## EGH421: Vibration and Control Modelling and Analysis # EGH421: Vibration and Control

## Our Real Student’s Score cards #### Question 1—Laplace Transforms and solving ODEs (5 marks) Answer the following:

• Solve (by hand) the following differential equation using Laplace transforms: xx + 2xx + xx = 5ee−tt + tt where xx(0) = 0, xx(0) = 1. While you must do this by hand, you are encouraged to check your work with MATLAB commands (e.g. partfrac or residue for the partial fraction expansion).
• A system has the following differential equation: yyâƒ› + 3yy + 5yy = xxâƒ› + 4xx + 6xx + 8xx find the ratio GG(ss) = XX(ss) YY(ss) assuming all initial conditions are zero. When xx(tt) = δδ(tt) (i.e. an “impulse”), what is the value of yy(tt) as tt → ∞? (Hint: use Final Value Theorem). Simulate GG(ss) with an impulse input in MATLAB and plot it. Comment on its agreement with the Final Value Theorem result.

#### Question 2 – Mechanical system modelling and analysis

Figure 1 shows a system with a cart with a mass on it. The interface between the cart and the masses has viscous friction. Do the following:

• Find the transfer function XXFF(SS)2(ss) . Do not plug in any numerical values yet.
• For the remainder of the question, use the parameters mm1 = 1 kg, mm2 = 0.2 kg, kk = 1000 N/m , and cc = 1 Ns/m. Use MATLAB to draw the Bode plot (i.e. the “frequency response”). Use this c) Use diagram MATLAB to to predict simulate the the steady-state system for amplitude the input FF(tt) of xx2= (tt) 10 when sin(28.4tt) the input for is a FF(tt) sufficient = 10 time sin(28.4tt). so that the steady-state oscillation is achieved (Hint: use lsim). Comment on the agreement with your answer in b).

Cholette Question 3—Electrical system modelling and analysis (6 marks) The circuit below shows a filter commonly used as the output stage of a power supply. In this application, vviiii(tt) is switched between 0 and a DC input supply voltage (13V in this case). The filter smooths the switched input so that it is close some desired output from 0-13V. In this exercise, you are interested in determining the start-up behaviour of the system. To this end, do the following:

• Find energised. the transfer function VVVVooooooiiii(ss)(ss) . You may assume that the capacitor and inductor are initially de-
• At L−1tt {VV= oooott0, (ss)}. vviiii(tt) Please is switched use partial to 13 fraction V (i.e. expansion vviiii(tt) = 13uu(tt) and the ). Laplace Find the Transform time response table vvto oooottconduct (tt) = the inversion(s), showing all of your work and explaining your logic clearly to receive full credit.
• How long will it take for vvoooott(tt) to reach 6V? You may use computer tools here, but please clearly explain your logic for full credit.

#### Question 4—Electromechanical system modelling and analysis (8 marks)

Consider the electro-mechanical system shown in Figure 2. The coordinates xx1 and xx2 are the positions of mm1 and mm2 with respect to the ground, respectively. The mass mm2 slides on top of the cart and this surface may be considered frictionless. Do the following:

• Find the transfer function GG(ss) = and do not plug in parameter values XXEEaa(ss) 2(ss) yet using (i.e. the the parameters transfer function denoted should in the include figure. mmIgnore 1, JJ, etc.). gravity,
• Go ahead and plug in the number to the transfer function (Table 1). Analytically determine the time response (i.e. xx(tt) = L−1{XX(ss)}) for a step input of eeaa(tt) = 24uu(tt) (i.e. a step of height 24). You may use MALTAB to do the partial fraction expansion, but you must show the code you used to do so. To receive full credit, you must do the inversion of the Laplace Transforms using the table, and clearly discus your logic. c) Simulate the system in MATLAB for an input eeaa(tt) = 24uu(tt). Plot your analytical function (answer to part b) on the same plot and compare. Comment on their agreement. d) Sketch (by hand) the Bode plot for GG(ss). As a part of this, please carefully compute and label:
• All asymptotes (e.g. low-frequency & high frequency)
• All corner frequencies
• The frequency where the phase crosses φφ ≈ −180âˆ˜
• Parameter Value Mass, mm1 (kg) 0 Mass, mm2 (kg) 5 Armature Resistance, RR (Ω) 75 Motor Torque Constant, KKtt (Nm/A) 4 Motor Back EMF Constant, KKbb (Vs/rad) 4 Pinion Rotational Inertia, JJ (Nms2/rad) 0 Pinion radius, rr (m) 0.1 Spring Constant, kk (N/m) 500

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