MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 7 pdf

, termed modal forces, are the components ofthe external force field Fer; t, as expressed in the modal coordinate system.. If it is the case,the mode shapes of the conservative system ca

4.3.1 Equations of motion projected onto a modal basis

Let us consider the forced dynamical problem governed by the linear partialdifferential equations of the general type:

K[
X] + C[ ˙
X] + M[ ¨
X] =
F(e)(
r; t)+ I C and B.C

&

[4.35]

I.C stands for the initial conditions and B.C for the conservative boundary

con-ditions The stiffness, damping and mass operators K[ ], C[ ], M[ ] can depend onthe vector position
r in the Euclidean space To the forced problem governed by[4.35], we associate the modal problem [4.36], which satisfies the same boundaryconditions:

[K(
r) − ω2M(
r)][
ϕ] = 0+ B.C

&

[4.36]solutions of which are defining the following modal quantities:

{
ϕn}; {ωn}; {Kn}; {Mn}; n = 1, 2, 3, [4.37]

As in the case of discrete systems, the mode shapes can be used to determine

an orthonormal basis with respect to the stiffness and mass operators, in which thesolution
X(
r, t) of [4.35] can be expanded as the modal series:

X(
r, t) =

∞(

n =1

qn(t )
ϕn(
r) [4.38]

where the time functions qn(t ) n= 1, 2, , termed modal displacements, are thecomponents of the displacement field
X(
r; t) in the modal coordinate system.Formal proof of such a statement is not straightforward and is omitted asalready mentioned in the introduction As the dimension of the functional vec-tor space is infinite, a delicate problem of convergence and space completenessarises Substitution of [4.38] into the equation of motion [4.35], leads to:

The system of equations [4.39] is projected on the k-th mode shape
ϕkby usingthe scalar product [1.43] and the orthogonality properties [4.2] The transformation

results into the following ordinary differential equation:

Kkqk+

∞(

n =1

Ckn˙qn+ Mk¨qk= Q(e)k (t )where Ckn= 
ϕk, C[
ϕn](V) and Q(e)k (t )= 
ϕk,
F(e)(V)

[4.40]

The subscript (V) accounts for the space integration domain involved in thescalar product Q(e)k (t ) k= 1, 2, , termed modal forces, are the components ofthe external force field
F(e)(
r; t), as expressed in the modal coordinate system

As already discussed in [AXI 04], Chapter 7, the result [4.40] can be drasticallysimplified if coupling through the damping operator is negligible If it is the case,the mode shapes of the conservative system can be assumed to be orthogonal withrespect to the C damping operator and [4.40] reduces to a single equation which

governs the forced vibration of an harmonic oscillator, called modal oscillator:

Mk(ω2kqk+ 2ωkςk˙qk+ ¨qk)= Q(e)k (t )where 
ϕk, C[
ϕn](V) =

The Laplace transform of the displacement field in the physical coordinatessystem is finally obtained by using the modal series [4.38]:

4.3.2 Deterministic excitations

To study the time response of continuous systems it is necessary to know both thespatial distribution of the excitations and their variation with time The deterministicexternal forcing functions are described by a vector
F(e)(
r; t) which may standeither for an external force field which fluctuates with time, or for a prescribedmotion assigned to some degrees of freedom of the mechanical system Firstly, it isuseful to make the distinction between forcing functions in which space and timevariables are separated and those where they are not It is convenient to start withthe former case which is more common

4.3.2.1 Separable space and time excitation

The general form of this kind of excitation can be written as:

Figure 4.22 Example of a travelling load

4.3.2.2 Non-separable space and time excitation

When the position vector used to describe the spatial distribution of the load

is time dependent it becomes impossible to separate the time and space variables

This is typically the case of the so called travelling loads, which are of practical

importance in many applications Let us consider, for instance, a train running on

a flexible bridge at a constant cruising speed V0, see Figure 4.22 One is interested

in analysing the response of the bridge loaded by the weight of the running train.Let 2λ designates the length of the train As a first approximation made here forthe sake of simplicity, the total weight
P = −M
g is assumed to be distributeduniformly along the train So, the forcing function is written as:

F(e)(x, t)=2λP
[U(x − V0t+ λ) − U(x − V0T − λ)]

