Balakumar Balachandran, Magrab E.B. Vibrations

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The constants A and B are determined from the expressions for the

displacement at X

and X

. At n  0, X

 0 and, therefore, from Eqs. (g),

A  0. From Eq. (c) and the fact that X

N1

 0, we ﬁnd that

(h)

Upon using Eq. (f) and Eqs. (g) with A  0, Eq. (h) becomes

(i)

After employing several trigonometric identities, Eq. (i) can be written as

(k)

We see that this equation is satisﬁed when

(l)

If we ignore the trivial value l  0, the values of b obtained from Eq. (l) and

Eq. (f) lead to

or, from Eq. (d), that

(m)

The corresponding mode shapes are determined by using Eq. (g) with

A  0 and Eq. (l); thus,

where B

is an arbitrary constant, the subscript l indicates the lth natural fre-

quency, and n indicates the displacement of the nth mass.

If we let N  4, then the four natural frequencies are

(n)

 2v

sin a

b 1.9021v

 2v

sin a

b 1.6180v

 2v

sin a

b 1.1756v

 2v

sin a

b 0.6190v

 B

sin a

nlp

N  1

n  1,2, . . ., N

l  1,2, . . ., N

 2v

sin a

21N  1 2

l  1, 2, . . ., N



 4sin

21N  1 2

l  1, 2, . . ., N

b 

N  1

l  0,1, . . ., N

sin1b1N  122 0

12  211  cos b 22sin1bN 2 sin1b 3N  142

12 

 X

N1

390 CHAPTER 7 Multiple Degree-of-Freedom Systems

and the corresponding matrix of modal vectors is

(o)

where the ﬁrst column corresponds to the ﬁrst mode and so on.

These numerical results also can be obtained from Eqs. (7.5a) and (7.5b).

In the present notation and with the assumptions stated above, these equations

become

Then, the eigenvalues can be determined from

Evaluation of this determinant

for the chosen number of masses and springs

yields the same results as those obtained from Eqs. (m) and the same nor-

malized results as given by Eq. (o).

EXAMPLE 7.19

Single degree-of-freedom system carrying a system of oscillators

Consider the multiple degree-of-freedom system shown in Figure 7.20. The

kinetic energy, potential energy, and the dissipation function are, respectively,

(a)

where x

is the absolute displacement of mass m

and is the absolute veloc-

ity of mass m

. Upon substituting expressions (a) into the Lagrange equations

D 1x

, x

, . . ., x

2



n2

 x

V1x

, x

, . . ., x

2



n2

 x

T1x

, x

,. . ., x

2



n2

det3K

4

40 0

3M 4 m3M

4 m D

1000

0100

0010

0001

3K 4 k3K

4 kD

2 100

1210

0 121

0012

£  D

0.5878 0.9511 0.9511 0.5878

0.9511 0.5878 0.5878 0.9511

0.9511 0.5878 0.5878 0.9511

0.5878 0.9511 0.9511 0.5878

7.3 Free Response Characteristics 391

The MATLAB function eig was used.

with q

 x

and Q

 0, n  1, 2, . . ., N, and performing the needed opera-

tions, we obtain

(b)

In order to determine the natural frequency equation, we assume that

(c)

Upon substituting Eqs. (c) into Eqs. (b), we obtain

(d)

We notice that from the second to Nth equations in Eqs. (d), we can solve for

, n  2, . . ., N in terms of X

. Thus,

(e)

Upon substituting Eqs. (e) into the ﬁrst of Eqs. (d) and solving for X

,we

obtain

(f)

where

(g)H

1j2

11  j2z

v

1  v



 j2z



n  2, 3, . . ., N

1 

 j2z



n 2

1j2 0



1jvc

 k

 m

 jvc

n  2, . . ., N

m

 jvc

 X

2 k

 X

2 0

m

 jvc

 X

2 k

 X

2 0

m

 jc



n2

v1X

 X

2 k



n2

 X

2 0

1t 2 X

jvt

n  1, 2, . . ., N

 c

 x

2 k

 x

2 0

 c

 x

2 k

 x

2 0

 c



n2

 x

2 k



n2

 x

2 0

392 CHAPTER 7 Multiple Degree-of-Freedom Systems

. . .

FIGURE 7.20

A single degree-of-freedom system carrying a system of oscillators.

and

Referring to Eq. (5.82), we see that 0H

( j)0, n  2, . . ., N, is the ampli-

tude response of each of the single degree-of-freedom systems excited by a

moving base.

