![A two degree of freedom system has a mass matrix, [M] and a non-normalized modal column matrix U given by 1 2 Find the normalized modal column matrix, U]n such that [U] [M][U]n-[ transpose and [4] is the identity matrix. I, where T denotes the](http://img.homeworklib.com/questions/fb483420-7ee8-11eb-98d0-9156fbcef532.png?x-oss-process=image/resize,w_560)
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A two degree of freedom system has a mass matrix, [M] and a non-normalized modal column...
4.62. Consider the following two-degree-of-freedom system and compute the response assuming modal damping ratios of (1...
4.62. Consider the following two-degree-of-freedom system and compute the response assuming modal damping ratios of (1 = 0.05 and 12 = 0.01 :
1. Consider the two degree of freedom system shown. (a) Find the natural frequencies for the system (b) Determine the modal fraction for each mode. (c) Draw the mode shapes for each mode and iden...
1. Consider the two degree of freedom system shown. (a) Find the natural frequencies for the system (b) Determine the modal fraction for each mode. (c) Draw the mode shapes for each mode and identify any nodes for each mode. (d) Demonstrate mode shape orthogonality. (e) If F- and the motion is initiated by giving the mass whose displacement is a velocity of 0.2 m/s when in equilibrium, determine 0) and ,0 (f) Determine the steady-state solution for both *)...
Homework 8: Modal and Direct Solution Approaches Figure 1 shows a system with two masses. The two...
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...
The vector-matrix form of the system model is: (18 0 18000 72001 x f(t) or Mx...
The vector-matrix form of the system model is: (18 0 18000 72001 x f(t) or Mx + Kx = f(t) 08.3. -7200 8000 | X, 3. (1) X= M = (18 0 08 K 18000 -7200 7200 8000 and f(t) = (1) 12(0)] [x₂(t) The system's eigenvalues, natural frequencies, and eigenvectors are: 1 2 = 400, 0, = 20 s', and v, 1.5 1.) = 1600, 0), = 40 s', and v, = -1.5 1 1 The inverse of modal...
The vector-matrix form of the system model is: (18 0 18000 72001 x f(t) or Mx...
The vector-matrix form of the system model is: (18 0 18000 72001 x f(t) or Mx + Kx = f(t) 08.3. -7200 8000 | X, 3. (1) X= M = (18 0 08 K 18000 -7200 7200 8000 and f(t) = (1) 12(0)] [x₂(t) The system's eigenvalues, natural frequencies, and eigenvectors are: 1 2 = 400, 0, = 20 s', and v, 1.5 1.) = 1600, 0), = 40 s', and v, = -1.5 1 1 The inverse of modal...
Governing equation of motion and modal transformation
A two DOF system consists of two equal masses, m, two springs having stiffnesses k and 2k, and two viscous dampers each having damping coefficient c, see the figure.Here; k = m = c =1 [N/m, kg, Ns/m].a) Write the governing equation of motion in matrix form.b) Calculate the undamped eigenfrequencies and their corresponding mass normalized mode shapes.c) Derive the governing equations (in matrix form) in the modal domain, ie. use modal transformation. Let p(t)=10 N. Set up the equations...
Consider an undamped system where the vector-matrix form of the system model is: F(t) [8 olx...
Consider an undamped system where the vector-matrix form of the system model is: F(t) [8 olx Mx + Kx = 0 18X, + 2000 -1800 x -1800 4500 I:1-[0] The system is initially at rest with x(0) = 0 and x,0)=0 when input F(t) = 84 sin15t is applied to the system. Use the modal decomposition method described in chapter 5 to find the system response. Some intermediate results (find these as part of your solution) are: The system's two...
For the following 2DOF linear mass-spring-damper system r2 (t) M-2kg K -18N/m C- 1.2N s/m i(t) - ...
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...
Consider an undamped system where the vector-matrix form of the system model is: [F(t) [8 orë...
