Page 812 - The Mechatronics Handbook

0066_Frame_C25 Page 11 Wednesday, January 9, 2002 7:05 PM

The response of a linear system to a sinusoidal input is referred to as the frequency response of the
system. An input signal, u(t) = U sin ωt, that is, a sine wave with amplitude U and frequency ω j , has a
Laplace transform

Uw
us() = ----------------.
2
s + w 2
Consequently, the response of the system is given by

Uw
ys() = Gs()----------------
2
s + w 2
and a partial fraction expansion of y(s) will result in terms that represent the (stable) transient behavior
of y(s) and the term associated to the sinusoidal input u(s). Elimination of the transient effects and
performing an inverse Laplace transform will yield a periodic time response y(t) of the same frequency
ω j given by

yt() = AUsin ( wt + f)

where the amplitude magnification A and the phase shift φ are given by

A = G s() s=iw , f = ∠ G s() s=iw (25.17)

By evaluating the transfer function G(s) along the imaginary axis s = iω, ω ≥ 0, the magnitude |G(iω)|
gives information on the relative amplification of the sinusoidal input, whereas the phase G∠ (iω) gives
information on the relative phase shift between input and output.
This analysis can be easily extended to discrete time systems by employing the relation between the
Laplace variable s and the z-transform variable in (25.11) to obtain the discrete time sinusoidal response

yk() = AUsin ( wk + f)

where the amplitude magnification A and the phase shift φ are given by

A = G z() i∆Tw , f = ∠ G z() (25.18)
z=e z=e i∆Tw

Due to the sampling nature of the discrete time system, the transfer function G(z) is now evaluated on
the unit circle

i∆Tw p
e ,0 ≤ w < -------
∆T
to attain information of the magnitude and phase shift of the sinusoidal response.
Plotting the frequency response of a dynamical system gives insight in the pole locations (resonance
modes) and zero locations of the dynamical system. As an example, the frequency response of the second
order system given in (25.16) has been depicted in Fig. 25.4. It can be seen from the figure that, as
expected, the second order system is less damped for smaller damping coefficients β and this results in
a larger amplitude response of the second order system at the resonance frequency ω n = 6 rad/s. It can
also be observed that the phase change at the resonance frequency becomes more abrupt for smaller
damping coefficients.

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