Two particle of respective masses mi = 4m and m2 = 3m are attached to two spring of %3D respective spring constants k and 3k as shown in the adjacent figure. k ; 4m 3k 3m a) Write down the Lagrangian of the system. b) Obtain Lagrange equations for each coordinate. c) Calculate the normal frequencies of the coupled oscillations. Deduce which frequency correspond to the symmetric and non-symmetric modes. d) Find the normal modes (eigenvectors) associated with the normal frequencies. e) Deduce a general solution for x1(t) and x2(1). f) Obtain a normal coordinate for each of the normal frequencies.

Two particle of respective masses mi = 4m and m2 = 3m are attached to two spring of %3D respective spring constants k and 3k as shown in the adjacent figure. k ; 4m 3k 3m a) Write down the Lagrangian of the system. b) Obtain Lagrange equations for each coordinate. c) Calculate the normal frequencies of the coupled oscillations. Deduce which frequency correspond to the symmetric and non-symmetric modes. d) Find the normal modes (eigenvectors) associated with the normal frequencies. e) Deduce a general solution for x1(t) and x2(1). f) Obtain a normal coordinate for each of the normal frequencies.

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

A 2 DOF mass-spring system shown below with n = 1, k = 8, and m initial conditions 2₁ (0) = 1, 2₂(0) = 0, ₁ (0) = 0, and ₂(0) = 1. (a) Find the natural frequencies and normalized modal matrix [X] for the system. T x₁ (1) (b) Using modal analysis and the Cartesian solution form, find the free vibration re- sponse of the system x₁(t) and x2(t). Ţ x2₂ (1) www. 00000 = m m₁ = 00000 k₁ = k 00000 k₂ = nk m₂ = m k3= k = 2 DOF mass-spring system. 2 with the
For the system below, the mass, m is attached to the end of a cantilever beam of length, l, elasticity, E, and moment of inertia, I. Derive the equation of motions and find the natural frequencies in two cases by lagrange energy method a) k1=k2=0 b) k1≠k2≠0
(1) The bar of mass, m= 10kg and length L=lm is fixed at the left end as shown below. Find; (a) The equations of motion of the system, k=2000N/m, k= 1000 N-m/rad. (b) The natural frequencies of the system (c) The response of the system if initial displacements are 01(0) =5rad and 0 2(0) = -5rad k, k k 30 kg Figure 1: spring - mass system
t. Design the following application case The mechanical vibration system is shown in Figure (a). When a step input with 3N amplitude is subjected to this system, the movement rule of the mass m is shown in Figure (b). Evaluate the values of mass m, damping constant c and spring constant k. m (a) F-3N XX 1.0 x(1)(cm) 0 2 (b) M,-0.095 Ⓒ
Q5:- in the fig M= 20 kg, c-7 N.s/m, k=10 N/M, the excitation is 10 sin5t, find natural frequency, excition frequency Excitation force, is the device in the resonse Frequency, Is the device under damping? Find the equation of motion A typical accelerometer Schematic of an accelerometer
3
i, Consider the following 1-DOF system, where k :0-3 kg, calculate the natural frequency in rad/s and Hz. Also find the period of scillations and the maximum displacement if the spring isinitially displaced 10 am with no initial velocity. 857.8 N/m and m = 49.2 x
Q3/ For the system shown in figure if the damping ratio is 0.3 derive the equation of motion using Newton's second law and find: a- The natural frequency. b- The damping constant. c- The ratio of any consecutive cycles. k=2 x 10 N/m m = 4.2 kg 10 cm 40 cm 60 cm ITTTI
A simple harmonic oscillator, of mass and natural frequency , experiences anoscillating driving force . Therefore, its equation of motion is where is its position. Given that at we have , , find thefunction . Describe the solution if is approximately, but not exactly, equal to .
A force of 20 newton stretches a spring 1 meter. A 5 kg mass is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 10 times the instantaneous velocity. 1) Let x denote the downward displacement of the mass from its equilibrium position. [Note that x>0 when the mass is below the equilibrium position. ] Assume the mass is initially released from rest from a point 3 meters above the equilibrium position. Write the differential equation and the initial conditions for the function x(t) 2) Solve the initial value problem that you wrote above. 3)Find the exact time at which the mass passes through the equilibrium position for the first time heading downward. (Do not approximate.) 4)Find the exact time at which the mass reaches the lowest position. The "lowest position" means the largest value of x
3
A particle of mass m attached to a rigid support by a support of constant k. At equilibrium, the spring hangs vertically. This mass-spring system is joined by another identical oscillator, whose spring is hung from the previous mass. Consider only the vertical movement. a) write the equations of motion of the coupled system b)Compute the normal mode frequencies for one-dimensional vertical oscillations and then show that the ratio of the two normal frequencies is √5+1/√5−1 c)Find the ratio of the amplitudes of the two masses in each separate mode. (You do not need to consider the force of gravity acting on the masses because it is independent of the displacements.)