U (x) is the Heaviside step function and the running abscissa V0t is taken at themiddle of the train The resultant and the abscissa xa(t )at which it is applied areobtained by using the relationships [4.45] and [4.46]:

Now, if the length of the train remains much smaller than the bridge spandenoted L, the load can be reasonably modelled as a concentrated load whichleads to the modal forces:

Q(e)

n = 
ϕn,
P δ(x− V0t ) =
P ·
ϕn(V t ) [4.48]According to the result [4.48], the whole series of modes is excited as timeelapses If the actual length of the train is accounted for, the modal forces are found

=2λ1

 V T +λ

V T −λ

So, the modal excitation is found to depend on the ratio of the modal wavelength

on the train length Assuming for instance that the bridge deck can be modelled as

a pinned-pinned beam, the modal force is expressed as:

be removed from the modal basis

4.3.3 Truncation of the modal basis

4.3.3.1 Criterion based on the mode shapes

The last result of the preceding subsection can be restated as a general rule,according to which all the mode shapes which are orthogonal to the spatial distri-bution of the excitation can be discarded from the modal model In other words, theonly modes which contribute to the response series [4 42] are those modes whichare not orthogonal to the spatial distribution of the excitation:


ϕn, (
r)
u(
r)(V)= n= 0 [4.50]where for convenience the criterion [4.50] is formulated in the case of a separatedvariables forcing function

It may be noted that a modal truncation based on the mode shape criterion issimilar to the elimination of some components of the physical displacements in asolid body, based on the orthogonality with the loading vector field

Figure 4.23 Pinned-pinned beam loaded by a transverse concentrated force

example –Beam loaded by a concentrated transverse force

The problem is sketched in Figure 4.23 The equilibrium equation is written as:

2, where ξ= x/L and c2= EI/ρS

The Laplace transform of the response is:

Z(ξ, ξ0; s)= 2 ˜f (s)M

b

∞(

n=1

sin(nπ ξ ) sin(nπ ξ0)(c2(nπ/L)4+ 2scςn(nπ/L)2)

If ξ0= 0.5, the non-vanishing terms of the series are related to the odd modes

n= 2k + 1, k = 0, 1, 2, only:

Z(0.5, ξ ; s)= 2 ˜f (s)M

b

∞(

Figure 4.24 Symmetric loading of a beam provided with symmetric supports

If the load is symmetrically, or skew symmetrically, distributed with respect

to the cross-section at mid-span of the beam and if the boundary conditionsare symmetric, the only modes which contribute to the response must verifythe same conditions of symmetry as the loading function, see Figure 4.24.Although this rule is correct from the mathematical standpoint, it is still necessary

to be careful when using it, because real structures present inevitably ial and geometrical defects which spoil the symmetry of the ideal model Onestriking example of the importance of such ‘small’ defects will be outlined insubsection 4.4.3.3

mater-4.3.3.2 Spectral criterion

The spectral considerations made in [AXI 04] Chapter 9, concerning the ical response of forced N -DOF systems, can be extended to continuous structures

dynam-to produce a very useful criterion for restricting the modal basis dynam-to a finite number

of modes Figure 4.25 is a plot of the power density spectrum of some excitationsignal, which in practice extends over a finite bandwidth, limited by a lower cut-offfrequency fc1and by a upper cut-off frequency fc2; that is, outside the interval

fc1, fc2the excitation power density becomes negligible The dots on the frequencyaxis mark the sequence of the natural frequencies of the excited structure, which

of course extends to infinity

Figure 4.25 Spectral domains of excitation versus structure response properties

Let us consider first the response of a single mode fn to the excitation signal.The response is found to be quasi-inertial if fn/fc1 ≪ 1, in the resonant range

if fc1 < fn < fc2, and quasi-static if fn/fc2 ≫ 1 Therefore, a finite number

of low frequency modes can lie in the quasi-inertial range, and infinitely manyother modes lie in the quasi-static range The contribution to the total response ofthe modes lying in the quasi-inertial range can be accounted for by neglecting thestiffness and damping terms of the modal oscillators and only a finite number ofsuch modal contributions are to be determined On the other hand, the contribution

to the total response of the modes lying in the quasi-static range can be accountedfor by neglecting the damping and inertial terms of the modal oscillators; howeverthere are still infinitely many modal contributions to be accounted for The methodfor avoiding the actual calculation of such an infinite series is best described startingfrom a specific example

