Damped and driven mass spring oscillator

# Damped and driven mass spring oscillator - Damped and...

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Damped and driven mass spring oscillator 1)A mass m is attached to the wall by a spring 2)The (horizontal) forces acting on the mass are F spring , F friction and whatever external force we apply F ext 3) Newton’s law tells us that F= F spring +F friction + F ext =ma

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If the position of the mass is measured in terms of y the displacement from the springs natural length, then Hooke’s law says that F spring = ky A reasonable approximation for the frictional force is F friction = by’ The frictional force will be proportional to the velocity and in the opposite direction. Acceleration is the second derivative of position a=y’’
The equation that describes the motion is my’’= ky by’+F ext or my’’+by’+ky=F ext From experience, we know that we can specify the starting position and velocity, y(0) and y’(0), and then the system will evolve. This is a second order differential equation and we expect that we can use two initial conditions to uniquely specify a solution. This is a linear , constant coefficient ODE.

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Linearity A second order differential equation is said to be linear if we can write it in the form
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Damped and driven mass spring oscillator - Damped and...

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