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10.7: Chapter 10 Homework

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  • Problems 10.1–10.4 are frequency response problems (inspired by Myklestad, 1944) about standard 2nd order mass-damper-spring systems. In each case, the mass is excited by sinusoidal force \(f_{x}(t)=F \cos \omega t\), and it responds with steady-state sinusoidal displacement \(x(t)=X \cos (\omega t+\phi)\), where excitation frequency in Hz is \(f=\omega / 2 \pi\). Some partial answers are given to help you see if you are on the right track; you still must calculate these values as if the answers were not given. To derive maximum benefit from these problems, use the appropriate equations for calculations, but also figure out how the results are represented in the FRF magnitude and phase graphs, Figures 10.1.1 and 10.2.1. Be sure to use the appropriate value for \(g\), either 386.1 inch/s2 or 9.807 m/s2.

    1. (Myklestad, 1944, p. 61) Suppose that damping is negligibly small. The measured weight of the mass is \(W\) = 19.0 lb, and the spring stiffness is \(k\) = 13.0 lb/inch. With \(F\) = 5.00 lb at a certain frequency of excitation, the response amplitude is \(X\) = 0.750 inch. Calculate the dimensionless dynamic amplification \(X /(F / k)\). There are two different excitation frequencies, \(f_{1}<f_{2}\), that can produce this particular dynamic amplification; calculate both frequencies (in Hz), and the response phase \(\phi\) (in degrees) at both frequencies. (partial answers: \(\phi_{1}\) = 0°, \(f_{2}\) = 3.18 Hz)
    2. Suppose that damping is negligibly small. The measured weight of the mass is \(W\) = 8.50 kgf, and the spring stiffness is \(k\) = 2.50 kN/m. With \(F\) = 5.00 N at a certain frequency of excitation, the response amplitude is \(X\) = 20.0 mm. Calculate the dimensionless dynamic amplification \(X /(F / k)\). There are two different excitation frequencies, \(f_{1}<f_{2}\), that can produce this particular dynamic amplification; calculate both frequencies (in Hz), and the response phase \(\phi\) (in degrees) at both frequencies. (partial answer: \(f_2\) = 2.86 Hz)
    3. The mass weighs \(W\) = 1.26 kN, and the viscous damping and spring constants are, respectively, \(c\) = 2,400 N-s/m and \(k\) = 1.12 MN/m. Calculate the undamped natural frequency \(f_{n}\) in Hz and the viscous damping ratio \(\zeta\). Suppose that the response amplitude is \(X\) = 2.50 mm when the phase \(\phi\) = −90.0°; calculate the corresponding amplitude \(F\) of the excitation force. (partial answer: \(F\) = 560 N)
    4. (Myklestad, 1944, p. 113) The mass weighs \(W\) = 2.36 lb, and the spring and viscous damping constants are, respectively, \(k\) = 37.8 lb/inch, and \(c\) = 0.0169 lb-s/inch. The force amplitude is \(F\) = 0.785 lb. Calculate the critical damping constant \(c_{c}\), the damping ratio \(\zeta\), the maximum possible amplitude of response \(X_{r}\), and the amplitude \(X\) and phase \(\phi\) (in degrees) for excitation at 13.0 Hz. (Note: to calculate \(\phi\) correctly, you must account carefully for the quadrant of the fraction in the arctangent argument; if you just calculate the fraction and use the arctan function on your calculator, you will get the wrong answer; if you need some help, see the atan2 command in MATLAB). Is the small-\(\zeta\) approximation appropriate for this system? (partial answers: \(X_{r}\) = 0.591 inch, \(\phi\) = −155°)
    5. A frequency response measurement is conducted on a standard 2nd order \(m\)-\(c\)-\(k\) system with known stiffness constant \(k\), so that the input pseudo-static displacement is calculated from the measured input force, \(u(t)=f_{x}(t) / k\). A measured graph of the input and output displacements at one particular excitation frequency is shown below. Calculate from the graph, with as much accuracy as the data permits, the following quantities: the frequency (in Hz); the FRF magnitude ratio \(X / U\); the FRF phase \(\phi\) (in degrees). What are the natural frequency \(f_n\) (in Hz) and the damping ratio \(\zeta\) of this system?
      clipboard_e983dc0907a2da52f2ae3437677c4bf46.png
      Figure \(\PageIndex{1}\)
    6. The mechanical system of homework Problem 9.1 [with \(f_{x}(t)=0\)] has base excitation input \(x_i(t)\) and output \(x(t)\). The ODE of motion is \(m \ddot{x}+\left(c_{1}+c_{2}\right) \dot{x}+k x=c_{1} \dot{x}_{i}+k x_{i}\), which includes right-hand-side (RHS) dynamics, where \(m\) is the mass, \(c_1\) and \(c_2\) are viscous damping constants, and \(k\) is the stiffness constant.
      clipboard_e561523cb1abc87557cfbc473be8a5968.png
      Figure \(\PageIndex{2}\)
      1. Derive for this system the transfer function \(\operatorname{TF}(s)=\left.L[x(t)]\right|_{I C s=0} / L\left[x_{i}(t)\right]\).
