292
7 Forced Vibration Analysis of Multiple Degrees of Freedom System
7.9 Direct Integration for Determining Response of MDF
Systems
So far the forced vibration analysis of MDF systems has been made by the mode
superposition method, in that the unknown displacements are expressed in terms
of normal coordinates. However, the solution of equations of motion can also be
obtained directly. But a closed bound solution for such cases is a very difficult
proposition. The problem can be tackled in a more elegant manner, by using numerical
techniques. We have already looked into the principles of some of the numerical
methods applied to SDF systems. All these methods can be extended to MDF systems.
In this section, we limit ourselves only to Newmark’s method of direct integration.
As mentioned earlier, the method is very popular and can be applied to nonlinear
systems as well.
Time T, over which one is interested in knowing the response, is divided into n
intervals, each of duration t. Knowing the displacement, velocity and acceleration
at the ith time instant, one can obtain those values at (i + 1)th time instant.
Let {x}, { ˙
x} and { ¨
x} be the displacement, velocity and acceleration matrices of
the masses. Equations (4.20) and (4.21) are extended to MDF systems.
{ ˙
x} i+1 = { ˙
x} i +
(1 − δ){ ¨
x} i + δ{ ¨
x} i+1
t
(7.64a)
{x} i+1 = {x} i + { ˙
x} i t +
1
2
− α
{ ¨
x} i + α{ ¨
x} i+1
t
2
(7.64b)
Discussion has been made in Sect. 4.2.3 regarding the choice of the values of α
and δ. To evaluate {x} and { ˙
x} at (i +1)th time instant, the information of acceleration
at (i+1)th time instant is needed. It can be obtained from the equation of motion at
(i+1)th time instant.
[M]{ ¨
x} i+1 + [C]{ ˙
x} i+1 + [K ]{x} i+1 = {F} i+1
(7.65)
An iterative scheme may be adopted for the solution of necessary quantities at
(i+1)th time instant from three sets of simultaneous equations given by Eqs. (7.64a–
7.64b) to (7.65). They, however, can be explicitly determined on the basis of the
solution of equations. The algorithm for Newmark’s method as given by Bathe n [8]
is appended below.
(1) The initial conditions of the systems are always specified. In most cases, the
system starts at rest.
(2) Based on the material, geometry and damping properties of the structure, the
stiffness matrix [K], mass matrix [M] and damping matrix [C] may be formed.
(3) Based on the chosen values of α and δ and the time interval t, the integration
constants may be determined. They are
7 Forced Vibration Analysis of Multiple Degrees of Freedom System
7.9 Direct Integration for Determining Response of MDF
Systems
So far the forced vibration analysis of MDF systems has been made by the mode
superposition method, in that the unknown displacements are expressed in terms
of normal coordinates. However, the solution of equations of motion can also be
obtained directly. But a closed bound solution for such cases is a very difficult
proposition. The problem can be tackled in a more elegant manner, by using numerical
techniques. We have already looked into the principles of some of the numerical
methods applied to SDF systems. All these methods can be extended to MDF systems.
In this section, we limit ourselves only to Newmark’s method of direct integration.
As mentioned earlier, the method is very popular and can be applied to nonlinear
systems as well.
Time T, over which one is interested in knowing the response, is divided into n
intervals, each of duration t. Knowing the displacement, velocity and acceleration
at the ith time instant, one can obtain those values at (i + 1)th time instant.
Let {x}, { ˙
x} and { ¨
x} be the displacement, velocity and acceleration matrices of
the masses. Equations (4.20) and (4.21) are extended to MDF systems.
{ ˙
x} i+1 = { ˙
x} i +
(1 − δ){ ¨
x} i + δ{ ¨
x} i+1
t
(7.64a)
{x} i+1 = {x} i + { ˙
x} i t +
1
2
− α
{ ¨
x} i + α{ ¨
x} i+1
t
2
(7.64b)
Discussion has been made in Sect. 4.2.3 regarding the choice of the values of α
and δ. To evaluate {x} and { ˙
x} at (i +1)th time instant, the information of acceleration
at (i+1)th time instant is needed. It can be obtained from the equation of motion at
(i+1)th time instant.
[M]{ ¨
x} i+1 + [C]{ ˙
x} i+1 + [K ]{x} i+1 = {F} i+1
(7.65)
An iterative scheme may be adopted for the solution of necessary quantities at
(i+1)th time instant from three sets of simultaneous equations given by Eqs. (7.64a–
7.64b) to (7.65). They, however, can be explicitly determined on the basis of the
solution of equations. The algorithm for Newmark’s method as given by Bathe n [8]
is appended below.
(1) The initial conditions of the systems are always specified. In most cases, the
system starts at rest.
(2) Based on the material, geometry and damping properties of the structure, the
stiffness matrix [K], mass matrix [M] and damping matrix [C] may be formed.
(3) Based on the chosen values of α and δ and the time interval t, the integration
constants may be determined. They are