2. Figure 2 shows a simplified model to simulate a recording head flying over a rough disk surfacein computer hard disk drives. The head has mass m and is supported by a suspension withstiffness k₁. Moreover, the moving disk surface will generate an air bearing lifting the headslightly above the disk surface (e.g., in the order of 20 nm). The air bearing is simplied asa linear spring with stiffness k2 and damping coefficient c. Let x(t) be the roughness of thedisk surface and serve as the input excitation to the head/suspension system. Moreover, y(t)is the relative displacement of the head to the disk. In real hard disk drive applications, wewant to keep y(t) almost constant, so that the head can follow the disk surface to performread/write operations.(a) Show that the equation of motion ismi+cy + (k1 + k2) y = −mä – k₁x(2)1(b) Derive the frequency response function. Plot the magnitude and phase of the frequencyresponse function. In plotting the frequency response function, let's definek1k1 + k2W₁ =W2=mm(3)Describe the motion of the head in the following three frequency ranges: 0 < w< W1,W1

a moving head in a recording head flying over a rough disk surface. The question asks you to derive…

Answered: 2. Figure 2 shows a simplified model to…

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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you are given the mass, damping, andspring constants of an undriven spring-mass systemmy'' + μy + ky = 0.You are also given initial conditions. Use a numerical solverto(i) provide separate plots of the position versus time (y vs.t) and the velocity versus time (v vs. t),(ii) provide a combined plot of both position and velocity versus time, and(iii) provide a plot of the velocity versus position (v vs. y) inthe yv phase plane. a. m = 1 kg, μ = 0 kg/s, k = 9 kg/s2, y(0) = 3 m,y'(0) = 2 m/s b.m = 4 kg, μ = 4 kg/s, k = 1 kg/s2, y(0) = 3 m,y'(0) = 1 m/s
2. Figure 2 shows a roller rolling and slipping at the same time on a slippery surface. The center of the roller moves with a displacement r(t) and the roller rotates with angle (t). The roller has mass m, mass moment of inertia I, and radius r. In addition, a layer of fluid with viscous damping coefficient c is present to lubricate the roller. Moreover, the roller is pulled by a spring under a given (i.e., prescribed) displacement u(t), whereas the spring has spring constant k. Also, the roller is subjected to an applied torque M(t). Figure 3 shows the free-body diagram of the roller. Answer the following questions. (a) Apply F = ma to derive an equation of motion governing the translation z(t). (b) Apply M = Ia to derive an equation of motion governing the rotation 0(t). (c) Eliminate the variable (t) from the two equations of motion derived in parts (a) and (b). You should obtain a third-order differential equation in x(t). (d) Initially (i.e., at t = 0), the roller has no…
Q4-6 please
Task 2 External Force Damper Damper mass mass m M m WWA Smooth Surface w No Friction Spring Spring Consider the above spring-mass-damper system. Let the mass m=1 kg, spring stiffness k= 1 N/m and damping coefficient b = 1 N.s/m. (a) What is the number of degrees of freedom? (b) If the damping coefficient, b, is zero and he external force is At) = sint, find the steady state position amplitude for the two masses.
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
Q1 A mechanical system with a rotating wheel of mass Mw (uniform mass distribution) is shown in Figure Q1. Springs and dampers are connected to wheel using a flexible cable without skip on wheel. (a) Determine all the mathematical modeling equations of the system for the translational and rotational motion. (b) Using the result in Q1(a), determine the translational motion equation in term of ? as a function of input motion ?. (c) By referring to standard second-order system form, determine the expressions for natural frequency and damping ratio of the system.
B6
Assume you have a 1U cubesat, which has a total mass of 2kg. It has one reaction wheel to control its orientation around its nadir pointing axis. The mass of the reaction wheel is 200g. The wheel itself has a diameter of 30mm and a thickness of 10mm. You may ignore any translational motion for the time being. (a) Write down the kinematic and dynamic equations of the system and compute all relevant constant values. (b) Where do you want to mount your wheel? Make a drawing and a qualitative argument.
A coil spring suspension is designed as a suspension for a mega truck that will be released in the future with a viscous damper. a.) What is the bumper's damping coefficient such that the system has a damping ratio of 1.25 when the bumper is engaged by a 20 tons railroad car and has a stiffness of 2 x 10^5 N/m? b.) Another load for the truck has been installed that has a mass of only 4500 kg. What are the natural frequency and a damping ratio of a system with the coil spring suspension when the load is being applied?
Example # 1 For the mechanical system shown, a force of 2 lb( step input) is applied to the system, the mass oscillates, as shown in fig. determi from this response curve. The displacement x is measured from the equilibrium position m, b, k of the system 0.0095 ft 0.1 ft 4 5 (b)
3 Problem Consider the following model of a mechanical system: The system has a mass m, a linear spring with stiffness k, and two identical dampers with damping constant b. The left wall generates an input motion xin (t) that causes the mass to undergo a displacement x(t) from its equilibrium position. The initial position and velocity are zero. • Find the ODE describing the motion of the system by drawing a free-body diagram and applying Newton's 2nd Law. Your answer should be in terms of the following variables: Xin Xin, X, X, x, b,k, m Show that the transfer function is G(s) = - X(s) Xin(s) = 2b m k s+ m k s²+ + m When taking the Laplace transform L[xin(t)] you may assume the initial (input) condition Xin(0) = 0.
Consider the double mass/double spring system shown below. - click to expand. Both springs have spring constants k, and both masses have mass m; each spring is subject to a damping force of F friction We can write the resulting system of second-order DEs as a first-order system, w'(t) = Aw(t), with w = (x₁, x₁, x₂, x₂) T For values of k = 4, m = 1 and c = 21,2 = 0.5±3.2i, V₁ 23,4= - 0.5 ± 1.13i, V3 1, the resulting eigenvalues and eigenvectors of A are -0.039 0.248i 0.813 0.024 +0.153i -0.502 -0.134-0.302i' 0.409 -0.216 - 0.489i 0.661 and (a) Find a set of initial displacements x₁(0), x₂(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero. (x₁(0), x₂(0)) Enter your answer using angle braces, (and ). cx (friction proportional to velocity). (b) At what frequency will the masses be oscillating in this mode? Frequency = rad/s

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