      2. Use the fundamental result \(T F(j \omega)=F R F(\omega)=\left[X(\omega) / X_{i}\right] e^{j \phi(\omega)}\) and the result from part 10.6.1 to derive the relatively simple algebraic equation for real magnitude ratio \(X(\omega) / X_{i}\). This equation should be in terms of physical parameters \(m\), \(c_1\), \(c_2\), and \(k\), and excitation frequency \(\omega\). [NOTE: You should implement complex division by the polar method described in Section 2.1; do not use the rectangular method, because it will produce an awful algebraic mess!] If you wish, you may derive the associated relatively simple algebraic/trigonometric equation for the real phase angle \(\phi(\omega)\) (in radians), but that is not required.
      3. Let the system constants be \(m\) = 128.5 kg, \(c_1\) = 1,600 N-s/m, \(c_2\) = 800 N-s/m, and \(k\) = 1.12 MN/m. Calculate and plot the frequency response (magnitude ratio \(X / X_{i}\), and phase \(\phi\) in degrees) over the range of excitation frequencies 0-30 Hz. Use the over-under graphical format of Figure 10.2.1, except plot both graphs on linear (not log) scales. As in the script that produced Figure 10.2.1, let MATLAB do most of the work for you by starting with the complex frequency response function \(F R F(\omega)\) from part 10.6.2, then using MATLAB's capability for performing complex arithmetic with the abs and angle functions. If you need some help with the array multiplications required for this FRF, see the "frf=..." command lines in the script that produced Figure 10.2.1. Submit your MATLAB script along with your FRF graphs.
    7. The drawing represents a very simplified model of a land vehicle driving over a wavy (washboard) road with constant forward velocity \(V\). The vehicle mass is \(m\), and it is connected to the rolling wheel through the shock strut that has viscous damping constant \(c\) and stiffness constant \(k\). The deviation of the road surface from flatness is denoted as \(y_{s}(x)=Y_{s} \cos (2 \pi x / \lambda)\), in which \(x\) is horizontal distance traveled from a reference position, \(Y_s\) is the amplitude (wave height at the crests), and \(\lambda\) is the wave length. We make the following assumptions for this model: the wheel itself is rigid, it rolls without slipping, and it maintains contact with the road at all times due to the weight of mass \(m\); also, wave height \(Y_s\) is small relative to wheel radius, so that, in effect, the contact point is always directly beneath the wheel axle, which means that \(y_s\) plays the role of base excitation, and essentially is a function of time: \(y_{s}(x)=Y_{s} \cos \omega t\).
      1. Assume for the moment that \(y_{s}(t)\) is arbitrary base excitation. Draw a dynamic freebody diagram of mass \(m\), and derive from your DFBD the ODE of vertical dynamic motion \(y(t)\), measured relative to the static equilibrium position for a flat road surface. First, write the ODE in terms of constants \(m\), \(c\), and \(k\); second, use the standard definitions from Equations 7.1.3 and 9.1.6 to write the ODE in terms of natural frequency \(\omega_{n}\) and viscous damping ratio \(\zeta\).
      2. Use the second ODE of part 10.7.1 to derive the complex frequency response function for arbitrary input \(y_{s}(t)\) and output \(y(t)\) as s\(F R F(\omega)=\frac{1+j 2 \zeta \beta}{\left(1-\beta^{2}\right)+j 2 \zeta \beta} \equiv \frac{Y(\omega)}{Y_{s}} e^{j \phi(\omega)}\), in which \(\beta \equiv \omega / \omega_{n}\). Even though this system has right-hand-side dynamics that influence response, let us consider the condition \(\omega=\omega_{n}\). Derive equations for FRF magnitude ratio \(Y\left(\omega_{n}\right) / Y_{s}\) and phase \(\phi\left(\omega_{n}\right)\), in terms of \(\zeta\). (NOTE: To obtain the most manageable algebra, implement complex division by the polar method described in Section 2.1, not by the rectangular method.) If \(\zeta\) is small, e.g., \(\zeta=0.05\), how close are your values of \(Y\left(\omega_{n}\right) / Y_{s}\) and \(\phi\left(\omega_{n}\right)\) to the values at \(\omega=\omega_{n}\) that would prevail if there were no righthand-side dynamics?
      3. For the washboard road surface, we use the kinematic relation \(x=V t\) to express the base excitation as a sinusoidal function: \(y_{s}(t)=Y_{s} \cos (2 \pi V t / \lambda) \equiv Y_{s} \cos \omega t\), in which \(\omega \equiv 2 \pi V / \lambda\). Write the algebraic equation, in terms of \(m\), \(k\), and \(\lambda\), for the velocity \(V_{n}\) at which base excitation frequency equals the undamped natural frequency, \(\omega=\omega_{n}\).
