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 → +∞

1. A system has a transfer function G(s) given by: G(s)=\frac{5}{(2 s+3)(s+4)} Showing all your working, determine the response of the system in the time domain to a step input at time t = 0 of magnitude 1. Hence state whether the system is stable or unstable.

Problem 2. (1) (2.5 points) DErive the Equations of motion of a quarter-car model shown below. (2) (2.5 points) Obtain the state-space model. (3) (2.5 points) Obtain the transfer function: G(s) = Y(s)/R(s).

1. Find Re and Im of: z=\frac{2+j}{1+j} Show your calculations.

Figure P10.3 shows a general closed-loop control system. The plant transfer function is G_{p}(s)=\frac{1}{s^{2}+6 s+8} a. Determine whether the closed-loop system is stable for control gain K, 2. b. Compute the controller gain Kp so that step response shows 25% overshoot. c. Estimate the settling time for a step reference input if the control gain is K, 0.5.

2. Given the differential equation n ÿ + 3ỷ + y = 5ġ – 2g, where g(t) is the input function and y(t) is the response (output), find the transfer function Y(s)/G(s).

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]

1. The machine in the picture is a turbine working at 50H7 To mitigate the vibrations transmitted to the ground, the turbine is mounted on an isolator. The total mass of the turbine is 100Kg. We need to design the isolator such that the transmissibility is 0.5, with a natural frequency of vibration equal to the half of the working frequency of 50H7.Assuming that the isolator can be schematized by a spring and a damper in parallel, findthe values of k and h that satisfy the above described requirements

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")

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