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Torsional Vibration of Turbomachinery

# Chapter 8: Forced Response Analyses

## Overview

Forced response analysis generally falls into two main areas: steady-state response to sinusoidal stimuli and transient response to any defined applied torque-time history. In general, in either case different stimuli can be applied at all the nodes in the mathematical model.

For steady-state and transient forced response analysis, the matrix equations of motion can be approximately uncoupled using modal transformation, and the power of this approach is demonstrated in the case studies in Chap. 9 with several application examples. For steady-state response, the set of second-order differential equations are solved in closed form. For transient response, each equation is evaluated using numerical integration.

The general equation of motion for forced response is

 (8.1)

where [ M], [ C], and [ K] are square matrices of order n n ( n = number of nodes in model) and are the inertia, damping, and stiffness matrices, respectively. The other ? matrices in the equation are vectors of order n representing the rotational acceleration, velocity, and displacement, and [ T A] is the applied torque vector.

It is common practice for solving Eq. (8.1) to change variables from the actual response physical values [ ? 1... ? n] to the modal coordinates [ q 1... q m], where n is the number of nodes in the model and m is the number of modal coordinates to be employed.

The modal transformation is

 (8.2)

[ R

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