    8. A particular device is known to be an LTI mass-damper-spring system. You (the engineer) are required to identify experimentally the mass \(m\), the viscous damping constant \(c\), and the stiffness constant \(k\). First, you apply a static force of 162 lb, and you observe that the mass deflects statically by 0.108 inch. Next, you run a stepped-sine frequency-response test, applying sinusoidal force onto the mass, with the frequency increasing in small increments from 8 to 20 Hz. You measure at each frequency the steady-state input force magnitude \(F\) (in lbs), the output translation magnitude \(X\) (in inches) and the phase of translation relative to force. The frequency response is plotted below. Use this information to calculate \(m\), \(c\), and \(k\) in consistent units and with as much accuracy as the data permits. Show all calculations.
      clipboard_ead10c7dc464f30882ac7ac254f131393.png
      Figure \(\PageIndex{3}\)
    9. A particular device is known to be an LTI mass-damper-spring system, but the mass \(m\), viscous damping constant \(c\), and stiffness constant \(k\) are unknown. It is required that the system parameters be identified experimentally with a combination of static and vibration testing. First, a static force of 52.0 N is applied, and the mass is measured to deflect statically by 1.13 mm. Next, sinusoidal force is applied to the mass by an electromagnetic shaker, with the frequency increased in small increments from 1 to 10 Hz The steady-state input force magnitude \(F\) and output translation magnitude \(X\) are measured at each frequency, as is the phase \(\phi\) of translation relative to force. The frequency-response translation-to-force magnitude ratio and phase are plotted at left. Use this information to calculate \(m\), \(c\), and \(k\) in consistent SI units and with as much accuracy as the data permits. Show all calculations.
      clipboard_e218e7b0037ec09a932c8a3cf147c9ef7.png
      Figure \(\PageIndex{4}\)
    10. You, the engineer, are asked to determine experimentally the electrical constants (inductance \(L\) and resistance \(R_L\)) of a small coil, based upon the simple model of inductor and resistor in series, to see if these constants match the design specifications. You elect to infer these constants from a frequency response test on an \(LRC\) circuit driven by a sinewave voltage generator, with the coil connected in series to a calibrated capacitor with capacitance \(C\) = 0.500 \(\mu\)F. The circuit is shown in the drawing (from the example in Section 9.2), and its ODE in standard 2nd order form is \(\ddot{e}_{o}+\frac{R}{L} \dot{e}_{o}+\frac{1}{L C} e_{o}=\frac{1}{L C} e_{i}(t)\). You run a stepped-sine frequency-response test, applying sinusoidal input voltage of constant magnitude \(E_{i}=1.00\) V, with the frequency increasing in small increments from 100 to 1,000 Hz. You measure at each frequency the steady-state output voltage magnitude \(E_{o}\), and the phase \(\phi\) of output voltage relative to input voltage. The frequency response is plotted on the next page. Use this information to calculate \(L\) and \(R_L\) in consistent units and with as much accuracy as the data permits. Show all calculations.
      clipboard_eb73d999118263784c2bff125e9021a67.png
      Figure \(\PageIndex{5}\)
      clipboard_e7a1941a9926ac48e021805f72b9ab93b.png
      Figure \(\PageIndex{6}\)
    11. Consider the \(RC\) band-pass filter described in Section 10.4, for which the frequency response function is \(F R F(\omega)=T F(j \omega)=\frac{j \omega \tau_{H}}{\left(j \omega \tau_{H}+1\right)\left(j \omega \tau_{L}+1\right)}\). The purpose of this problem is to demonstrate the practical function for which that filter is designed, by means of FRF graphs. Let \(\tau_{H}=\frac{1}{2 \pi \times 10}\) s and \(\tau_{L}=\frac{1}{2 \pi \times 500}\) s to make the high-pass and low-pass break frequencies be, respectively, 10 Hz and 500 Hz. Now, use MATLAB to calculate and plot the FRF magnitude ratio and phase (in degrees) over the frequency range 0.1 to 10,000 Hz. Use the graphical format described in homework Problem 4.3 (log-log for magnitude ratio, semilog for phase in degrees, magnitude ratio graph directly over phase graph). As in Problem 4.3, let MATLAB do most of the work for you by starting with the complex FRF representation, and then using MATLAB’s capability for performing complex arithmetic. Recall, in particular, that the MATLAB function abs calculates the absolute value (magnitude) of a complex number, and the MATLAB function angle calculates the angle in radians of a complex number.
    12. An idealized mechanical model for motion sensors of the seismic type is shown in the figure. Such a sensor is entirely mounted on a moving body (is said to be “structure-borne”), unlike, for example, the proximity displacement sensor shown in Figures 7.6.1 and 7.6.5. The mechanical part of the sensor is a mass-damper-spring system sealed within a sturdy case, which is attached firmly to the body whose motion is sensed. The stimulus to the sensor is base excitation: translation \(x_i(t)\) of the body and the case. The absolute translation of seismic mass \(m\) is \(x(t)\). A transducer within the case detects the relative translation \(z(t) \equiv x(t)-x_{i}(t)\) and generates an electrical signal1 proportional to \(z(t)\), which typically is displayed and/or recorded by a data-acquisition-and-processing system, and may also serve as an input to a control system.
