MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 7 pdf

218 Structural elements
4.3.1 Equations of motion projected onto a modal basis
Let us consider the forced dynamical problem governed by the linear partial
differential 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 on
the 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 boundary
conditions:
[K(r) −ω
2
M(r)][ϕ]=0
+ B.C.

[4.36]

solutions of which are defining the following modal quantities:
{ϕ
n
}; {ω
n
}; {K
n
}; {M
n
}; 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 the
solution

X(r, t) of [4.35] can be expanded as the modal series:

X(r, t) =
∞

n=1
q
n
(t) ϕ
n
(r) [4.38]
where the time functions q
n
(t) n = 1, 2, , termed modal displacements, are the
components of the displacement field


X(r; t) in the modal coordinate system.
Formal proof of such a statement is not straightforward and is omitted as
already mentioned in the introduction. As the dimension of the functional vec-
tor space is infinite, a delicate problem of convergence and space completeness
arises. Substitution of [4.38] into the equation of motion [4.35], leads to:
∞

n=1
{K[ϕ
n
(r)]q
n
(t) +C[ϕ
n
(r)]˙q
n
(t) +M[ϕ
n
(r)]¨q
n
(t)}=

F
(e)
(r; t)
+ I.C.








[4.39]
The system of equations [4.39] is projected on the k-th mode shape ϕ
k
by using
the scalar product [1.43] and the orthogonality properties [4.2]. The transformation
Modal analysis methods 219
results into the following ordinary differential equation:
K
k
q
k
+
∞

n=1
C
kn
˙q
n
+ M
k
¨q
k
= Q
(e)
k
(t)

where C
kn
=ϕ
k
, C[ϕ
n
]
(V)
and Q
(e)
k
(t) =ϕ
k
,

F
(e)

(V)
[4.40]
The subscript (V) accounts for the space integration domain involved in the
scalar product. Q
(e)
k
(t) k = 1, 2, , termed modal forces, are the components of
the 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 drastically
simplified 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 with
respect 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:
M
k
(ω
2
k
q
k
+ 2ω
k
ς
k
˙q
k
+¨q
k
) = Q
(e)
k
(t)
where ϕ
k
, C[ϕ
n
]
(V)

=

c
k
= 2ω
k
M
k
ς
k
> 0; if n = k
0; otherwise
[4.41]
By using the Laplace transform, the image of the modal displacement is
obtained as:
˜q
k
(s) =
˜
Q
k
(s)
M
k

ω
2
k
+ 2ω
k

ς
k
s + s
2

where
˜
Q
k
(s) =
˜
Q
(e)
k
(s) +˙q
k
(0) +(2ω
k
ς
k
+ s)q
k
(0)
The Laplace transform of the displacement field in the physical coordinates
system is finally obtained by using the modal series [4.38]:

˜
X(r, s) =
∞


k=1
˜
Q
k
(s) ϕ
k
(r)
M
k

ω
2
k
+ 2ω
k
ς
k
s + s
2

[4.42]
Here, the number of terms of the series is infinite, which was obviously not
the case for N-DOF systems. Thus, in practice, to compute modal expansions like
[4.42], the series must be truncated to a finite number of terms. Truncation criteria
which lead to acceptable errors are discussed in the next subsections.
220 Structural elements
4.3.2 Deterministic excitations
To study the time response of continuous systems it is necessary to know both the
spatial distribution ofthe excitations and their variation with time. The deterministic
external forcing functions are described by a vector


F
(e)
(r; t) which may stand
either for an external force field which fluctuates with time, or for a prescribed
motion assigned to some degrees of freedom of the mechanical system. Firstly, it is
useful to make the distinction between forcing functions in which space and time
variables are separated and those where they are not. It is convenient to start with
the 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:

F
(e)
(r, t) = F
0
(r)u(r)f(t) [4.43]
(r) is a function, or a distribution, which specifies the space distribution of the
loading and which complies with the norm condition:

(V)
|(r)|dV = 1 [4.44]
(V) is the domain in which  is defined. F
0
is the scale factor of the load magnitude.
The unit vector u(r) specifies the load direction in the Euclidean space and finally
f(t) describes the time evolution, or time-history of the loading.  and f(t)are
assumed to be both integrable and square integrable. The resultant of the loading
is given by:


R(t) =

(V)

F
(e)
(r, t)dV = f(t)

(V)
(r)u(r)dV [4.45]
The point r
a
at which the resultant is applied is time independent. It is given by
the barycenter of the function :
r
a
=

(V)
r dV

(V)
dV
[4.46]
The modal forces related to this type of excitation are written as:
Q
(e)
n
(t) =


ϕ
n
,

F
(e)
(r; t)

(V)
= F
0
f(t)
n
; with 
n
=ϕ
n
,  u
(V)
[4.47]
Modal analysis methods 221
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 V
0
, 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 for
the sake of simplicity, the total weight

