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Circuits
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Resistive displacement transducer (potentiometer) xM Motion to be measured xi M K B xo + Vo _ VB
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Figure 625 Seismic displacement transducer
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The equation of motion for the mass-spring-damper system may be obtained by summing all the forces acting on the mass M : Kxo + B dxo d 2 xi d 2 xo d 2 xM =M =M dt dt 2 dt 2 dt 2
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where we have noted that the motion of the mass is equal to the difference between the motion of the case and the motion of the mass relative to the case itself; that is, xM = xi xo If we assume that the motion of the mass is sinusoidal, we may use phasor analysis to obtain the frequency response of the transducer by de ning the phasor quantities Xi (j ) = |Xi |ej i and Xo (j ) = |Xo |ej o
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The assumption of a sinusoidal motion may be justi ed in light of the discussion of Fourier analysis in Section 61 If we then recall (from 4) that taking the derivative of a phasor corresponds to multiplying the phasor by j , we can rewrite the second-order differential equation as follows: M(j )2 Xo + B(j )Xo + KXo = M(j )2 Xi ( 2 M + j B + K)Xo = 2 MXi and we can write an expression for the frequency response: Xo (j ) 2 M = H (j ) = Xi (j ) 2 M + j B + K The frequency response of the transducer is plotted in Figure 626 for the component values M = 0005 kg and K = 1,000 N/m and for three values of B: B = 10 N s/m B = 2 N s/m and B = 1 N s/m (solid line) (dotted line) (dashed line)
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The transducer clearly displays a high-pass response, indicating that for a suf ciently high input signal frequency, the measured displacement
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6
Frequency Response and System Concepts
2 15
Amplitude
1 05 0 101 200 150 100 50 0 101
Radian frequency (logarithmic scale)
Phase, degrees
103 104 105 Radian frequency (logarithmic scale)
B=2 B = 10
Figure 626 Frequency response of seismic transducer
(proportional to the voltage Vo ) is equal to the input displacement, xi , which is the desired quantity Note how sensitive the frequency response of the transducer is to changes in damping: as B changes from 2 to 1, a sharp resonant peak appears around the frequency = 316 rad/s (approximately 50 Hz) As B increases to a value of 10, the amplitude response curve shifts to the right Thus, this transducer, with the preferred damping given by B = 2, would be capable of correctly measuring displacements at frequencies above a minimum value, about 1,000 rad/s (or 159 Hz) The choice of B = 2 as the preferred design may be explained by observing that, ideally, we would like to obtain a constant amplitude response at all frequencies The magnitude response that most closely approximates the ideal case in Figure 626 corresponds to B = 2 This concept is commonly applied to a variety of vibration measurements We now illustrate how a second-order electrical circuit will exhibit the same type of response as the seismic transducer Consider the circuit shown in Figure 627 The frequency response for the circuit may be obtained by using the principles developed in the preceding sections: Vo (j L)(j C) j L = (j ) = Vi R + 1/j C + j L j CR + 1 + (j L)(j C) = 2 L 2 L + j R + 1/C
Comparing this expression with the frequency response of the seismic
Part I
Circuits
+ ~ _
vi(t)
vo(t) _
Figure 627 Electrical circuit analog of the seismic transducer
transducer, Xo (j ) 2 M = H (j ) = Xi (j ) 2 M + j B + K we nd that there is a de nite resemblance between the two In fact, it is possible to draw an analogy between input and output motions and input and output voltages Note also that the mass, M, plays a role analogous to that of the inductance, L The damper, B, acts in analogy with the resistor, R; and the spring, K, is analogous to the inverse of the capacitance, C This analogy between the mechanical system and the electrical circuit derives simply from the fact that the equations describing the two systems have the same form Engineers often use such analogies to construct electrical models, or analogs, of physical systems For example, to study the behavior of a large mechanical system, it might be easier and less costly to start by modeling the mechanical system with an inexpensive electrical circuit and testing the model, rather than the full-scale mechanical system
Decibel (dB) or Bode Plots Frequency response plots are often displayed in the form of logarithmic plots, where the horizontal axis represents frequency on a logarithmic scale (to base 10) and the vertical axis represents the amplitude of the frequency response, in units of decibels (dB) In a dB plot, the ratio |Vout /Vin | is given in units of decibels (dB), where Vout Vin = 20 log10
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