Question #308544

3. An ideal shock absorption system would use a critically damped oscillator to absorb shock loads. The location of the absorbing piston (π‘₯) is described by π‘₯ = πœπ‘’βˆ’π›Ύπ‘‘ where:

- 𝜏 is the linear damping coefficient

- 𝛾 is the exponential damping constant

- 𝑑 is the time (𝑠)

- π‘₯ is the displacement of piston (π‘š)

The tasks are to:

a) Draw a graph of displacement against time for 𝜏 = 12 and 𝛾 = 2, between 𝑑 = 0𝑠 and 𝑑 = 10𝑠.

b) Calculate the gradient at 𝑑 = 2𝑠 and 𝑑 = 4𝑠.

 QD099_September_2017

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   c) Differentiate the function of π‘₯ and calculate the value of 𝑑π‘₯ at 𝑑 = 2𝑠 and 𝑑 = 4𝑠. 𝑑𝑑

d) Compare your answers for part b and part c. (M1)

e) Calculate the derivative for the velocity function(𝑑2π‘₯).


1
Expert's answer
2022-03-10T13:21:16-0500

Solution


Given that


x = \tau e - \gamma t\


Now for \tau = 12,\,\,\,\gamma = 2\ , we have


x = 12e - 2t\


The plot between t=0t = 0 and t=10t = 10 is 




Solution (b)


Since the plot is a straight line, therefore, the gradient is constant which is m=βˆ’2m=-2


Henec gradient when t=2t=2 s is m=βˆ’2m=-2


And gradient when t=4t=4 s is m=βˆ’2m=-2


Solution (c)


x = 12e - 2t\


\frac{{dx}}{{dt}} = 0 - 2(1) = - 2\


Hence t=2t=2 s , we have gradient \frac{{dx}}{{dt}} = - 2\


And

Hence t=4t=4 s , we have gradient \frac{{dx}}{{dt}} = - 2\



Solution (d)


From (b) and (c), we see that the gradients when t=2t=2 s and when t=4t=4 s is m=βˆ’2m=-2, fix that is the same.



Solution (e)


x = 12e - 2t\


velocity is v=\frac{{dx}}{{dt}} = 0 - 2(1) = - 2\


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