      1. Sketch and label appropriate free-body diagrams, then use your FBDs to derive an ODE of motion for \(z(t)\) [not \(x(t)\)] in terms of constants \(m\), \(c\), and \(k\), and variable input quantity \(x_i(t)\) or its derivatives. Convert your ODE into the standard form Equation 10.2.1, except with \(z(t)\) [not \(x(t)\)] as the dependent variable: \(\ddot{z}+2 \zeta \omega_{n} \dot{z}+\omega_{n}^{2} z=\omega_{n}^{2} u(t)\). Write an explicit equation for standard input quantity \(u(t)\) in terms of the appropriate constants and \(x_i(t)\) or its derivatives.
      2. Evaluate the use of this seismic sensor as a translational accelerometer, a sensor of the body’s acceleration in the direction defined by \(x(t)\) and \(x_i(t)\). First, use the results of part 10.12.1 to show that the pseudo-static response is \(z_{p s}(t)=-(m / k) \ddot{x}_{i}(t)\). This means that if the quantity \(\ddot{z}+2 \zeta \omega_{n} \dot{z}\) is small (or zero) in comparison with \(\omega_{n}^{2} z\), then the seismic sensor acts as an accelerometer, since then \(z(t)\) and its corresponding electrical signal are directly proportional to \(\ddot{x}_{i}(t)\)2. This pseudo-static response is reflected clearly in the FRF graphs of Figure 10.2.1: If the frequencies of motion within base excitation \(x_i(t)\) are substantially below the natural frequency of the sensor (\(\beta \equiv \omega / \omega_{n}=1\)), then both the magnitude and phase of response follow closely those of excitation. An accelerometer is a seismic sensor intended to measure frequencies that are considerably lower than the sensor’s own natural frequency. Suppose you are designing an accelerometer that will measure with reasonable engineering accuracy body accelerations whose frequencies are no higher than 25% of the sensor’s natural frequency. What is the highest value of damping ratio \(\zeta\) (to three significant digits) that your sensor can have in order that phase error will be less than 5°, and what is the maximum magnitude error for that \(\zeta\)? Use Figure 10.2.1 to find an estimate, then make trial-and-error iterative calculations (preferably with MATLAB) to determine more precisely the required \(\zeta\) and magnitude ratio.
      3. Evaluate the use of this seismic sensor as a translational seismometer, a sensor of the body’s translation \(x_i(t)\). Now, due to the right-hand-side second derivative \(\ddot{x}_{i}\), your first ODE from part 10.12.1 is not in the standard form Equation 10.2.1, so it is necessary to derive new response equations. The appropriate transfer function is defined as \(T F(s)=\left.L[z(t)]\right|_{I C_{\mathrm{S}=0} \div} L\left[x_{i}(t)\right]_{I C s=0}\). Derive this \(T F(s)\), then use it to derive the corresponding FRF:\[F R F(\omega) \equiv \frac{Z(\omega)}{X_{i}} e^{j \phi(\omega)}=\frac{\beta^{2}}{\left(1-\beta^{2}\right)+j 2 \zeta \beta}, \text { with } \beta=\frac{\omega}{\omega_{n}}=\frac{\omega}{\sqrt{k / m}}, \text { and } \zeta=\frac{c}{2 m \omega_{n}}\]Suppose that a seismometer has damping ratio \(\zeta = 0.22\), high enough to dampen resonant response quickly. Plot over the range \(0.1 \leq \beta \leq 10\) the frequency response magnitude ratio \(Z(\omega) / X_{i}\) and, separately, phase angle \(\varphi(\omega)\) in degrees. (If necessary, see for guidance the description of producing Figure 10.2.1 with MATLAB.) From these graphs, and if you wish from low- and high-frequency asymptotes of \(Z(\omega) / X_{i}\), you should recognize that: A translational seismometer is a seismic sensor intended to measure frequencies that are considerably higher than the sensor’s own natural frequency. Suppose this seismometer is intended to measure with reasonable engineering accuracy body translations whose frequencies are no lower than five (5) times that of the sensor’s natural frequency. Calculate the greatest values of sensor error, percent of magnitude ratio and degrees of phase angle3? Describe physically the motion of the seismic mass in response to base-motion frequencies that are considerably higher than the seismometer’s natural frequency.
