User talk:Hopeton87

For the spring dashpot system shown, draw the free body diagram and derive the equation of motion.

Problem 3


The equation of motion
 * $$my''+f_{i}=f(t)$$

where:


 * $$my''$$

is the inertial force


 * $$f_{i}$$

is the internal force


 * $$f(t)$$

is the applied force

this analysis assumes:

Assumptions:

 * Motion in the horizontal direction
 * Massless spring
 * Massless dashpot
 * massless connections

Solution:
To analyze the system we look at the kinematics and kinetics of the system

Kinematics: This involves the displacement which affects the mass. The displacement, represented by:
 * $$y$$  :    This is the total displacement of the spring plus the displacement of the dashpot.
 * $$y_{k}$$  :     represents the displacement of the spring
 * $$y_{c}$$  :     represents the displacement of the dashpot
 * $$y=y_{k}+y_{c}$$

Kinetics: this involves the forces associated with the displacements. The spring and dashpot are in series, therefore at any section in the series the internal for will be the same. This is denoted as:  $$y_{i}$$. The force in the dashpot is proportional to the first time derivative of displacement (velocity)


 * $$f_{k}=f_{c}=f_{i}=ky_{k}=cy'_{c}$$

Where :  $$k$$   is the spring constant.


 * $$c$$  is the damping coefficient.

and:  $$y'_{c}$$   is the velocity of the dashpot. From this we get :  $$y'_{c}=(k/c)y_{k}$$ which presents :  $$y'_{c}$$   in terms of :   $$y_{k}$$





The constitutive relations:

The spring force is equal to the spring constant times the displacement of the spring. The damping force from the dashpot is equal to the damping coefficient times the velocity.
 * $$y=y_{k}+y_{c}$$

this presents two unknown dependent variables by using the relation


 * $$ y=y_{k}+(y_{c}')' $$

Becomes:


 * $$ y=y_{k}+ \frac {k}{k} y_{k}' $$

now we can rewrite the equation of motion: $$my''+f_{i}=f(t)$$

as :$$ m[yk''+ \frac {k}{c} yk']+fi=f(t) $$ However >:  $$f_{i}=f_{k}$$

so we get: $$m[yk''+ \frac {k}{c} yk']+fk=f(t)$$


 * {| style="width:100%" border="0"


 * style="width:25%; padding:10px; border:2px solid #8888aa" |
 * $$ m[y_{k}''+ \frac {k}{c} y_{k}']+f_{k}=f(t) $$   :
 * $$ m[y_{k}''+ \frac {k}{c} y_{k}']+f_{k}=f(t) $$   :


 * 
 * }
 * }