P =−M g is assumed to be distributed
uniformly along the train. So, the forcing function is written as:

F
(e)
(x, t) =

P
2λ
[U(x − V
0
t +λ) −U (x − V
0
T −λ)]
U(x) is the Heaviside step function and the running abscissa V
0
t is taken at the
middle of the train. The resultant and the abscissa x
a
(t) at which it is applied are
obtained by using the relationships [4.45] and [4.46]:

R =

P
2λ


+∞
−∞
[U(x − V
0
t +λ) −U (x − V
0
t −λ)]dx
=

P
2λ


+∞
V
t−λ
dx −

+∞
V
t+λ
dx

=

P
x
a
(t) =

1
2λ

+∞
−∞
x[U(x − V
0
t +λ) −U (x − V
0
t −λ)]dx
=

Vt+λ
Vt−λ
xdx= V
0
t
222 Structural elements
Now, if the length of the train remains much smaller than the bridge span
denoted L, the load can be reasonably modelled as a concentrated load which
leads to the modal forces:
Q
(e)
n
=ϕ
n
,

Pδ(x − V
0

t)=

P ·ϕ
n
(V t) [4.48]
According to the result [4.48], the whole series of modes is excited as time
elapses. If the actual length of the train is accounted for, the modal forces are found
to be:
Q
(e)
n
=
1
2λ

L
0
(

P ·ϕ
n
(x))[U(x − V
0
t +λ) −U (x − V
0
T −λ)]dx
=
1
2λ


VT+λ
VT−λ
(

P ·ϕ
n
(x)) dx [4.49]
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:
Q
(e)
n
=
PL
2nπλ

cos

nπ(V
0
T −λ)
L

− cos

nπ(V
0
T +λ)
L


which is found to vanish for all the modes such that λ = mL/n where m is an
integer less than n. So, such modes do not contribute to the bridge motion and can
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, the
only modes which contribute to the response series [4. 42] are those modes which
are not orthogonal to the spatial distribution of the excitation:
ϕ
n
, (r)u(r)
(V)
= 
n
= 0 [4.50]
where for convenience the criterion [4.50] is formulated in the case of a separated
variables forcing function.
It may be noted that a modal truncation based on the mode shape criterion is
similar to the elimination of some components of the physical displacements in a
solid body, based on the orthogonality with the loading vector field.
Modal analysis methods 223
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:
EI
∂
4

Z
∂x
4
+ C
˙
Z + ρS
¨
Z = f(t)δ(x− x
0
); Z(0) = Z(L) = 0;
∂
2
Z
∂x
2




0
=
∂
2
Z
∂x
2





L
= 0
The beam being provided with pinned support conditions at both ends, the modal
quantities relevant to the problem are:
ϕ
n
(ξ) = sin(nπ ξ ); M
n
= ρSL/2 = M
b
/2; K
n
= (nπ)
2
EI
2L
3
;
ω
2
n
=
(nπ)
4
L
4
c
2
, where ξ = x/L and c
2

= EI/ρS
The Laplace transform of the response is:
Z(ξ, ξ
0
; s) =
2
˜
f(s)
M
b
∞

n=1
sin(nπξ ) sin(nπ ξ
0
)
(c
2
(nπ/L)
4
+ 2scς
n
(nπ/L)
2
)
If ξ
0
= 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
∞

k=0
(−1)
k
sin((2k + 1)πξ)
c
2
((2k + 1)π/L)
4
+ 2scς
2
n
((2k + 1)π/L)
2
example. – Beam symmetrically or skew symmetrically loaded
224 Structural elements
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 conditions
are symmetric, the only modes which contribute to the response must verify
the 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 mater-
ial and geometrical defects which spoil the symmetry of the ideal model. One

striking example of the importance of such ‘small’ defects will be outlined in
subsection 4.4.3.3.
4.3.3.2 Spectral criterion
The spectral considerations made in [AXI 04] Chapter 9, concerning the dynam-
ical response of forced N -DOF systems, can be extended to continuous structures
to produce a very useful criterion for restricting the modal basis to a finite number
of modes. Figure 4.25 is a plot of the power density spectrum of some excitation
signal, which in practice extends over a finite bandwidth, limited by a lower cut-off
frequency f
c1
and by a upper cut-off frequency f
c2
; that is, outside the interval
f
c1
, f
c2
the excitation power density becomes negligible. The dots on the frequency
axis 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
Modal analysis methods 225
Let us consider first the response of a single mode f
n
to the excitation signal.
The response is found to be quasi-inertial if f
n
/f
c1
≪ 1, in the resonant range

if f
c1
<f
n
<f
c2
, and quasi-static if f
n
/f
c2
≫ 1. Therefore, a finite number
of low frequency modes can lie in the quasi-inertial range, and infinitely many
other modes lie in the quasi-static range. The contribution to the total response of
the modes lying in the quasi-inertial range can be accounted for by neglecting the
stiffness and damping terms of the modal oscillators and only a finite number of
such 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 accounted
for by neglecting the damping and inertial terms of the modal oscillators; however
there are still infinitely many modal contributions to be accounted for. The method
for avoiding the actual calculation of such an infinite series is best described starting
from a specific example.
Let us consider again a vehicle of mass M, travelling at speed V
0
on a flex-
ible bridge. Assuming the bridge deck is modelled as an equivalent straight beam
provided with pinned supports at both ends and damping is neglected, the equations
of the problem are written as:
EI
∂
4

