User:Egm4313.sp12.team20.sablotsky.ca/Report 1

Statement
Derive the equation of motion of a spring-dashpot system in parallel, with a mass and applied force $$\displaystyle f(t). $$ The spring-dashpot system is shown in Figure 1.

Figure 1

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

$$ \displaystyle y = y_{k} = y_c$$
 * style="width:95%" |
 * (1.0)
 * }

Kinetics:
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f(t) = my'' + f_1 $$
 * style="width:95%" | $$ \displaystyle
 * (1.1)
 * }


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 * style="width:95%" | $$ \displaystyle f_1 = f_k + f_c $$
 * (1.2)
 * }

By Substitution of $$ \displaystyle f_1 $$ into equation (1.1) we obtain
 * {| style="width:100%" border="0"|-

f(t) = my'' + f_k + f_c $$ Constitutive Relations: -Force of Spring
 * style="width:95%" | $$ \displaystyle
 * (1.3)
 * }
 * {| style="width:100%" border="0"|-

f_k = ky_k $$
 * style="width:95%" | $$ \displaystyle
 * (1.4)
 * }

-Force of Dash-Pot
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f_c = cy'_c$$ Because of the kinematic relations where $$ \displaystyle y_c = y_k $$ we manipulate equation (1.4) and obtain
 * style="width:95%" | $$\displaystyle
 * (1.5)
 * }
 * {| style="width:100%" border="0"|-

f_k = ky_k = ky_c$$
 * style="width:95%" | $$\displaystyle
 * (1.6)
 * }

Therefore, by subsitution into equation (1.2)
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 * style="width:95%" | $$ f_1 = f_k + f_c \rightarrow f_1 = ky_c+ cy'_c$$
 * (1.7)
 * }

After finally substituting equation (1.7) into the original f(t) equation (1.1), the final equation equals