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 $f(t)$

Given

Spring-dashpot system in parallel

Solution

The kinematics of the system can be described as,

$x=x_k=x_c$

$⟶(1)$

The kinetics of the system can be described as,

$m\ddot{x}+f_I=f(t)$

$⟶(2)$

and,

$f_I=f_k+f_c$

$⟶(3)$

Given that,

$f_k=k{x}_k$

$⟶(3a)$

$f_c=c\dot{x_c}$

$⟶(3b)$

From (1), it can be found that,

$\dot{x}=\dot{x_k}=\dot{x_c}$

and,

$\ddot{x}=\ddot{x_k}=\ddot{x_c}$

From (3), it can be found that,

$f_I=c\dot{x_c}+k{x}_k$

Finally, it can be found that

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