Modal Analysis Lecture Notes

ME 461 Mechanical Vibrations

Modal Analysis Lecture Notes

by

Brian J. Olson

Contents

1 The EOM in Physical Coordinates 1

2 Modal Transformation of the EOM 12.1 The Generalized Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Orthonormalization of Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 The Expansion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Modal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Decoupling the EOM 33.1 Converting to Modal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Orthogonality of Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.2.1 Unnormalized Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2.2 Orthonormal (Normalized) Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.3 Decoupled Modal EOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3.1 Unnormalized Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3.2 Orthonormal (Normalized) Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Modal Solution 5

5 Transforming Initial Conditions 55.0.3 Unnormalized Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.0.4 Orthonormal (Normalized) Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6 Converting Back to Physical Coordinates 6

7 The Fundamental Steps 6


1 The EOM in Physical Coordinates

We begin with the matrix-vector differential equation describing undamped forced vibration of a lineardiscrete N degree of freedom (DOF) system, that is,

Mx+Kx = f(t), (1)

where M and K are the mass and stiffness matrices, respectively. The system solution and forcing vectorsare given by x(t) = (x1, x2, . . . , xN )

and f(t) = (f1, f2, . . . , fN )

. In general, the system is given some

initial displacements and velocities, which are described by the vectors x(0) = (x1(0), x2(0), . . . , xN (0))

and x(0) = (x1(0), x2(0), . . . , xN (0)).

2 Modal Transformation of the EOM

2.1 The Generalized Eigenvalue Problem

In order to perform a modal analysis, it is first necessary to solve the generalized eigenvalue problem, whichresults in the system natural frequencies i and mode shapes u

(i) (i = 1, 2, . . . , N). The eigenvalues, orsquare of each natural frequency, can be obtained from the N th order characteristic polynomial in 2

det(K 2M) = 0. (2)

Each corresponding eigenvector, or mode shape, can be obtained by employing the equation(K 2iM

)u(i) = 0 i = 1, 2, . . . , N (3)

2.2 Orthonormalization of Normal Modes

It is sometimes convenient to work with normalized, or orthonormal mode shapes. In particular, we wish toscale each mode shape such that

u(i)

Mu(i) = 1, (4)

where u(i) is the ith orthonormal mode. To accomplish this, we seek some constant i such that

u(i) = iu(i). (5)

By mapping Equation (5) into Equation (4), we obtain

iu(i)Miu

(i) = 2i u(i)Mu(i) = 1. (6)

Solving Equation (6) for the constant i, it follows that

i =

1

u(i)Mu(i)=

1

Mii, (7)

where Mii = u(i)Mu(i), a scalar, is the ith generalized mass. Hence, the ith orthonormal mode is given by

u(i) =

1

Miiu(i). (8)

Modal Analysis Lecture Notes 1 Revised August 14, 2004


2.3 The Expansion Theorem

It is noted that the component equations of motion represented by Equation (1) are, in general, coupled viathe mass matrix M (called dynamic coupling) and/or the stiffness matrix K (called static coupling). Thismeans that one cannot solve for the jth solution xj(t) without solving simultaneously for all of the othersolutions xi(t) (i = 1, 2, . . . , N 1). It is always possible, however, to find a set of modal coordinates qi(t)(also called principle or generalized coordinates) so that the equations of motion are uncoupled. This is aconsequence of the Expansion Theorem which can be stated as follows.

2.1 THEOREM. (Expansion Theorem). Let {v(1),v(2), . . . ,v(N)} be a set of N linearlyindependent vectors that are orthonormal with respect to some square matrix A. If x(t) is anarbitrary vector in N -dimensional space, it can written as the linear combination

x(t) = c1v(1) + c2v

(2) + . . .+ cNv(N) =

Ni=1

civ(i) = Vc, (9)

where c = [c1, c2, . . . , cN ]

is the N -vector of coordinates of x(t) with respect to the basisv(1),v(2), . . . ,v(N). The vector c can be obtained by premultiplying Equation (9) by VA andinvoking orthonormality relations. This results in

c = VAx. (10)

2.4 Modal Transformation

Since the system mode shapes are linearly independent and orthogonal with respect to the mass matrix, theExpansion Theorem guarantees that they can be used as a basis in N -dimensional space. To this end, thesolution vector x(t) of Equation (1) is expressed by a linear combination of the system mode shapes, eachmultiplied be a time-dependent generalized coordinate, say, qi(t). That is,

x(t) = u(1)q1(t) + u(2)q2(t) + . . .+ u

(N)qN (t) =

Ni=1

u(i)qi(t). (11)

If we define the N N modal matrix (sometimes called the fundamental matrix)

U =[u(1),u(2), . . . ,u(N)

], (12)

and let q(t) = (q1(t), q2(t), . . . , qN (t))(modal vector), then Equation (11) can be written in the matrix-

vector form1

x(t) =[u(1),u(2), . . . ,u(N)

]q1q2...qN

= Uq(t). (13)

Equation (13) is a coordinate transformation from physical coordinates x(t) to modal coordinates q(t).