Z
∂x
4
+ ρS
¨
Z =−Mgδ(x − V
0
t)
Z(0) = Z(L) = 0;
∂
2
Z
∂x
2




0
=
∂
2
Z
∂x
2




L

= 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
of the type:
ω
2
n
q
n
+¨q
n
=−g
2M
M
b
sin

nπV
0
t
L

; where ω
2
n
=

nπ
L


4
EI
ρS
It immediately appears that the dynamic response strongly depends upon the
cruising speed V
0
of the load. In particular, the resonant response of the n-th mode
occurs if the following condition is fulfilled:
nπV
n
L
=

nπ
L

2

EI
ρS
⇒ V
n
=
nπ
L

EI
ρS
Shifting to the spectral domain, the Fourier transform of the beam deflection is
found to be:

⌢
Z
(x, ω) =−g
M
M
b
∞

n=1
(δ(
n
− ω) − δ(
n
+ ω))
i(ω
2
n
− ω
2
)
sin

nπx
L

226 Structural elements
where 
n
=
nπV

0
L
.
which becomes infinite at the undamped resonances ω
n
= 
n
.
Nevertheless, it may be also realized that in most cases of practical interest even
the smallest cruising speed V
1
needed to excite the first resonance of the beam is
likely to be far beyond the realistic speed range of the vehicle. This is because a
bridge is designed to withstand large static transverse loads. As a consequence, the
dynamical response of the loaded bridge can be determined entirely by using the
quasi-static approximation:

nπ
L

4
EI
ρS
q
n
=−g
2M
M
b
sin


nπV
0
t
L

⇒ q
n
=−
gM
K
n
sin

nπV
0
t
L

;
where K
n
=
M
b
2

nπ
L


4
EI
ρS
and the bridge deflection is found to be:
Z(x; t) =−gM
∞

n=1
1
K
n
sin

nπV
0
t
L

sin

nπx
L

To restate the conclusions of this example as a general rule, one-dimensional
problems are considered for mathematical convenience. Further extension to the
case 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 outlined
above, 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 Fourier
transform of the response is written as:

⌢
Z
(x; ω) = F
0
⌢
f
(ω)
∞

n=1
ϕ
n
(x)
n
M
n

ω
2
n
+ 2iω
n
ως
n
− ω
2

[4.51]
The excitation spectrum S
ff

(ω) = 2
⌢
f
(ω)
⌢
f
*
(ω) is again assumed to be negli-
gible outside the finite interval ω
c1
, ω
c2
.Ifω
n
designates the natural frequencies of
the structure, ordered as an increasing sequence, the three following distinct cases
have to be discussed:
1. Resonant response range: ω
n
∈[ω
c1
, ω
c2
]
It is clearly necessary to retain all the modes whose frequencies lie within
the spectral range of the excitation, except if they are orthogonal to the spatial
distribution of the excitation, that is if ψ
n
vanishes. The contribution of such
modes to the response is given by the full expression [4.51]. The spectrum

Modal analysis methods 227
of the modal response can be conveniently related to the excitation spectrum
(cf. Volume 1 Chapter 9) as:
S
(n)
ZZ
(x; ω) =

ϕ
n
(x)
n
K
n

2

1
(1 −(ω/ω
n
)
2
)
2
+ 4(ως
n
/ω
n
)
2


S
ff
(ω)
[4.52]
Spectral relationships like [4.52] are especially useful to characterize the
magnitude of the vibration without entering into the detailed time-history of
the response. It is recalled that the mean square value of the modal response
can be inferred from the modal response spectrum S
(n)
ZZ
(x; ω) by integration
with respect to frequency as detailed in [AXI 04], Chapter 8.
2. Quasi-static responses: ω
c2
≪ ω
N
In the same way as in the travelling load example, the series [4.51] can be
reduced to the quasi-static form:
⌢
Z
(x; ω, N) = F
0
⌢
f
(ω)
∞

n=N
ϕ

n
(x)
n
K
n
[4.53]
The displacement is synchronous with the time evolution of the excitation.
Furthermore, the series [4.53] is found to converge, as appropriate from the
physical standpoint. Here, convergence can be immediately checked as the
modal stiffness coefficients K
n
are proportional to n
4
for bending modes and
to n
2
in the case of torsion and longitudinal modes. Furthermore, the series
[4.53] calculated from n = 1 instead of n = N, must converge to the static
solution Z
s
of the forced problem related to the system [4.35]:
K(x)Z
s
= F
0
(x)
+ B.C.