    13. The drawing at right is the prototypical idealized model for the subject of vibration isolation. Suppose that mass \(m\) in the drawing houses a reciprocating internal combustion engine with a single vertically oriented piston-cylinder pair, and that the damper-spring-base assembly attaches the mass to the floor. The engine exerts an oscillatory vertical force \(f(t)\) on mass \(m\), and the mass-damper-spring system transmits force through the rigid base to the stiff floor, which reacts the transmitted force with oscillatory vertical force \(f_{R}(t)\)4. For the benefit of the floor and its superstructure (building, vehicle, etc.) and for the comfort of the occupants, it is usually desirable to design the damper-spring support so that \(f_{R}(t)\) is as small as practical, in other words, to isolate the engine from the floor. In practice, the damper-spring support often is an elastomeric (rubber-like) padding.
      1. Use a dynamic free-body diagram (DFBD), if necessary, to write the ODE of dynamic motion \(y(t)\) relative to the static equilibrium position, in terms of excitation force \(f(t)\) and constants \(m\), \(c\), and \(k\). Take the Laplace transform of the ODE (assuming zero ICs) to obtain an equation for transform \(L[y(t)] \equiv Y(s)\), which equation will include as the input quantity transform \(L[y(t)] \equiv F(s)\). Next, write an equation for dynamic reaction force \(f_{R}(t)\) in terms of \(y(t)\) and its derivative and the appropriate constants, then take its Laplace transform to find an equation for transform \(\L\left[f_{R}(t)\right] \equiv F_{R}(s)). Next, combine the two transform equations appropriately to find the transfer function \(T F(s)=F_{R}(s) / F(s)\) relating the floor-reaction dynamic force to the excitation force. Finally, use the standard definitions of \(\omega_{n}\), \(\zeta\), and \(\beta\) to show that the corresponding complex frequency-response function, known as transmissibility, is \(T F(j \omega) \equiv F R F(\omega) \equiv \frac{F_{R}(\omega)}{F} e^{j \phi(\omega)}=-\frac{1+j 2 \zeta \beta}{\left(1-\beta^{2}\right)+j 2 \zeta \beta}\)5.
      2. Plot on a single graph the transmissibility magnitude ratio \(F_{R}(\omega) / F\) for the cases of zero damping, \(\zeta=0\), and the relatively high damping ratio \(\zeta=0.25\); plot these curves on linear (not logarithmic) scales over the frequency ratios \(0 \leq \beta \leq 4\), and display the magnitude ratio only over the range \(0 \leq F_{R}(\omega) / F \leq 2.5\). (If necessary, see for guidance the description of producing Figure 10.2.1 with MATLAB.) Explain how your graph shows (do not bother with any theory) that there is effective vibration isolation only if \(\beta>\sqrt{2}\). Suppose that you know mass \(m\) and excitation frequency \(\omega\); what is the range of spring constant \(k\) for which there is effective vibration isolation? Physically, should the springs be stiff or soft, and how stiff or soft? Does positive damping either increase or decrease the effectiveness of vibration isolation at all excitation frequencies, or is damping a mixed blessing, helpful in some range of frequencies, but not so much in others?
    14. The translational velocity \(v(t)\) of a point on an object is sometimes measured with use of a structure-borne accelerometer (see homework Problem 10.12). An accelerometer is a transducer that senses translational acceleration in one direction, e.g. \(a(t)=\ddot{x}(t)\), and converts the motion into an electrical voltage signal, \(e_{a}(t)\), which can be displayed, or recorded, or processed, or used in a control system. The transducer’s output signal is nominally \(e_{a}(t)=C_{e a} a(t)+n(t)+\varepsilon\): \(C_{e a}\) is the sensor’s calibration factor, with units such as volts per meter/s; \(n(t)\) is small, randomly varying electrical “noise”; and \(\mathcal{E}) is a small, constant offset voltage. Error voltage \(n(t)+\varepsilon\) is unrelated to the sensed motion; it is an unwanted but practically unavoidable by-product of a transducer’s circuitry. Acceleration is the rate of change of velocity, \(\dot{v}(t)=a(t)\), so the accelerometer’s output signal must be integrated to produce an electrical signal \(e_{v}(t)\) that is proportional to velocity. Thus, the basic ODE that an &(tv ) exact integrator (in analog circuitry or a digital algorithm) would solve is \(\dot{e}_{v}(t)=(1 / T) e_{a}(t)\), where \(T\) is a physical constant having dimensions of time (e.g., \(T=-R C\) for the op-amp-circuit integrator of homework Problem 5.6). However, in practice exact integration of an accelerometer’s signal is not desirable due to constant error \(\mathcal{E}), because \(\int e_{a}(t) d t=\int\left[C_{e a} a(t)+n(t)+\varepsilon\right] d t=\int C_{e a} a(t) d t+\int n(t) d t+\varepsilon t\). The term \(\int n(t) d t\) typically is negligible because random \(n(t)\) has average value of zero. On the other hand, the error term \(\mathcal{E} t\) is artificial drift that grows with time and distorts the exact integrator’s output signal, regardless of how small \(\mathcal{E}) is. Therefore, it is necessary to use approximate integration that is not vulnerable to artificial drift but is still sufficiently accurate for practical purposes. Perhaps the simplest approximate integrator, which we name the low-pass approximate integrator, is defined by the 1st order ODE \(\dot{e}_{v}+\Omega e_{v}= (1 / T) e_{a}(t)\); the static response to non-zero constant \(e_{a}\) is clearly non-zero, \(e_{v}=e_{a} /(\Omega T)\). This non-zero static response makes the low-pass approximate integrator unsuitable for applications that require the device’s output to have zero offset voltage. To avoid such an artificial constant static response, we have the band-pass approximate integrator defined by the 2nd order ODE \(\ddot{e}_{v}+\Omega \dot{e}_{v}+\Omega^{2} e_{v}=(1 / T) \dot{e}_{a}(t)\), which has right-hand-side dynamics.
