As seen in this chapter, the responses of linear, first, and second-order sys- tems to simple step and sinusoidal inputs are rather complex. Most ex- periments involve more than one instrument. Thus, the responses of most experimental measurement systems will be even more complex than the simple cases examined here.
When each instrument in a measurement system is linear, as described in Chapter 2, the total measurement system response can be calculated easily. For the overall system, [a] the static sensitivity is the product of all of the static sensitivities, [b] the magnitude ratio is the product of all of the magnitude ratios, and [c] the phase shift is the sum of all of the phase shifts. In the end, the most appropriate way to determine the dynamic response characteristics of a measurement system is through dynamic calibration. This can be accomplished by subjecting the system to a range of either step or sinusoidal inputs of amplitudes and frequencies that span the entire
FIGURE 4.8
The magnitude ratio of a second-order system responding to sinusoidal-input forcing.
range of those that would be encountered in an actual experiment. With this approach, the system’s dynamic errors can be quantified accurately.
Example Problem 4.5
Statement: A pressure transducer is connected through flexible tubing to a static pressure port on the surface of a cylinder that is mounted inside a wind tunnel. The structure of the flow local to the port is such that the static pressure, p(t), varies as
p(t) = 15sin2t,
in which t is time. Both the tubing and the pressure transducer behave as second-order systems. The natural frequencies of the transducer, ωn,trans, and the tubing, ωn,tube,
are 2000 rad/s and 4 rad/s, respectively. Their damping ratios are ζtrans= 0.7 and
ζtube= 0.2, respectively. Find the magnitude attenuation and phase lag of the pressure
signal, as determined from the output of the pressure transducer, and then write the expression for this signal.
Solution: Because this measurement system is linear, the system’s magnitude ratio, Ms(ω), is the product of the components’ magnitude ratios, and the phase lag, φs(ω), is
the sum of the components’ phase lags, where ω the circular frequency of the pressure. Thus,
Ms(ω) = Mtube(ω) × Mtrans(ω)
and
FIGURE 4.9
The phase shift of a second-order system responding to a sinusoidal-input forcing.
Also, ω/ωtube= 2/4 = 0.5 and ω/ωtrans= 2/2000 = 0.001. Application of Equa-
tions 4.62 and 4.64, noting ζtrans= 0.7 and ζtube= 0.2, yields φtube= −14.9◦, φtrans
= −0.1◦, Mtube= 1.29, and Mtrans= 1.00. Thus, φs(2) = −14.9◦+ −0.1◦= −15.0◦
and Ms(2) = (1.29)(1.00) = 1.29. The pressure signal, as determined from the out-
put of the transducer, is ps(t) = (15)(1.29)sin[2t − (15.0)(π/180)] = 19.4sin(2t − 0.26).
Thus, the magnitude of the pressure signal at the output of the measurement system will appear 129 % greater than the actual pressure signal and be delayed in time by 0.13 s [(0.26 s)/(2 rad/s)].
4.8
*Numerical Solution Methods
Differential equations governing a system’s response to input forcing may be nonlinear and not have exact solutions. Fortunately, methods are available to numerically integrate most ordinary differential equations and obtain the system response [24]. The basic solution approach is to reduce any higher- order ordinary differential equations to a system of coupled, first-order ordi- nary differential equations. Then, the first-order equations are solved using finite-difference methods. For example, the second-order ordinary differen- tial equation
d2y(t)
dt2 − cy(t) = d (4.65)
can be reduced to two first-order ordinary differential equations by using the substitution dy/dt = z(t), which yields the system of equations
dy(t)
dt = z(t) and
dz(t)
dt = cy(t) + d. (4.66)
Two initial conditions are needed to obtain a specific solution.
The numerical solution of a first-order ordinary differential equation can be obtained using various finite-difference methods [3]. The exact differen- tial, dy(t)/dt = f (y, t), is approximated by a finite difference. There are many ways to approximate dy(t)/dt. The choice depends upon the required accuracy and computation time. A straightforward finite-difference approx- imation for dy(t)/dt is the forward Euler expression
f (yn, tn) ≈yn+1− yn
∆t , (4.67)
where n and n + 1 denote the n-th and n + 1-th points. Equation 4.67 leads directly to
yn+1= yn+ ∆t f (yn, tn). (4.68)
The expression for f (y, t) is obtained from the governing first-order ordinary differential equation. An initial condition, y(0), also is specified. This permits the value of yn+1 to be computed for a fixed ∆t from Equation 4.68. This
algorithm is applied successively up to the desired final time.
