13.4: Chapter 13 Homework

1. Consider the standard ODE Equation 3.4.8 for a stable 1^st order system, \(\dot{x}+\left(1 / \tau_{1}\right) x=b u(t)\), in which \(u(t)\) is the input quantity, \(x(t)\) is the output quantity, \(\tau_{1}\) is the positive time constant, and \(b\) is a constant. Denote the Laplace transforms as \(L[u(t)]=U(s)\) and \(L[x(t)]=X(s)\). Derive and draw for this system a Laplace block diagram with feedback, a block diagram of the same type as Figure 13.2.3. The input signal on the left side of your block diagram is \(U(s)\), and the output signal on the right side is \(X(s)\).
2. With input and output voltages \(e_{i}(t)\) and \(e_{o}(t)\), respectively, the governing ODE for the electrical \(LRC\) series circuit is \(L \ddot{e}_{o}+R \dot{e}_{o}+(1 / C) e_{o}=(1 / C) e_{i}(t)\), as derived in Section 9.2. Denote the Laplace transforms as \(L\left[e_{i}(t)\right]=E_{i}(s)\) and \(L\left[e_{o}(t)\right]=E_{o}(s)\), then derive and draw for this system a Laplace block diagram with feedback, a block diagram of the same type as Figure 13.2.3. The input signal on the left side of your block diagram is \(E_{i}(s)\), and the output signal on the right side is \(E_{o}(s)\).
3. A certain 3^rd order system has excitation \(u(t)\) and response \(x(t)\), and its governing ODE is \(a_{1} \ddot{x}+a_{2} \ddot{x}+a_{3} \dot{x}+a_{4} x=b u(t)\), in which all \(a_i\) and \(b\) are constants. Denote the Laplace transforms as \(L[u(t)]=U(s)\) and \(L[x(t)]=X(s)\). Derive and draw for this system a Laplace block diagram with feedback, a block diagram of the same type as Figure 13.2.3, but somewhat more complicated. The input signal on the left side of your block diagram is \(U(s)\), and the output signal on the right side is \(X(s)\). Your block diagram should have three feedback branches.
4. Consider the higher-order (4^th order in this case), 2-DOF mechanical system of homework Problems 11.2 and 12.1, for which the coupled matrix equation of motion is\[\left[\begin{array}{cc}
   m & 0 \\
   0 & J_{H}
   \end{array}\right]\left[\begin{array}{c}
   \ddot{y}_{1} \\
   \ddot{\theta}_{2}
   \end{array}\right]+\left[\begin{array}{cc}
   k_{y} & -a k_{y} \\
   -a k_{y} & k_{\theta}+a^{2} k_{y}
   \end{array}\right]\left[\begin{array}{c}
   y_{1} \\
   \theta_{2}
   \end{array}\right]=\left[\begin{array}{c}
   0 \\
   a F(t)
   \end{array}\right] \nonumber \]
   1. From the matrix equation, write the two separate scalar ODEs that are coupled by the term \(-a k_{y} \theta_{2}\) in the “\(y_{1}(t)\) ODE” (i.e., the ODE that includes acceleration \(\ddot{y}_{1}\)), and the term \(-a k_{y} y_{1}\) in the “\(\theta_{2}(t)\) ODE.” This system has the single input, force \(F(t)\), but two output quantities, translation \(y_{1}(t)\) of the mass, and rotation \(\theta_{2}(t)\) of the rigid bar.
   2. Denote the Laplace transforms as \(L[F(t)]\), \(L\left[y_{1}(t)\right]=Y(s)\), and \(L\left[\theta_{2}(t)\right]=\Theta(s)\). Take the Laplace transforms of both ODEs, with zero ICs. Now, derive and draw for this system a Laplace block diagram (in the form of Figure 13.2.3, but with differences) that has feedback and also “feed-across” between the forward-flowing branch that develops from the \(y_{1}(t)\) ODE and the forward-flowing branch that develops from the \(\theta_{2}(t)\) ODE. The input signal on the left side of your block diagram is \(L[F(t)]\), and the two output signals on the right side are \(Y(s)\) and \(\Theta(s)\), one from each of the forward-flowing branches.