System Dynamics

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

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.

7:02 1 Done 5 -dub-prod.instructure.com AA ○ 20 points The figure below illustrates a two-degree-of- freedom system. Mass m₁ = 10 kg is located at the end of a clamped beam with a mass of my = 10 kg, a Flexural rigidity of EI = 4000 Nm², and a length of L = 2 m. Mass m₁ is connected to mass m2 = 20 kg using two springs, k₁ = 1000 N/m and k2 = 3000 N/m, as shown in the figure. Determine the second mode shape vector? beam mp (-3.75) (9) (-4.24) (-5.56) Return (-2.251 Submit 7:02 1 Done -dub-prod.instructure.com AA ○ freedom system. Mass m₁ = 10 kg is located at the end of a clamped beam with a mass of m, 10 kg, a Flexural rigidity of EI = 4000 Nm², and a length of L2 m. Mass m₁ is connected to mass m₂ = 20 kg using two springs, k₁ = 1000 N/m and k23000 N/m, as shown in the figure. Determine the second mode shape vector? (-3:75) (71²¹) (-4,24) (-5.56) E (-2,25) Return Clear my selection Submit - beam mp/n8:16 1 Done -dub-prod.instructure.com Tastrate a Live augice Un Mass m1 = Return 10 kg is located at the eam with a mass of my = 10 kg, of EI = 4000 Nm², and a length m₁ is connected to mass ; two springs, k1 = 1000 N/m /m, as shown in the figure. ond mode shape vector? beam mp (-3.75) (¹) (-4:24) (-5.56) (-2:25) Clear my selection .. k₁ k₂ 4G O AA Submit m₁ ↓ www X1 m₂ ↓ X2

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]

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]

3.) Derive the relationship between the height h, and the time for the hydraulic system shown in Fig 17.25, pp 437. Neglect inertance. [20 pts]

1. (8 points) the figure below shows the responses yi(t), y5 (t), ..., yž (t) of eight different processes to a unit impulse input. the transfer functions of the eight processes are listed below in a shuffled order. match the transfer function with the response and justify each using analytical mathematics (poles, time constant, gain, damping coefficient, and the first- and second-order impulse response we derived in class). i.e., while I encourage you to use Controlz.jl to check your answers, "the simulation matched" is not an acceptable justification. G1(s)= 6/3s+1 corresponds to panel (a,....B,....,h) because G2(s) = 5/2s+1 corresponds to panel ( a,b....h)______ because =. G3(s) = 12/6s+1 corresponds to panel (a,b,....h) because . G4(s) = 6 / 9/4s2+3/5s+1 corresponds to panel ( a,b,....h) because G5 (s) 6/9/4s2+3/2s+1 corresponds to panel (a,b,...h) because 6.G6(s) =6/9/4s2+ 9/2s+1 corresponds to panel (a,b, ....,h) because 6.G7(s) = 6/9s2+9s+1 corresponds to panel (a,b,....h) because 6.- G8s(s) = 6/9s2-3/10s+1 corresponds to panel ( a,b,....h) beacuse

Consider the closed-loop control system shown below: whcre K1 and K2 arc the positive constants. Derive the closed-loop sensitivity function: S(s) = E(s)/R(e). (2) (2.5 points) Determine K, and K2 such that wn = 4 rad/sec, and t, = 1 sec. Note:uhere u and t are the natural freguency and damning ratio respectivel: t_{s}=\frac{4}{\omega_{n}} \text {, where } \omega_{n} \text { and } \zeta \text { are the natural frequency and damping ratio, respectively. }

5.) Derive the differential equation for a motor driving a load through a gear system as shown in Figure 17.27 on page 438, which relates the angular displacement of the load with time.