      1. Using the definition of dimensionless excitation frequency \(\beta \equiv \omega / \Omega\), derive and show that the complex frequency-response functions of the exact integrator, the low-pass approximate integrator, and the band-pass approximate integrator can be written in the forms, respectively, \([\Omega T \times F R F(\omega)]_{\text {exact }}=1 /(j \beta)\), and \([\Omega T \times F R F(\omega)]_{\text {band-pass }}=j \beta /\left(1-\beta^{2}+j \beta\right)\).
      2. Plot on one graph the magnitudes versus \(\beta\) (at least over the range \(0.1 \leq \beta \leq 10\)) of \([\Omega T \times F R F(\omega)]_{\text {exact }}\), and \([\Omega T \times F R F(\omega)]_{\text {low-pass }}\), and \(\[\Omega T \times F R F(\omega)]_{\text {band-pass }}). Plot on another graph the phases versus \(\beta\) (over the same range) of these three frequency-response functions. Your graphs should clearly display features noted earlier such as the static responses of the low-pass and band-pass approximate integrators, and they should show that for \(\beta \equiv \omega / \Omega>6\) both approximate integrators have negligible magnitude error, relative to the exact integrator, and phase error less than 10°.
    15. Consider the reaction-mass actuator (RMA) drawn schematically below, which includes an \(m\)-\(c\)-\(k\) system that is augmented with an internal force generator, \(f_{i}(t)\). An external dynamic voltage signal \(e_{i}(t)\) commands the force generator to impose equal and opposite dynamic forces \(f_{i}(t)\) upon reaction mass \(m\) and the rigid interface, forces \(f_{i}(t)\) that are proportional to input \(e_{i}(t)\). The intended function of the RMA is to “react” against \(m\) in order to impose a dynamic force \(f_{a}(t)\) through the connecting rod onto the rigid wall drawn at left. However, due to the dynamics of the \(m\)-\(c\)-\(k\) system, the actuation force \(f_{a}(t)\) actually transmitted through the connecting rod is different than internal force \(f_{i}(t)\).
      clipboard_e262a2b93c07833c68c6392faeaab8aaf.png
      Figure \(\PageIndex{7}\): Reaction-Mass Actuator (RMA) in a calibration setup

      Your tasks in this problem are to derive the mathematical relationship between the output actuation force \(f_{a}(t)\) and the input voltage signal \(e_{i}(t)\), and then to calculate and plot frequency response that illustrates the character of RMA functioning.

      1. Derive the very simple ODE relating translation \(x(t)\) of reaction mass \(m\) to the actuation force \(f_{a}(t)\) shown acting onto the rigid interface. (The masses of the internal components—spring, damper, and force generator—are negligible relative to \(m\).) Then derive from that ODE the transfer function \(F_{a}(s) / X(s)\), where \(F_{a}(s)\) and \(X(s)\) are the Laplace transforms, respectively, of \(f_{a}(t)\) and \(x(t)\).
      2. The linear relationship between generated internal force \(f_{i}(t)\) and input voltage signal \(e_{i}(t)\) is \(f_{i}(t)=G e_{i}(t)\), where gain (calibration) constant \(G\) has units such as newtons/volt. Derive the ODE relating the translation \(x(t)\) of reaction mass m to the internal force \(f_{i}(t)\) shown acting onto \(m\), and then to input signal \(e_{i}(t)\). Then derive from that ODE the transfer function \(X(s) /E_{i}(s)), where \(E_{i}(s)\) is the Laplace transform of \(e_{i}(t)\).
      3. Multiply the transfer functions of parts 10.15.1 and 10.15.2 in order to show that the required transfer function relating the output actuation force \(f_{a}(t)\) to the input voltage signal \(e_{i}(t)\) is \(F_{a}(s) E_{i}(s)=G s^{2} /\left(s^{2}+2 \zeta \omega_{n} s+\omega_{n}^{2}\right)\), where, as usual, \(\omega_{n}^{2}=k / m\) and \(\zeta=c /\left(2 m \omega_{n}\right)\).