[4.54]
This is because the Hilbert space of the solutions is complete by definition.

So, the modal expansion of Z
s
is found to be:
Z
s
(x) = F
0
∞

n=1
ϕ
n
(x)ψ
n
K
n
[4.55]
As a consequence, the quasi-static part of the dynamical response [4.51]
may be conveniently written by using a finite number of terms only:
⌢
Z
(x; ω, N) = F
0
⌢
f
(ω)R
N
(x) [4.56]
where
R

N
(x) =
Z
s
(x)
F
0
−
N

n=1
ϕ
n
(x)ψ
n
K
n
228 Structural elements
R
N
(x) is termed the quasi-static mode or pseudo-mode. It gives the resultant
of theindividual quasi-static contributions to theresponse of the infinitely many
natural modes of vibration which lie in the quasi-static range, based on the
spectral criterion ω
c2
≪ ω
N
. Provided the solution of the static problem [4.54]
may be made available from a direct analytical or numerical calculation, the
pseudo-mode can be determined by using [4.56]. Then, the Fourier transform

of the solution to the dynamical problem [4.35] is expanded as:
⌢
Z
(x; ω) = F
0
⌢
f
(ω)

N

n=1
ϕ
n
(x)
n
M
n

ω
2
n
+ 2iω
n
ως
n
− ω
2

+ R

N
(x)

[4.57]
The quasi-static correction term R
N
(x) may also be interpreted in a slightly
distinct way, by making use of the concept of equivalent stiffness, defined here
as the force to displacement ratio:
K
eq
(x, ) =
F
0
Z
s
(x)
; where
1
K
eq
(x, )
=
∞

n=1
ϕ
n
(x)
n

K
n
[4.58]
K
eq
(x, ) depends on the position along the beam and on the spatial dis-
tribution (x) of the load through the static deflection Z
s
(x), as evidenced
in equation [4.55]. In agreement with [4.57], the equivalent stiffness coeffi-
cient which characterizes the truncation of the modal basis to the order N ,is
defined as:
1
K
N
(x, )
=
∞

n=N+1
ϕ
n
(x)
n
K
n
=
1
K
eq

(x, )
−
N

n=1
ϕ
n
(x)
n
K
n
[4.59]
This way of introducing the quasi-static corrective term for the truncated
model presents the physical interest to put clearly in evidence that by dropping
out the modes n>N, the truncated model is artificially stiffened, as expected
since the number of degrees of freedom is decreased. The effect of adding the
pseudo-mode contribution to the solution is precisely to provide the appropriate
correction by adding a spring ‘mounted in series’ with the truncated model.
Stated briefly, the inverse of the truncation stiffness accounts for the flexibility
of the neglected modes.
3. Quasi-inertial response: ω
p
≪ ω
c1
By analogy with the quasi-static approximation, it is also possible to
simplify the contribution of the modal responses related to the modes
whose frequencies are within the quasi-inertial range. The corrective term
Modal analysis methods 229
is defined by:
F

0
⌢
f
(ω)R
p
(x; ω); where R
p
(x; ω) =−
p

n=1
ϕ
n
(x)
n
ω
2
M
n
[4.60]
Following the same procedure as in the quasi-static case, it is possible to
define an equivalent mass of the system and an equivalent truncation mass
coefficient. Accordingly, the expansion [4.57] can be further simplified as:
⌢
Z
(x; ω) = F
0
⌢
f
(ω)




R
p
(x, ω) +
N

n=p+1
ϕ
n
(x)ψ
n
M
n

ω
2
n
+ 2iω
n
ως
n
− ω
2

+R
N
(x)




[4.61]
Modal truncation in the quasi-inertial range is of practical interest when a
large number of modes lie in the low frequency range ω
p
≪ ω
c1
.
4.3.4 Stresses and convergence rate of modal series
The modal series of the type [4.38] are found to converge for any realistic
excitation, as anticipated based on physical reasoning. This can be checked math-
ematically by noting that the infinite number of terms involved in the quasi-static
term R
N
(x) form a sequence decreasing at least as 1/n
2
. The convergence of the
stress 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 of
the 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 analysed
first 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
230 Structural elements
1. Local formulation
The solution of the static problem is immediate. The local equilibrium

equation and boundary conditions are:
−ES
d
2
X
dx
2
= F
0
δ(x −L)
X(0) = 0



⇔
−ES
d
2
X
dx
2
= 0
X(0) = 0; ES
dX
dx




L

= F
0
The solution is:
X
s
(ξ) =
LF
0
ES
ξ; where ξ =
x
L
The equivalent stiffness is K(ξ,1) = ES/(Lξ ) and the axial stress is N (ξ ) =
(ES/L)(dX/dξ) = F
0
.
2. Modal formulation
The modal basis complying withthe boundary conditionsof the problem are:
ϕ
n
(ξ) = sin ̟
n
ξ; ̟
n
= π(1 + 2n)/2; ω
n
=
c
0
̟

n
L
; where c
2
0
= E/ρ
M
n
= ρSL/2; K
n
π
2
ES
8L
(1 +2n)
2
The modal (or generalized) forces are:
Q
(e)
n
=ϕ
n
, F
0
δ(ξ − 1)
(L)
= F
0
sin ̟
n