In the absence of damping; that is, when c

 0, n  1, 2, . . ., N, we use

Eq. (f) to ﬁnd that the natural frequencies of the system are determined from

the solution to

(h)

In the special case where N  2, Eq. (h) becomes

(i)

where m

and v

are given by Eq.(7.41). We see that Eq. (i) is the same as

Eq. (7.45), when k

 0 in Eq. (7.45).

7.3.2 Undamped Systems: Properties of Mode Shapes

In this section, we examine the properties of the eigenvectors of undamped

systems described by Eq. (7.25), which, after setting l

v

,is

(7.55)

It is shown that the eigenvectors {X}

, and hence, the modal matrix , have

a special property called orthogonality, which will be described shortly. This

property follows from the properties of eigenvectors associated with real,

symmetric matrices. We shall take advantage of this orthogonality to solve the

coupled equations of the form given by Eq. (7.3) and other systems in the next

chapter. Another important property is that the eigenvectors {X}

form a lin-

early independent set. We address the orthogonality of the modes next, and

explain the linear independence of the modes at the end of this section.

Orthogonality of Modes

We see from Eq. (7.55) that

(7.56)

where the matrices and are symmetric and v

are the different eigen-

values and {X}

are the associated eigenvectors. We consider Eq. (7.56) for

two distinct frequencies v

and v

; thus, we have

(7.57)v

3M 45X6

 3K 45x6

3M 45X6

 3K 45X6

3K 43M 4

3M 45X6

 3K 45X6

for

j  1, 2, . . ., N

3£ 4

33K4 v

3M 445X6 506



 11  v

11  m

22

 v

 0

1 

c1 

n2

11  v



d 0







7.3 Free Response Characteristics 393

Pre-multiplying the ﬁrst equation of Eqs. (7.57) by {X}

and the second equa-

tion of Eqs. (7.57) by {X}

leads to

(7.58)

Taking the transpose

of the second equation of Eqs. (7.58), we ﬁnd that

(7.59)

since we have assumed that and are symmetric matrices. Then, Eqs.

(7.58) and (7.59) lead to

(7.60)

On subtracting one equation from the other in Eqs. (7.60), we arrive at

(7.61)

Since v

 v

, Eq. (7.61) implies that

(7.62)

Equation (7.62) provides a deﬁnition of orthogonality for the eigenvectors

(or eigenmodes or modes) of a system. Also, from Eqs. (7.60), it follows that

(7.63)

From Eqs. (7.62) and (7.63), we see that the modes are orthogonal with respect

to both the mass matrix and the stiffness matrix , and this important

property is one that we will make use of in the normal-mode approach for de-

termining responses of multiple degree-of-freedom systems in Section 8.2.

Equations (7.62) and (7.63) can be shown to be true for cases where the

eigenvalues are not distinct;

that is, when v

 v

. Thus, in general, we know

(7.64)

where v

and the corresponding {X}

are obtained from the solutions to,

respectively,

(7.65)v

3M 45X6

 3K 45X6

 0

for

j  1, 2, . . ., N

det 3v

3M 4 3K44 0

5X6

3K 45X6

 0

l  m

5X6

3M 45X6

 0

3K 43M 4

5X6

3K 45X6

 0

for

 v

5X6

3M 45X6

 0

for

 v

 v

25X6

3M 45X6

 0

5X6

3M 45X6

 5X6

3K 45X6

5X6

3M 45X6

 5X6

3K 45X6

3K 43M 4

5X6

3M 45X6

 5X6

3K 45X6

5X6

3M 4

5X6

 5X6

3K 4

5X6

35X6

3M 45X6

 35X6

3K 45X6

5X6

3M 45X6

 5X6

3K 45X6

5X6

3M 45X6

 5X6

3K 45X6

394 CHAPTER 7 Multiple Degree-of-Freedom Systems

From linear algebra, we know that See Appendix E.

D. C. Murdoch, Linear Algebra, John Wiley & Sons, NY, Chapter 6 (1970).

13A 43B 42

 3B 4

3A 4

Modal Mass, Modal Stiffness, and Modal Matrix

After pre-multiplying each side of Eq. (7.56) with , we arrive at

(7.66)

where the modal mass of the jth mode and the modal stiffness of the

jth mode are given by, respectively,

(7.67)

By using Eqs. (7.31) for the modal matrix [] and taking advantage of

Eqs. (7.62) and (7.67), we see that

(7.68a)

and in a similar manner, by making use of Eqs. (7.63) and (7.67), we obtain

the diagonal matrix

(7.68b)

