Step input of 3 units was applied to a system and the response of this system is shown in Figure Q2.2. Determine the transfer function of this system. 5 4.5 4 3.5 3 2.5 Amplitude 2 1.5 1 0.5 0 0 1 2 Time (seconds) Figure Q2.2 Step Response 3

A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): A. Use Laplace transform of the differential equation to determine the transfer function of the system.

nk int m The spring-mass-system shown in the figure has the following parameters: spring constant k = 4 N/m; mass m 6 %3D kg and the constant n = 1.6. M is the corresponding mass-matrix of the system. V1 and V2 are the eigenvectors associated with the smallest and largest natural frequencies of the system, respectively. If V,TV, = 1 and V2 V2 = 1, then what is value of V,™MV2 (in kg)? Answer:

For the unit step response shown find the transfer function of the system. 1.4 1.2 1.O 0.8 0.6 0.4 0.2 10 15 20 25 Time (seconds) Respanse

Harmonic oscillators. One of the simplest yet most important second-order, linear, constant- coefficient differential equations is the equation for a harmonic oscilator. This equation models the motion of a mass attached to a spring. The spring is attached to a vertical wall and the mass is allowed to slide along a horizontal track. We let z denote the displacement of the mass from its natural resting place (with x > 0 if the spring is stretched and x 0 is the damping constant, and k> 0 is the spring constant. Newton's law states that the force acting on the oscillator is equal to mass times acceleration. Therefore the differential equation for the damped harmonic oscillator is mx" + bx' + kr = 0. (1) k Lui Assume the mass m = 1. (a) Transform Equation (1) into a system of first-order equations. (b) For which values of k, b does this system have complex eigenvalues? Repeated eigenvalues? Real and distinct eigenvalues? (c) Find the general solution of this system in each case. (d)…

MA Figure (2) 3- The time response of a liner spring-mass-damper system with the mass of (m), the stiffness coefficient of (k), and a dampirg constant of (c) is given below in figure (2) with a set of coordinates marked by fillec circles. x(t) is in meters, it is in seconds. Indicate the units of all quantities involved in your solutions when calculate 0 -1- (0.53, 1.78) (0,0.8) 2 (2.1,-1.3) (3.7,0.9) L 4 (5.3,-0.7) - The damping ratio of the system. Its natural frequency w., damping corstart c, and stiffness coefficient k; for (m-1.25 kg). III- The amplitude of the motion if the phase angle is given as (0.4 rad). IV. Its initial conditions. (6.86, 0.5) 10 t[sec]

Given the vibrating system below: K4 Y(t) =Ysin30t where for = 30 and Y=20mm Find the following K1 K2 m C3 H C2 C1 C5 C4 1. Frequency Ratio 2. Displacement Transmissibility Ratio 3. Absolute displacement of the mass 4. Type of Damping 5. Equation of motion x(t). Assume Initial conditions for displacement and velocity 6. Graph 2 cycles of the vibrating system. You can use third party app for this. M = 10 kg K1=100 N/m K2= 80 N/m K3=75 N/m K4= 120 N/m C1 = 20Ns/m C2=40 Ns/m C3= 35Ns/m C4= 15 Ns/m C5= 10 Ns/m

össlg äbäi What is the the transfer function of the first-order system which has the impulse response curve as in * ?the following figure Impulse Response 1.8 1.6 1.4 1.2 System: sys Time (seconds): 0.333 Amplitude: 0.737 1 0.8 Slop 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (seconds) 1/(s+3) O 2/(s+3) O 6/(s+3) O No one of the answers in a, b, and c ( Amplitude 2.

'p' (liquid density 'A' (tank cross sectional area) 'Qin' (Input Flow Rate) "Que P 'R' (restriction coefficient) '' (head of liquid) dh Qin = A + dt Figure 1 The single tank system (Figure 1) has been modelled by the first order differential equation given as equation. The equation describes the relationship between the input flow rate entering the tank and the head of liquid in the tank. ph R 5 m equation The following constants are provided: R = 40 Kpa s m², 4 = 10 m², p= 1001 kg m², and g = 9.81 m s A pump is suddenly switched on and provides a step input flow rate of 0.5 m³s¹. (1) Using Laplace Transforms, solve equation and provide an expression to show how the level in the tank will vary in time after the step input has been applied.

Question 4 a) A control engineer has modelled the suspension system of a new model car using a 2nd order differential equation. Using Laplace Transform, the engineer has managed to work out the transfer function, which is given below: 0.001 s2 + 12s+81 What is the natural frequency, the damping ratio and the constant K of the system? Please show all calculations. b) The same engineer has studied the suspension system of a SUV vehicle and modelled it using again a 2nd order differential equation, which has resulted into the following 2nd order transfer function: Y(s) 32 U(s) 4s² +8s + 16 For this system first calculate the damping ratio and its natural frequency and state whether the system is underdamped, overdamped or critically damped. Then calculate the peak time, peak value, settling time and damped natural frequency of the system.