Steady-State Response

When subjected to a harmonic input, a linear system tends to respond at its natural frequencies as well as the frequency of the input.  If there is damping in the system, the natural response gradually decreases and the system approaches a steady-state condition in which it vibrates solely at the input frequency.  The steady-state response differs from the input only in amplitude and phase as shown in Figure 1.  Thus an amplitude ratio and a phase angle are the only two parameters required to describe the steady-state output of a linear system subjected to a harmonic input.

Figure 1. Amplitude and phase between input loading and steady-state response

Frequency Response Analysis

A frequency response analysis is a method used to compute steady-state response to harmonic inputs of constant amplitude and varying frequency. The results obtained from a frequency response analysis of a mechanical structure could include displacements, velocities, accelerations, forces, and stresses. The response at each frequency step is a complex number that is defined by a magnitude and a phase angle, or by real and imaginary components. Figure 2 gives an example of a frequency response plot that shows response magnitude and phase angle (relative to the input) versus frequency.

In the time domain, if the input F(t) is a sine wave,

then the steady-state response x(t) is given by

where Fo is the input amplitude, w is the input (and response) frequency, xo is the response amplitude, and q is the phase angle. The input F(t) and output x(t) may also be defined using the cosine function:

Figure 2. Frequency Response Plot

References

Craig, R. R. (1981), Structural Dynamics, John Wiley & Sons (New York).

Ogata, Katsuhiko (1998), System Dynamics (Third Edition), Prentice-Hill (New Jersey).

Thomson W.T., and Dahleh, M. D. (1997), Theory of Vibrations with Applications (Fifth Edition), Prentice Hall (New Jersey). 

Woods, R. L., Lawrence, K. L. (1997), Modeling and Simulation of Dynamic Systems, Prentice Hall (New Jersey).