1It should be noted that the equality on the RHS of Equation (13) is not a trivial step. Its proof requires invoking thedefinitions of a transpose of a matrix and matrix multiplication.



3 Decoupling the EOM

3.1 Converting to Modal Coordinates

Mapping Equation (13) into Equation (1), we arrive at the intermediate relationship

MUq(t) +KUq(t) = f(t). (14)

We now multiply Equation (14) through by U to obtain

UMUq(t) +UKUq(t) = Q(t), (15)

whereQ(t) = Uf(t) (16)

is the generalized force vector.

3.2 Orthogonality of Normal Modes

It can be shown that Equation (15) represents a set of N decoupled ordinary differential equations (ODEs)in modal space. This important result follows directly from orthogonality of mode shapes, which is discussednext in terms of both unnormalized and orthonormal (normalized) modes.

3.2.1 Unnormalized Modes

Given two unnormalized normal modes u(i) and u(j), their orthogonality properties can be described by therelationships

u(i)

Mu(j) =

{0 i 6= j

Mii i = j(17)

u(i)

Ku(j) =

{0 i 6= j

Kii = Mii2i i = j

(18)

where the constants Mii and Kii are called the generalized mass and stiffness coefficients, respectively.Equation (17) and Equation (18) indicate that the modal vectors u(i) and u(j) are orthogonal with respectto both the mass and stiffness matrices. Two important consequences of these orthogonality properties are

UMU =

u(1)

Mu(1) u(1)

Mu(2) . . . u(1)

Mu(N)

u(2)

Mu(1) u(2)

Mu(2) . . . u(2)

Mu(N)

......

. . ....

u(N)

Mu(1) u(N)

Mu(2) . . . u(N)

Mu(N)

=

M11 0 . . . 00 M22 . . . 0...

.... . .

...0 0 . . . MNN

= diag(Mii)

(19)and

UKU =

u(1)

Ku(1) u(1)

Ku(2) . . . u(1)

Ku(N)

u(2)

Ku(1) u(2)

Ku(2) . . . u(2)

Ku(N)

......

. . ....

u(N)

Ku(1) u(N)

Ku(2) . . . u(N)

Ku(N)

=

K11 0 . . . 00 K22 . . . 0...

.... . .

...0 0 . . . KNN

= diag(Kii) (20)

where U =[u(1),u(2), . . . ,u(N)

]is the modal matrix and N is the order of the square matrices M and K.



3.2.2 Orthonormal (Normalized) Modes

If the mode shapes are normalized according to Equation (8), then their orthogonality properties can bedescribed by the relationships

u(i)

Mu(j) =

{0 i 6= j

1 i = j(21)

u(i)

Ku(j) =

{0 i 6= j

2i i = j(22)

where i is the ith natural frequency. Equation (21) and Equation (22) indicate that the orthonormal modal

vectors u(i) and u(j) are orthogonal with respect to both the mass and stiffness matrices. Two importantconsequences of these orthogonality properties are

UMU =

u(1)

Mu(1) u(1)

Mu(2) . . . u(1)

Mu(N)

u(2)

Mu(1) u(2)

Mu(2) . . . u(2)

Mu(N)

......

. . ....

u(N)

Mu(1) u(N)

Mu(2) . . . u(N)

Mu(N)

=

1 0 . . . 00 1 . . . 0...

.... . .

...0 0 . . . 1

I = diag(1) (23)

and

UKU =

u(1)

Ku(1) u(1)

Ku(2) . . . u(1)

Ku(N)

u(2)

Ku(1) u(2)

Ku(2) . . . u(2)

Ku(N)

......

. . ....

u(N)

Ku(1) u(N)

Ku(2) . . . u(N)

Ku(N)

=

21 0 . . . 00 22 . . . 0...

.... . .

...0 0 . . . 2N

= diag(2i ) (24)

where U =[u(1), u(2), . . . , u(N)

]is the normalized modal matrix.

3.3 Decoupled Modal EOM

It is now shown that, due to orthogonality of normal modes, Equation (15) is actually decoupled.


In light of the structure of Equation (19) and Equation (20) the decoupled equations of motion follow fromEquation (15) and are given by

diag(Mii)q(t) + diag(Kii)q(t) = Q(t). (25)

Equation (25) represents a set of N decoupled, second-order, ordinary differential equations in the modalcoordinates qi(t). The i

th decoupled equation is given by

Miiqi(t) +Kiiqi(t) = Qi(t), (26)

where Mii (generalized mass) and Kii (generalized stiffness) are the ith diagonal elements of UMU and

UKU, respectively. (See Equation 19 and Equation 20.) The modal forcing, or generalized force Qi(t) isgiven by the ith element of Q(t) (see Equation 16), or it may be computed using2

Qi(t) = u(i)f(t). (27)

2Note that Qi(t) = u(i) f is a dot product so that Qi(t) is a time-dependent scalar.




