8.12: Chapter 8 Homework

1. A “shaped”, 10-millisecond force pulse in pounds is described by the equation \[\begin{aligned}
   f_{x}(t)=&\left[50 \sin ^{2}(100 \pi t)\right] \mathrm{lb} \\
   & \times[H(t)-H(t-0.01 \mathrm{sec})]
   \end{aligned} \nonumber \] which is graphed at right. In a calculation of system response to this input force, we wish to approximate the force as an ideal impulse function, \(f_{x}(t) \cong I_{F} \delta(t)\). What is the value (with units) of \(I_{F}\)? (Hint: before embarking on a long integration, examine the graph carefully for symmetries that will make the calculation almost trivial.)

   

   Figure \(\PageIndex{1}\)

2. Consider an undamped mass-spring system. The mass weighs 20.0 lb including an air-jet thruster that is attached to it, and the spring is calibrated to stretch 0.279 inch when subjected to 100 lb of tension. The air-jet thruster has been designed and calibrated to provide 0.767 lb of thrust in a nearly perfect flat pulse. Let us study the dynamic position response \(x(t)\) of the mass, starting from zero ICs, to this force input from the thruster.
   1. Calculate the natural frequency of the mass-spring system in rad/s and Hz, and the natural period of the system in seconds and milliseconds. Calculate the impulse magnitude \(I_{F}\) delivered by the air-jet thruster in a blast of duration \(t_{d}\) = 4.00 milliseconds. (Partial answers: \(T_{n}\) = 75.53 ms, \(I_{F}\) = 3.068 × 10-3 lb-s)
   2. Model the 4.00-ms air-jet blast as a perfect flat pulse of force. For this input, write equations that describe the real pulse response \(x(t)\) (in inches); for help, see Equation 8.1.6 and Equation 7.1.4; do not re-derive anything that is already available. Use MATLAB to make an accurate time history plot of this real pulse response \(x(t)\) (in inches) over the time interval \(0 \leq t \leq 40\) ms.
   3. Approximate the air-jet force input as an ideal impulse with the same impulse magnitude \(I_{F}\) as calculated in part 8.2.1. Use Equation 8.7.5 to write the numerical equation for the ideal impulse response \(x(t)\) in inches.
   4. Use MATLAB to make two accurate time history plots, on the same graph, of \(x(t)\) (in inches) over the time span \(0 \leq t \leq 40\) ms: the real pulse response of part 8.2.2 and the ideal impulse response of part 8.2.3.
3. Consider the mass-spring system drawn below with \(m\) = 8.03 kg and \(k\) = 317 N/m. This system is initially at rest in the static equilibrium position when it is hit by a flat force pulse \(f_{y}(t)=F\left[H(t)-H\left(t-t_{d}\right)\right]\) of magnitude \(F=4.50\) N and short duration \(t_d\) = 0.05 s. In this problem, we analyze the dynamic displacement of the mass relative to the initial static equilibrium position.

   

   Figure \(\PageIndex{2}\)

   1. The following succinct response equation can be derived easily with use of transform Equation 8.1.2 and the general Laplace transform of a function that is translated in time: \(L\left[f\left(t-t_{d}\right) H\left(t-t_{d}\right)\right]=e^{-s t_{d}} F(s)\).\[y(t)=\frac{F}{k}\left[\left(1-\cos \omega_{n} t\right) H(t)-\left(1-\cos \omega_{n}\left(t-t_{d}\right)\right) H\left(t-t_{d}\right)\right] \nonumber \] Show that this equation is fully equivalent to the response equations in the two-part solution Equation 8.1.6. Note from Equation 7.1.4 that \(U=F / k\).

      Next, use MATLAB to plot the response curve of \(y(t)\) versus \(t\) from \(t\) = 0 to \(t\) = 1 s, the period of the mass-spring system.

