User:Egm4313.s12.team14.kimiagarov/HW 1

Problem R1.2

Given:

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FBD:

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 * $$\displaystyle{F_k = ky}$$
 * $$\displaystyle (Eq. 1) $$
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 * $$\displaystyle{F_c = cy'}$$
 * $$\displaystyle (Eq. 2) $$
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From the FBD:


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 * $$\displaystyle{ma = -ky - cy' + r(t)}$$
 * $$\displaystyle (Eq. 3) $$
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Plug (Eq. 1) and (Eq. 2) into (Eq 3)

$${my'' + ky + cy' = r(t)}$$

Problem R1.6

water level $${h}$$

$${h' = -k\sqrt{h}}$$

Homogeneous:


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 * $$\displaystyle{h'_h + k\sqrt{h_h} = 0}$$
 * $$\displaystyle (Eq. 1) $$
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Particular:


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 * $$\displaystyle{h'_p + k\sqrt{h_p} = 0}$$
 * $$\displaystyle (Eq. 2) $$
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Add (Eq. 1) and (Eq. 2):

$${h'_h + h'_p +k(\sqrt{h_h} +\sqrt{h_p}) = 0}$$

It could be said that:   $${h'_h + h'_p = h'}$$

But, $${\sqrt{h_h} + \sqrt{h_p} \ne \sqrt{h_h + h_p} = \sqrt{h}}$$

Therefore, superposition cannot be used.