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 frictionWe can write the resulting system of second-order DEs as a first-order system, w'(t) = Aw(t), with w = (x₁, x₁, x₂, x₂) TFor values of k = 4, m = 1 and c=21,2 = 0.5±3.2i, V₁23,4= - 0.5 ± 1.13i, V31, the resulting eigenvalues and eigenvectors of A are-0.039 0.248i0.8130.024 +0.153i-0.502-0.134-0.302i'0.409-0.216 - 0.489i0.661and(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

Given:To find:Initial displacementsFrequency

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 ´(1) = Aw(1), with w = (x₁, x₁, x₂, x₂) ™ For values of k = 4, m = 1 and c = 1, the resulting eigenvalues and eigenvectors of A are 21.2 -0.5±3.2i, V₁ = -0.039-0.248i 0.813 0.024 +0.153i -0.502 -0.134-0.302i 0.409 23.4 = -0.5 ± 1.13i, V3 = -0.216 - 0.489i 0.661 and - cx (friction proportional to velocity). (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. (0), x₂(0)) = Enter your answer using angle braces, (and). (b) At what frequency will the masses be oscillating in this mode? Frequency = rad/s

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 ´(1) = Aw(1), with w = (x₁, x₁, x₂, x₂) ™ For values of k = 4, m = 1 and c = 1, the resulting eigenvalues and eigenvectors of A are 21.2 -0.5±3.2i, V₁ = -0.039-0.248i 0.813 0.024 +0.153i -0.502 -0.134-0.302i 0.409 23.4 = -0.5 ± 1.13i, V3 = -0.216 - 0.489i 0.661 and - cx (friction proportional to velocity). (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. (0), x₂(0)) = Enter your answer using angle braces, (and). (b) At what frequency will the masses be oscillating in this mode? Frequency = rad/s

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

See similar textbooks

Similar questions

Q2: For mass-spring system shown. The mass is given an initial displacement x(0)= 0.1 m, and released from rest. Find: 1- The position of the mass after 2 seconds. 2- The velocity of the mass after 2 seconds. 3- Plot the response for three cycles and label the result from 1 & 2. 4- What is the period of oscillation? 5- What is the acceleration of mass (m) after 5 second? Į x(t) k=100 N/m m = 4 kg
WERK the equivalent spring constant for below if the constant spring is 80 ?N/m k1 m3 k3 k6 m1 k4 m6 m2 k5 840 Non one of them 540 480 O k2
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
6. The electro-mechanical system shown below consists of an electric motor with input voltage V which drives inertia I in the mechanical system (see torque T). Find the governing differential equations of motion for this electro-mechanical system in terms of the input voltage to the motor and output displacement y. Electrical System puthiy C V V₁ R bac (0) T bac T Motor - Motor Input Voltage - Motor Back EMF = Kbac ( - Motor Angular Velocity - Motor Output Torque = K₂ i Kbacs K₁ - Motor Constants Mechanical System M T Frictionless Support
For the shown spring-mass-damper system. Let m =1 kg, b = 2 N.s/m, k = 2 N/m and f(t) = sin2t N. (a) Write the equation of motion. (b) Find the amplitude of the steady state oscillations. (c) Find the steady state maximum acceleration.
Solve the following IVP's for the undamped (b= 0) spring-mass system. Describe, in words, the meaning of the initial conditions. Also, state the period and frequency and describe their meaning in layman's terms. Assume we are using the metric system.
4 QUESTION 20 The car bridge in Figure Q20 can be modelled as a damped-spring oscillator system with mass M = 10000 kg, spring coefficient k = 50000 N-m-1 and damping constant c = 50000 N-s-m-1. Cars cross the bridge in a periodic manner such that the bridge experiences a vertical force F (N) expressed by F = mg sin(10t) where m = 1136 kg is the average mass of passing cars, g = 9.81 m-s-2 is the gravitational acceleration and t (s) is the time. Determine the maximum force magnitude transmitted to the foundation (see Figure Q20) during the steady-state oscillatory response of the system. Provide only the numerical value (in Newtons) to zero decimal places and do not include the units in the answer box. E m M foundation Figure Q20: Vibrating car bridge.
A mass of 2 kilograms is on a spring with spring constant k newtons per meter with no damping. Suppose the system is at rest and at time t = 0 the mass is kicked and starts traveling at 2 meters per second. How large does k have to be to so that the mass does not go further than 3 meters from the rest position? use 2nd order differential equations to solve (mechanical vibrations)
Tutorial 4 Determine the normalized dynamic matrix for the system as shown in Figure 4. The values of K,=200 kN/m, K2=150 kN/m, K3=100 kN/m, K,=Ks= 50 kN/m, Kg=200 kN/m, m,= 100 kg, m2= 50 kg, m3= 20 kg, & m4= 200 kg. K1 K3 K2 m2 X2 m3 X3 K. m4
A 3-kilogram mass hangs from a spring with a constant of 4 newtons per meter. The mass is set into motion by giving it a downward velocity of 3 meters per second. Damping in newtons equal to five times the velocity in meters per second acts on the mass during its motion. At time t = 6 seconds, it is struck upwards with a hammer imparting a unit impulse force. Set up the initial-value problem to compute the displacement of the mass as a function of time. Do not solve the equation.
Q3) A 16 Ib cylinder stretches a spring 2/27 inches by itself. The spring is initially displaced 1/2 ft upwards from its equilibrium position and given an initial velocity of 12 inches/sec downward. Find the displacement at any time. The damping force on the system is 6 times the instantaneous velocity.
3. A 100-pound weight is suspended from a vertical spring whose module is 100 pounds per foot. The weight is pulled downward 9 inches from its equilibrium position and released. Viscous fluid exerts 20 Ibs of force when the plunger moves through it at 0.5 ft/sec; 3.a What is the damping condition? Write the equation which describes the motion of the weights. Show graph. 3.b displacement 3.c velocity 3.d acceleration