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

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

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