ρω2u

(One can show that u = gradxϕ(R), Dx(u) = ((u′R−uR/R)eR⊗eR + IduR/R), divxu = ∆ϕ = R−2∂R(R2uR)) = R−1∂2

RR(Rϕ)and finally Divx(Dx(u)T ) = ∆u)

Answer :

ε = Dx(u) = Dx(u)T divxu = trε = ∆ϕ

Divxσ = Divx(λ(divxu)Id + 2µε) = λgrad divxu + 2µDivx(Dx(u)T )= (λgradx(∆ϕ) + 2µ∆gradxϕ)

0 = gradx

((λ + 2µ)∆ϕ + ρω2ϕ

)∆ϕ = −k2ϕ ϕ =

sin(kR)k3R

with k = ω√

ρλ+2µ (the cos solution being singular at the center).

The boundary condition at R = Ro reads :

σeR = (−λok2ϕ + 2µo∂

2RRϕ)eR = 0

and knowing that ∂RRϕ = −k2ϕ− 2∂Rϕ/R it gives :

Y

1− κY 2= tg(Y )

with Y = kRo et 4κ = λ+2µµ . A first order development around

Y = π finally gives :

To ≈2Ro

cpo

(1− 1

κπ2

)

22

(b) For the earth Ro = 6400km, ρo = 6000kg/m3, the wave velocities

mesured during earthquake giving cp =√

λ+2µρ = 10km/s et

cs =√

µρ = 5km/s. Give a estimate of the largest period To.

Answer : To = 19mn. However these spherical modes are diffi-cult to observe since the seismic sources do not create volumetricchanges and thus do not excite these modes. The mode havingthe largest period is a torsional mode but the equation in this caseare much more difficult to handle. They give a period about twicebigger than the one obtained here since the sher wave velocity cso

is about half the pressure wave velocity cpo.

2. We now account for the gravity field. Show that the static stress stateinduced by these forces is isotropic σref = −prefId and compare itsamplitude to the elastic moduli (The gravitational constant is G =6.6410−11Nkg−2m2) :

Answer : We only perform a stress computation with vanishing ac-celeration in the actual configuration. We then keep x as a spacevariable knowing that it will become the reference configuration whenvibrations will be studied. One has :

Divσo + ρo go︸︷︷︸Dx(Vo)

= 0 and ∆xVo = −4πGρo

Taking σo = −PoId and the boundary condition Ro leads to :

Vo = −2π

3GρoR

2 Po = ρo(Vo(R)− Vo(Ro))

At the center Po = 200GPa and since λ + 2µ = 600GPa with µ =150GPa, the initial stress state has to be accounted for.

3. We now look for a small amplitude spherical free vibrating mode of aself gravity sphere. The reference configuration is the static self grav-ity sphere subjected to the stress field −PoId. Around this state thesphere is assumed to have a linear isotropic elastic behavior charac-terized by the two LamE’s coefficients λ = λo − Po and µ = µo + Po

and the displacement u from this configuration is assumed to be small(compared to Ro with a small derivative (compared to 1).

(a) give the linearized balance of momentum and boundary conditionsatisfied by u with a particular attention to the follower forcesdue to gravity and simplify this equation when u = grad ϕ cos(ωt)

23

noticing that in this case Divx(Dx(u)T − divxuId) = 0 and(Dx(u)T − divxuId)er = 2u/r.

Answer : The linearized equations 24 and the constitutive equa-tion gives :

Divx

(λotr(ε)Id + 2µoε + (Dx(u)T − divxuId)Pref

)+ ρDx(g)u = ρu

with Dx(g) = 4π3 GρId. The particular case where u = grad ϕ

leads to :

(λo + 2µo)∆ur = −(ω2 + 4πGρ)ρur

(b) Giving the linearized boundary condition find the new first nat-ural period of the self graviting earth.

Answer : From equation (25) we keep the same equation as forthe first question :

σeR = (−λok2pϕ + 2µo∂

2RRϕ)eR = 0

The first modified natural period T ∗o is thus given by :

T ∗o = To

√1

1−GρoT 2o /π

≈ 21mn30s

Chapter 1

Virtual Power Principle andenergy balance

1.1 Introduction

In chapter ??, we have derived the basic equations modeling the small vibra-tions around a equilibrium state for continuum mechanics systems, underthe hypothesis of linear elastic constitutive behavior. Further, free vibratingsolutions have been defined, and described in some particular cases. Energybalance obviously has something to do with the existence of these specialsolutions, and it is the main objective of this chapter to discuss this issue.For this, we will introduce the Virtual Power Principle (VPP) in section 1.2,which will also be used to build approximate solutions to our dynamic prob-lem.

