Friday, February 14, 2014

EJSS SHM model with a vs x and v vs x

EJSS SHM model with a vs x and v vs x graph
EJSS simple harmonic motion model with a vs x and v vs x graph
based on models and ideas by

  1. lookang http://weelookang.blogspot.sg/2010/06/ejs-open-source-simple-harmonic-motion.html?q=SHM
  2. Wolfgang Christian and Francisco Esquembre http://www.opensourcephysics.org/items/detail.cfm?ID=13103

http://weelookang.blogspot.sg/2014/02/ejss-shm-model-with-vs-x-and-v-vs-x.html
EJSS simple harmonic motion model with a vs x and v vs x graph
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMaxvx/SHMaxvx_Simulation.html
source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMaxvx.zip
author: lookang
author of EJSS 5.0 Francisco Esquembre
notes:
http://weelookang.blogspot.sg/2014/02/ejss-shm-model-with-vs-x-and-v-vs-x.html
EJSS simple harmonic motion model with a vs x and v vs x graph
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMaxvx/SHMaxvx_Simulation.html
source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMaxvx.zip
author: lookang
author of EJSS 5.0 Francisco Esquembre


The equations that model the motion of the spring mass system are:

Mathematically, the restoring force $ F $ is given by 


$ F = - k x $

where $ F $  is the restoring elastic force exerted by the spring (in SI units: N), k is the spring constant (N·m−1), and x is the displacement from the equilibrium position (in m).

Thus, this model assumes 

$ \frac{\delta x}{\delta t} = v_{x} $


$ \frac{\delta v_{x}}{\delta t} = -\frac{k}{m}(x-l) - \frac{bv_{x}}{m} + \frac{A sin(2 \pi f t)}{m} $

where the terms

$ -\frac{k}{m}(x-l) $ represents the restoring force component as a result of the spring extending and compressing.

$ - \frac{bv_{x}}{m}$ represents the damping force component as a result of drag retarding the mass's motion.

$ + \frac{A sin(2 \pi f t)}{m} $ represents the driving force component as a result of a external periodic force acting the mass $ m $.

What is SHM?

Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law. The motion is sinusoidal in time and demonstrates a single resonant frequency. In order for simple harmonic motion to take place, the net force of the object at the end of the pendulum must be proportional to the displacement. In other words, oscillations are periodic variations in the value of a physical quantity about a central or equilibrium value.

Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, it accelerates and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, at x = 0, the mass has momentum because of the impulse that the restoring force has imparted. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then tends to slow it down, until its velocity reaches zero, whereby it will attempt to reach equilibrium position again.

As long as the system has no energy loss, the mass will continue to oscillate. Thus, simple harmonic motion is a type of periodic motion.


Definition of SHM:

A periodic motion where the acceleration a of an object is always directed towards a fixed equilibrium position and is proportional to its displacement x from that fixed point.

Spring mass system with $ xo = 0 , vo = 2 $, showing the graph of $ a = - \omega^{2} x $
$ a = - \omega^{2} x $


Spring mass system with $ xo = 0 , vo = 2 $, showing the graph of $ v = \pm \omega \sqrt{ ( x_{o}^{2}-x^{2})}  $


$ v = \pm \omega \sqrt{ ( x_{o}^{2}-x^{2})}  $



If motion starts at the equilibrium position and starts to move to the positive direction solutions to the defining equation are:

$ x = x_{o} sin ( \omega t ) $
motion starts at the equilibrium position and starts to move to the positive direction, defining equation follows  $ x = x_{o} sin ( \omega t ) $


$ v = x_{o} \omega cos ( \omega t ) $

 motion starts at the equilibrium position and starts to move to the positive direction, defining equation follows $ v = x_{o} \omega cos ( \omega t ) $



$ a = - x_{o} \omega^{2} sin ( \omega t ) $
motion starts at the equilibrium position and starts to move to the positive direction, defining equation follows $ a = - x_{o} \omega^{2} sin ( \omega t ) $



The variation with time of x, v and a seen together graphically is as follows:
Note that
(1) the velocity of the body is deduced from the gradient of the x-t (displacement-time) graph and
(2) the acceleration of the body is deduced from the gradient of the v-t (velocity-time) graph.

If the motion starts to the negative amplitude position:


$ x = - x_{o} cos ( \omega t ) = x_{o} sin ( \omega t -  \frac{\pi }{2} )$

motion starts at the negative position and starts to move to the positive direction, defining equation follows $ x = - x_{o} cos ( \omega t ) = x_{o} sin ( \omega t - \frac{\pi }{2} )$

$ v = x_{o} \omega sin ( \omega t ) = x_{o} \omega cos ( \omega t -  \frac{\pi }{2} )$

motion starts at the negative position and starts to move to the positive direction, defining equation follows $ v = x_{o} \omega sin ( \omega t ) = x_{o} \omega cos ( \omega t - \frac{\pi }{2} )$


$ a =  x_{o} \omega^{2} cos ( \omega t ) = - x_{o} \omega^{2} sin ( \omega t -  \frac{\pi }{2} )$

motion starts at the negative position and starts to move to the positive direction, defining equation follows $ a = x_{o} \omega^{2} cos ( \omega t ) = - x_{o} \omega^{2} sin ( \omega t - \frac{\pi }{2} )$


Spring mass system with $ xo = -1 , vo = 2 $, showing the graph of $ a = - \omega^{2} x $

Spring mass system with $ xo = -1 , vo = 2 $, showing the graph of $ v = \pm \omega \sqrt{ ( x_{o}^{2}-x^{2})}  $


therefore , in general:

$ x = x_{o} sin ( \omega t - \phi ) $


$ v = x_{o} \omega cos ( \omega t - \phi ) $

$ a = - x_{o} \omega^{2} sin ( \omega t - \phi ) $