6.6 Approximation Methods for the Fundamental Frequency

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The smallest natural frequency, known as the fundamental frequency, of a multi-DoF system is often of

greater interest than the high natural frequencies because its forced response in many cases is the largest.

One approach to this problem is to extend the Rayleigh method to matrix problems. We will see that the

Rayleigh frequency approaches the fundamental frequency from the high side.

6.6.1 Rayleigh Method

Let M and K be the mass and stiffness matrices, respectively, and x is the assumed displacement vector for

the amplitude of vibration. For harmonic motion, the maximum kinetic energy is

Tmax ¼ 1=2vxt Mx

and the maximum potential energy is

Umax ¼ 1=2xt Kx

Since the maximum kinetic energy equals the maximum potential energy, these two quantities are

equal. Equating these two and solving for v2 gives the Rayleigh quotient:

v2 ¼

xt Kx

xt Mx

It can be shown (Thomson and Dahleh, 1998) that this quotient approaches the lowest natural frequency

from above and it is somewhat insensitive to the choice of amplitudes.

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6.6.2 Dunkerley’s Formula

Dunkerley’s formula produces a lower bound for the fundamental frequency and can be used in

conjunction with the Rayleigh method to get a good approximation for the fundamental frequency.

Dunkerley’s formula is based on the characteristic equation for the flexibility coefficients. The flexibility

influence coefficient, aii, is defined as the displacement at i due to a unit force applied at j with all other

forces equal to zero. This concept is most easily understood through an example.

6.6.2.1 Computation of the Flexibility Matrix

The procedure for computing the flexibility matrix and in particular, the computation of the flexibility

matrix for the three spring – mass matrix systems shown in Figure 6.6, are discussed.

Example

First, one applies a unit force to mass 1 with no other forces present, i.e., f1 ¼ 1; f2 ¼ f3 ¼ 0: The

displacements are located in the first column of the flexibility matrix.

This gives

x1

x2

x3

2

664

3

775

¼

1=k1 0 0

1=k1 0 0

1=k1 0 0

2

664

3

775

1

0

0

2

664

3

775

In this case, springs k2 and k3 are not stretched and are displaced equally with mass 1. Now, a unit force is

applied to mass 2 and there are no other forces. This allows us to write the second column of the matrix to

get

x1

x2

x3

2

664

3

775

¼

0 1=k1 0

0 1=k1 þ 1=k2 0

0 1=k1 þ 1=k2 0

2

664

3

775

0

1

0

2

664

3

775

This time, the unit force is transmitted through k1 and k2: The spring k3 is not stretched. Finally, the force

is applied to mass 3 and there are no other forces present. This gives the third column of the matrix:

x1

x2

x3

2

664

3

775

¼

0 0 1=k1

0 0 1=k1 þ 1=k2

0 0 1=k1 þ 1=k2 þ 1=k3

2

664

3

775

1

0

0

2

664

3

775

Since the flexibility matrix is the sum of the three previous matrices, it is given by

x1

x2

x3

2

664

3

775

¼

1=k1 1=k1 1=k1

1=k1 1=k1 þ 1=k2 1=k1 þ 1=k2

1=k1 1=k1 þ 1=k2 1=k1 þ 1=k2 þ 1=k3

2

664

3

775

f1

f2

f3

2

664

3

775

An interesting feature of the flexibility matrix is that it is symmetric about the diagonal. For simplicity of

notation, let the ij element of the flexibility matrix be given by aijmj: Dunkerley’s formula is obtained

from the characteristic equation of the flexibility matrix, which is obtained by computing

the determinant of the following matrix:

a11m1 2 1=v2 a12 a13

a21 a22m2 2 1=v2 a23

a31 a32 a33m3 2 1=v2

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

¼ 0

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A third-degree equation in 1=v2 is obtained by expanding the determinant. One obtains the following

cubic equation:

ð1=v2Þ3 2 ða11m1 þ a22m2 þ a33m3Þð1=v2Þ2 þ · · · ¼ 0

A cubic equation has three roots that are denoted by ð1=v2i

Þ for i ¼ 1; 2; 3: This allows the cubic equation

to be factored:

ð1=v2 2 1=v21

Þð1=v2 2 1=v22

Þð1=v2 2 1=v23

Þ ¼ 0

The highest two powers of this equation are given by

ð1=v2Þ3 2 ð1=v21

þ 1=v22

þ 1=v23

Þð1=v2Þ2 þ · · · ¼ 0

The coefficient of the second highest power is equal to the sum of the roots of the characteristic equation,

which is also equal to the sum of the diagonal elements of the matrix A21: This relationship is not just

true for n ¼ 3 but is more generally true for n greater than or equal to 3. For the general n-DoF system

1=v21

þ 1=v22

þ · · ·1=v2

n ¼ a11m1 þ a22m2 þ · · · þ annmn

The fundamental frequency is the smallest natural frequency. Since v2; v3; … are larger than v1; the

reciprocal of these frequencies is smaller then the reciprocal of the fundamental frequency. An estimate

for the fundamental frequency is obtained by neglecting all of the higher modes in the left-hand side of

the above equation. This estimate gives a value for v1 that is smaller then the true value of the

fundamental frequency. Dunkerley’s formula is a lower bound for the fundamental frequency and it is

given by

1=v21

, a11m1 þ a22m2 þ · · · þ annmn

6.6.3 Summary of Approximations for the Fundamental Frequency

1. Rayleigh method

v2 ¼

xt Kx

xt Mx

2. Dunkerley’s formula 1=v21

, a11m1 þ a22m2 þ · · · þ annmn:

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