# University of Florida/Egm4313/s12.team7/Report1

## R.1.1: Equation of Motion for a Spring-dashpot Parallel System in Series with Mass and Applied Force

### Problem Statement

Given a spring-dashpot system in parallel with an applied force, find the equation of motion.

### Background Theory

For this problem, Newton's second law is used,

$\sum F=ma$ (1-1)

and applied to the mass at the end of a spring and damper in parallel.

### Solution

Assuming no rotation of the mass,

$y=y_{m}=y_{k}=y_{c}$ (1-2)

Therefore, the spring and damper forces can be written as, respectively,

$f_{k}=ky$ (1-3)

and

$f_{c}=cy'$ (1-4)

And the resultant force on the mass can be written as

$F=ma=my''$ Now, from equations (1-3 ) and (1-4 ) all of the forces can be substituted into Newton's second law, and since each distance is equal,

$\sum F=my''=-f_{k}-f_{c}+f(t)$ (1-5)

Substituting these exact force equations,

$my''=-ky-cy'+f(t)$ A little algebraic manipulation yields

$my''+cy'+ky=f(t)$ Finally, dividing by the mass to put the equation in standard form gives the final equation of motion for the mass:

$y''+{\frac {c}{m}}y'+{\frac {k}{m}}y={\frac {1}{m}}f(t))$ (1-6)