Chapter 3.1, Problem 67E

To determine

To find : piston velocity, acceleration and jerk when frequency is double.

Answer to Problem 67E

Doubling the frequency leads, velocity to two times, acceleration to four times and jerk to eight times.

Explanation of Solution

Given information :

The piston is moving up and down and its position is given by

  $s=A\cos \left(2\pi bt\right)$

where A is amplitude and b is frequency.

Formula needed :

Velocity

  $v=\frac{ds}{dt}$, acceleration

  $a=\frac{dv}{dt}$

and jerk

  $j=\frac{da}{dt}$.

For velocity, differentiate

  $s=A\cos \left(2\pi bt\right)$

with respect to

  $t$.

  $\begin{array}{c}\frac{ds}{dt}=\frac{d}{dt}A\cos \left(2\pi bt\right)\\ v=-A\sin \left(2\pi bt\right)\cdot \frac{d}{dt}\left(2\pi bt\right)\\ =-A\sin \left(2\pi bt\right)\cdot \left(2\pi b\right)\\ =-2\pi bA\sin \left(2\pi bt\right)\end{array}$

For acceleration, differentiate

  $v=-2\pi bA\sin \left(2\pi bt\right)$

with respect to

  $t$.

  $\begin{array}{c}\frac{dv}{dt}=-2\pi bA\cdot \frac{d}{dt}\sin \left(2\pi bt\right)\\ a=-2\pi bA\cdot \cos \left(2\pi bt\right)\frac{d}{dt}\left(2\pi bt\right)\\ =-2\pi bA\cdot \cos \left(2\pi bt\right)\left(2\pi b\right)\\ =-{\left(2\pi b\right)}^{2}A\cdot \cos \left(2\pi bt\right)\end{array}$

For jerk, differentiate

  $a=-{\left(2\pi b\right)}^{2}A\cdot \cos \left(2\pi bt\right)$

with respect to

  $t$.

  $\begin{array}{c}\frac{da}{dt}=-{\left(2\pi b\right)}^{2}A\cdot \frac{d}{dt}\cos \left(2\pi bt\right)\\ j={\left(2\pi b\right)}^{2}A\cdot \sin \left(2\pi bt\right)\frac{d}{dt}\left(2\pi bt\right)\\ ={\left(2\pi b\right)}^{2}A\cdot \sin \left(2\pi bt\right)\left(2\pi b\right)\\ ={\left(2\pi b\right)}^{3}A\cdot \sin \left(2\pi bt\right)\end{array}$

Now double the frequency means replace b with 2b .

The velocity is

  $\begin{array}{c}v=-2\pi \left(2b\right)A\sin \left(2\pi \left(2b\right)t\right)\\ =-4\pi bA\sin \left(4\pi bt\right)\end{array}$

The acceleration is

  $\begin{array}{c}a=-{\left(2\pi \left(2b\right)\right)}^{2}A\cdot \cos \left(2\pi \left(2b\right)t\right)\\ =-4{\left(2\pi b\right)}^{2}A\cdot \cos \left(4\pi bt\right)\end{array}$

The jerk is

  $\begin{array}{c}j={\left(2\pi 2b\right)}^{3}A\cdot \sin \left(2\pi 2bt\right)\\ =8{\left(2\pi b\right)}^{3}A\cdot \sin \left(4\pi bt\right)\end{array}$

When frequency become double, velocity become double, acceleration become four times and jerk become eight times.