Therefore, Eqs. (7.66) can be written in matrix form as

(7.69a)3v

4 3M

1

43M

4 3K

3£ 4

3K 43£ 4 D

0 K

oo∞o

T 3K

 D

0 M

oo∞o

T 3M

 D

5X6

3M 45X6

5X6

3M 45X6

5X6

3M 45X6

5X6

3M 45X6

5X6

3M 45X6

5X6

3M 45X6

oo∞

5X6

3M 45X6

5X6

3M 45X6

5X6

3M 45X6

 µ

5X 6

∂b3M45X6

3M 45X6

3£ 4

3M 43£ 4 µ

5X 6

5X6

∂3M4b5X6

5X6

 5X6

3K 45X6

j  1, 2, . . ., N

 5X6

3M 45X6

j  1, 2, . . ., N

 K

j  1, 2, . . ., N

5X6

3M 45X6

 5X6

3K 45X6

5X6

7.3 Free Response Characteristics 395

which means that

(7.69b)

in agreement with Eqs. (7.66).

Mass-Normalized Modes

Equation (7.68a) can also be used for normalizing the mode shapes in lieu of

Eqs. (7.29) and (7.30) or Eq. (7.34). If one normalizes the mode shapes so that

for the ﬁrst mode

(7.70a)

and for the second mode

(7.70b)

and so forth, then the mode shapes are said to be mass-normalized, and these

normalized mode shapes are also referred to as ortho-normal modes. In gen-

eral, for a system with N degrees of freedom, Eq. (7.62) together with Eqs.

(7.70a) and (7.70b) are expressed as

(7.71a)

where is the identity matrix. It follows from Eqs. (7.60) and (7.63) that

(7.71b)

We take advantage of the orthogonality of the modes in developing a solution

for the response of multiple degree-of-freedom systems in Section 8.2.

Linear Independence of Eigenvectors

We now consider the fact that the eigenvectors {X}

form a linearly inde-

pendent set. This means that for a system with N degrees of freedom with N

modes {X}

,any N-dimensional vector is constructed as a linear combination

of these eigenvectors. In physical terms, the implication is that any vibratory

3£ 4

3K 43£ 4 3v

3I 4

3£ 4

3M 43£ 4 3I 4

63M 4d

t 1

63M 4d

t 1

 D

0 K

oo∞o

0 v

oo∞o

T D

1/M

01/M

oo∞o

1/M

0 K

oo∞o

396 CHAPTER 7 Multiple Degree-of-Freedom Systems

motion of a system is viewed as a weighted sum of oscillations in the indi-

vidual modes. This observation, along with the orthogonality of the modes,

forms the basis of the normal-mode approach discussed in Section 8.2. Math-

ematically, linear dependence of the eigenvectors means that

(7.72)

for non-zero constants C

. The orthogonality properties given by Eqs. (7.64)

is used to show that Eq. (7.72) is true only if the C

are all zero, thus verify-

ing that the eigenvectors are not linearly dependent, and hence, they form a

linearly independent set.

EXAMPLE 7.20

Orthogonality of modes, modal masses, and modal stiffness

of a spring-mass system

We now illustrate how the orthogonality of the modes of a vibratory system

is examined, and how the modal masses and modal stiffness are computed.

We consider the two degree-of-freedom system treated in Example 7.14

where m

 1.2 kg, m

 2.7 kg, k

 10 N/m, k

 20 N/m, and k



15 N/m. Then, from Eq. (7.38a)

(a)

Since the mass and stiffness matrices are real and symmetric matrices,

the modes will be orthogonal to the mass and stiffness matrices. We shall now

show this through numerical calculations. To this end, we recall the modal

matrix from Eq. (f ) of Example 7.14, which is

(b)

To check the orthogonality of the mode shapes, we note that

(c)

To determine the modal masses, we ﬁnd that

(d)M

 5X6

3M 45X6



52.518 16

1.2 0

0 2.7

2.518

f 10.311

 5X6

3M 45X6



50.893 16

1.2 0

0 2.7

0.893

f 3.658

5X6

3K 45X6



2.518 1

30 20

20 35

0.893

f 0

5X6

3M 45X6



2.518 1

1.2 0

0 2.7

0.893

f 0

3£ 4 c

0.893 2.518

3£ 4

3M 4 c

1.2 0

0 2.7

d kg

and

3K 4 c

30 20

20 35

d N/m

5X6

 C

5X6



 C

5X6

 506

7.3 Free Response Characteristics 397

and to determine the modal stiffness associated with each mode, we compute

(e)

In addition, to compare these results with the results provided in Example 7.14

for the natural frequencies, we carry out the following. From Eq. (7.69b) and

the computed quantities above, we ﬁnd that

(f)

and

(g)

which agree with the values presented in Eq. (d) of Example 7.14.

