System Dynamics

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

6. Do Problem 9.20 from the textbook. [Statement: A smooth, flat plate of length I= 6m and width b = 4 m is placed in water with an upstream velocity of U = 0.5 m/s. Determine the boundary layer thickness and the wall shear stress at the center and the trailing edge of the plate. Assume a laminar boundary layer.]

.A Mars rover autonomous vehicle includes a robotic arm for collecting rock samples. The dynamics of the robotic arm system have been analysed, and a root locus obtained for changes in a controller gain k varying from 0 to oo. The root locus is shown in the attached Root Locus.pdf document, and available for download from the MEC321 Blackboard course pages. a) Write down the start points and end points of the root locus diagram. b) Describe in detail how the system's transient response changes as k is increased. c) The designers wish to achieve a damping ratio of 0.423 from the system, but with the fastest possible settling time. (i) Use the magnitude condition to determine the required value for k, noting the need for the fastest possible settling time. d) Write down the fastest possible settling time for the system, and briefly explain why this is the maximum possible value.

: Consider the following spring-mass-damper system: Draw free body diagrams for each mass. i) Write the equations of motion for each mass as differential equations in the time domain. iii) Convert the equations of motion for each mass into algebraic equations using the Laplacetransform. (Assume zero initial conditions.) (3 points) iv) Solve for the transfer function G(s) = X2(s)/F(s). (You do not need to simplify your answer orconvert the transfer function back to the time domain.) (3 points) v) Find a state-space representation of the equations of motion.

5 Given the transfer function G(s)=\frac{2 s+3}{3(s+8)} compute the frequency response y (r for the input u(t) = 0,3 cos 5t.

4. Find the inverse Laplace transform of: F(s)=\frac{5(s+2)}{(s+1)(s+3) s^{2}} Show your calculations.

Given the 1/0 equation 2 y+10 y=3 u compute the frequency response y,,() for the input u(r) 18 sin 4r.

The thin homogeneous 300 lb plate is hanging from a cable attached to point O when it is subjected to an impulse of -20k lb.s at the corner A. Determine the angular velocity vector of the plate immediately after the impulse occurs.

Solve the following differential equation by the trial solution: \ddot{y}+25 y=1 It has initial conditions y(0)=1 / 2 and \dot{y}(0)=1 / 10 The answer will have the form A * \cos 5 t+B * \sin 5 t+C What is

s Use MATLAB 10 plot the Bode diagram for the 1-DOF mechanical system in Problem 9.11 (Fig. P9,11). Estimate the frequency response for the position input v) = 0.04 sin 50r m by reading the Bode diagram(indicate the frequency response parameters on the plot of the Bode diagram). Obtain a more accurate answer by using MATLAB's bode command with Ieft-handside arguments for computing magnitude and phase angle.