      4. During the 1990s, a team of aerospace companies designed a specialized RMA and fabricated several units of the device for use as vibration-control actuators in experiments on a test bed that simulated a space-satellite laser-beam-director structure [Dettmer, 1995, which uses another common name, proof-mass actuator (PMA) for the device]6. Imagine that these RMAs are restored for a new project and that you, as an instrumentation engineer, are assigned to perform system identification and calibration tests on one of them. First, you run free-vibration tests with the force generator turned off, \(f_{i}(t)=0\). You excite vibration by displacing reaction mass \(m\) from \(x = 0\), then releasing it (twang testing, see homework Problem 9.4). You measure the frequency of the resulting free vibration of \(m\) to be 5.00 Hz, with very light inherent damping that reduces the amplitude of vibration in half in exactly 9 full cycles [in Equation 9.5.5, \(r_{1 / 2}= 9.00\)]. The RMA has a flange allowing you to bolt a calibrated mass of 3.00 kg firmly to reaction mass \(m\); so you run a second twang test, from which you measure the frequency of free vibration of reaction mass plus added mass, \(m\) + 3.00 kg, to be 4.23 Hz (see homework Problem 7.9). Next, you remove the 3.00-kg added mass and turn on the force actuator. In static testing, you measure a linear relationship between applied voltage \(e_{i}\) and translation \(x\), with maximum voltage \(e_{i}=+10.0) V moving \(m\) the distance \(x=-6.00\) mm (and, conversely, −10.0 V producing +6.00 mm). Use your measured data to infer values for the components \(m\), \(c\), \(k\), and \(G\) of the RMA system (Partial answer: \(G\) = 4.47 N/V). Finally, calculate the frequency-response function corresponding to transfer function \(F_{a}(s) /E_{i}(s)\) of part 10.15.3, and plot it in the format of Figure 10.2.1 [log(magnitude ratio) vs. log(frequency in Hz), linear(phase in degrees) vs. log(frequency in Hz)] over the excitation-frequency range 1-100 Hz. These graphs should indicate that the RMA produces output actuation force \(f_{a}(t)\) nicely representative of input voltage signal \(e_i(t)\) for frequencies above about twice the RMA’s natural frequency, but that \(f_{a}(t)\) is not at all representative of \(e_i(t)\) for frequencies near and below the RMA’s natural frequency.
    16. A distributed-parameter shear-building structural model used in an instructional laboratory is shown in the drawing and the photograph, and is described in detail in
      clipboard_ed1d6ed8c310c7fb7edc8f26e6a153580.png
      Figure \(\PageIndex{8}\): Side and Front Views

      homework Problem 7.10. Measurements by students give the effective vibrating mass as \(m_{E}= 0.00473\) lb-s2/inch, and the lateral stiffness due to the two \(L =\) 12-inch aluminum beams as \(k_{E}= 43.6\) lb/inch. The inherent structural and aeroacoustic damping of the vibrating beams and mass is very, very small, but a supplementary dashpot (near top left in the photo) attached between the mass and ground increases the total measured effective viscous damping constant to \(c_{E}=0.0309\) lb-s/inch

      clipboard_eb94e0084d727206e5e375c7f20e290e8.png
      Figure \(\PageIndex{9}\)
      1. Calculate the undamped natural frequency \(f_{n}\) in Hz and the damping ratio \(\zeta\). (Answers: \(f_{n}=15.3\) Hz, \(\zeta=0.0340\))
      2. A small electromagnetic shaker7 is attached to the right-hand side of the mass, and it produces the time-varying force \(f_{x}(t)\) labeled on the drawing. Any real force actuator has a maximum stroke, the range of motion of the device’s moving parts, over which:
        1. the output force \(f_{x}(t)\) is proportional to the input electrical command signal, and/or
        2. the moving parts of the device can deflect without damaging themselves or other parts within the device.
        It is important to recognize that the moving parts of the shaker are attached directly to the structure, so they experience the same deflection as the attachment point on the structure. Hence, the structural vibration can actually cause the maximum shaker stroke to be exceeded, especially at or near resonance. Suppose you want to run an experiment to measure an FRF of the shear-building model, in a stepped-sine sweep of excitation frequency \(f\) in Hz from \(0.5 f_{n}\) to \(1.5 f_{n}\). The shaker will drive the mass with sinusoidal force \(f_{x}(t)=F \cos (2 \pi f t)\), and the mass will respond with steady-state sinusoidal translation \(x(t)=X \cos (2 \pi f t+\varphi)\). Suppose the shaker in this case has the maximum stroke of \(\pm\)0.05 inch and maximum force magnitude \(F_{max}\) = 2 lb (typical values for a small commercial off-the-shelf shaker). If you were to run the stepped-sine sweep so as to measure directly the mechanical admittance (also known as dynamic flexibility) by holding the force magnitude constant at 80% of its capability, \(F = 0.8F_{max}\), what are the theoretically predicted magnitudes \(X\) of structural (and shaker moving parts) translation at the frequencies \(0.5 f_n\), \(1.0 f_n\), and \(1.5 f_n\)? (Partial answer: \(X\left(f_{n}\right)=0.539\) inch) Why would this dynamic-flexibility approach not work in real life?