In section 1.3 this principle is then used to build a countable set ofeigenmodes and eigenfrequencies, in the case of small linear elastic vibrationsaround a non pre-stressed state.

Similar results are then derived in section 1.4 in the case of pre-stressedstates, and with follower forces, but neglecting Coriolis forces.

Finally, the section 1.5 discusses issues related to energy dissipation,using the same principle.

1.2 Virtual power principle

In the framework of continuum mechanics, conservation laws for the massand the momentum have been separated into two sets: on the one hand,local partial differential equations, and on the other hand, boundary andinitial conditions. This approach has two major drawbacks: it mixes twosets of equations of different nature, and it introduces high-order derivativesthrough the divergence theorem.

24

25

Figure 1.1: Small transformation in a Galilean frame

Weak formulations have been proposed to overcome these difficulties.They basically consist in multiplying the local volume equations by an ad-missible virtual field, defined on the domain of analysis, and in integratingthe result over the entire domain. Therefore, the basic equations are notenforced anymore at each and every point, but rather in a mean sense, for aset of admissible virtual fields. Further, using integration by parts, some ofthe partial derivatives are transfered to the virtual fields, and the boundaryconditions are integrated into a common formulation.

1.2.1 Weak formulation and Virtual Power Principle

Applying this approach to the balance of momentum in a moving frame, onthe current configuration Vt at time t, given by equation (1), leads to:∫

Vt

(Divxσ + fv − ργ) ·wdV = 0,

for any virtual field w defined on Vt. Since divx(σT w) = (Divxσ)·w+σ :Dx(w), where A : B = tr(ABT ) is the scalar product on second ordertensors, the integration by part leads to:∫

∂Vt

w.(σn)dS −∫

Vt

σ : Dx(w)dV +∫

Vt

(fv − ργ) ·wdV = 0

The final steps consist in defining an admissible field as a field that vanisheson Γu, and in accounting for the Neumann boundary condition σn|

RΓσ

=text. This gives the equivalent weak formulation:

Find u in V with u = uext on Γu such that for all w defined on V withw = 0 on Γu :∫

Vt

σ : Dx(w)dV︸ ︷︷ ︸−Pint(w)

+∫

Vt

ργ ·wdV︸ ︷︷ ︸Pkin(w)

=∫

Γσ

text ·wdS +∫

Vt

fv ·wdV︸ ︷︷ ︸Pext(w)

(1.1)

26

Considering w as a velocity field brings a physical meaning to the threeterms arising in equation (1.1). Indeed, the right-hand side represents thepower Pext(w) of the external forces (surface forces on Γσ and body forces onVt) in a velocity field equal to w. It is worth noticing that, since w vanisheson Γu, the traction vector on this part of the boundary brings no externalpower, even though it does not vanish itself. The terms on the left-handside are the powers of the internal forces Pint(w) and of the inertial forcesPkin(w).

1.2.2 Power balance

Taking Γu = ∅ one can view the power of the inertial forces in the truevelocity field as the time derivative of the kinematical energy Ekin(v). Indeed,choosing w = v in (1.1), leads to1:

Pkin(v) =d

dt

(∫V

ρ‖v‖2

2dV

)︸ ︷︷ ︸

Ekin(v)

,

since dvdt = γ either in a Galilean or a moving frame2. The kinematic energy

Ekin is related to the positive symmetric mass bilinear form M defined as:

M(w,w′) =∫

Vt

ρw ·w′dV (1.2)

since Ekin(w) = M(w,w).Finally, the Power balance of the structure in the moving frame reads :

dEkin(v)dt

= Pint (v) + Pext (v) (1.3)

1.2.3 Free energy without initial stresses

When assuming small transformations around a non-pressed state in a mov-ing frame, an isotropic linear elastic behavior of the material, and no followerforces, the virtual power of inner forces reads:

Pint(w) = Pe(w) = −Ke(u,w) = −∫

V(λdivxudivxw + 2µε(u) : ε(w))dV,

1It should be reminded here that the property:

d

dt

ZV

ρ‖v‖2

2dV =

ZV

ρd

dt

‖v‖2

2dV

is satisfied because of the mass conservation,2In a moving frame the total derivative d

dtshould not be confuse with the partial

derivative ∂t denoted by ˙ elsewhere in these textbook. Indeed :

d

dt= ∂t + Ω

27

where the symmetry of the stress and strain tensors has been accounted for,together with the identification between the actual volume Vt and the initialone V . Ke is the stiffness bilinear form, which is related to the so-calledinternal free energy Eint(u), a positive quadratic form of the strain tensorε(u):

Eint(u) =12Ke(u,u) =

12

∫V

(λtr(ε)2 + 2µtr(ε2))dV ≥ 0.

