ADVANCED VIBRATION Lecture #1 Asst. Prof. Dr. Mahir Hameed Majeed ©2018

contents Multi-degree of freedom system 1.General 3-D.O.F system 2.Lagrange's equation 3.Methods of solving Eigenvalue problems 12/22/2018lecture #12

objectives 1.To know how to deal and analyze the multi-degree of freedom vibration 2.To evaluate the useful equations and methods of analyzing such case of vibration 3.To solve different cases of multi-degree of freedom systems 12/22/2018lecture #13

principles A multi degrees of freedom (D.O.F) system requires two or more coordinates to describe its motion. These coordinates are called generalized coordinates when they are independent of each other and equal in number to the degrees of freedom of the system. Normal mode vibrations are free vibrations that depend only on the mass and stiffness of the system and how they are distributed. 12/22/2018lecture #14

General 3 D.O.F system Many engineering vibration problems can be treated by the theory of one-degree-of-freedom systems. More complex systems may possess several degrees of freedom. The standard technique to solve such systems, if the degrees of freedom are not more than three, is to obtain the equations of motion by Newton's law of motion, by the method of influence coefficients, or by Lagrange's equations. Then the differential equations of motion are solved by assuming an appropriate solution. Solving of differential equations of motion becomes increasingly more laborious as the number of degrees of freedom increases. While the solution is straight forward for an undamped multi-degree-of-freedom system, it becomes much more complex for a damped system. Analytical/closed-form solutions can be established for 2degrees of freedom systems. But for more degree of freedom systems numerical analysis using computer is required to find natural frequencies (eigenvalues) and mode shapes (eigenvectors). 12/22/2018lecture #15

= × No. of D.O.F No. of masses (m) of the system No. of possible types of motion of each mass 12/22/2018lecture #16

Flexibility Matrix: 12/22/2018lecture #17

Example 1: 12/22/2018lecture #18

Problem 1: 12/22/2018lecture #19

Stiffness Matrix: 12/22/2018lecture #110

Example 2: Find stiffness matrix for the spring-mass system shown in the figure below. 12/22/2018lecture #111

Solution: 12/22/2018lecture #112

12/22/2018lecture #113

Hence, the stiffness matrix can be given by: Problem 2; Obtain the stiffness matrix for the following system: 12/22/2018lecture #114

Lagrange Equation 12/22/2018lecture #115

12/22/2018lecture #116

Example 3 (Spring pendulum): Consider a pendulum made of a spring with a mass m on the end as showing in the Figure. The spring is arranged to lie in a straight line (which we can arrange by, say, wrapping the spring around a rigid massless rod). The equilibrium length of the spring is l, Let the spring have length l + x(t), and let its angle with the vertical be θ(t). Assuming that the motion takes place in a vertical plane, find the equations of motion for x and θ. 12/22/2018lecture #117

Solution: The kinetic energy may be broken up into the radial and tangential parts, so we have; The potential energy comes from both gravity and the spring, so we have; The Lagrangian is therefore; 12/22/2018lecture #118

By partial differentiation for the two variables, x and θ 12/22/2018lecture #119

Problem 3: Two massless sticks of length 2r, each with a mass m fixed at its middle, are hinged at an end. One stands on top of the other, as shown in the figure. The bottom end of the lower stick is hinged at the ground. They are held such that the lower stick is vertical, and the upper one is tilted at a small angle ε with respect to the vertical. They are then released. At this instant, what are the angular accelerations of the two sticks? Work in the approximation where ε is very small. 12/22/2018lecture #120

Methods of solving Eigenvalue problems 12/22/2018lecture #121

12/22/2018lecture #122

(1) Adjoint Matrix method: 12/22/2018lecture #123

Example 4: Find the normal modes for torsional vibration of a shaft with two rotors as shown in figure below: 12/22/2018lecture #124

Solution 12/22/2018lecture #125

12/22/2018lecture #126

12/22/2018lecture #127

Other methods: 2. The Householder-QR inverse(HQRI) solution is the most efficient for calculation of all eigenvalues and eigenvectors. 2. The determinant search technique (polynomial root solver) is most appropriate to determine the lowest eigenvalue and corresponding eigenvectors of systems with a small bandwidth. 3. The subspace iteration solution is effective for the lowest eigenvalue and eigenvectors in very large systems with a large bandwidth. 12/22/2018lecture #128

Problem 4: solve the last example using one the listed methods, then compare the results with that obtained by the Adjoint matrix method. 12/22/2018lecture #129

References K. J. Bathe, and E. L. Wilson, Numerical Methods in Finite Element Analysis, Englewood Cliffs, N.J., Yousef Saad, Numerical Methods For Large Eigenvalue Problems, Society for Industrial and Applied Mathematics, David Morin, The Lagrangian Method, harvard.edu (draft version), 2007 Owus M. Ibearugbulem, Osasona, E. S. and Maduh, U. J., Iterative Determinant Method for Solving Eigenvalue Problems, International Journal of Computational Engineering Research (IJCER), ISSN (e): 2250 – 3005 || Vol, 04 || Issue, 9 || September – 2014 ||. How, Deyst, Lagrange's Equations, Massachusetts Institute of Technology, (Based on notes by Blair 2002), /22/2018lecture #130