System Dynamics

Consider the following IVP: with initial condition x(to) = -10 and to = 0. x*+2x=0 1. What is the particular solution, x(t)? 2. What is the value of x as time t → ∞? A. x → →∞0 B. x → -10 C. x→0 D. x → +10 E. x → +∞

For each of the following ODEs determine if the eigenvalues are (a) real and distinct, (b) repeated, (c) complex conjugate pairs:

For each of the following linear, time-invariant, 2nd order homogeneous ODEs solve for the particular solution that satisfies the initial values given.

Match each one of the responses shown below (Cases 1-4) with one of the following IVPS

1. What is maximum force in the tow cable and when does it occur? Does this maximum force occur when the submarine goes through its maximum acceleration? Explain your anwer. 2. What is the elongation in the tow cable due to the drag of the submarine at steady state? 3. What is the maximum velocity of the submarine and when does it occur? You recommend that the senior officer should select a cable length that would minimize the value of the peak force in the cable, while keeping the cable from being unnecessarily long. Therefore, you suggest that the system should have a damping ratio = 0.707., 1, 1.3 1. Find the length of the cable for damping ratio = 0.707., 1, 1.3 2. Adjust the gain values of your block diagram to agree with this cable length. 3. Simulate and determine the maximum velocity and maximum cable force. Compare the results with the 300 ft. cable. 4. What is the physical significance of = 0.707., 1, 1.3 in a dynamical system in relation to the shape of the response? Why is it the advantage of the cable with = 0.707. over 1, 1.3?

Analyze the problem statement to prepare a free body diagram of the submarine (shown in Figure 1). Use the prepared free body diagram to find the equation of motion for the submarine. Prepare diagrams of the model under various conditions in Simulink and run each simulation for 30 seconds. Use the given damping coefficient values to determine the stiffness values. Run each simulation using the stiffness values calculated in step 4. Determine the maximum velocity of the submarine and the maximum force and elongation in the towing cable for the system using each calculated stiffness value. Plot all results, then compare the velocity, towing force, and elongation for each set of conditions.

1.) Derive an equation relating the input force F, with the output, displacement x, for the systems described in the textbook by Fig 17.20 (both systems a) and b) )on page 437. [20 pts]

2.) Derive the relationship between the output, the potential difference across the resistor R (vR),and the input v for the series LCR circuit shown in Fig 17.23 pp 437. Solve the same problem if the output is considered to be the voltage drop across the inductor. [20 pts]

Consider the differential equation below. m \ddot{y}=k(z(t)-y)+b \frac{d}{d t}(z(t)-y) where y(t) = Yejwt and z(t) = Zejwt Ya) Solve for the frequency response, i.e., the ratio Y/Z ) Assuming that the damping ratio is 3 = 0.05 and that m = 1, plot the magnitude of the transfer function vs. frequency of the response of part (a), i.e., plot Y/Z vs.(W/WN). Use a logarithmic scale for the x-axis. ) Use the plot in part (b) to explain for which range of frequencies one can ignore z(t) in the original differential equation (i.e. set z(t) to zero). Assuming that the damping ratio is = 0.05 and that m = 1, plot the phase angle of the transfer function vs. frequency of the response of part (a) (i.e. the phase angle of Y/Z vs. (W/WN). Use a logarithmic scale for the x-axis. (e) Assuming that 3 = 0.1, and m = 1, use the plots in parts (b) and (d) to explain for which range of frequencies one can ignore all the derivative terms in the original differential equation. ) With the same assumptions as in part (e), use the solution for part (a) to find y(t)when z(t)10sin(3t).= Consider a car moving at a constant speed Vo on a wavy road where the "waves"have a peak-to-peak distance (wavelength) of L, and an amplitude of Z, inches.Show that this system will result in the same differential equation provided at the beginning of the problem.

4.) Figure 17.26 on page 438 shows a thermal system involving two compartments, with one containing a heater. If the temperature of the compartment containing the heater is T1, the temperature of the other compartment is T, and the temperature surrounding the compartments is T3, develop equations describing how the temperature T,and T2 vary with time. All the walls of the containers have the same thermal resistance and negligible capacity. The two containers have the same thermal capacitance C. [20 pts]