3.9 Damped Systems

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Let us examine the possibility of extending the modal analysis results to damped systems. Damping is an

energy dissipation phenomenon. In lumped-parameter models of vibrating systems, damping force can

be represented by a resisting force at each lumped mass. In view of Equation 3.1, then, the system

equations for a damped system can be written as

My€ þ Ky ¼ f ðtÞ 2 d ð3:57Þ

in which d is the damping force vector. Modeling of damping is usually quite complicated. Often, a linear

model, whose energy dissipation capacity is equivalent to that of the actual system, is employed. Such a

model is termed as an equivalent damping model. The most popular model for damping is the linear

viscous model, in which the damping force is proportional to the relative velocity. In lumped-parameter

dynamic models, (linear) viscous damping elements may be assigned across pairs of DoF or across a DoF

and a fixed reference. The damping force for such a model may be expressed as

d ¼ Cy_ ð3:58Þ

in which C is the damping matrix. The resulting damped system equation is

My€ þ Cy_ þ Ky ¼ f ðtÞ ð3:59Þ

To determine the elements of C for a given system model by the influence coefficient approach, the same

procedure outlined previously for obtaining the elements of K may be used, except that the velocities y_i

should be used in place of the displacements yi: The coefficients cij are termed as damping influence

coefficients.

Response

y

(Normalized

with f0 )

Time t

(Normalized with w 2)

Quadratic

Growth

0 10 20 30 40 50 60 70

500

1000

1500

2000

y1

y2

0

mw 2

2

FIGURE 3.11 Forced response obtained through modal analysis.

3-32 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

3.9.1 Proportional Damping

Because the modal vectors are orthogonal with respect to both M and K, the transformation 3.43 will

decouple the undamped system equation 3.41. It should be clear, however, that the same transformation

will not diagonalize the C matrix in general. As we have observed in the previous section, decoupling or

modal decomposition, is a convenient tool in response analysis. This is because each uncoupled

modal equation is a simple oscillator equation with a well-known solution that can be transformed back

(or, recombined) to obtain the total response. This simple procedure cannot be used for damped systems

unless the modal vectors are orthogonal with respect to C as well.

In modal vibration, all DoF move in the same displacement proportion, as given by the modal vector.

This type of synchronous motion may not be possible in damped systems. Another way to state this is

that most damped systems do not possess real modes. If we try to excite such a damped system at one of

its natural frequencies, we will notice that the constant proportion, given by the modal vector (for the

undamped system), is violated during motion. Furthermore, if the undamped system has a (fixed) node

point in the particular mode, that point would not remain fixed but would move in a cyclic manner when

set to vibrate at the natural frequency of the mode.

Note that viscous damping is just a model for energy dissipation. If we become rather restrictive in

choosing the parameters of viscous damping, we will be able to develop an equivalent damping matrix,

with respect to which, the modal vectors of the undamped system would be orthogonal. In other words,

we require that the transformed damping matrix C􀀊 be a diagonal matrix; thus

CTCC ¼ diag½C1; C2; …; Cn􀀉 ¼ C􀀊 ð3:60Þ

In this case, the corresponding viscous damping model is termed proportional damping or Rayleigh

damping (after the person who first identified this simplification).

Modal decomposition of Equation 3.59, assuming proportional damping using the transformation

3.43, results in the canonical form (uncoupled modal equation)

M􀀊 q€ þ C􀀊 q_ þ K􀀊 q ¼ f􀀊ðtÞ ð3:61Þ

or

Miq€i þ Ciq_i þ Kiqi ¼ f􀀊iðtÞ for i ¼ 1; 2; …; n

Equation 3.61 can be written in the standard form for a damped simple oscillator:

q€i þ 2ziviq_i þv2i

qi ¼ f􀀊iðtÞ for i ¼ 1; 2; …; n ð3:62Þ

in which, the modal matrix is assumed M-normal (i.e., Mi ¼ 1). It can be concluded that a

proportionally damped system possesses real modal vectors that are identical to the modal vectors of its

undamped counterpart. The damped natural frequency, however, is smaller than the undamped natural

frequency and is given by

vdi ¼

ffiffiffiffiffiffiffiffi

1 2 z2i

q

vi ð3:63Þ

One way to guarantee proportional damping is to pick a damping matrix that satisfies

C ¼ cmM þ ckK ð3:64Þ

From modal transformation (see Equation 3.61), it follows that Equation 3.64 corresponds to

Modal damping constant Ci ¼ cmMi þ ckKi

or

2zivi ¼ cm þ ckv2i

Modal Analysis 3-33

© 2005 by Taylor & Francis Group, LLC

or

Modal damping ratio zi ¼

1

2

cm

vi þ ckvi

􀀏 􀀐

ð3:64aÞ

Equation 3.64, however, is not the only way to achieve real modes in a damped system (Equation 3.60).

The first term on the right-hand side (RHS) of Equation 3.64 is termed the inertial damping matrix. The

corresponding damping force on each lumped mass in the model will be proportional to the momentum.

This term may represent the energy loss associated with a momentum and is termed momentum

damping. Physically, this is incorporated by assigning a viscous damper between each DoF and its fixed

reference, with the damping constant proportional to the mass concentrated at that location.

The second term in Equation 3.64 is termed the stiffness damping matrix. The corresponding damping

force is proportional to the rate of change of the local deformation forces (stresses) in flexible structural

members and joints. It may be interpreted as a simplified model for structural damping. Physically, this

model is realized by assigning a viscous damper across every spring element in the model, with the

damping constant proportional to the stiffness. It is known that rate of change of stresses or rate of

change of strains will give rise to viscoelastic damping, which is associated with plasticity and

viscoelasticity. This type of damping is known as strain-rate damping.