Let us consider again a vehicle of mass M, travelling at speed V0 on a ible bridge Assuming the bridge deck is modelled as an equivalent straight beamprovided with pinned supports at both ends and damping is neglected, the equations

flex-of the problem are written as:

EI∂

4Z

∂x4 + ρS ¨Z = −Mgδ(x − V0t )Z(0)= Z(L) = 0; ∂

= 0

By projecting this system on the pinned-pinned modal basis, we get the system

of uncoupled ordinary differential equations, comprising an infinite number of rows

⌢

Z(x, ω)= −gMM

b

∞(

n =1

(δ(n− ω) − δ(n+ ω))i(ω2n− ω2) sinnπ x

L



where n= nπ V0

L which becomes infinite at the undamped resonances ωn= n

Nevertheless, it may be also realized that in most cases of practical interest eventhe smallest cruising speed V1needed to excite the first resonance of the beam islikely to be far beyond the realistic speed range of the vehicle This is because abridge is designed to withstand large static transverse loads As a consequence, thedynamical response of the loaded bridge can be determined entirely by using thequasi-static approximation:

and the bridge deflection is found to be:

Z(x; t)= −gM

∞(



To restate the conclusions of this example as a general rule, one-dimensionalproblems are considered for mathematical convenience Further extension to thecase of two or three-dimensional problems is straightforward as it suffices to deal

in the same way with each of the Euclidean dimensions of the problem As outlinedabove, it is relevant to discuss the relative importance of the modal expansion terms

of the response in relation to the spectral content of the excitation The Fouriertransform of the response is written as:

negli-1 Resonant response range: ωn∈ [ωc1, ωc2]

It is clearly necessary to retain all the modes whose frequencies lie withinthe spectral range of the excitation, except if they are orthogonal to the spatialdistribution of the excitation, that is if ψnvanishes The contribution of suchmodes to the response is given by the full expression [4.51] The spectrum

of the modal response can be conveniently related to the excitation spectrum(cf Volume 1 Chapter 9) as:

2 Quasi-static responses: ωc2≪ ωN

In the same way as in the travelling load example, the series [4.51] can bereduced to the quasi-static form:

⌢Z(x; ω, N )= F0

⌢

f (ω)

∞(

to n2in the case of torsion and longitudinal modes Furthermore, the series[4.53] calculated from n= 1 instead of n = N, must converge to the staticsolution Zs of the forced problem related to the system [4.35]:

K(x)Zs = F0(x)+ B.C

&

[4.54]

This is because the Hilbert space of the solutions is complete by definition

So, the modal expansion of Zs is found to be:

Zs(x)= F0

∞(

n=1

ϕn(x)ψn

Kn

RN(x)is termed the quasi-static mode or pseudo-mode It gives the resultant

of the individual quasi-static contributions to the response of the infinitely manynatural modes of vibration which lie in the quasi-static range, based on thespectral criterion ωc2≪ ωN Provided the solution of the static problem [4.54]may be made available from a direct analytical or numerical calculation, thepseudo-mode can be determined by using [4.56] Then, the Fourier transform

of the solution to the dynamical problem [4.35] is expanded as:

coeffi-1

KN(x, ) =

∞(

of the neglected modes

3 Quasi-inertial response: ωp≪ ωc1

By analogy with the quasi-static approximation, it is also possible tosimplify the contribution of the modal responses related to the modeswhose frequencies are within the quasi-inertial range The corrective term

n =1

ϕn(x)n

ω2Mn [4.60]Following the same procedure as in the quasi-static case, it is possible todefine an equivalent mass of the system and an equivalent truncation masscoefficient Accordingly, the expansion [4.57] can be further simplified as:

4.3.4 Stresses and convergence rate of modal series

The modal series of the type [4.38] are found to converge for any realisticexcitation, as anticipated based on physical reasoning This can be checked math-ematically by noting that the infinite number of terms involved in the quasi-staticterm RN(x)form a sequence decreasing at least as 1/n2 The convergence of thestress and strain expansions is also granted almost everywhere along the beam.However, convergence rate of stress or strain series is significantly slower than that

of the displacement series because they are obtained through a space derivation ofthe latter It is worth illustrating this important point by taking an example.example –Stretching of a straight beam

The static response of a cantilevered beam loaded by an axial force is analysedfirst by solving directly the boundary value problem (local formulation) and then

by using the modal expansion method (Figure 4.26)

Figure 4.26 Cantilevered beam loaded by an axial force

#L

= F0The solution is:

Xs(ξ )= LFES0ξ; where ξ =LxThe equivalent stiffness is K(ξ , 1)= ES/(Lξ) and the axial stress is N (ξ) =(ES/L)(dX/dξ )= F0

2 Modal formulation

The modal basis complying with the boundary conditions of the problem are:

ϕn(ξ )= sin ̟nξ; ̟n= π(1 + 2n)/2; ωn= c0L̟n; where c02= E/ρ

Mn= ρSL/2; Kn

π2ES8L (1+ 2n)2The modal (or generalized) forces are:

Q(e)n = ϕn, F0δ(ξ− 1)(L) = F0sin ̟n= (−1)nF0, n= 0, 1, 2 The modal expansion of the type [4.55] gives the displacement field as theFourier series:

X(ξ, 1)=FES0Lπ82

∞(

n=0

(−1)nsin[π(1/2 + n)ξ]

The series is found to converge rather quickly to the correct result as shown

in Figure 4.27, in which the displacement is plotted versus the modal cut-offindex N at ξ = 0.75 and ξ = 0.1 Incidentally, this kind of calculation is aconvenient method to determine the value of a fairly large number of series.For instance, here it is found that:

∞(

n =0

1(1+ 2n)2 =π

28

Figure 4.27 Convergence of modal series of displacement

Figure 4.28 Convergence of modal series of stress

To determine the stress field, the displacement series is differentiated term byterm, which results in:

Nn(ξ, 1)= −F04

π

∞(

n=0

(−1)ncos[π(1/2 + n)ξ]

It may be immediately noticed that convergence of the series is not uniform,

as all the terms vanish at ξ = 1 This is a direct consequence of the mode shapesused for the expansion, which refer to a free end condition at ξ= 1 Anywhereelse the series converges in an alternative way to the correct value as shown inFigure 4.28

{
ϕn(
r)} is the related modal basis If other elastic supports are added to the structure,the equation becomes:

K(
r)
X(
r; t) + Ka(
r)
X(
r; t) + M(
r)
¨X(
r; t) = 0 + elastic B.C [4.65]where Ka(
r) stands for the stiffness operator of the additional supports, which can

be either concentrated at some discrete locations, or continuously distributed atthe boundary of the structure It is possible to approximate the new modal basis{
φn(
r)} by projecting the system [4.65] on {
ϕn(
r)}; which gives a truncated modalexpansion of the type:

φn(
r) =

N(

j =n

Kj(a)qj = 0 [4.67]

The coupling terms result from the lack of orthogonality of the
ϕn(
r) with respect

to the operator Ka(
r) The solution of [4.67] produces N new mode shapes
φn(
r)

as expressed in the {
ϕn(
r)} basis They can be written in terms of the physicalcoordinates
r by using [4.66] The method is illustrated by some examples in thenext subsections As could be anticipated, it will be shown that the higher thetruncation order and the less the order of the calculated modes, the more accurate

is the method

4.4.1.1 Beam in traction-compression with an end spring

In this first example, the beam is fixed at one end and supported at the other end

by a linear spring, as shown in Figure 4.29 The spring is successively modelled

as an elastic impedance, i.e a boundary condition, or as a connecting force, that is

an additional loading term in the dynamic equation In both cases, the object is toderive the longitudinal modes of the modified system

Figure 4.29 Longitudinal modes of a beam fixed at one end supported by a spring at the

other end

1 Spring modelled as an elastic impedance:

The modal system is written as:

So the modal system is written as:

¯

X′′+ ̟2X¯ = 0; with ¯X(0)= 0; ¯X′(1)= γLX(1)¯Again, the general solution of the differential equation is X(ξ ) =

asin(̟ ξ )+ b cos(̟ ξ) and the boundary conditions give the characteristicequation:

ωn= nπ; ϕn(ξ )= sin[ωnξ](c) Spring stiffness coefficient has a nonzero finite value, for instance: γL=π/4

