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measurement device in terms of ideal lumped elements of stiffness, mass, and damping is given in
Figure 3.3b and the corresponding free-body diagram in Figure 3.3c. Let y denote the position of the
small mass within the accelerometer and x denote the displacement of the body. Solving Newton’s
second law for the free body yields the second-order linear, ordinary differential equation
m
d
2
y
dt 2 þ c
dy
dt
þ ky ¼ c
dx
dt
þ kx
Since the displacement y is the pertinent output from the accelerometer due to displacement x, the
equation has been written such that all output terms, that is, all the y terms, are on the left side. All
other terms are to be considered as input signals and are placed on the right side. Comparing this to
the general form for a second-order equation (n ¼ 2; m ¼ 1) from Equation 3.1,
a
2 d
2
y
dt 2 þ a 1
dy
dt
þ a 0 y ¼ b 1
dx
dt
þ b 0 x
we can see that a 2 ¼ m, a 1 ¼ b 1 ¼ c, a 0 ¼ b 0 ¼ k, and that the forces developed due to the velocity and
displacement of the body become the inputs to the accelerometer. If we could anticipate the waveform of
x, for example, x(t) ¼ x 0 sin vt, we could solve for y(t), which gives the measurement system response.
Fortunately, many measurement systems can be modeled by zero-, first-, or second-order linear,
ordinary differential equations. More complex systems can usually be simplified to these lower
orders. Our intention here is to attempt to understand how systems behave and how such response is
closely related to the design features of a measurement system; it is not to simulate the exact system
behavior. The exact input–output relationship is found from calibration. But modeling guides us in
choosing specific instruments and measuring methods by predicting system response to signals, and
in determining the type, range, and specifics of calibration. Next, we examine several special cases
of Equation 3.1 that model the most important concepts of measurement system behavior.
3.3 SPECIAL CASES OF THE GENERAL SYSTEM MODEL
Zero-Order Systems
The simplest model of a measurement systems and one used with static signals is the zero-order
system model. This is represented by the zero-order differential equation:
a 0 y ¼ F t
ð Þ
Dividing through by a 0 gives
yðtÞ ¼ KF t
ð Þ
ð3:2Þ
where K ¼ 1/a 0 . K is called the static sensitivity or steady gain of the system. This system property was
introduced in Chapter 1 as the relation between the change in output associated with a change in static
input. In a zero-order model, the system output is considered to respond to the input signal instantaneously.
If an input signal of magnitude F(t) ¼ A were applied, the instrument would indicate KA, as modeled by
Equation 3.2. The scale of the measuring device would be calibrated to indicate A directly.
For real systems, the zero-order system concept is used to model the non–time-dependent
measurement system response to static inputs. In fact, the zero-order concept appropriately models
any system during a static calibration. When dynamic input signals are involved, a zero-order model
3.3 Special Cases of the General System Model 83
15:24:52 Page 83
measurement device in terms of ideal lumped elements of stiffness, mass, and damping is given in
Figure 3.3b and the corresponding free-body diagram in Figure 3.3c. Let y denote the position of the
small mass within the accelerometer and x denote the displacement of the body. Solving Newton’s
second law for the free body yields the second-order linear, ordinary differential equation
m
d
2
y
dt 2 þ c
dy
dt
þ ky ¼ c
dx
dt
þ kx
Since the displacement y is the pertinent output from the accelerometer due to displacement x, the
equation has been written such that all output terms, that is, all the y terms, are on the left side. All
other terms are to be considered as input signals and are placed on the right side. Comparing this to
the general form for a second-order equation (n ¼ 2; m ¼ 1) from Equation 3.1,
a
2 d
2
y
dt 2 þ a 1
dy
dt
þ a 0 y ¼ b 1
dx
dt
þ b 0 x
we can see that a 2 ¼ m, a 1 ¼ b 1 ¼ c, a 0 ¼ b 0 ¼ k, and that the forces developed due to the velocity and
displacement of the body become the inputs to the accelerometer. If we could anticipate the waveform of
x, for example, x(t) ¼ x 0 sin vt, we could solve for y(t), which gives the measurement system response.
Fortunately, many measurement systems can be modeled by zero-, first-, or second-order linear,
ordinary differential equations. More complex systems can usually be simplified to these lower
orders. Our intention here is to attempt to understand how systems behave and how such response is
closely related to the design features of a measurement system; it is not to simulate the exact system
behavior. The exact input–output relationship is found from calibration. But modeling guides us in
choosing specific instruments and measuring methods by predicting system response to signals, and
in determining the type, range, and specifics of calibration. Next, we examine several special cases
of Equation 3.1 that model the most important concepts of measurement system behavior.
3.3 SPECIAL CASES OF THE GENERAL SYSTEM MODEL
Zero-Order Systems
The simplest model of a measurement systems and one used with static signals is the zero-order
system model. This is represented by the zero-order differential equation:
a 0 y ¼ F t
ð Þ
Dividing through by a 0 gives
yðtÞ ¼ KF t
ð Þ
ð3:2Þ
where K ¼ 1/a 0 . K is called the static sensitivity or steady gain of the system. This system property was
introduced in Chapter 1 as the relation between the change in output associated with a change in static
input. In a zero-order model, the system output is considered to respond to the input signal instantaneously.
If an input signal of magnitude F(t) ¼ A were applied, the instrument would indicate KA, as modeled by
Equation 3.2. The scale of the measuring device would be calibrated to indicate A directly.
For real systems, the zero-order system concept is used to model the non–time-dependent
measurement system response to static inputs. In fact, the zero-order concept appropriately models
any system during a static calibration. When dynamic input signals are involved, a zero-order model
3.3 Special Cases of the General System Model 83