EA3_springblock1 - EA3: Systems Dynamics Spring-block...

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EA3: Systems Dynamics Spring-block oscillator Sridhar Krishnaswamy 1 III The spring-block oscillator Let us consider a simple spring-block system in the absence of any friction or air drag. We assume that the block is rigid and has mass, and the spring is massless but is deformable. This is what is called a lumped parameter idealization of a simple mechanical system, where all the “springiness” (deformability) is attributed to the spring and all the “massiveness” is attributed to the block. In reality, real springs do have mass, and real blocks are deformable. However, ours is a useful approximation (if, for instance, the mass of the block is a lot more than that of the spring, and the stiffness of the block is a lot higher than that of the spring). Also, we can actually model useful real life systems with such simple idealized elements. Spring constant K (N/m) mass m r x Frictionless, No gravity Figure 3.1 : Undamped spring-block oscillator Let the position of the block when the spring is unstretched ('relaxed' position of the spring) be the origin of our coordinate system (see Fig.3.1). If the block is now moved by an amount r x as shown, the spring stretches (or compresses) and therefore starts to pull (or push) on the block with a force F x = - Kr x (we are considering a linear elastic spring which you have seen in EA2). From the free-body diagram of the block, we find that the equations of motion of block are given by: dv x dt = 1 m F x = - K m r x dr x dt = v x (3.1)
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EA3: Systems Dynamics Spring-block oscillator Sridhar Krishnaswamy 2 Let us say that at time t=0, the block is gently moved to r x0 and let go with zero velocity at this time, ie r x ( t = 0) = r xo ; v x ( t = 0) = v xo = 0 . Given these initial conditions, we want to find the subsequent motion of the block. You can modify one of the MATLAB m-files we have used previously to integrate the above numerically. MATLAB spits out the following graphs for the position and velocity of the block: Remarks: 1. There is something fishy about this numerical solution. It appears that the oscillation amplitude increases with time! This has to be incorrect as it goes counter to what we see physically. If anything, the oscillations will die down in reality, but that has to do with friction and energy dissipation which we have not included here. As we will see 0 10 20 30 40 50 60 -0.4 -0.2 0 0.2 0.4 Position Time 0 10 20 30 40 50 60 -0.2 -0.1 0 0.1 0.2 Velocity Time Figure 3.2 : MATLAB solution for the position and velocity of the spring- block oscillator. Forward Euler algorithm used.
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EA3: Systems Dynamics Spring-block oscillator Sridhar Krishnaswamy 3 shortly, for our frictionless undamped spring-block system, theoretically the oscillations should go on and on forever, neither growing nor decreasing in amplitude. 2.
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EA3_springblock1 - EA3: Systems Dynamics Spring-block...

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