= (−1)
n
F
0
, n = 0, 1, 2
The modal expansion of the type [4.55] gives the displacement field as the
Fourier series:
X(ξ,1) =
F
0
L
ES
8
π
2
∞

n=0
(−1)
n
sin[π(1/2 + n)ξ]
(1 +2n)
2
[4.62]
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-off
index N at ξ = 0.75 and ξ = 0.1. Incidentally, this kind of calculation is a
convenient 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
2
8
Modal analysis methods 231
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 by
term, which results in:
N
n
(ξ,1) =−F
0
4
π
∞

n=0
(−1)
n
cos[π(1/2 + n)ξ]
(1 +2n)
[4.63]
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 shapes

used for the expansion, which refer to a free end condition at ξ = 1. Anywhere
else the series converges in an alternative way to the correct value as shown in
Figure 4.28.
4.4. Substructuring method
4.4.1 Additional stiffnesses
Let us consider first the case of a single structure described by the dynamic
equation:
K(r)

X(r; t) + M(r)

¨
X(r; t) = 0 +elastic B.C. [4.64]
232 Structural elements
{ϕ
n
(r)}is the related modal basis. If other elastic supports are added to the structure,
the equation becomes:
K(r)

X(r; t) + K
a
(r)

X(r; t) + M(r)

¨
X(r; t) = 0 +elastic B.C. [4.65]
where K
a

(r) stands for the stiffness operator of the additional supports, which can
be either concentrated at some discrete locations, or continuously distributed at
the boundary of the structure. It is possible to approximate the new modal basis
{

φ
n
(r)}by projecting the system [4.65] on
{
ϕ
n
(r)
}
; which gives a truncated modal
expansion of the type:

φ
n
(r) =
N

j=1
q
j
ϕ
j
(r) [4.66]
Substituting [4.66] into [4.65] and projecting the result onto the mode ϕ
n
(r), the

current row of the coupled modal system is obtained as:

K
n
+ K
(a)
n
− ω
2
M
n

q
n
+
N

j=n
K
(a)
j
q
j
= 0 [4.67]
The coupling terms result from the lack of orthogonality of the ϕ
n
(r)with respect
to the operator K
a
(r). The solution of [4.67] produces N new mode shapes


φ
n
(r)
as expressed in the
{
ϕ
n
(r)
}
basis. They can be written in terms of the physical
coordinates r by using [4.66]. The method is illustrated by some examples in the
next subsections. As could be anticipated, it will be shown that the higher the
truncation 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 to
derive 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
Modal analysis methods 233
1. Spring modelled as an elastic impedance:
The modal system is written as:
−ES
d
2
X

dx
2
− ω
2
ρSX = 0; X(0) = 0; ES
d
2
dx
2




L
= K
L
X(L) [4.68]
To discuss the effect of the additional support it is found convenient to use
dimensionless variables based on a few pertinent scaling factors, as follows:
ξ = x/L;
¯
X = X/L; ̟ = ω/ω
0
; γ
L
= K
L
L/ES; ω
0
=

1
L

E
ρ
[4.69]
So the modal system is written as:
¯
X
′′
+ ̟
2
¯
X = 0; with
¯
X(0) = 0;
¯
X
′
(1) = γ
L
¯
X(1)
Again, the general solution of the differential equation is
X(ξ) =
a sin(̟ ξ) + b cos(̟ ξ) and the boundary conditions give the characteristic
equation:
b = 0; ̟
n
cot(̟

n
) = γ
L
[4.70]
It is of interest to examine the following specific cases for further
discussion:
(a) Spring stiffness coefficient is zero: γ
L
= 0
̟
n
= (1 + 2n)
π
2
; ϕ
n
(ξ) = sin(̟
n
ξ); n = 0, 1, 2
(b) Spring stiffness coefficient is infinite: 1/γ
L
= 0
ω
n
= nπ; ϕ
n
(ξ) = sin[ω
n
ξ]
(c) Spring stiffness coefficient has a nonzero finite value, for instance: γ

L
=
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:
−ES
d
2
X
dx
2
− ω
2
ρSX =−K
L
Xδ(x −L); with X(0) = 0 [4.71]
234 Structural elements
Using the dimensionless quantities already defined above, the modal system
is written as:

−
¯
X
′′
− ̟
2
¯
X =−γ
L
¯
Xδ(ξ − 1);
¯
X(0) = 0 [4.72]
The mode shapes φ
n
are expanded as a linear combination of the fixed-free
beam modes:
φ
n
(ξ) =
∞

j
q
j
ϕ
j
(ξ)
The projection of [4.72] onto ϕ
n

results in:

̟
2
n
− ̟
2

q
n
=−2γ
L
(−1)
n

j
q
j
[4.73]
Then the solution is obtained by solving a linear homogeneous coupled
system. The modal frequencies of [4.73] are expected to lie within the range:
̟
n
(γ
L
= 0)<̟
n
(γ
L
= π/4)<̟

n
(γ
L
→∞) ⇒
(2n −1)π
2
<̟
n
<nπ;
n = 1, 2,
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:



2γ
L
+ ̟
2
1
− ̟
2
−2γ
L
2γ
L
−2γ
L
−2γ

L
2γ
L
+ ̟
2
2
− ̟
2
−2γ
L
2γ
L
2γ
L
−2γ
L
2γ
L
+ ̟
2
3
− ̟
2
−2γ
L
−2γ
L
2γ
L
−2γ

L
2γ
L
+ ̟
2
4
− ̟
2





q
1
q
2
q
3
q
4


=


0
0
0
0



by solving this system, the following values of the modified natural frequencies
are obtained numerically:
̟
1
= 1.9652; ̟
2
= 4.8788; ̟
3
= 7.9571; ̟
4
= 11.071
If the projection is restricted to the first two modes, the values become:
̟
1
= 1.9784; ̟
2
= 4.8889
As a consequence of the stiffening effect due to modal truncation, these
values are found to be higher (though by a small amount only) than the exact
values. On the other hand, it is also interesting to check the convergence of the
method when γ
L
becomes so large as to enforce practically a fixed boundary
condition; the numerical results presented below refer to a spring stiffness
coefficient hundred times larger than the first modal stiffness. Six modes are
Modal analysis methods 235
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
= 12.665
The 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 sixteen
for 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 make
use of the quasi-static corrective term [4.59] than to increase the size of the
modal basis.
3. Correction of the modal truncation
It is possible to improve the model presented above by adding to it a stiff-
ness coefficient K
T
which accounts suitably for the truncation of the modal
basis. This numerical spring is mounted in series with the physical spring,
as sketched in Figure 4.30, in such a way that the equivalent stiffness of the
support is decreased, as given by:
K
e
=
K
T
K
L

K
T
+ K
L
⇒
1
γ
e
=
1
γ
T
+
1
γ
L
[4.74]
Using the relation [4.59], the dimensionless flexibility of truncation is
obtained as:
1/γ
T
= 1 −
8
π
2
N−1

n=0
1
(2n +1)

2
Including the equivalent flexibility [4.74] in the model, and adopting
γ
L
= 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 two
first modes only, the results are still sufficiently accurate for most applications:
γ
e
= 0.7285 and ̟
1
= 1.9538; ̟
2
= 4.8755. Finally, if the end spring stiff-
ness is so large as to be practically equivalent to a fixed condition, satisfactory
results can be obtained by selecting N = 6, which gives γ
e
= 29.67.
4.4.1.2 Truncation stiffness for a free-free modal basis
Here, the beam is supported by a spring K
L

at one end and left free at the other.
Again, we want to compute the longitudinal vibration modes of the supported beam
236 Structural elements
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 the
context 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 elastic
impedance, the following characteristic equation for the natural frequencies is γ
L
=
̟
n
tan(̟
n
). Selecting for instance γ
L
= 1, the first reduced frequencies are ̟
1
=
0, 86035; ̟
2
= 3, 4256; ̟
3
= 6, 4373. Now if the modal projection method is
used, a difficulty arises in determining the truncation flexibility since it is a priori
infinite, due to the presence of the free rigid mode. However, let us discuss the
problem a little further. The constrained beam is governed by:
−
¯
X

′′
− ̟
2
¯
X = γ
L
¯
Xδ(ξ);
¯
X
′
(1) = 0
By setting γ
L
= 0, the free-free modal basis is obtained:
̟
(ff )
n
= nπ; a
n
= 0 ⇒ ϕ
n
(ξ) = cos(nπ ξ ); n = 0, 1, 2
If four modes are retained in the projection, the following homogeneous system to
be solved is:



2(γ
L

+ ̟
2
) 2γ
L
2γ
L
2γ
L
2γ
L
2γ
L
+ π
2
− ̟
2
2γ
L
2γ
L
2γ
L
2γ
L
2γ
L
+ 4π
2
− ̟
2

