2.9 Modal Analysis

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Mechanical structures exhibit eigenmodes. These are oscillating homogeneous solutions of

the linear (or linearized) governing equations. Their amplitude can be freely scaled. The

corresponding frequency of the oscillation is called the eigenfrequency. Modal analysis,

that is, the determination of the eigenfrequencies and eigenmodes is important in structural

analysis since the eigenmodes are the preferred shapes a structure will assume when subject

to loading. Indeed, one way of calculating dynamic response is by assuming that it is a

linear combination of the lowest eigenfrequencies (Meirovitch 1967).

2.9.1 Frequency calculation

A frequency analysis starts from the governing equation (2.27) in homogeneous form

_M_ _ ¨U _ + _K_ _U_ = _0_ (2.243)

with initial conditions

_U_t=t0

= _U0_ (2.244)

_ ˙U _t=t0

= _V0_ . (2.245)

To obtain the eigenmodes, a solution in the form

_U_ = _Uj _ eiωj t (2.246)

is proposed (separation of the space and time variables). Consequently, Equation (2.243)

yields

_K_ _Uj _ = ω2

j _M_ _Uj _ . (2.247)

This is a classical generalized eigenvalue problem with well-known properties. Since _K_

is symmetric and _M_ is symmetric and positive-definite, the eigenvalues are real and the

eigenmodes are orthogonal with respect to _M_. Indeed, suppose that λj := ω2

j is complex

with eigenvector _Uj _, then, taking the complex conjugate of Equation (2.247)

_K_ _Uj _ = λj _M_ _Uj _ (2.248)

reveals that λj must also be an eigenvalue with eigenvector _Uj _. Premultiplying

Equation (2.248) by _Uj _T and Equation (2.247) by _Uj _T yields

_Uj _T _K_ _Uj _ = λj _Uj _T _M_ _Uj _ (2.249)

_Uj _T _K_ _Uj _ = λj _Uj _T _M_ _Uj _ . (2.250)

Taking the transpose of Equation (2.249) and subtracting the results from Equation (2.250)

leads to

0 = (λj − λj ) _Uj _T _M_ _Uj _ . (2.251)

LINEAR MECHANICAL APPLICATIONS 107

Since _M_ is positive definite, we have (Greenberg 1978)

_Uj _T _M_ _Uj _ >0 if_Uj _ _= 0. (2.252)

Accordingly,

λj = λj (2.253)

and λj = ω2

j is real, and so are the corresponding eigenmodes.

Now, let _Ui_ and _Uj _ be two different solutions:

_K_ _Ui_ = λi _M_ _Uj _ (2.254)

_K_ _Uj _ = λj _M_ _Uj _ . (2.255)

Multiplying Equation (2.254) by _Uj _T and Equation (2.255) by _Ui_T, taking the transpose

of Equation (2.255), subtracting both and taking the symmetry of _K_ and _M_ into

account, one obtains

(λi − λj ) _Uj _T _M_ _Ui_ = 0. (2.256)

For λi _= λj one has

_Uj _T _M_ _Ui_ = 0 (2.257)

which shows that the eigenmodes are orthogonal indeed. They are generally normed such

that

_Uj _T _M_ _Uj _ = 1. (2.258)

Premultiplying Equation (2.247) by _Uj _T yields

_Uj _T _K_ _Uj _ = λj _Uj _T _M_ _Uj _ . (2.259)

The matrix _M_ is positive definite. If _K_ is positive definite as well, λj = ω2

j is not

only real but also strictly positive. This implies that for each eigenvalue λj there are two

real eigenfrequencies: ωj and −ωj . They correspond to the solutions _Uj (X)_ eiωj t and

_Uj (X)_ e−iωj t , or, alternatively, to _Uj (X)_ cos(ωj t) and _Uj (X)_ sin(ωj t). Notice that

these homogeneous solutions are bounded by Uj (X).

If λj = 0, then ωj = 0 is a double root. The solution of Equation (2.243) now amounts to

_Uj (X)_ and _Uj (X)_ t. For λj < 0 the eigenfrequencies are imaginary: ωj = ±i_(−λj )

leading to the solutions _Uj (X)_ e−

√

(−λj t) and _Uj (X)_ e

√

(−λj t). Accordingly, for λj ≤ 0,

at least one of the solutions is not bounded.

Eigenvalue problems such as Equation (2.247) are usually solved with dedicated numerical

packages such as ARPACK (Lehoucq et al. 1998). A continuous system has infinitely

many eigenmodes. Usually, only the lowest ones (10 up to maybe 100) are practically

important. In what follows, ωj will be assumed to be positive.

