Eigenvalue Problem and Calculation of Natural Frequencies of a MDOF System

Eigenvalue Problem and Calculation of Natural Frequencies of a MDOF System

In the previous section, the general equations of motion of a general MDOF system was mentioned to be

2 Md 2 u/dt + C du/dt + K u = f(t) …………………….….(8.6) The free vibration condition for the dynamic motion of MDOF system is obtained by setting f(t) = 0; i.e.,

2 Md 2 u/dt + C du/dt + K u = 0 …………………….….(9.1) In order to obtain the natural frequency of the undamped system, if C is also set equal to zero, the equations of motion reduce to

2 Md 2 u/dt +Ku=0 …………………….….(9.2)

If the displacement vector can be chosen as the summation of a number (equal to the DOF) of variable

separable vectors u(t) = q r (t) r

………………………..(9.3) where q r (t) is a time-dependent scalar and r is a space-dependent vector.

With q(t) = A i nrt r e , or q r (t) = C 1r cos ( nr t) + C 2r sin ( nr t) .……………...………..(9.4)

2 Eq. (9.2) can be written as [

nr M + K] q r (t) r =0

nr M] r =0 …………………….….(9.5) Since the vector u is not zero, Eq. (9.5) turns into the following eigenvalue problem

[K 2

nr M =0 …………………….….(9.6) i.e., the determinant of the matrix (K 2

nr M) is zero.

Eq. (9.6) is satisfied for different values of the ‘natural frequency’ nr , which implies that there can be several natural frequencies of a MDOF system. In fact, the number of natural frequencies of the system is equal to the degrees of freedom of the system, i.e., size of the displacement vector. However, consideration of only the first few can adequately model the structural behavior of a dynamic system.

There are several ways to solve the eigenvalue problem of Eq. (9.6), the suitability of which depends on the size of the matrices and the number of eigenvalues required to represent the system accurately.

For each value of th nr , the vector r is called a modal vector for the r mode of vibration. Once a natural frequency is known, Eq. (9.5) can be solved for the corresponding vector r to within a multiplicative constant. The eigenvalue problem does not fix the absolute amplitude of the vectors r , only the shape of the vector is given by the relative values of the displacements.

Thus the vector r (i.e., the eigenvectors, also called the natural mode of vibration, normal mode, characteristic vector, etc.) physically represents the modal shape of the system corresponding to the natural frequency. The relative values of the displacements in the vector r indicate the shape that the structure would assume while undergoing free vibration at the relevant natural frequency.

The undamped natural frequencies and modal shapes calculated from the above procedure usually prove to be adequate in the subsequent dynamic analyses, since the damped natural frequencies are often quite similar to the damped natural frequencies for typical (undamped) systems, as mentioned in the discussion on SDOF systems.

However, the damped natural frequencies and modal shapes can also be calculated by the methods mentioned before. For that, q i nrt

r (t) = A r e will lead to the following equation

The solution of Eq. (9.7) provides the natural frequencies of the system, from which the natural modes can also be obtained.

Example 9.1 Calculate the natural frequencies and determine the natural modes of vibration of the 2-storied building system shown in Figs. 8.1 and 8.2, whose governing equations of motion are given by Eq. (8.5). Assume,

1 =k 2 = 25 k/ft, m 1 =m 2 = 1 k-sec /ft, c 1 =c 2 = 0 (i.e., the same stiffnesses and masses as used for the SDOF system before are used here for an undamped 2-DOF system).

Solution The mass and stiffness matrices of the system are given by M and K as follows

while the damping matrix C = 0

Thus the eigenvalue problem is given by

So that the natural frequencies can be obtained from the equation

2 2 2 4 (50 2 – nr ) (25 – nr ) – (–25) (–25) = 0 1250 – 75 nr + nr –625 = 0 nr = 9.55, 65.45 nr = 3.09, 8.09 rad/sec

f nr = nr /2 = 0.492, 1.288 cycle/sec; T nr = 1/f nr = 2.033, 0.777 sec

The two values of the natural frequency indicate the first and second natural frequency of the system. n1 = 3.09 and n2 = 8.09 rad/sec for this system. [Recall that the natural frequency n was equal to 5 rad/sec (i.e., f n = 0.796 cycle/sec) for the SDOF system in Example 2.1, which is greater than n1 but less than n2 ]

Once the natural frequencies are known, modal shapes can be determined from the eigenvalue equation. For the first natural frequency, the eigenvalue equations are

from both these equation, 1.618 1,1 – 2,1 =0

For the second natural frequency, the equations are

First Mode

Second Mode

Fig. 9.1: Modal Shapes of the system –25

from both these equations, – 1,2 – 1.618 2,2 =0

Thus, the first two modal shapes are as shown in Fig. 9.1.