If the mode shapes are normalized according to Equation (8) we let U = U in Equation (15) and Equa-tion (16). Then in light of the structure of Equation (23) and Equation (24) the decoupled equations ofmotion are given by

diag(1)q(t) + diag(2i )q(t) = Q(t), (28)

orIq(t) +q(t) = Q(t), (29)

where I is the identity matrix. Equation (29) represents a set of N decoupled, second-order, ordinarydifferential equations in the modal coordinates qi(t). The i

th decoupled equations is given by

qi(t) + 2i qi(t) = Qi(t), (30)

where, since the modes are normalized, Mii = 1, Kii = 2i , and

Qi(t) = u(i)f(t). (31)

4 Modal Solution

Equation (26) and Equation (30) are recognized to be of the same form as those describing the dynamicsof an undamped single DOF harmonic oscillator. We simply employ the methods of Chapters 2-4 to solvethem. In particular,

qi(t) = qtransienti + q

forcedi , (32)

The transient, or free vibration, is described by

qtransienti = qi(0) cosit+qi(0)

isinit. (33)

where i =

KiiMii

is the ith natural frequency of vibration. The nature of qforcedi depends on the nature of

the excitation (i.e., harmonic, periodic, arbitrary, etc.).

5 Transforming Initial Conditions

In order to specify the constants qi(0) and qi(0) in Equation (33) the initial conditions must be transformedfrom physical coordinates to modal coordinates.


From Equation (13) we obtain

x(0) = Uq(0) q(0) = U1x(0)

x(0) = Uq(0) q(0) = U1x(0)(34)



where

q(0) = (q1(0), q2(0), . . . , qN (0))

q(0) = (q1(0), q2(0), . . . , qN (0))

and

x(0) = (x1(0), x2(0), . . . , xN (0))

x(0) = (x1(0), x2(0), . . . , xN (0))

Note that this requires the determination of U1, which can be computationally expensive.


If the mode shapes are normalized, then

x(0) = Uq(0) (35)

x(0) = Uq(0) (36)

Premultiplying both sides of Equation (35) and Equation (36) by UM we obtain the relations

UMx(0) = UMUq(0) (37)

UMx(0) = UMUq(0) (38)

But in light of orthogonality with respect to the mass matrix (namely, Equation 23), it follows that

q(0) = UMx(0) (39)

q(0) = UMx(0) (40)

6 Converting Back to Physical Coordinates

Once the N modal solutions qi(t) have been obtained the physical solutions xi(t) are obtained by employingthe modal transformation given by Equation (13).

7 The Fundamental Steps

1. Formulate the equations of motion (EOM) and cast into the matrix-vector form

Mx+Kx = f(t).

2. Solve the generalized eigenvalue problem.

(a) Obtain the system natural frequencies i (i = 1, 2, . . . , N) using

det(K 2M) = 0.



(b) Obtain the corresponding system mode shapes u(i) (i = 1, 2, . . . , N) using

(K 2iM)u(i) = 0.

3. (Optional) Normalize the system mode shapes. The ith orthonormal mode is give by

u(i) = iu(i),

where

i =

1

u(i)Mu(i)=

1

Mii.

4. Assemble the modal matrix.

U =[u(1),u(2), . . .u(N)

](for unnormalized modes)

U =[u(1), u(2), . . . u(N)

](for orthonormal modes)

5. (Forced Vibration Problems) Transform the forcing vector f(t) to obtain the modal force vector.

Q(t) =

{Uf(t) (for unnormalized modes)

Uf(t) (for orthonormal modes)

6. (Free Vibration Problems) Transform the initial conditions.

q(0) =

{U1x(0) (for unnormalized modes)

UMx(0) (for orthonormal modes)

q(0) =

{U1x(0) (for unnormalized modes)

UMx(0) (for orthonormal modes)

7. Write the set of N decoupled, second-order ODEs in modal coordinates.

Miiqi(t) +Kiiqi(t) = Qi(t) (for unnormalized modes)

qi(t) + 2i qi(t) = Qi(t) (for orthonormal modes)

8. Obtain the solution in modal coordinates using the methods of Chapters 2-4. Each solution qi(t) in

the modal vector q(t) = (q1(t), q2(t), . . . , qN (t))is given by

qi(t) = qtransienti + q

forcedi ,

where

qtransienti = qi(0) cosit+qi(0)

isinit.

The nature of qforcedi depends on the nature of the excitation.

9. Convert back to physical coordinates using

x(t) =

{Uq(t) (for unnormalized modes)

Uq(t) (for orthonormal modes)


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Modal Analysis Lecture Notes