E1C03 09/14/2010
15:24:53 Page 92
With m ¼ À0:194 ¼ À1=t, the time constant is calculated as t ¼ 5:15 seconds.
COMMENT If the experimental data were to deviate significantly from first-order behavior,
this would be a clue that either our assumptions do not fit the real-problem physics or that the test
conduct has control or execution problems.
Simple Periodic Function Input
Periodic signals are commonly encountered in engineering processes. Examples include vibrating
structures, vehicle suspension dynamics, biological circulations, and reciprocating pump flows.
When periodic inputs are applied to a first-order system, the input signal frequency has an important
influence on measuring system time response and affects the output signal. This behavior can be
studied effectively by applying a simple periodic waveform to the system. Consider the first-order
measuring system to which an input of the form of a simple periodic function, F(t) ¼ A sin vt, is
applied for t ! 0
þ :
t_ y þ y ¼ KA sin vt
with initial conditions y(0) ¼ y 0 . Note that v in ½rad=s ¼ 2pf with f in [Hz]. The general solution to
this differential equation yields the measurement system output signal, that is, the time response to
the applied input, y(t):
y t
ð Þ ¼ Ce
Àt=t
þ
KA
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ vt
ð Þ
2
q
sin vt À tan
À1
vt
À
Á
ð3:8Þ
where the value for C depends on the initial conditions.
So what has happened? The output signal, y(t), of Equation 3.8 consists of a transient and a
steady response. The first term on the right side is the transient response. As t increases, this term
decays to zero and no longer influences the output signal. Transient response is important only
during the initial period following the application of the new input. We already have information
about the system transient response from the step function study, so we focus our attention on the
second term, the steady response. This term persists for as long as the periodic input is maintained.
From Equation 3.8, we see that the frequency of the steady response term remains the same as the
input signal frequency, but note that the amplitude of the steady response depends on the value of the
applied frequency, v. Also, the phase angle of the periodic function has changed.
Equation 3.8 can be rewritten in a general form:
y t
ð Þ ¼ Ce
Àt=t
þ B v
ð Þsin vt þ F
½
B v
ð Þ ¼
KA
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ vt
ð Þ
2
q
F v
ð Þ ¼ Àtan
À1
vt
ð Þ
ð3:9Þ
where B(v) represents the amplitude of the steady response and the angle FðvÞ represents the phase
shift. A relative illustration between the input signal and the system output response is given in
Figure 3.11 for an arbitrary frequency and system time constant. From Equation 3.9, both B and f are
frequency dependent. Hence, the exact form of the output response depends on the value of the frequency
92 Chapter 3 Measurement System Behavior
15:24:53 Page 92
With m ¼ À0:194 ¼ À1=t, the time constant is calculated as t ¼ 5:15 seconds.
COMMENT If the experimental data were to deviate significantly from first-order behavior,
this would be a clue that either our assumptions do not fit the real-problem physics or that the test
conduct has control or execution problems.
Simple Periodic Function Input
Periodic signals are commonly encountered in engineering processes. Examples include vibrating
structures, vehicle suspension dynamics, biological circulations, and reciprocating pump flows.
When periodic inputs are applied to a first-order system, the input signal frequency has an important
influence on measuring system time response and affects the output signal. This behavior can be
studied effectively by applying a simple periodic waveform to the system. Consider the first-order
measuring system to which an input of the form of a simple periodic function, F(t) ¼ A sin vt, is
applied for t ! 0
þ :
t_ y þ y ¼ KA sin vt
with initial conditions y(0) ¼ y 0 . Note that v in ½rad=s ¼ 2pf with f in [Hz]. The general solution to
this differential equation yields the measurement system output signal, that is, the time response to
the applied input, y(t):
y t
ð Þ ¼ Ce
Àt=t
þ
KA
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ vt
ð Þ
2
q
sin vt À tan
À1
vt
À
Á
ð3:8Þ
where the value for C depends on the initial conditions.
So what has happened? The output signal, y(t), of Equation 3.8 consists of a transient and a
steady response. The first term on the right side is the transient response. As t increases, this term
decays to zero and no longer influences the output signal. Transient response is important only
during the initial period following the application of the new input. We already have information
about the system transient response from the step function study, so we focus our attention on the
second term, the steady response. This term persists for as long as the periodic input is maintained.
From Equation 3.8, we see that the frequency of the steady response term remains the same as the
input signal frequency, but note that the amplitude of the steady response depends on the value of the
applied frequency, v. Also, the phase angle of the periodic function has changed.
Equation 3.8 can be rewritten in a general form:
y t
ð Þ ¼ Ce
Àt=t
þ B v
ð Þsin vt þ F
½
B v
ð Þ ¼
KA
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ vt
ð Þ
2
q
F v
ð Þ ¼ Àtan
À1
vt
ð Þ
ð3:9Þ
where B(v) represents the amplitude of the steady response and the angle FðvÞ represents the phase
shift. A relative illustration between the input signal and the system output response is given in
Figure 3.11 for an arbitrary frequency and system time constant. From Equation 3.9, both B and f are
frequency dependent. Hence, the exact form of the output response depends on the value of the frequency
92 Chapter 3 Measurement System Behavior