EA3_mechsystems_2

EA3_mechsystems_2 - EA3 Systems Dynamics Mechanical Systems...

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EA3: Systems Dynamics Mechanical Systems Sridhar Krishnaswamy 16 VI.4 DAMPED OSCILLATIONS Consider the {spring-damper}-mass system shown. Figure 6.11 : Damped spring-mass system There is a slight complication to this system. Is this one-dimensional? Actually, in this case, we have the spring and the damper in parallel (whereas thus far the elements were all connected in series). However the system is still one-dimensional in that the ties (the big vertical lines connecting the nodes of the elements in parallel) are assumed not to rotate, but only to translate in the x-direction. Therefore the entire tie connector can be thought of as just an extended node, and this is how we will treat them. Let us now crank our analysis machinery. State Variables: X = r sp 1 v m 3 Geometric Relations: ( i ) x 1 = 0 ( ii ) r sp 1 = x 2 - x 1 = x 2 ( iii ) r D 2 = x 2 - x 1 = x 2 v D 2 = ˙ x 2 ( iv ) x 3 - x 2 = constant ˙ x 3 = ˙ x 2 = v m 3 1 2 3 1:K 1 2:C 2 3:m 3

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EA3: Systems Dynamics Mechanical Systems Sridhar Krishnaswamy 17 Equilibrium Relations: ( v ) - f sp 1 - f D 2 = m 3 ˙ v m 3 Constitutive Relations: ( vi ) f sp 1 = K 1 r sp 1 ( vii ) f D 2 = C 2 v D 2 State Equations: (by now you should be able to do this) ˙ r sp 1 ˙ v m 3 = 0 1 - K 1 m 3 - C 2 m 3 r sp 1 v m 3 (†) Feeding this to an appropriate MATLAB m-file with K1=20, m3=10, C2=2, and initial condition X(0)={0, 0.2}’, we find: 0 5 10 15 20 25 30 35 40 45 50 -0.2 -0.1 0 0.1 0.2 Stretch of Spring 1 0 5 10 15 20 25 30 35 40 45 50 -0.2 -0.1 0 0.1 0.2 Velocity of Mass 3 Time Figure 6.13 : Damped oscillation of a spring-damper system 2 3 f sp1 f D2 m 3 ˙ v m 3
EA3: Systems Dynamics Mechanical Systems Sridhar Krishnaswamy 18 Note that this system exhibits some oscillatory behavior (like the spring-mass oscillator), but because of the energy dissipating damper, the amplitude of oscillations decays down with time (compare with the result of the spring-damper system which exhibited an exponential behavior). We were able to deduce the analytical solution to the spring-damper and the spring-mass systems. Now, try to guess the form of the analytical solution for the spring-damper-mass system above before reading on. Analytical solution: Just as we did for the undamped spring-mass oscillator of Chapter 3, I am going to swap out one of the state variables (v m3 ) in favor of the stretch of the spring (r sp1 ). Note that for this case, the stretch of the spring is the same as that of the position of the mass. This requires some simple algebra. The second of our state-equations (†) reads:

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This note was uploaded on 04/17/2008 for the course GEN_ENG 203 taught by Professor Krishnaswamy during the Fall '08 term at Northwestern.

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EA3_mechsystems_2 - EA3 Systems Dynamics Mechanical Systems...

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