1. Consider the simplified model we discussed in class about a car on a bumpy road; see Fig. 1.The car has a mass m supported by stiffness k and damping c. The road gives a displacementexcitation R(t) to the car. The transfer function from R(t) to the car displacement y(t) isHy(s) ==Y(s)R(s)=cs+kms² + cs+k(1)Let the acceleration of the car be a(t) = ÿ(t). Determine the frequency response functionGa(w) from R(t) to a(t). Plot |Ga(w) as a function w. Hint: Analyze |Ga(w)| for w << wn,w≈wn, and w >> wn.my(t)Suspensionk₁y(t)HeadmkAirBearingx(t)k2R(t)Disk SurfaceFigure 1: A simple model ofa car on a bumpyroadFigure 2: Suspension in computerhard disk drives

The question asks for the frequency response function relating the road excitation to the car's…

Answered: 1. Consider the simplified model we…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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The physical system shown below consists of a mass, viscous damping, and two parallel springs. Do the following: a) Neatly draw a proper free body diagram b) Find the differential equation of motion that describes the system. c) Find the transfer function X(s) / F(s). x(t) ki k2 m f(t) b
You are requested to design an automotive suspension or shock absorber system. In order to simplify the problem to one dimensional multiple mass-spring-damper system, a quarter vehicle model is used. The system parameters and free-body diagram of such system is shown below. M₁: Automobile body mass M₂: Wheel and suspension mass K₁: Spring constant of suspension system K2: Spring constant of wheel and tire B: Damping constant of shock absorber (a) Obtain the transfer function of X₁ (s) F(s) T₁(s) = = 2500 kg = 320 kg and T₂(s) = = 80,000 N/m = 500,000 N/m = 350 N-s/m Automobile- Suspension system Wheel- M₁ M₂ X₁ (s) - X₂ (S) F(s) in terms of the parameters of mass, damper and elastance (M, B and K). (b) Express the T₁ (s) and T₂ (s) with numerical values. c) Plot the x₁ (t) and x₁ (t) = x₂(t) outputs of this passive suspension system for the input torque f(t) = 2,000 N. fit) K₂ x₂(t) -Tire
Get the equation of motion by drawing the free body diagram of the given systems. a) Get the system's transfer function and find the unit digit answer. Show all decals in detail. m = 1 kg b= 20 Ns/m k = 125 N/m F ww k b) get the transfer function of the system. X(s)/Pg(s) =? Show all decals in detail. resistance R k Pg massless piston area (A) capacitance
mechanical vibrations 3m i+4cx+ 2kx = 4cj+3ky For the system given above, obtain the state-s pace representation,
1. Suppose you are riding your bicycle on a bumpy road having a surface profile that varies harmonically with +/- 6 cm undulations. The distance between consecutive peaks of these undulations is 2 m. When you sit on the seat of your bike for your casual ride, the springs deflect 5 cm. When you are seated, the damper under the seat provides an equivalent linear viscous damping of 10% of the critical damping. A simple representation of your ride on the "never-ending" rough road is shown below. (a) If you are riding your bicycle at a horizontal speed of 2.5 m/sec, how much bumping up and down will you experience? (b) Next day, you are carrying a backpack which increases your on-seat weight by 20%. Assuming that you are still able to ride with same speed, will this "loaded" ride be more or less comfortable than your previous, "no backpack" ride? M k/23 k/2 6 cm 2 m
Equation of motion of a suspension system is given as: Mä(t) + Cx(t) + ax² (t) + bx(t) = F(t), where the spring force is given with a non-linear function as K(x) = ax²(t) + bx(t). %3D a. Find the linearized equation of motion of the system for the motion that it makes around steady state equilibrium point x, under the effect of constant F, force. b. Find the natural frequency and damping ratio of the linearized system. - c. Find the step response of the system ( Numerical values: a=2, b=5, M=1kg, C=3Ns/m, Fo=1N, xo=0.05m
1 An object of mass 125 kg is released from rest from a boat into the water and allowed to sink. While gravity is pulling the object down, a buoyancy force of times the weight of the object is pushing the object up (weight = mg). If we assume that water 40 resistance exerts a force on the object that is proportional to the velocity of the object, with proportionality constant 10 N-sec/m, find the equation of motion of the object. After how many seconds will the velocity of the object be 90 m/sec? Assume that the acceleration due to gravity is 9.81 m/ sec2. Find the equation of motion of the object. X(t) = %3D
2. For the system below, find the transfer function fromfi to x (driving point receptance) and from f. to ä, (driving point accelerance). What is the acceleration response of mass m, if m; = 2 kg, m; = 4 kg, k, = 40 N/m, k =100 N/m, and k; = 200 N/m, fi(t) = 20 cos(3t) N and f:(r) = 0? WW m, WW m W
A mass-spring-dashpot system with constants k = 0,6, c = 0,7 and m = 100 g makes mechanical vibration. The system is exposed to no external force. If the system is at rest, initial position and velocity are 2 m and 6 m/sn, respectively. Under these circumstances, find the general equation of certain motion using the "Method of Laplace Transforms". 1. %3D
a) A vehicle circulates on a road as shown in Figure Q2a. The road profile can be modelled as the input u(t). The vehicle is modelled as a quarter car of mass m, and the suspension has a spring stiffness coefficient k = 2 Nm and a damper of coefficient c = 2 Nm s. Find the position of the mass, y(t) and any time t if the road profile is a unit step. m 一 Road Figure Q2a b) In Figure Q2b, a disk flywheel J of mass m 32 kg and radius r = 0.5 m is driven by an electric motor that when it is working produces an oscillating torque of Tin = 10sin(wt) N- m. The shaft bearings may be modelled as viscous rotary dampers with a damping coefficient of BR = 0.4 N-m-s/rad and stiffness coefficient KR = 2 Nm If the flywheel is at rest at t 0 and the power is suddenly applied to the motor, do the following [Hint:J = mr²/2]: (i) Find the natural frequency of oscillation of the disk expressed in Hz. (ii) Find the damping ratio for this system. (iii) Describe in no more than 3-4 lines the behaviour of the…
The figure that is attached illustrates a system in which a force F is applied to a mass m2 that is connected to another mass m1 via a spring and a damper, and mass m1 is connected to a wall via a damper. The equations of motion that govern the time evolution of the mass displacements, y1(t)and y2(t), are given below. A) Define 4 state variables of the system (xi, i = 1,…,4) as phase variables, and define the control input u. Convert the two equations of motion above into four 1st-order ODEs that are functions of the state variables and the control input.(b) Write the four 1st -order ODEs from part (a) in state-space form,x = Ax+Bu.
A simple pendulum whose string measures 1 = 2 m and whose mass is 5 kg is at rest in its position of equilibrium and is subjected to a harmonic forcing of intensity F₁ = 2 N and frequency w rad/s. find: a) The frequency @ for which the system is in resonance. b) Make a graph of angular position v/s time where the amplitude and period of the system if the frequency w is equal to twice the resonant frequency.

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