2γ
L
2γ
L
2γ
L
2γ
L
2γ
L
+ 9π
2
− ̟
2





q
1
q
2
q
3
q
4


=



0
0
0
0


The first row of the modal equation is the projection of the constrained local
equation onto the rigid body mode n = 1. By solving numerically the system for
γ
L
= 1, the following natural frequencies are obtained:
̟
1
= 0.87901; ̟
2
= 3.4412; ̟
3
= 6.4770; ̟
4
= 9.5387
Although accounting for the truncation stiffness is not formally feasible, nothing
prevents us in practice from providing the system with a fictitious spring of stiffness
coefficient K
f
, see Figure 4.31. It is appropriate to select for K
f
a value much
smaller than the modal stiffness related to the first non-rigid mode of the free-free

beam, in such a way that the perturbation remains negligibly small, except in so
far as the frequency of the rigid mode is concerned. The static response to an axial
force is thus performed with this support, artificially introduced in the system to
make the calculation possible. The erroneous contribution of the rigid mode can be
eliminated afterwards. The procedure can be described starting from the equivalent
flexibility as determined in the initial and then in the perturbed system. For the
initial system:
1
γ
e
=
“

1
0

”
+
2
π
2
∞

n=1
1
n
2
=
“


1
0

”
+
1
3
where “1/0” stands for the free rigid mode singular contribution.
Modal analysis methods 237
Figure 4.31. Beam provided with an elastic support at one end
For the perturbed system, in which perturbation of the non-rigid modes is
assumed to be negligible:
1
γ
′
=
1
γ
f
+
1
3
where 1/γ
f
is the finite flexibility of the fictitious support.
Then the truncation flexibility is given by:
1
γ
T
=

1
γ
′
−
1
γ
f
−
2
π
2
N

n=2
1
n
2
=
1
3
−
2
n
2
N

n=2
1
n
2

where 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:
γ
e
= 0.9465; ̟
1
= 0.8607; ̟
2
= 3.4263; ̟
3
= 6.4386; ̟
4
= 9.5322
which 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 P
0
. The buckling load P
c
= (π/L)
2
EI was
already determined in subsection 4.2.5.2. The related buckling mode shape was
found to be:
ψ
n
(x) =
1
2


cos

nπx
L

− 1

238 Structural elements
However, as already indicated, the vibration modes cannot be expressed in a
closed analytical form. So the modal projection method is applied to solve the
problem in terms of a truncated series by using the modal basis of the unloaded
beam. The equation of the prestressed beam is thus written as:
ρS
¨
Z + P
0
∂
2
Z
∂x
2
+ EI
∂
4
Z
∂x
4
= 0
provided with the boundary condition:


P
0
(∂Z/∂x) + EI(∂
3
Z/∂x
3
)



L
= 0
It is emphasized again that this condition depends on the axial force, which is
not accounted for in the modal basis. Accordingly, the projection is carried out
directly as a weighting integral, producing a non-symmetrical prestress operator
denoted K
(0)
, which couples the modal degrees of freedom:

L
0
ϕ
n
(x)

ρS
¨
Z + P
0

∂
2
Z
∂x
2
+ EI
∂
4
Z
∂x
4

dx = 0
leading to:
(K
n
− ω
2
M
n
)q
n
+ P
0

K
(0)
n,j
q
j

= 0
The results obtained by using the first four modes are fairly accurate, as shown
in Figure 4.32. Here, the mode shapes are normalized with respect to the largest
deflection magnitude. It is noted that the analytical buckling mode shape is nicely
fitted by the modal approximation as soon as the preload is sufficiently large.
4.4.2 Additional inertia
The methods presented in the last subsection can be transposed without any dif-
ficulty in the case of ‘inertial connections’ i.e. when concentrated masses are added
to the original system. To illustrate this point we take the example often encountered
in rotating machinery of a shaft provided with a fly wheel of large inertia.
Figure 4.33 is a simple model of a rotor which includes a shaft simply supported
at both ends ( pinned-pinned conditions) and a fly wheel, modelled as a rigid circular
disk, located at the distance x
0
from the left end. The bending moment of inertia
I
b
of the shaft is small in comparison with that of the fly wheel, denoted I
w
. The
equilibrium equation could be obtained by adding the kinetic energy of the disk
induced by bending and axial displacements to the Bernoulli–Euler Lagrangian.
Another means to derive such an equation is to add directly the inertia forces and
Modal analysis methods 239
Figure 4.32. Shape and frequency of the first mode as function of the prestress
Figure 4.33. Shaft and fly wheel
moments due to the disk in the force and moment balance equations [2.18], [2.21].
Following the last procedure, the equations are:
∂Q
z

∂x
+ M
w
¨
Zδ(x − x
0
) +ρS
¨
Z = 0
∂M
z
∂x
+ Q
z
− I
w
∂
¨
Z
∂x




x
0
δ(x −x
0
)
[4.75]

