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Unformatted text preview: tarting with the original secondorder differential equation. f (t ) = mx + cx + kx Augment the equation with the identity relationship 0 = mx − mx . Lecture Notes 70 06/16/06 12:25 PM Mechanical Vibrations I f (t ) = mx + cx + kx 0 = mx − mx Rearranging the equations yields, mx = f (t ) − cx − kx mx = mx Converting to a vector/matrix equation set yields, mx f (t ) − cx − kx = mx mx x Substituting the state vector, y = , into the above expression yields, x x f (t ) − cx − kx d x d m = m = m { y} = m { y} = mx dt x dt x f (t ) − cx − kx m { y} = mx Dividing by m and manipulating the righthand side similarly yields,
k f (t ) c k f (t ) c k f (t ) c − x − x − x − x − x − { y} = m m m = m + m m = m + m m x 0 0 1 0 x x k k f (t ) c f (t ) c − x − − − { y} = m + m m = m + m m { y} x 0 1 0 0 1 0 Therefore, the statespace expansion of the original equation becomes,
k f (t ) c − − { y} = m + m m { y} or 0 1 0 k f (t ) c − x d x − m = m + m x dt x 0 0 1 At this point, the original secondorder differential equation has been manipulated into two, simultaneous, firstorder differential equations. In this form, it is suitable for use in
Lecture Notes 7106/16/06 12:25 PM Mechanical Vibrations I the numerical integration scheme presented. (Actually, it can be used with any firstorder numerical integration scheme.) Numerical Solution
Combining the results of the two preceding sections, allows the development of an iteration scheme for numerically evaluating the response of the system to an arbitrary input (force). Constructing the state vector from the current conditions, { y}n = xn xn Evaluating the state equation of motion yields the current derivative, { y}n f (tn ) c − = m + m 0 1 − k m { y}n 0 Finally, evaluating the numerical integration rule (in this case, Euler’s Method) yields, { y}n+1 = { y}n + ∆t { y}n or xn+1 xn xn = + ∆t xn+1 xn xn This provides a simple marching scheme to numerically integrate the equations of motion applying both initial conditions and an arbitrary force. In general, a more precise numerical integration rule may be substituted for the last step*. * TODO: Add 2DOF nonlinear example. 7206/16/06 12:25 PM Lecture Notes Mechanical Vibrations I #15 – Multiple Degree of Freedom – Detailed Overview
In general, this section will not be covered during the course Mechanical Vibrations I. Instead, it is provided as a reference for those who wish to know a little more about MDOF systems, but who for various reasons may not end up taking the two following courses, Mechanical Vibrations II & III. General Solution Approach
The matrix equation of motion for a general multidegreeoffreedom system can be written as (Time Domain): [ M ]{ x(t )} + [C ]{ x(t )} + [ K ]{ x(t )} = { f (t )}
The solution of this linear, constant matrix coefficient, second order differential equation follows the solution approach for the simpler single degree of freedom problem. The solution take...
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This note was uploaded on 09/29/2013 for the course MECHANICAL ME taught by Professor Regalla during the Fall '11 term at Birla Institute of Technology & Science, Pilani  Hyderabad.
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