This leads to the classical power balance, which states that the variation,with time, of the sum of the internal free energy and the kinematic energyis the (real) power of the external forces :

d

dt(Eint + Ekin) = Pext

(du

dt

).

This property is quite useful to relate the VPP and the Hamilton’s principle.

Hamilton’s Principle The Lagrangian functional L of any displacement fieldu on V is defined as :

L =∫ T

o

(Ekin(u) + Eint(u) + Pext(u)) dt,

where Pext(u) is the mechanical work, at time t, made by the external forces.The Hamilton’s principle states that, given the initial and final values, this

Hamiltonian functional L is stationary for the solution of the dynamic problem.Since L is defined on a functional space, its derivative is defined in the Frechet

sense. The first order expansion, in terms of a fluctuation δu around the solutionu, gives:

L(u + δu) = L(u) + DuL(δu) + o(‖δu‖).

Therefore:

DuL(δu) =∫ T

o

(∫V

ρu · δudV + Pint(δu) + Pext(δu))

dt.

Integrating the first term by parts, and noticing that δu = 0 at t = 0, T becausethe initial and final values are given, the VPP (1.1) leads to :

DuL(δu) =[∫

V

ρu · δudV

]T

0

+∫ T

o

(−Pkin(δu) + Pint(δu) + Pext(δu)) dt = 0

As a consequence, the solution of a dynamical problem with a quadratic innerenergy and given initial and final states is the solution that minimizes L - theshortest distance in the state variable space. This classical result is hardly used tosolve dynamical problem but it is of major importance in the static case. Indeedin this case one obtains that the static solution of an elastic problem is the oneminimizing Eint(w) + Pext(w) in Vo when uext = 0.

28

1.2.4 Mathematical framework

The results obtained until now are all conditioned on the mathematical existenceof the quantities defined, and in particular on the boundedness of the integrals.In fact, this defines the mathematical framework in which the weak formulation iswell-posed. At all time, the displacement and the velocity fields must be in thespace Vo of fields with finite energy3 :

Vo =w(x),x ∈ V | ‖w‖2Vo

= Eint(w) + ω2refEkin(w) < +∞;

w(x) = 0,x ∈ Γu (1.4)

where the dimension of the coefficient ωref > 0 is that of a circular frequency. Whenthe domain is bounded, this functional space, equipped with its norm ‖ · ‖2Vo

, isshown to be a Hilbert space. Therefore, it possesses the nice properties of hav-ing a countable basis and being a closed space. These properties are particularlyimportant when an approximate solution is sought for because many convergenceproperties and error estimation are then available.

1.3 Free vibrating modes of non pre-loaded struc-tures

The mathematical framework given in section (1.2.4) has a direct implicationon the existence of free vibrating solutions and leads to the following:

Theorem 1 A bounded structure, with a linear elastic behavior around anon-pressed state, in a galilean frame, and without follower forces, has acountable set of free vibrating modes, or eigenmodes, un and associatednatural frequencies, or eigenfrequencies, ωn verifying:

• Ke(un,w) = ω2nM(un,w) ∀w ∈ Vo

• 0 ≤ ωo ≤ ... ≤ ωn... limn→+∞

ωn = +∞

• M(un, un′) = δnn′Eref

ω2ref

Ke(un, un′) = δnn′Erefω2

n

ω2ref

• unn∈N is a basis of Vo

A schematic proof of this theorem can be obtained making use of theRayleigh quotient which is also of major importance to build approximatesof the eigenfrequencies and the eigenmodes. Lazy readers could jump overthe next section to concentrate on modal expansion of the general solution.

3Actually, mathematicians work with dimensionless quantities and the norm they useis the so-called H1 norm corresponding to ρ = 1, µ = 1, ωref = 1, λ = 0, tr(ε2) beingreplaced by the norm on the gradient Dx(u) : Dx(u).