108 LINEAR MECHANICAL APPLICATIONS

2.9.2 Linear dynamic analysis

A general linear dynamic analysis starts from Equation (2.27):

_M_ _ ¨U _ + _K_ _U_ = _F_ . (2.260)

Frequently, a damping term linear in the velocity _ ˙U _ is added

_M_ _ ¨U _ + _C_ _ ˙U _ + _K_ _U_ = _F_ . (2.261)

If the damping is of the Rayleigh type, _C_ is defined as a linear combination of _M_ and

_K_:

_C_ = α _M_ + β _K_ . (2.262)

The quintessence of modal dynamic analysis is the fact that the response _U_, solution

of Equation (2.261), can be written as a linear combination of the eigenmodes _Ui_, which

is the solution of Equation (2.247). This relates to the fact that the eigenmodes constitute

an orthogonal basis for the solution space of Equation (2.260). Accordingly,

_U(t)_ =_

i

bi (t) _Ui_ . (2.263)

Notice that only the coefficients bi (t) are a function of time, the eigenmodes _Ui_ are not. In

reality, only a finite number of eigenmodes is calculated and the series in Equation (2.263)

is truncated. The truncated series is an approximation of _U(t)_. The quality of the approximation

depends on the number of eigenmodes and the frequency content of the loading.

Substituting Equation (2.263) into Equation (2.261), one obtains

_

i

_M_ _Ui_ ¨bi (t) +_

i

_C_ _Ui_ ˙bi (t) +_

i

_K_ _Ui_ bi (t) = _F(t)_ . (2.264)

Premultiplying by _Uj _ and using Equations (2.247) and (2.262) yields

_

i

_Uj _T _M_ _Ui_ _¨bi (t) + (α + βω2

i )˙bi (t) + ω2

i bi (t)_ = _Uj _T _F(t)_ (2.265)

and because of the orthogonality condition, Equation (2.257), and norming condition,

Equation (2.258),

¨b

j (t) + (α + βω2

j )˙bj (t) + ω2

j bj (t) = _Uj _T _F(t)_ . (2.266)

Equation (2.266) is the central equation for modal dynamics. It can be written for each

mode and constitutes a linear inhomogeneous second-order ordinary differential equation

with constant coefficients. The key point is that due to the choice of _C_, the modes are

LINEAR MECHANICAL APPLICATIONS 109

independent of each other. The differential equations have to be complemented by the initial

conditions bj (0) and ˙bj (0) obtained from _U0_ := _U(t = 0)_ and _V0_ := _ ˙ U(t = 0)_

(Equation (2.263))

_

i

bi(0) _Ui_ = _U0_ ⇒ bj (0) = _Uj _T _M_ _U0_ (2.267)

and

_

i

˙b

i (0) _Ui_ = _V0_ ⇒ ˙bj (0) = _Uj _T _M_ _V0_ . (2.268)

Equation (2.266) is frequently written as

¨b

j (t) + 2ζjωj

˙b

j (t) + ω2

j bj (t) = _Uj _T _F(t)_ (2.269)

where ζj is the friction coefficient defined by

ζj :=

α + βω2

j

2ωj

. (2.270)

Notice that because of Equation (2.270), the friction coefficient depends on the eigenvalues.

A large α-coefficient leads to low frequency damping and a large β-coefficient to highfrequency

damping.

The solution of Equation (2.269) basically depends on the character of the discriminant,

defined by

ωjd := ωj_1 − ζ 2

j . (2.271)

It arises in the solution of the quadratic equation obtained by substituting eλt in the homogeneous

differential equation. One obtains the following cases:

1. ωjd ∈ R

+

bj (t) = 1

ωjd

_ t

0

_Uj _T _F(τ)_ e−ζjωj (t−τ) sin[ωjd(t − τ)] dτ

+ e−ζjωj t



cos[ωjdt ] + ζj

_1 − ζ 2

j

sin[ωjdt ]



bj (0)

+ _ 1

ωjd

e−ζjωj t sin[ωjdt ]_ ˙bj (0). (2.272)

110 LINEAR MECHANICAL APPLICATIONS

To obtain Equation (2.272), formula 2.663.1 and 2.667.5 in (Gradshteyn and Ryzhik

1980) were used. The solution is called subcritical and consists of oscillatory functions.

2. ωjd = 0

bj (t) = _ t

0

_Uj _T _F(τ)_ e−ζjωj (t−τ)(t − τ) dτ

+ e−ζjωj t [1 + ζjωj t ]bj (0) + te−ζjωj t ˙bj (0). (2.273)

To obtain Equation (2.273), formula 2.322.1 in (Gradshteyn and Ryzhik 1980) was

used. The solution is called critical and exhibits an exponential nonoscillatory behavior.