Consider an undamped system where the vector-matrix form of the system model is: [F(t) [8 orë Mx + x = LO 183, 2000 -1800 x (-1800 45001 The system is initially at rest with X (0) = 0 and 2,0)=0 when input F(t) = 84 sin 15t is applied to the system. Use the modal decomposition method described in chapter 5 to find the system response. Some intermediate results (find these as part of your solution) are: The system's two...
4.22. Consider the vibrating system described by 42 -2 1 Compute the mass-normalized stiffness matrix, the eigenvalues, the normalized eigen- vectors, the matrix P, and show that PTMP I and PTKP...
4.22. Consider the vibrating system described by 42 -2 1 Compute the mass-normalized stiffness matrix, the eigenvalues, the normalized eigen- vectors, the matrix P, and show that PTMP I and PTKP is the diaggaal matrix of eigenvalues Л
4.22. Consider the vibrating system described by 42 -2 1 Compute the mass-normalized stiffness matrix, the eigenvalues, the normalized eigen- vectors, the matrix P, and show that PTMP I and PTKP is the diaggaal matrix of eigenvalues Л
4.62. Consider the following two-degree-of-freedom system and compute the response assuming modal damping ratios of (1 = 0.05 and 12 = 0.01 :
1. Consider the two degree of freedom system shown. (a) Find the natural frequencies for the system (b) Determine the modal fraction for each mode. (c) Draw the mode shapes for each mode and identify any nodes for each mode. (d) Demonstrate mode shape orthogonality. (e) If F- and the motion is initiated by giving the mass whose displacement is a velocity of 0.2 m/s when in equilibrium, determine 0) and ,0 (f) Determine the steady-state solution for both *)...
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...
The vector-matrix form of the system model is: (18 0 18000 72001 x f(t) or Mx + Kx = f(t) 08.3. -7200 8000 | X, 3. (1) X= M = (18 0 08 K 18000 -7200 7200 8000 and f(t) = (1) 12(0)] [x₂(t) The system's eigenvalues, natural frequencies, and eigenvectors are: 1 2 = 400, 0, = 20 s', and v, 1.5 1.) = 1600, 0), = 40 s', and v, = -1.5 1 1 The inverse of modal...
The vector-matrix form of the system model is: (18 0 18000 72001 x f(t) or Mx + Kx = f(t) 08.3. -7200 8000 | X, 3. (1) X= M = (18 0 08 K 18000 -7200 7200 8000 and f(t) = (1) 12(0)] [x₂(t) The system's eigenvalues, natural frequencies, and eigenvectors are: 1 2 = 400, 0, = 20 s', and v, 1.5 1.) = 1600, 0), = 40 s', and v, = -1.5 1 1 The inverse of modal...
Consider an undamped system where the vector-matrix form of the system model is: F(t) [8 olx Mx + Kx = 0 18X, + 2000 -1800 x -1800 4500 I:1-[0] The system is initially at rest with x(0) = 0 and x,0)=0 when input F(t) = 84 sin15t is applied to the system. Use the modal decomposition method described in chapter 5 to find the system response. Some intermediate results (find these as part of your solution) are: The system's two...
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...
Consider an undamped system where the vector-matrix form of the system model is: [F(t) [8 orë Mx + x = LO 183, 2000 -1800 x (-1800 45001 The system is initially at rest with X (0) = 0 and 2,0)=0 when input F(t) = 84 sin 15t is applied to the system. Use the modal decomposition method described in chapter 5 to find the system response. Some intermediate results (find these as part of your solution) are: The system's two...
4.22. Consider the vibrating system described by 42 -2 1 Compute the mass-normalized stiffness matrix, the eigenvalues, the normalized eigen- vectors, the matrix P, and show that PTMP I and PTKP is the diaggaal matrix of eigenvalues Л
4.22. Consider the vibrating system described by 42 -2 1 Compute the mass-normalized stiffness matrix, the eigenvalues, the normalized eigen- vectors, the matrix P, and show that PTMP I and PTKP is the diaggaal matrix of eigenvalues Л
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