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₯ = ππβπΎπ‘ where:
- π is the linear damping coefficient
- πΎ is the exponential damping constant
- π‘ is the time (π )
- π₯ is the displacement of piston (π)
The tasks are to:
a) Draw a graph of displacement against time for π = 12 and πΎ = 2, between π‘ = 0π and π‘ = 10π .
b) Calculate the gradient at π‘ = 2π and π‘ = 4π .
QD099_September_2017
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c) Differentiate the function of π₯ and calculate the value of ππ₯ at π‘ = 2π and π‘ = 4π . ππ‘
d) Compare your answers for part b and part c. (M1)
e) Calculate the derivative for the velocity function(π2π₯).
Solution
Given that
x = \tau e - \gamma t\
Now for \tau = 12,\,\,\,\gamma = 2\ , we have
x = 12e - 2t\
The plot between and is
Solution (b)
Since the plot is a straight line, therefore, the gradient is constant which is
Henec gradient when s is
And gradient when s is
Solution (c)
x = 12e - 2t\
\frac{{dx}}{{dt}} = 0 - 2(1) = - 2\
Hence s , we have gradient \frac{{dx}}{{dt}} = - 2\
And
Hence s , we have gradient \frac{{dx}}{{dt}} = - 2\
Solution (d)
From (b) and (c), we see that the gradients when s and when s is , fix that is the same.
Solution (e)
x = 12e - 2t\
velocity is v=\frac{{dx}}{{dt}} = 0 - 2(1) = - 2\
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