EJSS simple harmonic motion model with a vs x and v vs x graph
based on models and ideas by
- lookang http://weelookang.blogspot.sg/2010/06/ejs-open-source-simple-harmonic-motion.html?q=SHM
- Wolfgang Christian and Francisco Esquembre http://www.opensourcephysics.org/items/detail.cfm?ID=13103
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| 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 |
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| 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:
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).
δxδt=vx
δvxδt=−km(x−l)−bvxm+Asin(2πft)m
where the terms
−km(x−l) represents the restoring
force component as a result of the spring extending and
compressing.
−bvxm represents the damping
force component as a result of drag retarding the mass's
motion.
+Asin(2πft)m
represents the
driving force component as a result of a external periodic
force acting the mass m .
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.
a=−ω2x
v=±ω(x2o−x2)−−−−−−−−√
therefore , in general:
x=xosin(ωt−ϕ)
v=xoωcos(ωt−ϕ)
a=−xoω2sin(ωt−ϕ)
Thus, this model assumes
where the terms
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.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
|
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| Spring mass system with
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If motion starts at the equilibrium position and starts to move to the positive direction solutions to the defining equation are:
x=xosin(ωt)
motion
starts at the equilibrium position and starts to
move to the positive direction, defining
equation follows x=xosin(ωt)
v=xoωcos(ωt)
motion
starts at the equilibrium position and starts to
move to the positive direction, defining
equation follows v=xoωcos(ωt)
a=−xoω2sin(ωt)
motion
starts at the equilibrium position and starts to
move to the positive direction, defining
equation follows a=−xoω2sin(ωt)
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.
|
| motion
starts at the equilibrium position and starts to
move to the positive direction, defining
equation follows |
|
| motion
starts at the equilibrium position and starts to
move to the positive direction, defining
equation follows |
|
| motion
starts at the equilibrium position and starts to
move to the positive direction, defining
equation 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=−xocos(ωt)=xosin(ωt−π2)
motion
starts at the negative position and starts to
move to the positive direction, defining
equation follows x=−xocos(ωt)=xosin(ωt−π2)
v=xoωsin(ωt)=xoωcos(ωt−π2)
motion
starts at the negative position and starts to
move to the positive direction, defining
equation follows v=xoωsin(ωt)=xoωcos(ωt−π2)
a=xoω2cos(ωt)=−xoω2sin(ωt−π2)
motion
starts at the negative position and starts to
move to the positive direction, defining
equation follows a=xoω2cos(ωt)=−xoω2sin(ωt−π2)
|
| motion
starts at the negative position and starts to
move to the positive direction, defining
equation follows |
|
| motion
starts at the negative position and starts to
move to the positive direction, defining
equation follows |
|
| motion
starts at the negative position and starts to
move to the positive direction, defining
equation follows |
 |
| Spring mass system with
|
 |
| Spring mass system with
|
therefore , in general: