Consider the mechanical system shown below. The system is at rest for t <0. The input force u is given at t = 0. The displacement x is the output of the system and is measured from the equilibrium position. Obtain the transfer function. You have to give the details of your work. a) Set up the governing equations for the system. b) Obtain the transfer function (s)/u(s) c) Use Force-Voltage analogy to get the electrical equivalent system equations. k3

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1. Consider the simplified model we discussed in class about a car on a bumpy road. The car has a mass m supported by stiffness k and damping c. The road gives a displacement excitation R(t) to the car. The transfer function from R(t) to the car displacement y(t) is Hy(s) - Y(s) R(s) cs + k ms² + cs+k (1) (a) Determine the driving point impedance ZR(s), which is the ratio of the velocity R(t) to the force acting on the wheel from the road. (b) Engineer X is testing the car in the Vehicle Dynamics Lab of a start-up company GF.com. Because of insufficient cash flow, Engineer X only has the equipment to mea- sure ZR(s), and identify the zeros and poles of ZR(s). But eigineer X wants to find the poles of Hy(s). To do so, should Engineer X use the zeros or poles of ZR(s) instead? Explain why? m y(t) R(t) Figure 1: A simple model of a car on a bumpy road

Q2: Derive the 1st order transfer function of a liquid-level system shown fig. (1) to determine the response of the outlet flow (q.(t)) to unit pulse change in the inlet flow (qin (t)). q (t) I Ih(t) 1 Fig.(1) R go (t)

2) Consider the spring-mass-damper system hanging down from a ceiling. Initially, the system is at rest. Then, we gently place another mass (M) on top the original one (m). First, please obtain the equation of motion of the mass m in the system (before we placed the other object). Assign the displacement references yourself and obtain the equation accordingly. From the equation of motion, please obtain the transfer function, G(s), between an external force on the mass m , f(t), and the displacement of the mass m, x(t). Then, please obtain the motion of the object M, using the transfer function G(s). Don't forget that G(s) belongs to the system shown on the top! To solve this, represent the force applied by the added mass (M) as a function of time. How would you describe the act of adding a mass as an external force input? Then solve for x(t). c = 10 Ns/m k = 50 N/m m = 5 kg M M = 25 kg F(s) G(s) X(s)

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The ratio of output to input of a system in Laplace domain is known as Transfer function . Select one: True False

For the mechanical system shown in Fig-1, u₁, u₂ are the inputs and y₁, y2 are the outputs YI Y Y2Y k₁ mi UI U2 + m2 Fig-1 a) Write the differential equations that model this system b) Find a state space representation of the system.