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

**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:

- The force from the spring. This is equal to
upwards. (If either x*k·(y+x*_{in}),_{in}or y increases the spring elongates resulting in an upward force) - The force from the friction on the left side of the mass. This
is equal to
, upwards (v*k·(v*_{y}+v_{xin})_{y}is the first derivative of y with respect to time, i.e., the velocity; v_{xin}is the derivative of x_{in})). - The force from the friction on the right side; this is also equal to
.*k·(v*_{y}+v_{xin}) - D'Alembert's force is upward and equal to
, where a*m·a*_{y}_{y}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=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.

© Copyright 2005 to 2019 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College