29

1.3.1 The Rayleigh quotient

Due to the balance of energy, a free-vibrating solution u(x, t) = cos(ωt)u(x, ω)with Pext(w) = 0 satisfies :

ω2 = Q(u) =Eint(u)Ekin(u)

=Ke(u, u)M(u, u)

where Q(u) is the so-called Rayleigh quotient. This already proves the ex-istence of at least one free vibrating mode namely the fondamental mode.Indeed the Rayleigh quotient being positive it has on Vo a minimal valueω2

o . Since this space is complete this value is reached for at least one givenfield uo ∈ Vo. This field is defined up to a multiplicative constant. It can benormalized so as to have a fixed energy and the current practice consists intaking the kinematic energy4 as a reference ω2

refEkin(uo) = Eref . We finallyobtain :

ω2o = ω2

ref

Eint(uo)Eref

= infw∈Vo

Q(w)

Moreover the corresponding dynamic field uo(x, t) = cos(ωot)uo(x) satisfiesthe VPP with no external forces.

Proof: indeed, since the Rayleigh quotient reaches its lower limit for u = uo,and taking any small deviation uo + αw, one must have:

0 ≤ Q(uo + αw)− ω2o = 2αω2

ref

Ke(uo,w)− ω2oM(uo,w))

Eref+ O(α2)

Since α is unsigned, one must have for all w ∈ Vo:

Ke(uo,w) = ω2oKe(uo,w).

When multiplied by cos(ωt), this is nothing but the PPV, for uo(x, t), and with noexternal loads

Using a recursive procedure, a countable set of natural circular frequen-cies ωn are defined, with their associated free-vibrating modes:

ω2n = ω2

ref

Eint(un)Eref

= infw∈Vn

Q(w) Vn−1 = Vn ⊕⊥ L(un−1)

where L(un−1) is the one-dimensional subspace of Vo whose basis vector isun−1, and where ⊕⊥ indicates the sum of two orthogonal subspaces5. It is

4We will see in chapter ?? that such a scaling is not accurate for higher modes and ascaling with respect to the total energy ‖w‖2Vo

can be more efficient.5In practice, this means that for all wn−1 ∈ Vn−1 there exist a unique wn ∈ Vn and

αn−1 ∈ R such that:

wn−1 = wn + αn−1un−1 Ekin(wn−1) = Ekin(wn) + α2n−1Ekin(uo)

30

easy to show that ωn is an increasing series6.Showing that:

limn→+∞

ωn = +∞

is a much more involving issue and is basically related to the fact that thekinematic energy is a weaker norm that the internal free energy -very fastoscillating solution can be bounded without having a bounded derivative.Mathematically speaking this results comes from some compactness argu-ments which are beyond the scope of the course.

1.3.2 Eigenmode expansion

Since unn∈N is an orthogonal basis of the solution space Vo, any displace-ment field u(x, t), under any loading, can be expanded on this basis:

u(x, t) =+∞∑0

qn(t)un(x) + ur(x, t) (1.5)

where ur(x, t) is any regular enough field satisfying ur(x, t) = uext on Γu

and where the time dependent modal amplitudes qn(t) are given by :

qn(t) =ω2

ref

Eref

∫V

ρ(u(x, t)− ur(x, t)) · un(x)dV (x) (1.6)

when u(x, t)− ur(x, t) are known.Using each of the eigenmodes as virtual fields in the VPP leads to inde-

pendent equations for the coefficients qn(t) of that expansion. These equa-tions read, for all n ≥ 0:

ω2nqn + qn = fn(t), (1.7)

fn(t) def=ω2

ref

Eref(Pext(un)−Ke(ur, un)−M(ur, un)) , (1.8)

together with the following initial conditions:

qn(0) =ω2

ref

ErefM(uo − uro, un) qn(0) =

ω2ref

ErefM(vo − vro, un) (1.9)

where uro(x) = ur(x, t = 0) and vro(x) = ur(x, t = 0).The three equations (1.7-1.9) are the classical equations for a Single

Degree of Freedom (SDOF) oscillator, the solution of which is given inbox (1.3.3).

