MATHEMATICAL REPRESENTATIONS
3.4 INPUT–OUTPUT MODELS AND FREQUENCY RESPONSE
3.4.1 Frequency Response
An input–output representation of an LTI dynamic system leads to the concept of the frequency response. Consider an LTI dynamic system modeled as a matrix of transfer
functions,
y(s)=H(s)w(s), (3.73)
where H(s) is obtained from equation (3.63):
H(s)=C(sI−A)−1B+D. (3.74)
The matrix H(s) is the matrix of input–output transfer functions as defined by the state variable representation of the system. Consider a harmonic input of the form
wj(t)=Wjsinωt, (3.75)
whereWj is an amplitude of the jth input andωis the frequency of the harmonic excitation. All other inputs are assumed to be equal to zero. The expression for the ith output can be written as
yi(s)=Hi j(s)wj(s). (3.76) Assuming that the system is asymptotically stable, the steady-state output to a har- monic excitation can be written as
yi(t→ ∞)= |Hi j(jω)|Wjsin(ωt+ Hi j(jω)). (3.77) The term Hi j(jω) is a complex-valued expression that can be written in real and imaginary terms as
Hi j(jω)=
Hi j(jω) +
Hi j(jω)
. (3.78)
The magnitude and phase can then be determined from
|Hi j(jω)| =
{Hi j(jω)}2+ {Hi j(jω)}2 (3.79)
Hi j(jω)=tan−1 {Hi j(jω)}
Hi j(jω). (3.80)
Equation (3.77) illustrates three important results in linear system theory:
1. The steady-state response of an asymptotically stable system oscillates at the same frequency as the frequency of the input.
2. The amplitude of the output is scaled by the magnitude of the transfer function evaluated ats= jω.
3. The phase of the output is shifted by the phase of the transfer function evaluated ats= jω.
These results emphasize the importance of the magnitude and phase of the input–
output frequency response. Evaluating the input–output transfer function ats= jω allows us to determine the amplitude and phase of the output relative to the input. For an asymptotically stable system, this explicitly determines the steady-state response of the system.
Example 3.3 Show that the frequency responseku(jω)/f(jω) in Example 3.2 can be written as a nondimensional function of the damping ratio and the frequency ratio =ω/ωn.
Solution The solution to Example 3.2 can be written in the Laplace domain as y(s)
f(s)
=
1/m s2+2ζωns+ω2n
1
f(s). (3.81)
The input–output transfer functionu(jω)/f(jω) is equal to the first row of the transfer function matrix:
u(s)
f(s) = 1/m
s2+2ζωns+ωn2
. (3.82)
The frequency response is obtained by substitutings= jωinto expression (3.82) and combining terms:
u(jω)
f(jω) = 1/m
ωn2−ω2+j2ζωnω. (3.83) Substituting the parameter=ω/ωninto the expression and dividing through byω2n
yields
u(jω) f(jω) = 1
mωn2
1
1−2+j2ζ. (3.84)
Recalling thatω2n =k/m, we can multiply both sides by the stiffness to produce the nondimensional expression
ku(jω)
f(jω) = 1
1−2+j2ζ. (3.85)
This result demonstrates that the right-hand side of the expression is a nondimensional function of the damping ratio and the frequency ratio.
Example 3.4 Plot the magnitude and phase of the frequency response ku(jω)/
f(jω) forζ =0.01,0.05,0.10,0.30,and 0.707. Discuss the nature of the response for <<1,≈1, and >>1.
Solution Plotting the frequency response requires that we obtain an expression for the magnitude and phase of the transfer function. This is obtained by first writing the transfer function in real and imaginary components as shown in equation (3.78).
Multiplying the transfer function by the complex conjugate of the denominator yields ku(jω)
f(jω) = 1−2 1−22
+4ζ22 −j 2ζ 1−22
+4ζ22. (3.86) The magnitude of the frequency response is obtained from equation (3.79) and the phase is obtained from equation (3.80):
ku(jω) f(jω)
= 1 1−22
+4ζ22
(3.87) ku(jω)
f(jω) = −2ζ
1−2. (3.88)
A plot of the frequency response magnitude and phase is shown in Figure 3.2.
The frequency response can be separated into three distinct regions. At excitation frequencies well below the resonance frequency of the system (1), the magnitude of the frequency response function is flat and the phase is approximately equal to 0◦.