Other methods can be used to determine an expression analogous to Equation 4.67. All these methods are easy to implement. The following more commonly used methods replace f (yn, tn) in Equation 4.68 by
f (yn+1, tn+1) (4.69)
for the backward Euler method, and
f (yn, tn) + f (yn+1, tn+1)
2 (4.70)
for the improved Euler method. The improved Euler method is more ac- curate than the forward and backward Euler methods. The fourth-order Runge-Kutta method replacement for Equation 4.67 is
k1+ 2k2+ 2k3+ k4
FIGURE 4.10
The response of the system dy(t)dt − 2y(t) = F (t) = 0.5 − t to forcing as determined by the MATLAB M-file odeint.m.
where k1 = f (yn, tn), k2 = f (yn+ ∆t 2 k1, tn+ ∆t 2 ), k3 = f (yn+ ∆t 2 k2, tn+ ∆t 2 ), and k4 = f (yn+ ∆t k3, tn+ ∆t). (4.72)
The fourth-order Runge-Kutta method is more accurate than any Euler method. It is the method used most frequently and is quite sufficient for most numerical integrations [3].
Example Problem 4.6
Statement: A first-order system is described by the equation dy(t)
dt − 2y(t) = F (t) = 0.5 − t,
with the initial condition y(0) = 1. Solve this differential equation numerically and analytically. Use four numerical methods: [1] forward Euler, [2] backward Euler, [3] improved Euler, and [4] fourth-order Runge-Kutta. Use a step size of 0.05 s. Plot all the results for comparison.
Solution: The MATLAB M-file odeint.m can be used for this purpose. The results are presented in Figure 4.10. The fourth-order Runge-Kutta and improved Euler solu- tions follow the exact solution closely. The backward Euler method underestimates the response. The forward Euler method overestimates the response.
4.9
Problem Topic Summary
Topic Review Problems Homework Problems
System Basics 1, 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 19, 21, 22, 23 First-Order 2, 8, 11, 12, 18, 20 1, 2, 3, 4, 5, 8, 11, 13, 17 Second-Order 9, 10 6, 7, 9, 10, 12, 14, 15, 16, 18, 19, 20 TABLE 4.2
Chapter 4 Problem Summary
4.10
Review Problems
1. Does a smaller diameter thermocouple or a larger diameter thermocou- ple have the large time constant?
2. The dynamic error in a temperature measurement using a thermocouple is 70 % at 3 s after an input step change in temperature. Determine the magnitude ratio of the thermocouple’s response at 1 s.
3. Determine the % dynamic error of a measurement system that has an output of 3 sin(200t) for an input of 4 sin(200t).
4. Determine the attenuation (reduction) in units of dB/decade for a mea- surement system that has an output of 3 sin(200t) for an input of 4 sin(200t) and an output of sin(2000t) for an input of 4 sin(2000t). 5. Is a strain gage in itself classified as a zero, first, second, or higher-order
system?
6. Determine the damping ratio of a RLC circuit with LC = 1 s2 that
has a magnitude ratio of 8 when subjected to a sine wave input with a frequency of 1 rad/s.
7. Determine the phase lag in degrees for a simple RC filter with RC = 5 s when its input signal has a frequency of 1/π Hz.
8. A first-order system is subjected to a step input of magnitude B. The time constant in terms of B equals (a) 0.707B, (b) 0.5B, (c) (1 − 1e)B,
or (d) B/e.
9. A second-order system with ζ = 0.5 and ωn = 2 rad/s is subjected to a
step input of magnitude B. The system’s time constant equals (a) 0.707 s, (b) 1.0 s, (c) (1 − 1e) s, or (d) not enough information.
10. A second-order system with ζ = 0.5 and ωn = 2 rad/s is subjected to
a sinusoidal input of magnitude Bsin(4t). The phase lag of the output signal in units of degrees is (a) −3, (b) −146, (c) −34, or (d) −180. 11. A first-order system is subjected to an input of Bsin(10t). The system’s
time constant is 1 s. The amplitude of the system’s output is approxi- mately (a) 0.707B, (b) 0.98B, (c) (1 −1
e)B, or (d) 0.1B.
12. A first-order system is subjected to an input of Bsin(10t). The system’s time constant is 1 s. The time lag of the system’s output is (a) −0.15 s, (b) −0.632 s, (c) −π s, or (d) −84.3 s.
13. What is the static sensitivity of the calibration curve F = 250W + 125 at W = 2?
14. The magnitude of the static sensitivity of the calibration curve V = 3 + 8√F at F = 16 is (a) 0, (b) 1, (c) 3, (d) 4, or (e) 8.
15. What is the order of each of the following systems? (a) Strain gage, (b) pressure transducer, (c) accelerometer, (d) RC circuit, (e) thermocouple, (f) pitot-static tube.