3. ωjd = iω

∗

jd, ω

∗

jd

∈ R

+

bj (t) = 1

ω

∗

jd

_ t

0

_Uj _T _F(τ)_ e−ζjωj (t−τ) sinh[ω

∗

jd(t − τ)] dτ

+ e−ζjωj t



cosh[ω

∗

jdt ] + ζj

_ζ 2

j

− 1

sinh[ω

∗

jdt ]



bj (0)

+

_ 1

ω

∗

jd

e−ζjωj t sinh[ω

∗

jdt ]

_ ˙bj (0). (2.274)

The solution is supercritical and exhibits an exponential nonoscillatory behavior. It

can also be written as

bj (t) = 1

2ω

∗

jd

_ t

0

_Uj _T _F(τ)_ _eω

−

(t−τ) − e−ω

+

(t−τ)_ dτ

+ 1

2

_eω

−

t + e−ω

+

t _ bj (0)

+



ζj

2_ζ 2

j

− 1

bj (0) + 1

2ω

∗

jd

˙b

j (0)



_eω

−

t − e−ω

+

t _ (2.275)

where

ω

− := ω

∗

jd

− ζjωj (2.276)

ω

+ := ω

∗

jd

+ ζjωj . (2.277)

In Equations (2.272), (2.273) and (2.275), the right-hand side loading is written inside

an integral sign. For pointwise linear loading, the integral can be evaluated exactly. Indeed,

LINEAR MECHANICAL APPLICATIONS 111

let the interval [0, t] be split in subintervals [ti−1, ti ] in which the loading is linear

in time

_Uj _T _F(τ)_ = aij + bij τ for τ ∈ [ti−1, ti ] (2.278)

with t0 = 0 and tn = t , and let

σ := t − τ (2.279)

then (use formulas 2.663.1 and 2.667.5 in (Gradshteyn and Ryzhik 1980))

_ t

0

_Uj _T _F(τ)_ e−ζjωj (σ ) sin ωjd(σ ) dτ

=

n

_

i=1



[aij + bij t ]

_e−ζjωj σ (−ζjωj sin[ωjdσ ] − ωjd cos[ωjdσ ])

ζ 2

j ω2

j

+ ω2

jd

_t−ti−1

t−ti

−bj

_ e−ζjωj σ

ζ 2

j ω2

j

+ ω2

jd

__

−ζjωjσ −

(ζ 2

j ω2

j

− ω2

jd)

(ζ 2

j ω2

j

+ ω2

jd)

_

sin[ωjdσ ]

−

_

ωjdσ + 2ζjωjωjd

(ζ 2

j ω2

j

+ ω2

jd)

_

cos[ωjdσ ]

__t−ti−1

t−ti



(2.280)

in Equation (2.272), (use formulas 2.322.1 and 2.322.2 in (Gradshteyn and Ryzhik 1980))

_ t

0

_Uj _T _F(τ)_ e−ζjωj (σ )(σ ) dτ

=

n

_

i=1



[aij + bij t ]

_

e−ζjωj σ _

− σ

ζjωj

− 1

ζ 2

j ω2

j

__t−ti

t−ti−1

−bj

_

e−ζjωj σ _

− σ 2

ζjωj

− 2σ

ζ 2

j ω2

j

− 2

ζ 3

j ω3

j

__t−ti

t−ti−1



(2.281)

in Equation (2.273) and (use formulas 2.311 and 2.322.1 in (Gradshteyn and Ryzhik 1980))

_ t

0

_Uj _T _F(τ)_ _eω

−

(σ ) − e−ω

+

(σ )_ dτ

=

n

_

i=1



[aij + bij t ]

_eω

−

σ

ω−

+ e−ω

+

σ

ω+

_t−ti

t−ti−1

−bj

_eω

−

σ _ σ

ω−

− 1

(ω−)2

_ + e−ω

+

σ _ σ

ω+

+ 1

(ω+)2

__t−ti

t−ti−1

'

(2.282)

112 LINEAR MECHANICAL APPLICATIONS

in Equation (2.275). Consequently, the solution for piecewise-linear loading can be written

down explicitly.

A special case of loading is the harmonic excitation satisfying

_F(t)_ = _FR + iFI _ ei_t . (2.283)

FR and FI are the in-phase and out-of-phase amplitude, respectively, and _ is the frequency

of the excitation. Now, Equation (2.266) reads

¨b

j (t) + (α + βω2

j )˙bj (t) + ω2

j bj (t) = _Uj _T _FR + iFI _ ei_t . (2.284)

Inspired by the form of the right-hand side, we assume a complex solution in the form

bj (t) = (bjR + ibjI ) ei_t (2.285)

the derivatives of which yield

˙b

j (t) = i_(bjR + ibjI ) ei_t (2.286)

¨b

j (t) = −_2(bjR + ibjI ) ei_t . (2.287)

Substitution into Equation (2.284) leads to

−_2(bjR + ibjI ) + i(α + βω2

j)_(bjR + ibjI ) + ω2(bjR + ibjI ) = _Uj _T _FR + iFI _ .