10–2 10–1 100 101 102
10–4 10–3 10–2 10–1 100 101 102
Ω = ω/ωn
|kx(jω)/u(jω)|
10–2 10–1 100 101 102
–180 –160 –140 –120 –100 –80 –60 –40 –20 0
(a) (b)
Ω = ω/ωn
phase (degrees)
ζ = 0.01 ζ = 0.05 ζ = 0.10 ζ = 0.30 ζ = 0.707
ζ = 0.01 ζ = 0.05 ζ = 0.10 ζ = 0.30 ζ = 0.707
Figure 3.2 (a) Magnitude and (b) phase of a mass–spring–damper system as a function of nondimensional frequency and damping ratio.
In this frequency region,
ku(jω) f(jω)
≈1 1; (3.89)
therefore, the amplitudeu(jω)/f(jω)≈1/kwhen the excitation frequency is much less than the natural frequency. Furthermore, the input and output waveform are approximately in phase at these frequencies. Notice that the damping ratio, and hence the damping coefficient and mass, do not influence the amplitude of the response at frequencies well below resonance. The magnitude of the response becomes amplified when the excitation frequency approaches the natural frequency of the system. The magnitude and phase at this frequency are
ku(jωn) f(jωn)
= 1
2ζ (3.90)
ku(jωn)
f(jωn) → ±∞. (3.91) These results demonstrate that the damping ratio strongly influences the amplitude at resonance. Smaller values ofζ will produce a larger resonant amplitude.
The relationship between damping and the response at resonance is due to the com- peting physical processes within the system. At resonance, the force due to the spring stiffness and the force due to the inertial acceleration of the mass cancel one an- other out. Thus, the only force that resists motion at the resonance frequency is the force associated with the damping in the system. This result is general and highlights an important relationship between system response and damping. Energy dissipa- tion strongly influences the response of the system near resonance frequencies. The smaller the energy dissipation, the larger the resonant amplification.
As the excitation frequency becomes much larger than the resonant frequency (1), the magnitude and phase approach
ku(jω) f(jω)
→ 1
2 (3.92)
ku(jω)
f(jω) → −2ζ
−. (3.93)
The expression for the magnitude approaches a small number as the frequency ratio becomes large, indicating that the displacement amplitude will decrease as the exci- tation frequency becomes large with respect to the natural frequency. As shown in Figure 3.2b, the phase approaches−180◦as the frequency ratio becomes much larger than 1. (Note that the signs in the phase expression have been retained to emphasize quadrant associated with the inverse tangent.) The phase response illustrates that the input and output will be of opposite sign at frequencies much higher than the resonant frequency.
A second interpretation of the frequency response of an LTI system is obtained by considering an impulse input applied to the system at time zero. In this case, the input to the system is modeled as a delta function
wj(t)=δ(t). (3.94)
The Laplace transform of an impulse input at time zero is
L{δ(t)} =1. (3.95) Substituting this result into equation (3.76) yields the result
yi(s)=Hi j(s). (3.96)
The time response yi(t) can be determined from the inverse Laplace transform of equation (3.96),
yi(t)=L−1{yi(s)} =L−1{Hi j(s)}. (3.97) This result demonstrates that the impulse response of an LTI system is equivalent to the inverse Laplace transform of the input–output transfer function. This result is used continuously; therefore, we generally assign the symbol
hi j(t)=L−1{Hi j(s)} (3.98) to designate theimpulse responsebetween theith output and jth input.
The importance of this result can be understood by examining the inverse Laplace transform of equation (3.76). If we apply the inverse Laplace transform to equa- tion (3.76), we obtain the following expression for yi(t) through the convolution theorem:
yi(t)=L−1{yi(s)} =L−1{Hi j(s)wj(s)} = t
0
hi j(t−λ)wj(λ)dλ. (3.99) Note that equation (3.99) applies toanydeterministic functionwj(t). Thus, the re- sponse to any deterministic input can be determined by convolving the impulse re- sponse and the input function. As discussed above, the impulse response is related to the frequency response of the dynamic system; therefore, having an expression for the frequency response is equivalent to saying that the response to any deterministic input can be obtained from convolution. This is a very powerful result for LTI systems.
Comparing this result to equation (3.45), we see that the impulse response can be determined directly from the state variable representation as
hi j(t)=CieAtBj. (3.100)
This result provides a link between the state variable representation and the impulse response of an LTI system.