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 → +∞

A particle which moves in two-dimensional curvilinear motion has coordinates in millimeters which vary with time / in seconds according to x 5i2+4 and y 2r3 +6. For time t= 3 s,determine the radius of curvature p of the particle path and the magnitudes of the normal and tangential accelerations.

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.

7. The sports car and the driver have a total mass of 2500kg. The projected front area of the car is 0.78m?. The car is traveling at50km/h when the driver puts the transmission into neutral and allows the car to freely coast until after 150s its speed reaches 40km/h. Determine the drag coefficient for the car, assuming its values is constant. Neglect rolling and other mechanical resistance. (Hint: Watch"Chapter 9-Finding CD Value.mp4")

Problem 5b (10 points total): In class, you have derived the response of a first-ordersystem to a unit-step input. Given a first-order system of the form G(s) = K/(1+ Ts),where T is the time-constant, and K is the constant, find:%3D i) The time-response to a unit-ramp input r(t) = t. (7 points) ii) The steady-state error for error measured as e(t) = r(t) - c(t). (Hint: the steady-state error's measured as t tends to infinity). (3 points)

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.

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]

.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.

5.1Derive the state-variable equations for the system that is modeled by the following ODES where a, w, and z are the dynamic variables and v is the input. 0.4 \dot{\alpha}-3 w+\alpha=0 0.25 z+4 z-0.5 z w=0 \ddot{w}+6 w+0.3 w^{3}-2 \alpha=8 v

1. The equations of motion of this system are +3y + 4y -32-4Z = 0 2 +52 +62-5ý-6y = f(t)