A closed form solution is not practical and numerical

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Unformatted text preview: tarting with the original second-order 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 right-hand 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 state-space 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 second-order differential equation has been manipulated into two, simultaneous, first-order 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 first-order 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 non-linear 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 multi-degree-of-freedom 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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