16. What is the magnitude ratio that corresponds to −6 dB?
17. What is the condition for an RLC circuit to be underdamped, critically damped, or overdamped?
18. A large thermocouple has a time constant of 10 s. It is subjected to a sinusoidal variation in temperature at a cyclic frequency of 1/(2π) Hz. The phase lag, in ◦, is approximately (a) −0.707, (b) −3, (c) −45, or
(d) −85.
19. What is the sensitivity of the linear calibration curve at E = 0.5 exp (10/T ) at (a) T = 283 K, (b) T = 300 K, and (c) T = 350 K. (d) What type of temperature sensor might result in such an exponential calibration curve?
20. Consider a first-order system where the frequency of the sinusoidal forc- ing function is 10 Hz and the system response lags by 90◦. What is the
Time (ms) Temperature ( C) 0 24.8 40 22.4 120 19.1 200 15.5 240 13.1 400 9.76 520 8.15 800 6.95 970 6.55 1100 6.15 1400 5.75 1800 5.30 2000 5.20 2200 5.00 3000 4.95 4000 4.95 5000 4.95 6000 4.95 7000 4.95 TABLE 4.3
Thermocouple Response Data
21. The signal 10sin(2πt) passes through a filter whose magnitude ratio is 0.8 and then through a linear amplifier. What must be the gain of the amplifier for the amplifier’s output signal to have an amplitude of 16?
22. An electronic manometer is calibrated using a fluid based manometer as the calibration standard. The resulting calibration curve fit is given by the equation V = 1.1897P − 0.0002, where the unit of P is inches of H20 and V is volts. The static sensitivity (in V/in. H20) is (a) 0.0002,
(b) 1.1897P2 - 0.0002P , (c) 1.1897, or (d) −0.0002.
23. Determine the static sensitivity at x = 2.00 for a calibration curve having y = 0.8 + 33.72x + 3.9086x2. Express the result with the correct number
4.11
Homework Problems
1. A first-order system has M (f = 200 Hz) = 0.707. Determine (a) its time constant (in milliseconds) and (b) its phase shift (in degrees).
2. A thermocouple held in room-temperature air is suddenly immersed into a beaker of cold water. Its temperature as a function of time is recorded. Determine the thermocouple’s time constant by plotting the data listed in Table 4.3, assuming that the thermocouple behaves as a first-order system. A more accurate method of determining the time constant is by performing a least-squares linear regression analysis (see Chapter 8) after transforming the temperatures into their appropriate nondimensional variables.
3. A first-order system with a time constant equal to 10 ms is subjected to a sinusoidal forcing with an input amplitude equal to 8.00 V. When the input forcing frequency equals 100 rad/s, the output amplitude is 5.66 V; when the input forcing frequency equals 1000 rad/s, the output amplitude is 0.80 V. Determine (a) the magnitude ratio for the 100 rad/s forcing case, (b) the roll-off slope (in units of dB/decade) for the ωτ = 1 to ωτ = 10 decade, and (c) the phase lag (in degrees) for the 100 rad/s forcing case.
4. The dynamic error in a temperature measurement using a thermometer is 70 % at 3 s after an input step change in temperature. Determine (a) the magnitude ratio at 3 s, (b) the thermometer’s time constant (in seconds), and (c) the magnitude ratio at 1 s.
5. A thermocouple is immersed in a liquid to monitor its temperature fluc- tuations. Assume the thermocouple acts as a first-order system. The temperature fluctuations (in degrees Celsius) vary in time as T (t) = 50 + 25 cos(4t). The output of the thermocouple transducer system (in V) is linearly proportional to temperature and has a static sensitivity of 2 mV/◦C. A step-input calibration of the system reveals that its rise
time is 4.6 s. Determine the system’s (a) time constant (in seconds), (b) output E(t) (in millivolts), and (c) time lag (in seconds) at ω = 0.2 rad/s.
6. A knowledgeable aerospace student selects a pressure transducer (with ωn = 6284 rad/s and ζ = 2.0) to investigate the pressure fluctuations
within a laminar separation bubble on the suction side of an airfoil. Assume that the transducer behaves as an over-damped second-order system. If the experiment requires that the transducer response has M (ω) ≥ 0.707 and |φ(ω)| ≤ 20◦, determine the maximum frequency
(in hertz) that the transducer can follow and accurately meet the two criteria.
7. A strain gage system is mounted on an airplane wing to measure wing oscillations and strain during wind gusts. The system is second order, having a 90 % rise time of 100 ms, a ringing frequency of 1200 Hz, and a damping ratio of 0.8. Determine (a) the dynamic error when subjected to a 1 Hz oscillation and (b) the time lag (in seconds).
8. In a planned experiment a thermocouple is to be exposed to a step change in temperature. The response characteristics of the thermocou- ple must be such that the thermocouple’s output reaches 98 % of the final temperature within 5 s. Assume that the thermocouple’s bead (its sensing element) is spherical with a density equal to 8000 kg/m3, a spe-
cific heat at constant volume equal to 380 J/(kg·K), and a convective heat transfer coefficient equal to 210 W/(m2·K). Determine the maxi-
mum diameter that the thermocouple can have and still meet the desired response characteristics.