      3. Suppose you run a stepped-sine sweep, with the same shaker properties as defined in part 10.16.2, in such a way as to measure directly the dynamic stiffness, instead of the dynamic flexibility. For example, imagine that at each discrete excitation frequency in the sweep you adjust the shaker force magnitude so that the response magnitude equals 80%, \(X = 0.04\) inch, of the maximum shaker stroke. Calculate the theoretically predicted magnitudes \(F\) of required shaker force at the excitation frequencies \(f = 0\) (the static condition), \(0.5 f_n\), \(1.0 f_n\), and \(1.5 f_n\). (Partial answer: \(F\left(f_{n}\right)=0.119\) lb) For what (if any) range of excitation frequencies would this dynamic-stiffness approach work in real life?
      4. Calculate and plot the theoretically predicted dynamic stiffness (magnitude and phase) of this structural system for excitation frequency over the range \(0 \leq f \leq 1.5 f_{n}\).
    17. Evaluate Equation 10.6.9 numerically (using MATLAB or your choice of software) to calculate and plot the time response from rest ICs to a suddenly applied sinusoidal (SAS) input into the electronic analog computer circuit at right. The ODE relating output voltage \(e_o(t)\) to input voltage \(e_i(t)\) derived in homework Problem 9.17, is, in standard 2nd order form, \(\ddot{e}_{o}+\frac{1}{R_{c} C} \dot{e}_{o}+\left(\frac{1}{R_{b} C}\right)^{2} e_{o}=\left(\frac{1}{R_{b} C}\right)^{2} \frac{R_{b}}{R_{a}} e_{i}(t)\). The input voltage is zero for time \(t < 0\) and \(e_{i}(t)=E_{i} \sin (2 \pi f t)\) for \(t \geq 0\), in which \(E_{i}=3.20\) V and \(f = 14.2\) Hz. The circuit component values are: \(R_{a}=R_{d}=50.0 \\ k\Omega\), \(R_{c}=500\\ k\Omega\), and \(C=2.04 \mu\)F. Calculate and plot both excitation and response over the time interval \(-0.1 \leq t \leq 2.4\) s, at increments of 0.001 s (to produce suitable graphical resolution). This circuit was “patched” (programmed with plug-in wires) on a Comdyna GP-10 analog computer, and the given input voltage was produced by a sine-wave generator; graphs of the actual measured time-varying input excitation and output response are on the next page. Do the numerically simulated excitation and response that you calculate look very similar to the measured excitation and response? (If not, they should!) Although the circuit is lightly damped, there is still clear beating behavior. Show that the period and frequency of beating evident in the time-response graph are what you would expect, based upon Equation 10.6.5 for beating of an undamped 2nd order system.
      clipboard_e64f8433a414539a41bee07c514e56b86.png
      Figure \(\PageIndex{10}\)
      clipboard_ee960a4d59f3b862b288e5596d2469392.png
      Figure \(\PageIndex{11}\)

    1For most commercially available sensors, the electrical signal passes through wires connected to the sensor, but some modern sensors send the signal wirelessly. The size of such a sensor ranges typically from that of an aspirin tablet to that of a large soda bottle.

    2Although the theoretical constant of proportionality is \(-m/k\), the true constant, including polarity, of any real sensor is always measured by laboratory calibration.

    3For the phase-angle error, regard a 180° phase difference as being equivalent to 0°, because it is simply a sign difference that is corrected in calibration and/or data processing.

    4The floor’s true reaction is distributed force, i.e. stress, acting against the base over the area of contact. Discrete force \(f_{R}(t)\) in the drawing is the equivalent resultant of the distributed force, that is, fR(t) has the magnitude of the total force distribution and its point of application is the center of the distribution.

    5This FRF applies for a vibration isolator with viscous damping. However, the internal-friction mathematical model of structural damping is more realistic than viscous damping. In Appendix B, Section 19.5, the internal-friction model is described and applied for vibration isolation.

    6The schematic drawing for this problem shows the connecting rod attached to a rigid wall, which is a configuration suitable for calibrating an RMA; however, in any end-user practical application, the connecting rod would be attached to a flexible structure. RMAs are applied in practice to impose dynamic forces upon structures, both to provide excitation for vibration testing, and to suppress unwanted vibration as actuators in control systems. An RMA can be structure-borne, meaning it can be completely supported by the structure onto which it imposes dynamic force. For example, large RMAs have been installed in the top floors of tall buildings in order to reduce bending vibration excited by wind and ground tremors; an RMA in this civil engineering application is often called an active tuned-mass damper (ATMD). The internal force generators of ATMDs are hydraulic motors, whereas those of small RMAs (having total weight on or under the order of 100 lb) are electromagnetic linear motors.