7.3.3 Characteristics of Damped Systems

In Sections 7.3.1 and 7.3.2, we treated the natural frequencies and mode shapes

of undamped multi-degree-of-freedom systems with symmetric inertia and sym-

metric stiffness matrices. The eigenvalues and eigenvectors of such systems are

real-valued quantities with a physical interpretation. Here, we examine the

eigenvalues and eigenvectors for damped multiple degree-of-freedom systems.

We revisit Eq. (7.3), set the external forces to zero, replace q

by x

, and

arrive at

(7.73)

Since we have a linear system of ordinary differential equations with

constant coefﬁcients, we can assume a solution of the form given by Eq.

(7.23a). On substituting Eq. (7.23a) into Eq. (7.73), the result is

(7.74)

Eigenvalue Problem

Noting that Eq. (7.74) should be satisﬁed for all time t, we arrive at

(7.75)

Although the trivial solution

(7.76)

satisﬁes Eq. (7.75), we are looking for nontrivial solutions of this system,

which has N algebraic equations in the (N  1) unknowns l, X

, X

, . . ., X

5X6 506

3M 4 l13C4 3G42 3K4 3H 445X6 506

3M 4 l13C4 3G42 3K4 3H 445X6e

 506

3M 45x

6 33C4 3G 445x

6 33K4 3H 445x6 506

0 v

d c

2.519 0

0 5.623

0 v

d c

6.345 0

0 31.618

4 c

0 K

d c

23.21/3.658 0

0 326.01/10.311

 5X6

3K 45X6



2.51 1

30 20

20 35

2.518

f 326.01

 5X6

3K 45X6



50.893 1

30 20

20 35

0.893

f 23.209

398 CHAPTER 7 Multiple Degree-of-Freedom Systems

The special values of l for which we have a nontrivial solution are called

eigenvalues.

The eigenvalues of Eq. (7.75) are given by the roots of the characteristic

equation

(7.77)

which is a polynomial in l of order 2N. Since all of the matrices in Eq. (7.77)

are real-valued, we end up with 2N roots for this characteristic equation of the

N degree-of-freedom system. For mechanical systems known as lightly

damped systems, these roots are in the form of N complex conjugates pairs,

and they are given by

(7.78)

For systems that do not fall under the category of lightly damped systems, one

or more of the eigenvalues is real. The associated eigenvectors are determined

by solving the algebraic system

(7.79)

Damped Systems Without Gyroscopic and Circulatory Forces

For the cases where the gyroscopic and circulatory forces are absent, Eq.

(7.73) is of the form

(7.80)

Following the steps that were used to obtain Eqs. (7.77) and (7.79), the eigen-

values associated with Eq. (7.80) are given by the roots of

(7.81)

and they have the form of Eqs. (7.78). The corresponding eigenvectors are

determined from

(7.82)

Proportional Damping

For the case of proportional damping, the damping matrix is given by

(7.83)3C 4 a3M4 b3K4

3C 4

3M 4 l

3C 4 3K445X6

 506

k  1, 2, . . ., N

det 3l

3M 4 l3C4 3K 44 0

3M 45x

6 3C45x

6 3K45x6 506

3M 4 l

13C4 3G 42 3K4 3H 445X6

 506

k  1, 2, . . ., N

 d

 jv

k  1, 2 , . . ., N

det3l

3M 4 l13C4 3G42 3K4 3H 44 506

7.3 Free Response Characteristics 399

L. Meirovitch, 1980, ibid. and P. C. Müller and W. O. Schiehlen, Linear Vibrations: A Theo-

retical Treatment of Multi-Degree-of-Freedom Vibrating Systems, Martinus Nijhoff Publishers,

Dordrecht, The Netherlands, Chapters 4 and 6 (1985).

In Section 8.3, we shall see why the state-space form of Eq. (7.73) is convenient to use and to

interpret the eigenvalues given by Eqs. (7.78) and the eigenvectors determined from Eqs. (7.79).

In Sections 8.2 and 8.3, we shall use the normal-mode formulation and the state-space formu-

lation to interpret the eigenvalues given by Eqs. (7.78) and the eigenvectors determined from

Eqs. (7.82). While the state-space formulation is applicable to arbitrary forms of the damping ma-

trix , the normal mode approach is applicable only to systems with proportional damping.3C 4