31

Figure 1.2: The undamped Single Degree Of Freedom oscillator

1.3.3 general solution for an undamped single oscillator

The general solution u(t) of an undamped SDOF with natural circular fre-quency ωo, initial conditions u(0) = uo and u(0) = vo, and applied forcesf(t) satisfying :

ω2ou + u = f(t) (1.10)

is given by :

u(t) = uo cos(ωot) +vo

ωosin(ωot) +

∫ t

0

sin(ωo(t− τ))ωo

f(τ)dτ (1.11)

1.3.4 Conclusion

We have been able to built a general solution to the small forced vibrationsof a bounded elastic non pressed structure. However, this solution cannotbe used in practice for the three following reasons:

• We first need to compute approximates of the eigenmodes and eigen-frequencies. To this aim, a general Ritz-Galerkin procedure - and moreparticularly the Finite Element Method - will be given in chapter ??.Incidentally we have already observed that the Rayleigh quotient andthe orthogonalization procedures can be very useful to give coarse ap-proximates.

• Until now, neither the geometric stiffness terms nor the inertial termshave been accounted for. However, we will see in section 1.4 that mostof them lead to symmetrical terms that can be cast in the formalismgiven here.

6Since Vn ⊂ Vn−1,

ωn−1 = infw∈Vn−1

Q(w) ≤ infw∈Vn

Q(w) = ωn

32

Figure 1.3: Initial and current configuration for small transformations in amoving frame

• Finally, damping - or energy dissipation - always occurs in real dynamicsystems and this damping is of primary importance in their responseto external loads. The proposed formalism fails at accounting for thisphenomenon in a general setting, and section 1.5 will give some cluesto it.

1.4 Free vibrating modes of loaded structures inmoving frames

As the VPP is based on a weak formulation of the field equations, it canbe applied straightforwardly to the linearized equations (24-25) obtained inchapter ?? on the reference configuration. This leads to the same VirtualPower Balance equation as (1.1) :

Pkin(w) = Pint(w) + Pext(w)

33

where the virtual powers of inner forces, inertial forces and external forcesare given by :

Pint(w) = −∫

Vσ : Dx(w)dV + Pg(w) (1.12)

Pg(w) = −∫

V(Dx(u)σref) : Dx(w)dV (1.13)

−∫

Γσ

w ·(pref(Idtr(ε)−Dx(u)T ) + gradpref · u

)ndS

+∫

Vw · (Dx(f)u) dV

Pext(w) =∫

Vfd ·wdV +

∫Γσ

td ·wdS (1.14)

Pkin(w) =∫

Vρ(u + 2Ωu + Ω2u) ·wdV (1.15)

Assuming an isotropic linear elastic behavior, the VPP reads, for any ad-missible field w in the structure :

Ke(u,w)+Kg(u,w)−M(Ωu,Ωw)+Ca(u,w)+M(u,w) = Pext(w) (1.16)

Kg(u,w) =∫

Vtr(Dx(u)σrefD

Tx(w)

)dV (1.17)

−∫

VDx(f ref) : (u⊗s w)dV (1.18)

+∫

Γσ

(u⊗s w) : (gradpref ⊗s n)dS (1.19)

−12

∫Γσ

pref(DRx (u)w + DR

x (w)u) · ndS +Kns(u,w)(1.20)

Kns(u,w) =12

∫Γσ

rot (prefw ∧ u) · ndS =12

∫∂Γσ

pref(w ∧ u) · ndl = 0

Ca(u,w) = 2M(Ωu,w) = 2∫

VρΩ · (u ∧w)dV (1.21)

where DRx (u) = Dx(u) − Iddivu and satisfies DR

x (u)a = rot (a ∧ u) forall constant vectors a. Tt has been assumed that the body forces satisfyrot f = 0.

We finally obtain a symmetric formulation, except for the Corialis termCa which is skew-symmetric. The new stiffness symmetric bilinear formKg(u,w) incorporates all the geometrical terms coming from the initialstress state (1.29), the follower body forces (1.18), the follower pressureon Γσ (1.19) and the surface deformation (1.20)7. Moreover the inertialterm −M(Ωu,Ωw) plays the role of a negative stiffness term, which maypossibly bring instability into the system.

7This term can be elaborated on a bit further, by integrating by part the body force

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1.4.1 Energy balance

Since Kg is symmetric, the energy balance reads:

d

dt(Eint(u) + Eg(u) + Ekin(v)) = Pext(v)−M(γo,v)

with v = u + Ωu and 2Eg = Kg(u,u) and :

2Ekin(v) = M(u, u) +M(Ωu,Ωu) + 2M(u,Ωu)

Ekin does not stand for the total kinematic energy, but only for its quadraticpart. The linear part, which corresponds to the power of inertial forces, isaccounted for in the power of external forces as M(γo,v), and the remainingconstant term M(vo,vo) corresponds to the kinematic energy related to themoving reference configuration.

