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,


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$$\displaystyle x = x_k = x_c $$
 *  $$\longrightarrow(1) $$
 * }
 * }

The kinetics of the system can be described as,


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$$\displaystyle m\ddot{x} + f_I = f(t) $$ and,
 *  $$\longrightarrow(2) $$
 * }
 * }
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$$\displaystyle f_I = f_k + f_c $$
 *  $$\longrightarrow(3) $$
 * }
 * }

Given that,


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$$\displaystyle f_k = kx_k $$
 *  $$\longrightarrow(3a) $$
 * }
 * }


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$$\displaystyle f_c = c\dot{x_c} $$
 *  $$\longrightarrow(3b) $$
 * }
 * }

From (1), it can be found that,
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\dot{x} = \dot{x_k} = \dot{x_c} $$ and,
 * $$\displaystyle
 * }
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\ddot{x} = \ddot{x_k} = \ddot{x_c} $$
 * $$\displaystyle
 * }

From (3), it can be found that,
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f_I =c\dot{x_c} + kx_k $$
 * $$\displaystyle
 * }

Finally, it can be found that
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$$\displaystyle m\ddot{x} + c\dot{x} + kx = f(t) $$
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 * style="width:10%; padding:10px; border:2px solid #8888aa" |
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 * }
 * }