240 Structural elements
which leads to the force balance:
EI
b
∂
4
Z
∂x
4
+
(
ρS +M
w
δ(x −x
0
)
)
¨
Z + ρI
w
∂
¨
Z
∂x




x
0

δ
′
(x − x
0
) = 0 [4.76]
The modal basis is described by the following quantities:
ϕ
n
= sin(nπ ξ ); K
n
= n
4
K
0
; K
0
=
EI
b
π
4
2L
3
;
M
n
=
ρSL
2
=

M
b
2
; ω
2
n
= n
4
ω
2
0
; ω
2
0
=
K
0
M
n
Projecting [4.76] on this basis, a coupled system of equations is obtained, which
is written in terms of reduced quantities as:
(n
4
− ̟
2
)q
n
−̟
2
µ

R
∞

j=1

j cos

nπx
0
L

cos

jπx
0
L

q
j

− ̟
2
µ
Z
∞

j=1

sin


nπx
0
L

sin

jπx
0
L

q
j

= 0
where ̟ =
ω
ω
0
; µ
R
=
2π
2
I
w
M
b
L
2
; µ

Z
=
M
w
M
b
[4.77]
Inertia of the disk induces coupling both in translation and rotation. If the disk
is at mid-span the odd order modes are sensitive to translation only, whereas the
even order modes are sensitive to rotation only. So all the modes are perturbed
by the presence of the flywheel. The shapes of the two first modes are shown in
Figure 4.34, in the case µ
z
= 2 and µ
R
= µ
R
/4. The solid lines refer to the
flywheel mounted on the shaft and the dashed lines to the shaft alone. As expected,
the effect of the wheel is important both on the mode shapes and on the natural
frequencies. Here, the modal basis is truncated to N = 23 but N = 6 would
produce very similar results.
4.4.3 Substructures by using modal projection
4.4.3.1 Basic ideas of the method
Many mechanical structures encountered in practice may be conveniently
described as an assembly of more simple substructures that are denoted here
(S
1
), (S
2

) etc., attached to each other at various places by connecting elements,
Modal analysis methods 241
Figure 4.34. The first two modes of the shaft with a flywheel computed by using the modal
projection method
for instance linear or nonlinear springs, as depicted schematically in Figure 4.35.
Numerical studies of the dynamic behaviour of such structures, taken as a whole,
may need a large number of DOF, especially if the finite element method is used.
In particular, when building the finite element model of the whole structure, it is
often difficult to take full advantage of the peculiarities of each substructure to sim-
plify the model. Furthermore, it is often difficult to optimize the size of the mesh
with respect to the space resolution and frequency range of physical interest. In
many applications, the alternative method described here has the decisive advant-
age of saving a large number of DOF and time steps and providing the analyst
with pertinent information on the physical features of the problem to be solved.
The method consists of projecting the equations of the whole system on a modal
basis made up of a set of modes which refer to each substructure taken individually,
that is in the absence of the elements connecting it to the others. The modes to be
retained in the model can be judiciously selected in close relation to the physical
particularities of the problem to be solved, by using the criteria already discussed
in subsection 4.3.3. The procedure can be conveniently described by considering
an assembly of two substructures only. In the absence of connecting elements, the
substructure (1) satisfies the equation:
K
1

X + C
1
˙

X + M

1
¨

X =

F
(e)
1
(r, t)
+ elastic B.C.



[4.78]
242 Structural elements
Figure 4.35. Assembly of distinct substructures
The related modal basis is:
{ϕ
(1)
n
}; ω
n
; M
n
; K
n
[4.79]
The substructure (2) satisfies the equation:
K
2


Y + C
2
˙

Y + M
2
¨

Y =

F
(e)
2
(r, t)
+ elastic B.C.



[4.80]
The related modal basis is:
{ϕ
(2)
n
}; ω
′
n
; M
′
n

; K
′
n
[4.81]
The connection between two points of the two substructures is modelled as
interaction forces, which can depend, linearly or not, on displacements, velocities
and accelerations of the connected points and be explicitly time dependent. The
coupled system of equations is thus written as:
K
1

X + C
1
˙

X + M
1
¨

X =

F
(e)
1
(r; t) +
J

j=1

F

L
j
(

X,

Y ,
˙

X,
˙

Y ,
¨

X,
¨

Y ; t)
+ elasticB.C.
K
2

Y + C
2
˙

Y + M
2
¨


Y =

F
(e)
2
(r; t) −
J

j=1

F
L
j
(

X,

Y ,
˙

X,
˙

Y ,
¨

X,
¨


Y ; t)
+ elastic B.C.
[4.82]
where F
L
j
stands for the force induced by the j-th connection.
The projection of [4.82] on the basis {
n
}={ϕ
(1)
n
}∪{ϕ
(2)
n
} produces a differ-
ential system of modal oscillators coupled by the connecting forces. Depending on
their nature, different techniques can be used to solve the problem. The method is
further described in the next subsections by taking a few examples.