(2.288)

Separating the real and imaginary parts of the equation yields two real equations:

(−_2bjR − (α + βω2

j)_bjI + ω2

j bjR = _Uj _T _FR_

−_2bjI + (α + βω2

j)_bjR + ω2

j bjI = _Uj _T _FI _

(2.289)

which is equivalent to

_ (ω2

j

− _2) −(α + βω2

j)_

(α + βω2

j)_ (ω2

j

− _2)

_ )bjR

bjI

* =

(_Uj _T _FR_

_Uj _T _FI _

'

(2.290)

the solution of which reads

bjR =

_Uj _T _FR_ (ω2

j

− _2) + _Uj _T _FI _ (α + βω2

j)_

(ω2

j

− _2)2 + (α + βω2

j )2 _2

(2.291)

bjI =

_Uj _T _FI _ (ω2

j

− _2) − _Uj _T _FR_ (α + βω2

j)_

(ω2

j

− _2)2 + (α + βω2

j )2_2

. (2.292)

2.9.3 Buckling

Buckling calculations are a special case of frequency calculations with preload. In

Equation (2.243), it was emphasized that frequency calculations are essentially homogeneous.

However, the eigenfrequencies of a structure do depend on the loading. For instance,

LINEAR MECHANICAL APPLICATIONS 113

a rotating blade has other eigenfrequencies in comparison to a static blade. This effect

manifests itself through a modified stiffness of the structure due to the stresses and displacements.

This is explained in Chapter 3 where the following modified stiffness matrix

is defined (Equation (3.17)):

_K_mod

= _K_LE

+ _K_ST

+ _K_LD . (2.293)

Here, _K_LE is the linear elastic stiffness matrix, _K_ST is the stress stiffness contribution

and _K_LD is the large deformation stiffness. By replacing _K_ in Equation (2.247) by

_K_mod, the loading is taken into account in the frequency calculation. The buckling load

can be defined as the load for which the lowest eigenfrequency reaches zero. Then, a

small perturbation will lead to buckling. Indeed, in Section 2.9.1 it was shown that a zero

eigenvalue, or equivalently a zero eigenfrequency, leads to an unbounded homogeneous

solution: the system is unstable.

As an example, look at the beam in Figure 2.20, loaded by a point force at its end.

Figure 2.21 shows the lowest eigenfrequency ω of the beam. It corresponds with a bending

mode with zero displacements at its fixed end. As the tensile force decreases, the lowest

eigenvalue ω2 decreases until it is zero and buckling occurs. Notice that as the eigenvalue

ω2 becomes zero, the eigenfrequency ω is zero too and the solution is unbounded

(Section 2.9.1).

Suppose that the structure is loaded by a static force system 1 and a buckling load

system 2. The static load system is defined as a system that is permanently acting and the

magnitude of which is not changing. The buckling load system varies in magnitude and the

basic question is at what value of the buckling load system will the collapse occur. To this

end, the buckling load system is scaled with a factor λ and the problem is reduced to the

question: at what value of λ is the lowest eigenvalue of the system zero? Equation (2.247)

is now equivalent to

__K_LE

+ _K_ST1

+ _K_LD1

+ _K_ST2λ

+ _K_LD2λ_ _U_ = 0. (2.294)

Index 1 stands for load system 1, index 2λ for λ times load system 2. _U_ is only nonzero

if the total stiffness matrix is singular which also implies that the matrix is not positivedefinite.

Equation (3.17) reveals that _K_ST is linear in the load but _K_LD is not (notice

the quadratic term in the displacement). Accordingly,

_K_ST2λ

= λ _K_ST2 (2.295)

_K_LD2λ

_= λ _K_LD2λ . (2.296)

F

h

8 h 1.5 h

Figure 2.20 Geometry of the beam

114 LINEAR MECHANICAL APPLICATIONS

−5 −4 −3 −2 −1 0

0

1 2 3 4 5

5

10

15

20

25

103 F

Eh2

103ωh_ρ

E

Figure 2.21 Lowest eigenfrequency for a beam under tension

Therefore, for linear buckling calculations, _K_LD2λ will be neglected leading to the following

eigenvalue problem:

__K_LE

+ _K_ST1

+ _K_LD1_ _U_ = −λ _K_ST2 _U_ . (2.297)

This is again a generalized eigenvalue problem with symmetric matrices similar to

Equation (2.247) except that _K_ST2 is not positive-definite. It is the governing buckling

equation and can be solved using the ARPACK package (Lehoucq et al. 1998).

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