Finally, it is worth noticing that since the Coriolis term Ca is skew-symmetric, it does not play any role in the Power balance.

1.4.2 Stability in a galilean frame

In this subsection we will assume that Ω = 0. On the contrary to the elasticenergy which is always positive, the geometrical terms can be negative.Indeed, the initial stress term (1.29) can be either positive, for a tractionstate, negative, for a compression state, or unsigned, for a mixed state suchas shear. The same remark holds for the other terms.

As a consequence, the system becomes unstable when :

infw∈Vo

Eint(w) + Eg(w)Ekin(w)

< 0

When this criterium is reached, solutions with exponential growth in timecan occur and the structure becomes instable for static loading. This is,for example, the case for the buckling of compressed structures. It is worthnoticing that, when this criterion is reached, the real structure might stillbe stable, but not anymore under the hypothesis of small transformationsaround the reference state. A full non-linear analysis is then required toassess the stability in this range.

term:

Kg(u, w) =

ZV

tr“Dx(u)σrefD

Tx (w)

”dV −

ZV

ρ(Ωu) · (Ωw)dV

−Z

V

f ref ·Divx(u⊗s w)dV

+

ZΓσ

(u⊗s w) : ((f ref − gradpref)⊗s n)dS

−1

2

ZΓσ

pref(DRx (u)w + DR

x (w)u) · ndS (1.22)

Hence, this avoids the computation of the gradient of the external force density.

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1.4.3 Eigenmodes in a galilean frame

Once the Coriolis forces are neglected, the eigenmode analysis and the eigen-mode expansion for linearized elastic linear structure can be conducted fol-lowing the same lines as given in section 1.3. This defines the new Rayleighquotient :

Q =Eint(w) + Eg(w)

Ekin(w)(1.23)

and the previous properties still hold in this case.

1.4.4 Eigenmodes in a moving frame

The stability analysis and the existence of eigenmodes in the moving frame is outof the scope of this course but we can notice that a sufficient stability condition is :

Ke(w,w) +Kg(w,w)−M(Ωw,Ωw) ≥ 0 ∀w ∈ Vo

where a negative centrifuge term has been added, leading to an additional sourceof instability.

When this condition is fulfilled eigenmodes and eigenfrequencies can be exhib-ited but not directly using the Rayleigh quotient and this will be studied in thefinite dimension case in chapter ??.

1.5 Damping

Damping or energy dissipation is a major issue in structural dynamics butit is also the one for which the mechanical and numerical modeling is theless rigourous. Damping is often used to hide the lack of knowledge onehas on the actual physical phenomena. It represents in a mean sense theenergy losses observed in the experiments. These energy losses may comefrom conversion to heat, creation of micro cracks, permanent displacement,acoustic emission, radiation in the supporting structures... In this coursewe will restrict ourselves to linear viscosity. From this point of view, theVPP is a very efficient tool. Indeed, it allows to describe the energy lossesthrough the virtual power of these dissipation forces, namely:

Pdis(w),

which is a linear function of w. Moreover, the second thermodynamic prin-ciple states that, under an isothermal process, the real power of these dissi-pative forces must be negative:

Pdis(v) ≤ 0 (1.24)

Since we are interested in a linear equivalent model Pdis(w) at time t issought as a linear function of the displacement fields u(x, τ) for all τ ≤ t

36

in order to ensure causality. In this course, we will restrict the analysisto memoryless dissipation processes. This implies that Pdis(w) is a linearfunction of the time derivatives of u at time t. In turn, this implies that,to ensure the positiveness of the real power for all possible values of theadmissible velocity field v, one must have:

Pdis(v) = −D(v,v) ≤ 0 (1.25)

where D is a bounded positive symmetric bilinear form on Vo. This type ofdamping is consistent with constitutive model for the material around thereference state of the Kelvin-Voight type. In the linear isotropic case, thismodel reads:

σ = σe + σν (1.26)σν = λνdivv + 2νD (1.27)

where σe is given by the classical Hooke’s Law and D = ε8. More complexdamping models can be accounted for in the Laplace or Fourier formalismproposed in chapter ??.

