Problem 2) For a 2 DOF system the equations of motion are given as: m9 [m...
Problem 2) For a 2 DOF system the equations of motion are given as: [mi 0 0 m2 (X2 mig L -m29 L -m29 L m29 L Where m1 =m2 =m g=gravity and L =length a) Determine the frequencies and mode shapes. b) Verify that the natural modes are orthogonal. c) Determine the response fX:(0) Note: x1(t) = xo , x2(t) = 0 and xi(t) = xo , iz(t) = 0 d) If the system is excited by a harmonic...
4. (30%) Consider a 2-dof system with 8 -2 1 0 [m] = 0 4 [k] = -2 2 (a) Find the natural frequencies and mode shapes of the system. (b) Find the uncoupled modal equations. (c) If the system has proportional damping and it is known that the damping ratios are both 0.05 for the 2 modes, find the damping matrix of the system.
2. For the following 3-DOF spring-mass system: (a) Derive the equations of motion. (b) Assuming ki-k2-k3-k and mi-m2-m3-m, determine the natural frequencies and mode shapes. rt
Homework 7: Undamped, 2-DOF System 1. A system with two masses of which the origins are at the SEPs is shown in Figure 1. The mass of m2 is acted by the external force of f(t). Assume that the cable between the two springs, k2 and k3 is not stretchable. Solve the following problems (a) Draw free-body diagrams for the two masses and derive their EOMs (b) Represent the EOMs in a matrix fornm (c) Find the undamped, natural frequencies...
For the following 2DOF linear mass-spring-damper system r2 (t) M-2kg K -18N/m C- 1.2N s/m i(t) - 5 sin 2t (N) f2(t)-t (N) l. Formulate an IVP for vibration analysis in terms of xi (t) and x2(t) in a matrix form. Assume that the 2. Solve an eigenvalue problem to find the natural frequencies and modeshape vectors of the system 3. What is the modal matrix of the system? Verify the orthogonal properties of the modal matrix, Ф, with system...
Find the total response of a single-DOF system with m = 10 kg, c = 20 N-s/m, k = 4000 N/m, xo = 0.01m and v0 = 0 when an external force F(t) = 400cos(5t) acts on the system. Assignment 2 1. Find the total response of a single-DOF system with m = 10 kg, c = 20 N-s/m, k = 4000 N/m, x, = 0.01m and Vo=0 when an external force F(t) = 400cos(51) acts on the system
Homework 8: Modal and Direct Solution Approaches Figure 1 shows a system with two masses. The two coordinates of which the origins are set up at the unstretched spring positions are also shown in Fig. 1. The system is excited by the force f(t) 1. (a) Draw the FBDs for the system and show that the EOMs can be written as (b) Find the undamped, natural frequencies and the corresponding mode shapes of the system for the given system parameters...
MEMB343 MECHANICAL VIBRATIONS ASSIGNMENT l. For the system shown in Figure 1, where mi=5 kg, m,-10 kg, ki=1000 N/m, k2-500 N/m, k, 2000 N/m, fi-100sin(15t) N and f-0, use modal analysis to determine the amplitudes of masses m, and m2. The equations of motion are given as sin(15t), wth natura frequencies 5 01[i, 0 10 500-500x, 500 2500jx, x,[100 ω,-14.14 rad's and a, = 18.71 rad/s, and mode shapes, Φ',, and Φ' k, Im Figure 1 MEMB343 MECHANICAL VIBRATIONS ASSIGNMENT...
4. (35 pts) Consider the system defined by: xit 5x1-2x2-R (1) #2-2x, +2x2 F) a) Compute the natural frequencies and the mode shapes. /dland -JS -2N5 b) Calculate the response for F(t)-F(t)-0 and initial conditions xo- e) Calculate the response for F-cosr, F,(o)-0 and initial conditions and -0. 0 d) Calculate Bi and B2 such that the system: -2x1 + 2x2-B2cos/6t does not experience resonance. 4. (35 pts) Consider the system defined by: xit 5x1-2x2-R (1) #2-2x, +2x2 F) a)...
Problem: Find the natural frequencies of the system shown in Figure. Take m 2 kg ma 2.5 kg ms 3.0 kg me = 1.5 kg 914 Given: Four degree of freedom spring-mass system with given masses an stiffnesses. Find: Natural frequencies and mode shapes. Approach: Find the eigenvalues and eigenvectors of the dynamical matrix. 1. Determine [m] and [k] matrices of the vibrating system with all details 2. Determine [DI matrix. 3. Determine Natural frequencies and mode shapes analytically 3....