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 x_in 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 x_in 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+x_in), upwards.  (If either x_in 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·(v_y+v_xin), upwards (v_y is the first derivative of y with respect to time, i.e., the velocity; v_xin is the derivative of x_in)).
3. The force from the friction on the right side; this is also equal to  k·(v_y+v_xin).
4. D'Alembert's force is upward and equal to m·a_y, where a_y is the second derivative of y with respect to time.

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

 

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For comparison the original system is shown below.

Original System

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