User:Egm4313.s12.team3.chaffee

Statement
Derive the equation of motion of the spring-mass-dashpot in Fig. 53, in Kreyszig 2011 p.85, with an applied force $$f(t)$$ on the ball.

Solution

 * Kinematics:
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$$\displaystyle y=y_c=y_k $$     (2.1)
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 * Kinetics:
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$$\displaystyle ma+{{f}_{c}}+{{f}_{k}}=f(t) $$     (2.2)
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 * Constitutive relations:
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$$\displaystyle y=y_k=y_c $$     (2.3)
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$$  \displaystyle {{f}_{k}}=k{{y}_{k}} $$     (2.4)
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$$  \displaystyle {{f}_{c}}=c{{y}_{c}}' $$     (2.5)
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$$\displaystyle ma=my'' $$     (2.6)
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 * Combining equations (2.2-2.6):
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$$\displaystyle my''+c{{y}_{c}}'+k{{y}_{k}}=f(t) $$     (2.7)
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$$  \displaystyle y=y_k=y_c $$     (2.8)
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 * Final equation of motion:
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$$ my''+cy'+ky=f(t) $$ (2.9)
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Author
Solved and typed by - --Egm4313.s12.team3.chaffee 11:36, 30 January 2012 (UTC) Reviewed By -

Edited By - --Egm4313.s12.team3.chaffee 00:00, 30 January 2012 (UTC)

Solution

 * Free body diagrams:

From the FBD of the spring, the spring creates a force opposing the direction of motion and the wall must always react in equal magnitude and opposite direction. Summation of the forces in the x-direction leads to:
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$$\displaystyle k{{y}_{k}}=k{{y}_{k}} $$     (3.1) From the FBD of the dashpot, the spring force acts on one side of the dashpot while the damping force acts on the other side. Summation of the forces in the x-direction leads to:
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$$\displaystyle f_k=f_c $$     (3.2)
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$$\displaystyle k{{y}_{k}}=cv=c{{y}_{c}}' $$     (3.3) From the FBD of the ball, the dashport force and the inertial forceacts on one side of the ball while the forcing function acts in the opposite direction. Summation of the forces in the x-direction leads to:
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$$\displaystyle f(t)=ma+c{{y}_{c}}' $$     (3.4) From the above equations we see that the spring force and damping force are equal and can all be considered internal forces.
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$$\displaystyle k{{y}_{k}}=c{{y}_{c}}'=f_{I} $$     (3.5) Knowing that the total displacement is:
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$$\displaystyle y={{y}_{k}}+{{y}_{c}} $$     (3.6)
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$$\displaystyle a=y'' $$     (3.7) We can combine equations (3.3), (3.5), and (3.6) to find:
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(3.8)
 * $$ my'' + f_{I} = f(t) $$
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Author
Solved and Typed By - Egm4313.s12.team3.chaffee 11:30, 25 January 2012 (UTC)

Reviewed By -

Edited By - --Egm4313.s12.team3.chaffee 15:51, 30 January 2012 (UTC)