Consider the mass-spring system in Figure 1. For the systemshown in Figure 1, suppose that k₁ = k, k₂ = k3 = 2k, andm₁ = m₂ = m. Obtain the equations of motion in terms ofX₁ and x₂. Assume that x₂ > x₁.X1A. Draw the necessary free-body diagrams and derive thedifferential equations of motion.B. Determine the transfer functions X₂ (s)/F(s). All the initialconditions are assumed to be zero.x2k3k₂m1m2wwwf(t)Figure 1k₁

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Answered: Consider the mass-spring system in…

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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(c) The system shown in figure task 2c is displaced from its static equilibrium position to the right a distance of 0.01m. An impulsive force acts towards the left on the mass at the instant of its release to give it an initial velocity Vo in that direction. If the system has the following parameters K=157OON/m, c=1570N-sec/m, m=9.8kg Figure: TASK 20 (a) Derive the expression for the displacement from the equilibrium position in terms of time t and initial velocity Vo (b) What value of Vo would be required to make the mass pass the position of the static equilibrium 1/100 sec after it is applied
Figure below shows the single degree of freedom control surface/actuator system. The control surface has a moment of inertia J = 656 kg.m2 about the hinge, the effective actuator stiffness and damping values are k= 6 MN/m and c = 106 kNs/m respectively and the rotation of the control surface is A rad. The actuator lever arm has length a = 0.57 m. It is assumed that the main surface of the wing is fixed rigidly as shown in the figure. Assume that the system is hit by stone and hence move from rest position by an initial velocity of vo = 3.51 rad/s. Maximum amplitude response is: 0(t) k. d O 1.7 degrees O 1.25 degrees O 334 degrees O 2.56 degrees O 2.05 degrees
4. This question relates to the static and dynamic balance of an unbalanced shaft that is supported by bearings at its two ends, as shown in Figure Q4. The shaft rotates at 300 rev/min and the out-of-balance loads are given below. m1(kg) = 1.6 m2(kg) =2.4 m3 = 2 kg Z O 4₁ X 13 Figure Q4 0₁ 02 03 30° 120° 315° r1(m)= 0.7 r2(m)=0.5 r3(m)= 0.6 11(m)=1.4 12(m)= 2.0 13(m)=2.5 a) Using the data provided in Table Q4, take moments about the left-hand end of the shaft and so construct the mrl diagram for the system. Calculate the angular position and the mass of material needed to close the mrl diagram, if the mass is to be added at a point 1.7 m from the left-hand end of the shaft at a radius of 1 m. b) Construct the mr diagram the left-hand end of the shaft at a radius of 1 m to close the mr diagram and so balance the forces acting on the shaft.
The torsional system of Figure Q6 oscillates about the centre of the disc. A linear spring and a linear damper are attached to the disc at radius r. A linear damping element is also attached at the edge of the disc at radius R. For this system assume that r = 0.15 m, R = 0.25 m, mass of disc m = 5 kg, stiffness coefficient k = 1200 N/m, and damping coefficient c = 10 Ns/m. Set up the differential equation of motion for this mechanical system. k Fin с R Figure Q6 C Disc 0
Two identical thin bars, each of mass m and length L, are welded together at a right angle as shown in figure 1. This T-shaped assembly is pinned to ground at point 0. Block B is given a prescribed forced displacement xg = b sin øt. Friction and damping are neglected. a) Derive the equation of motion for the system in terms of the angular position of the bars, 6, and parameters m, L, k, and g; b) Derive an expression for the natural frequency of the system; c) Derive an expression for the forced angular response of the system; L/2 L/2 L Хв
Problem 11
A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c ). The mass is set in motion with initial position xo and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) =C, e - pt cos (0,t-a,). Also, find the undamped position function u(t) = Co cos (@ot - ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c= 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). C = 1 m = - 4 c= 3, k= 8, x, = 7, vo = 0 x(t) = 14 e -4t - 7 e -8t , which means the system is overdamped. (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater…
PROBLEM3: - Derive the equation of motion and find the steady-state response of the system shown in Figure 3 for rotational motion about the hinge O for the following data: k₁ = k₂ = 5000 N/m, a = 0.25 m, b = 0.5 m, 1 = 1 m, M = 50 kg, m 500 N, w = 1000rpm. 10 kg, Fo F(t)= Fo sin cot Uniform rigid bar, mass m T esssst reelle Figure 3 |k₂ M
Figure below shows the single degree of freedom control surface/actuator system. The control surface has a moment of inertia J = 30 kg.m2 about the hinge, the effective actuator stiffness and damping values are k = 3000 N/m and c = 2610 N.s/m respectively and the rotation of the control surface is 0 rad. The actuator lever arm has length a = 0.2 m. A force f (t) = 108 (t) N is applied to the control surface at a distance d = 1 m from the hinge. It is assumed that the main surface of the wing is fixed rigidly as shown in the figure. Assume small angle approximation. Determine the tip impulse response (displacement at the location of f (t)) att = 0.087 s 0(t) f(t) d 37.34 mm 12.45 mm 24.90 mm 8.30 mm 49.79 mm
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 Ffriction -cz' (friction proportional to velocity). We can write the resulting system of second-order DEs as a first-order system, t' (t) = Au(t), with = (₁, 21, 22, 2₂) I For values of k = 4, m = 1 and c = 1, the resulting eigenvalues and eigenvectors of A are -0.039-0.248i 0.813 A₁2=-0.5±3.2i, v₁ = 0.024 +0.153i -0.502 -0.134-0.302i 0.409 -0.2160.489 0.661 (a) Find a set of initial displacements (0), 2(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero. A3,4 -0.5± 1.13i, z = and (2₁ (0), ₂(0)) = Enter your answer using angle braces, (and). (b) At what frequency will the masses be oscillating in this mode? Frequency rad/s
Pls see the question in the attcahed image and solve using either Newton’s method or Lagrange Energy method .
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.

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