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Unformatted text preview: ∑F x f − k x − f f = mx y ) N − mg = 0 ∑M cg ) f r + f f r = J cg x r Step 5: Evaluate the set of simultaneous differential equations to yield the equations of motion. [Mechanics II, Kinematics & Dynamics] Lecture Notes -706/16/06 12:25 PM Mechanical Vibrations I ∑F y ) N = mg ∑M ∑F x cg ) & ∑ M cg ) J f f = − f − cg x 2 r J f − k x − − f − cg r2 x = mx Step 5a: Identify the actual equation of motion. J cg m+ 2 x + kx = 2 f r Step 6: Solve the simultaneous set of differential equations of motion. The total solution ( xT (t ) ) involves the sum of two parts: the particular solution ( x p (t ) ) and the homogeneous solution ( xh (t ) ). [Differential Equations] Step 6a: The particular solution involves knowing the exact form of the forcing function ( f (t ) ). [Differential Equations] Step 6b: A second order homogeneous constant coefficient differential equation has complex exponentials as its solution. [Differential Equations, Numerical Methods] xh (t ) = Ceλt and therefore xh (t ) = C λ eλt & xh (t ) = Cλ 2eλt J cg 2 λt m + 2 λ + k Ce = 0 r therefore the system poles (eigenvalues) are: λ1,2 = ± j k m+ J cg r2 and the homogeneous solution is: xh (t ) = C1eλ1t + C2eλ2t Step 6c: Formulate the complete solution ( xT (t ) = x p (t ) + xh (t ) ) and evaluate the initial conditions to eliminate the constants of integration. [Differential Equations] Lecture Notes -8- 06/16/06 12:25 PM Mechanical Vibrations I Note: if this example had involved more than one independent dynamic coordinate, the solution would have additionally required the solution of an eigenvalue/eigenvector problem. [Differential Equations, Numerical Methods] Lecture Notes -9- 06/16/06 12:25 PM Mechanical Vibrations I #2 - Newton’s Method ( -or- d'Alembert’s Method ) Force Balance ∑F = Mq Moment Balance G ( -or- ∑F −Mq G =0 ) P ∑ M P = J Pθ + rG / P × M qP ( -or- ∑ M P − J Pθ − rG / P × M qP = 0 ) G Typical problems with Newton’s ( -or- d'Alembert’s ) formulation: • Be sure to establish the number of degrees of freedom first and formulate all terms in only those variables. Clearly identify which degrees of freedom are relative coordinates versus absolute coordinates. Also, clearly identify what will be the positive direction of motion for each coordinate. Watch out for rotational/translational problems. State any constraint relationships that relate independent and dependent coordinates. • Evaluate the static balance for the problem in order to determine whether the orientation of the system in the gravitational field will effect the equations of motion (Are the weights of the objects balanced by an initial static deflection in the springs?). When in doubt, perform a static force balance to determine the appropriate constraint equation. • For displacement, velocity and acceleration terms, be sure to develop absolute or relative displacement, velocity and accelerations of appropriate points as required. Watch...
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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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