1.5.1 VPP with viscous damping

The Virtual Power Principle keeps the same form but the power of innerforces now reads :

Pkin(w) = Pint(w) + Pext(w)Pint(w) = Pe(w) + Pg(w)−D(v,w) (1.28)

1.5.2 Proportional viscous damping

In order to prevent the coupling between the modal amplitudes throughoutthe damping term proportional damping is often assumed in practice. Thismeans that :

D(v,w) = α(Ke(v,w) +Kg(v,w)) + βM(v,w)

with α and β two positive numbers.However there are very few rational background to assess the hypothesis

but the simplification it allows in the solution process.8It is worth noticing that the classical Kelvin-Maxwell constitutive model consisting

in splitting the strain into and elastic strain rate De depending on the stress rate σ anda viscous strain Dν depending on the stress σ cannot be accounted for by the proposedmodel since it is not a memoryless model. Anyhow this model is of small importancein structural dynamics since it is devoted to visco-elastic fluids than cannot sustain nonhydrostatic stresses in the long range.

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1.6 Important formulae

• The Virtual Power Principle on the current configuration:

−∫

Vt

σ :Dx(w)dV︸ ︷︷ ︸Pint(w)

+∫

Γσ

text ·wdS +∫

Vt

fv ·wdV︸ ︷︷ ︸Pext(w)

=∫

Vt

ργ ·wdV︸ ︷︷ ︸Pkin(w)

• Virtual Power Principle for elastic structures in a moving frame withinitial stresses, follower forces and damping:

−P int(w)︷ ︸︸ ︷(Ke +Kg)(u,w) + (D+

Pkin(w)︷ ︸︸ ︷Ca)(u,w) +M(u,w)−Ki(u,w) = Pext(w)

• the stiffness bilinear form : The symmetric positive elastic stiffnessbilinear form Ke (isotropic case):

Ke(u,w) =∫

V(λdivudivw + 2µε(u) : ε(w))dV

• The symmetric positive mass bilinear form M:

M(γ,w) =∫

Vt

ργ ·wdV,

• the geometric stiffness bilinear form :

Kg(u,w) =∫

V

tr(Dx(u)σrefD

Tx(w)

)dV

+∫

V

Dx(f ref) : (u⊗s w)dV

−∫

Γσ

(u⊗s w) : (gradpref ⊗s n)dS

+12

∫Γσ

pref(DRx (u)w + DR

x (w)u) · ndS

• The skew-symmetric gyroscopic bilinear form:

Ca(u,w) = 2M(Ωu,w) = 2∫

V

ρ(Ωu) ·wdV

• the symmetric positive inertial bilinear form:

Ki(u,w) = M(Ωu,Ωw) =∫

V

ρ(Ωu) · (Ωw)dV

• the symmetric positive damping bilinear form:

D(w,w) ≥ 0

38

• The Rayleigh quotient and stability :

0 ≥ Q(w) =Ke(w,w) +Kg(w,w)−M(Ωw,Ωw)

M(w,w)

• The eigenbasis (without damping and gyroscopic terms):

(Ke +Kg −Ki)(un,w) = ω2nM(un,w)

1.7 Exercises

1.7.1 Rayleigh Quotient

For the string or the drum try to compute an upper bound using the Rayleighquotient for the fundamental natural frequency and compare to exact solu-tion.

Compute the first natural frequency of a shear beam using the staticdeflection (to be compared with future results).

1.7.2 The string with a moving load

Compute the modal forces fn(t) for a moving force f(x, t) = δ(x− vt)ey ona string.

1.7.3 Rigid structure

Apply the VPP to find the equations for a rigid structure.

1.7.4 Response of an undamped SDOF

To an harmonic force starting at 0 and show that even at resonance thesolution is finite at finite time.

1.7.5 Approximate fundamental frequency of the self grav-iting earth

• Approximate eigenfrequency without gravity with u(x) = αrer,w =er :

Dx = αId = ε σ = α(3λ + 2µ)Id

Ke(u,w) = V tr(σ) = 3V α(3λ + 2µ), Me(u,w) = ρV α3R2

5

ωo ≈1R

√5(3λ + 2µ)

ρ

39

• The initial stress due to gravity: σref = −Pref(r)Id

grad (ρVo−Pref) = 0, Pref(R) = 0, V (r) = −2π

3Gρr2 ⇒ Pref = ρ(Vo(r)−Vo(R))

• Approximate eigenfrequency with gravity :

Bibliography

[1] D. Aubry. Mecanique. Ecole Centrale Paris, 2007.

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