9. Determine by calculation the damping ratio value of a second-order sys- tem that would be required to achieve a magnitude ratio of unity when the sinusoidal-input forcing frequency equals the natural frequency of the system.
10. The pressure tap on the surface of a heat exchanger tube is connected via flexible tubing to a pressure transducer. Both the tubing and the transducer behave as second-order systems. The natural frequencies are 30 rad/s for the tubing and 6280 rad/s for the transducer. The damping ratios are 0.45 for the tubing and 0.70 for the transducer. Determine the magnitude ratio and the phase lag for the system when subjected to a sinusoidal forcing having a 100 Hz frequency. What, if anything, is the problem in using this system for this application?
11. Determine the percent dynamic error in the temperature measured by a thermocouple having a 3 ms time constant when subjected to a tem- perature that varies sinusoidally in time with a frequency of 531 Hz. 12. The output of an under-damped second-order system with ζ = 0.1 sub-
jected to step-input forcing initially oscillates with a period equal to 1 s until the oscillation dissipates. The same system then is subjected to sinusoidal-input forcing with a frequency equal to 12.62 rad/s. Deter- mine the phase lag (in degrees) at this frequency.
13. A thermocouple is at room temperature (70 ◦F) and at equilibrium
when it is plunged into a water bath at a temperature of 170 ◦F. It
takes the thermocouple 1 s to read a temperature indication of 120 ◦F.
What is the time constant of the thermocouple-fluid system? This same thermocouple is used to measure a sinusoidally varying temperature. The variation in degrees Fahrenheit is given by the equation
FIGURE 4.11
Current-pulse RL circuit.
What temperature does the thermocouple indicate after steady state conditions are reached?
14. A pressure transducer that behaves as a second-order system is sup- posed to have a damping ratio of 0.7, but some of the damping fluid has leaked out, leaving an unknown damping ratio. When the transducer is subjected to a harmonic input of 1850 Hz, the phase angle between the input and the output is 45◦. The manufacturer states that the natural frequency of the transducer is 18 500 rad/s. (a) What is the dynamic error in the transducer output for a harmonic pressure signal of 1200 Hz? (b) If the transducer indicates a pressure amplitude of 50 psi, what is the true amplitude of the pressure?
15. The output of an under-damped second-order system with ζ = 0.1 sub- jected to step-input forcing initially oscillates with a period equal to 1 s until the oscillation dissipates. The same system then is subjected to sinusoidal-input forcing with a frequency equal to 12.62 rad/s. Deter- mine the phase lag (in degrees) at this frequency.
16. Consider the RL circuit shown in the Figure 4.11, where the source is the current pulse Is(t) = 6 [u(t) − u(t − 1)] A, R = 5 Ω, and L = 5 H.
What is the current response of the circuit, i(t)?
17. For an RC circuit (R = 2 Ω; C = 0.5 F) with step-input forcing from 0 V to 1 V, determine (a) the voltage of the circuit at 1 s, (b) the voltage of the circuit at 5 s, and (c) the dynamic error at 1 s.
18. For an RLC circuit (R = 2 Ω; C = 0.5 F; L = 0.5 H) with sinusoidal- input forcing of the form F (t) = 2 sin(2t), determine (a) the phase lag in degrees, (b) the phase lag in seconds, and (c) the magnitude ratio. 19. For an RLC circuit, (a) what are the mathematical relationships involv-
ing R, L, and C for the system to be under-damped, critically damped, or over-damped? (b) What is the equivalent time constant of this sys- tem?
FIGURE 4.12 Simple RL circuit.
20. Consider the simple RL circuit shown in Figure 4.12 in which R = 10 Ω and L = 5 H. (a) What is the governing equation for the current in this circuit? Is it first order or second order? (b) What is the time constant for this system? (c) If the voltage source has a sinusoidal input of 5sin(10t) V, what is the solution to the governing equation? What is the magnitude ratio? What is the phase lag (in seconds)? (d) Plot the current response versus time.
[1] Boyce, W.E. and R.C. Di Prima. 1997. Elementary Differential Equations and Boundary Value Problems. 6th ed. New York: John Wiley and Sons. [2] Press, W.H., Teukolsy, S.A., Vetterling, W.T. and B.P. Flannery. 1992.
Numerical Recipes. 2nd ed. New York: Cambridge University Press. [3] S. Nakamura. 1995. Numerical Analysis and Graphic Visualization with
MATLAB. New York: Prentice-Hall.
Probability
CONTENTS
5.1 Chapter Overview . . . 142