   2. In order to compare with the real response, plot [on the same graph as part 8.3.1] the ideal impulse response of the system, \(y(t)=\left(I_{F} / m \omega_{n}\right) \sin \omega_{n} t\), Equation 8.7.5, using for \(I_{F}\) the value of the actual impulse from given data. Label clearly which curve is the real response and which is the ideal impulse response; title and label your graph appropriately.
4. In this exercise, let us apply the initial-value theorem to the problem of Section 8.7: the standard undamped 2^nd order system that has non-zero initial conditions and is disturbed by an ideal impulse, \(u(t)=I_{U} \delta(t)\).
   1. Determine the post-impulse initial value, \(x\left(0^{+}\right)\), by using Equation 8.7.2 for in the initial-value theorem. Is your result the same as that found directly from Equation 8.7.3?
   2. Use Equation 8.7.2 for \(L[x(t)]\) to find the Laplace transform \(L[\dot{x}(t)]\) of the “velocity” function; be sure to include both terms of Equation 8.6.3 for the transform of a derivative.
   3. Determine the post-impulse initial “velocity”, \(\dot{x}\left(0^{+}\right)\), by using your \(L[\dot{x}(t)]\) from part 8.4.2 in the initial-value theorem. Is your result the same as that of Equation 8.8.2?
5. Consider the standard 1^st order LTI-ODE (of a stable physical system) for dependent variable \(x(t)\): \(\dot{x}+\left(1 / \tau_{1}\right) x=b u(t)\), with IC \(x(0)=0\), and with excitation by an ideal impulse, \(u(t)=I_{U} \delta(t)\).
   1. Infer from the results of Section 8.5 the unit-impulse-response function (IRF), \(h(t)\).
   2. Use the result from part 8.5.1 to write the Duhamel integral response solution for the standard stable 1^st order system, which is comparable to Equation 8.11.2 for the standard undamped 2^nd order system.
   3. The object of this part is to compare numerically the ideal impulse response derived in Section 8.5 with the response to a real, flat pulse that has the same impulse magnitude as the ideal impulse and a pulse duration that is short relative to the system time constant. Use MATLAB to make two accurate time history plots, on the same graph, of the dimensionless output \(x(t) / b I_{U}\) versus dimensionless time \(t / \tau_{1}\) over the time interval \(0 \leq t / \tau_{1}\leq 1\). One plot will be the response to the ideal impulse. The other plot will be the response to a flat pulse having dimensionless duration \(t_{d} / \tau_{1}=0.2\); for this plot, use the response equations derived in homework Problem 6.2. You should find that the ideal impulse response is much easier to calculate than the real-pulse response. However, is the ideal impulse response a sufficiently accurate approximation to the real response, and, if so, is it accurate over all time, or over just some portion of the response time?
6. If you compare Figure 2.4.1 for the Heaviside unit-step function \(H(t)\) with Figure 3.4.2 for step response of a stable 1^st order system, you can see that as time constant \(\tau_{1}\) becomes progressively smaller, the 1^st order step response looks progressively more like the unit-step function. Using Equation 2.4.1 for \(H(t)\) as a model, let us define the “exponential-unit-step” function \(H_{e}(t)\): \[H(t)=\left\{\begin{array}{ll}
   0 & \text { for } t<0 \\
   1 & \text { for } t>0
   \end{array} \Rightarrow H_{e}(t) \equiv\left\{\begin{array}{c}
   0 \quad \text { for } t<0 \\
   1-e^{-t / \tau_{1}} \quad \text { for } t>0
   \end{array}\right.\right. \nonumber \] It is clear that \(H(t)\) can be defined as a limit of \(H_{e}(t)\): \(H(t)=\lim _{\tau_{1} \rightarrow 0} H_{e}(t)\). Now, let us define the “exponential-unit-impulse” function \(\delta_{e}(t)\): \[\delta_{e}(t) \equiv\left\{\begin{array}{c}
   0 \quad \text { for } t<0 \\
   \frac{d}{d t}\left(1-e^{-t / \tau_{1}}\right)=\frac{1}{\tau_{1}} e^{-t / \tau_{1}} \quad \text { for } t>0
   \end{array}\right. \nonumber \] Sketch by hand an over-and-under pair of graphs, the upper graph being \(H_{e}(t)\) vs. \(t\) and the lower graph being \(\delta_{e}(t)\) vs. \(t\). Sketch conceptually the two exponential functions for a few values of time constant \(\tau_{1}\); show in particular how \(H_{e}(t)\) and \(\delta_{e}(t)\) evolve as \(\tau_{1}\) becomes progressively smaller. Describe the character of \(\delta_{e}(t)\) as \(\tau_{1} \rightarrow 0\). Is it plausible physically to define the Dirac delta function as \(\delta(t)=\lim _{\tau_{1} \rightarrow 0} \delta_{e}(t)\), which, with the definition \(H(t)=\lim _{\tau_{1} \rightarrow 0} H_{e}(t)\), is equivalent to \(\delta(t)=d H / d t\)? [This is one of many possible limit-process definitions of \(\delta(t)\); Equation 8.4.1 is a more commonly used definition.]
7. Consider the mass-spring system (with damping neglected) of Figure 1.10.1, for which the ODE of motion is \(m \ddot{x}+k x=f_{x}(t)\), and the system parameters are \(m\) = 2.20 kg and \(k\) = 770 N/m. Initial conditions are zero: \(x(0)\) = 0 and \(\dot{x}(0)\) = 0. The excitation is a ramped exponential force pulse, \(f_{x}(t)=F_{m}\left(t / t_{m}\right) \exp \left(1-t / t_{m}\right)\), with \(F_m\) = 6.0 N and \(t_m\) = 1/12 s. Write a MATLAB program, or adapt the code of Convolution-sum Example 2 in Section 8.11, to calculate and plot an approximate numerical solution for \(x(t)\) over the time interval \(0 \leq t \leq 1\) s. Adjust the time-step size \(\Delta t\) and number of time steps over the 1 s interval in your code until the graph of your approximate solution appears very similar to that on Figure 1.10.2 of the corresponding exact solution. Submit your MATLAB code and your final graph of response.