Usually, structural damping is most appropriately modeled as being present across (lumped) stiffness

elements. Coulomb damping is modeled as acting between an inertia element and its fixed reference

point. These intuitive observations also support the damping model given by Equation 3.64. Some

terminology and properties of damped systems are summarized in Box 3.5.

Example 3.8

Let us examine the lumped-parameter damped model shown in Figure 3.12 to determine whether it has

real modes.

Note that if we modify the model as in Figure 3.13, the damping matrix will become proportional to

the stiffness matrix. This will be a case of proportional damping and the modified (damping) system will

possess real modes with modal vectors that are identical to those for the corresponding undamped

system. It should be verified that the equations of motion for this modified system (Figure 3.13) are

m 0

0 m

" #

y€ þ

2c 2c

2c 2c

" #

y_ þ

2k 2k

2k 2k

" #

y ¼

0

f ðtÞ

" #

Notice the similarity between the stiffness matrix and the damping matrix.

Another damped system that possesses classical modes is shown in Figure 3.14. In this model, the

damping matrix is proportional to the mass (inertia) matrix. It may be easily verified that the system

equations are

m 0

0 m

" #

y€ þ

c 0

0 c

" #

y_ þ

2k 2k

2k 2k

" #

y ¼

0

f ðtÞ

" #

Here, notice the similarity between the inertia matrix and the damping matrix.

Returning to the original system shown in Figure 3.12, the equations of motion can be written as

m 0

0 m

" #

y€ þ

c1 þ c2 2c2

2c2 c2

" #

y_ þ

2k 2k

2k 2k

" #

y ¼

0

f ðtÞ

" #

From these equations, it is not obvious whether this system possesses real modes. The undamped natural

frequencies are given by the roots of the characteristic equation

det

v2m 2 2k k

k v2m 2 2k

" #

¼ 0

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© 2005 by Taylor & Francis Group, LLC

The natural frequencies are v1 ¼

ffiffiffiffiffi

k=m p and v2 ¼

ffiffiffiffiffiffi

3k=m p : The corresponding mode shapes are given by

the nontrivial solution of

v2i

m 2 2k k

k v2i

m 2 2k

" #

c1

c2

" #

i¼

0

0

" #

for i ¼ 1; 2

Also, let us normalize the modal vectors by choosing the first element of each vector to be unity.

Then, by following the usual procedure of modal analysis, we will obtain the normalized modal vectors

c1 ¼

1

1

" #

and c2 ¼

1

21

" #

Box 3.5

TERMINOLOGY AND PROPERTIES

OF DAMPED SYSTEMS

Characteristic equation:

det½Ms2 þ Cs þ K􀀉 ¼ 0

Roots s are the eigenvalues li

ImðliÞ ¼ vdi ¼ damped natural frequencies

llil ¼ vi ¼ undamped natural frequencies

2ReðliÞ=vi ¼ zi ¼ modal damping ratios

ffiffiffiffiffiffiffiffiffiffi

1 2 2z2i

q

vi ¼ vri ¼ resonant frequencies

ffiffiffiffiffiffiffiffi

1 2 z2i

q

vi ¼ vdi

Existence of real modes:

* Condition for existence of modes that are identical to undamped modes

CTCC ¼ diagonal matrix

where

C ¼ modal matrix of undamped modes

* Another (equivalent) condition

M21CM21K ¼ M21KM21C ðProveÞ

i.e., M21C and M21K must commute

* Special case:

Proportional damping C ¼ cmM þ ckK

) Modal damping constant Ci ¼ cmMi þ ckKi

Modal damping ratio zi ¼

1

2

cm

vi þ ckvi

􀀏 􀀐

Modal Analysis 3-35

© 2005 by Taylor & Francis Group, LLC

The modal matrix is

C ¼ ½c1; c2􀀉 ¼

1 1

1 21

" #

Consequently, we obtain

CTMC ¼ M􀀊 ¼

2m 0

0 2m

" #

CTKC ¼ K􀀊 ¼

2k 0

0 6k

" #

CTCC ¼ C􀀊 ¼

c1 c1

c1 c1 þ 4c2

" #

and

CTfðtÞ ¼ f􀀊ðtÞ ¼

f ðtÞ

2f ðtÞ

" #

Notice that the transformed damping matrix C􀀊 is

not diagonal in general and, consequently, real

modes will not exist. This is to be expected

without the condition of proportional damping.

Proportional damping is realized in this model

if c1 ¼ 0: Then, C􀀊 will be diagonal and the

transformed systems equations will be uncoupled.

The uncoupled model equations are

2mq€1 þ 2kq1 ¼ f ðtÞ

2mq€2 þ 4c2q_2 þ 6kq2 ¼ 2f ðtÞ

The first mode is always undamped for this choice of damping model. This confirms that, in the case of

proportional damping, it is not generally possible to pick an arbitrary structure for the damping matrix.

For this reason, proportional damping is sometimes an analytical convenience rather than a strict

physical reality.

Modal analysis and response analysis of a system with general viscous damping can also be

accomplished using the state-space concepts. In this case, modal analysis is carried out in terms

of the eigenvalues and eigenvectors of the system matrix of a suitable state-variable model. These

“eigen results,” if complex, will occur in complex conjugate pairs and then the modes are said to be

complex. This approach is outlined next.

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