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

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B6
5. A mass on a spring is shown below. Let y(t) be the displacement of the spring as a function of time t. Position A is the equilibrium position corresponding to y = 0. As was done in class, we are taking the positive direction to be downward. The graph of the solution to an initial value problem for this spring is shown. Answer the following questions about this system. A m B A: y=0 C (a) The spring's starting position (at t = 0) could be at i. Position A (Equilibrium) ii. Position B (Above Equilibrium) iii. Position C (Below Equilibrium) iv. Not enough information to tell (b) The spring's starting velocity (at t = 0) is i. positive ii. zero iii. negative iv. Not enough information to tell (c) Immediately after being set in motion at t = 0) the spring
For the mechanical system shown, a- Find the equivalent mass, equivalent spring stiffness and equivalent damping coefficient when x (the displacement of the 2 kg block) is used as the generalized coordinate. b- Derive the equation of motion of the system in terms of equivalent system parameters (equivalent mass, equivalent spring stiffness and equivalent damping coefficient). 3000 N/m 2 kg 200 N.s/m /=0.04 kg-m² r = 10 cm 1000 N/m 1 kg I' 400 N.s/m ial
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
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.
S=2 U=0 T=0
The following data is given for the spring-mass-damper system shown in Figure • K1 = 250 N/m, K2 = 350 N/m • C1 = 150 N/ (m/sec), C2 = 200 N/ (m/sec) m = (25 +6) kg Determine: i. The undamped natural frequency. ii. The damping ratio. iii. The damped natural frequency. x(t) k 1 k 2 m C1 C2 Figure
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
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2 – Response of the two-degree-of-freedom system shown in to an initial displacements of uj 0.915 and u2 = 2.0 is such that q2 = 0. Find the mode shapes and a, where ki = a k2. Normalize each mode so that the modal mass has unit value. Write the equation for the force in the springs. k, 8m kz 2m U2 ui
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.