# University of Florida/Egm4313/s12.team6.hill/R1

## Problem 1: Spring-dashpot system in parallel

Derive the equation of motion of a spring-dashpot system in parallel, with a mass and applied force ${\displaystyle f(t)}$

### Given

Spring-dashpot system in parallel

### Solution

The kinematics of the system can be described as,

 ${\displaystyle \displaystyle x=x_{k}=x_{c}}$ ${\displaystyle \longrightarrow (1)}$

The kinetics of the system can be described as,

 ${\displaystyle \displaystyle m{\ddot {x}}+f_{I}=f(t)}$ ${\displaystyle \longrightarrow (2)}$

and,

 ${\displaystyle \displaystyle f_{I}=f_{k}+f_{c}}$ ${\displaystyle \longrightarrow (3)}$

Given that,

 ${\displaystyle \displaystyle f_{k}=kx_{k}}$ ${\displaystyle \longrightarrow (3a)}$
 ${\displaystyle \displaystyle f_{c}=c{\dot {x_{c}}}}$ ${\displaystyle \longrightarrow (3b)}$

From (1), it can be found that,

 ${\displaystyle \displaystyle {\dot {x}}={\dot {x_{k}}}={\dot {x_{c}}}}$

and,

 ${\displaystyle \displaystyle {\ddot {x}}={\ddot {x_{k}}}={\ddot {x_{c}}}}$

From (3), it can be found that,

 ${\displaystyle \displaystyle f_{I}=c{\dot {x_{c}}}+kx_{k}}$

Finally, it can be found that

 ${\displaystyle \displaystyle m{\ddot {x}}+c{\dot {x}}+kx=f(t)}$