The first roots of the transcendental equation x cot(x)−π/4 are found

to be:

̟1= 1.9531; ̟2= 4.8722; ̟3= 7.9524; ̟4= 11.067

2 Modal projection of the constrained system

The equilibrium equation of the constrained system is:

−ESd

2X

dx2 − ω2ρSX= −KLXδ(x− L); with X(0) = 0 [4.71]

Using the dimensionless quantities already defined above, the modal system

is written as:

− ¯X′′− ̟2X¯ = −γLXδ(ξ¯ − 1); X(0)¯ = 0 [4.72]The mode shapes φnare expanded as a linear combination of the fixed-freebeam modes:

φn(ξ )=

∞(

So, if the system [4.73] is truncated to the order N , accuracy is expected

to degrade progressively as n increases up to N Taking for instance N = 4,[4.73] is written as follows:

Figure 4.30 Added truncation spring

retained in the basis The four first pulsations are found to be:

̟1= 3.1426; ̟2= 6.2915; ̟3= 9.4551; ̟4= 12.665The relative error with respect to the exact values is an excess of about

3 percent Further, if the number of selected modes is increased, to sixteenfor instance, it is found that the relative error is reduced to about 1.3 percent.However, as indicated below, to improve accuracy it is more efficient to makeuse of the quasi-static corrective term [4.59] than to increase the size of themodal basis

3 Correction of the modal truncation

It is possible to improve the model presented above by adding to it a ness coefficient KT which accounts suitably for the truncation of the modalbasis This numerical spring is mounted in series with the physical spring,

stiff-as sketched in Figure 4.30, in such a way that the equivalent stiffness of thesupport is decreased, as given by:

Ke= KKTKL

T + KL ⇒ γ1

e = γ1

T +γ1L

n =0

1(2n+ 1)2Including the equivalent flexibility [4.74] in the model, and adopting

γL = π/4 and N = 4, the following natural frequencies are obtained:

̟1= 1.9532; ̟2 = 4.8725; ̟3 = 7.9531; ̟4= 11.068, which are found

to be very close to the exact values Even if the basis is reduced to the twofirst modes only, the results are still sufficiently accurate for most applications:

γe= 0.7285 and ̟1= 1.9538; ̟2= 4.8755 Finally, if the end spring ness is so large as to be practically equivalent to a fixed condition, satisfactoryresults can be obtained by selecting N = 6, which gives γe= 29.67

stiff-4.4.1.2 Truncation stiffness for a free-free modal basis

Here, the beam is supported by a spring KLat one end and left free at the other.Again, we want to compute the longitudinal vibration modes of the supported beam

by using a truncated basis of the vibration modes in the free-free configuration, i.e.without end springs As unnatural as it may appear, the procedure is useful in thecontext of the substructuring method, as made clear in subsection 4.4.3.

If a direct calculation is performed by modelling the elastic support as an elasticimpedance, the following characteristic equation for the natural frequencies is γL=

̟ntan(̟n) Selecting for instance γL= 1, the first reduced frequencies are ̟1=

0, 86035; ̟2 = 3, 4256; ̟3 = 6, 4373 Now if the modal projection method isused, a difficulty arises in determining the truncation flexibility since it is a prioriinfinite, due to the presence of the free rigid mode However, let us discuss theproblem a little further The constrained beam is governed by:

Figure 4.31 Beam provided with an elastic support at one end

For the perturbed system, in which perturbation of the non-rigid modes isassumed to be negligible:

1

γ′ =γ1

f +13where 1/γf is the finite flexibility of the fictitious support

Then the truncation flexibility is given by:

n =2

1

n2where n= 2 is the order of the first non-rigid mode of the free-free beam.Using this corrective term, a truncation to N = 5 gives:

γe= 0.9465; ̟1= 0.8607; ̟2= 3.4263; ̟3= 6.4386; ̟4= 9.5322which are very close to the exact values

4.4.1.3 Bending modes of an axially prestressed beam

A beam clamped at one end and provided with a sliding support at the other,

is prestressed by a compressive load P0 The buckling load Pc = (π/L)2EI was

already determined in subsection 4.2.5.2 The related buckling mode shape wasfound to be:

ψn(x)=12cosnπ x

L



− 1