Rework of "Example: System Defined by Relative Displacements"

Example: Rework of "Example: System Defined by Relative Displacements" without relative displacements

Differences between this example and the original are given as bold italics.

In the system below a mass, m, is hung from a rectangular frame by a spring (k).  There is viscous friction between the frame and the mass on either side (b).  The distance xin positive upwards) is measured from a fixed reference and defines the position of the frame.  The distance y (measured downwards) is the position of the mass relative to a fixed reference

Reworked system

Let's draw a free body diagram of the reworked system.  Again, we only need one free body diagram becasue the position xin is known; only y is unknown.  There are then four forces in the free body diagram:

1. The force from the spring.  This is equal to k·(y+xin), upwards.  (If either xin or y increases the spring elongates resulting in an upward force)
2. The force from the friction on the left side of the mass.  This is equal to k·(vy+vxin), upwards (vy is the first derivative of y with respect to time, i.e., the velocity; vxin is the derivative of xin)).
3. The force from the friction on the right side; this is also equal to  k·(vy+vxin).
4. D'Alembert's force is upward and equal to m·ay, where ay is the second derivative of y with respect to time.

These are shown below, along with the resulting equations of motion.

For comparison the original system is shown below.

Original System

Examing the two depictions of the system we can clearly see that z=xin+y (if xin increases and y remains constant, z increases; if y increases while xin remains constant, z also increases), or y=z-xin.  Replacing y by z-xin by the result above yields the original result.

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References