User:Egm4313.s12.team6.berthoumieux

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Statement[edit]

For each ODE in Fig.2 in K 2011 p.3 (except the last one involving a system of 2 ODEs), determine the order, linearity (or lack of), and show wether the principle of superposition can be applied.

Given[edit]

,

Solution[edit]


Order:2nd order. The highest derivative is a 2nd derivative on the y.
Linearity:Linear.
Superposition:Yes.


Order:1st order. The highest derivative is a 1st on the v.
Linearity:Non-linear.
Superposition:No.


Order:1st order
Linearity:Non-linear.
Superposition:No.


Order:2nd order.
Linearity:Linear.
Superposition:Yes.

,
Order:2nd order
Linearity:Linear.
Superposition:Yes.


Order:2nd order
Linearity:Linear.
Superposition:Yes.


Order:4th oder
Linearity:Linear.
Superposition:Yes.


Order:2nd order
Linearity:Non-linear.
Superposition:Yes.

Statement[edit]

Realize spring-mass-dashpot systems in series as shown in the figure.


With characteristic:


And with double real root at

Find the values of k, c, m.

Solution[edit]

First, we write down the equations we can figure out from kinematics and from kinetics.
Kinematics:
Kinetics:

Next, we can find any relations that exist:


Since we already established that , we can say that


We also established that so we can say that


Next, we plug in the value for :


Finally, plug back into the kinetics equation:


We are given that so the characteristic equation must be:


This relates to the second order ODE:


Comparing this to the following equation:


We find that