EJSS simple harmonic motion model with $x vs t$, v vs t and a vs t graphs

based on models and ideas by

EJSS simple harmonic motion model with x vs t, v vs t and a vs t graphs showing all 3 graphs https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMxva/SHMxva_Simulation.html source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMxva.zip author: lookang author of EJSS 5.0 Francisco Esquembre |

- 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

EJSS simple harmonic motion model with x vs t, v vs t and a vs t graphs showing all 3 graphs https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMxva/SHMxva_Simulation.html source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMxva.zip author: lookang author of EJSS 5.0 Francisco Esquembre |

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

$ 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.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.

## Types of Oscillations:

### Free oscillations

Free oscillations occur when a system is disturbed momentarily and then allowed to move without restraint. In the absence of damping caused by friction and viscous forces like air drag force, free

oscillations will last forever.

### Damped oscillations

Damped oscillations are free oscillations that decay with time as a result of frictional and viscous forces.

### Forced oscillations

Forced oscillations are oscillations that are subjected to a periodic driving force provided by an external agent such as motor or a push by a person etc.

Resonance is an interesting phenomenon that occurs when driving force frequency matches that of the system's natural oscillating frequency resulting in a motion that reaches some maximum amplitude.

## 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.
$ a = - \omega^{2} x $

## 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.

therefore , in general:

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} )$

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 ) $

## Damping:

If no frictional forces act on an oscillator (e.g. simple pendulum system, mass-spring system, etc.), then it will oscillate indefinitely.

In practice, the amplitude of the oscillations decreases to zero as a result of friction. This type of motion is called damped harmonic motion. Often the friction arises from air resistance (external damping) or internal forces (internal damping).

### No Damping:

System oscillates forever without coming to rest. Amplitude and thus total energy is constant.

no damping, b =0, SHM starts at equilibrium position $ x_{o} = 0 $ with $ v_{o} = 2 $ |

no damping, b =0, SHM starts at position $ x_{o} = 1 $ with $ v_{o} =2 $ |

no damping, b =0, SHM starts at position $ x_{o} = -1 $ with $ v_{o} = 1 $ |

### Light Damping:

System undergoes several oscillations of decreasing amplitude before coming to rest.

light damping, b =0.1, SHM starts at equilibrium position $ x_{o} = 0 $ with $ v_{o} = 2 $ |

light damping, b =0.1, SHM starts at position $ x_{o} = 2 $ with $ v_{o} = 2 $ |

light damping, b =0.1, SHM starts at position $ x_{o} = -1 $ with $ v_{o} = 2 $ |

### Critical Damping

System returns to equilibrium in the minimum time, without overshooting or oscillating about the equilibrium position.

#### Applications of Critical Damping

Car suspension

The spring of a car’s suspension is critically damped so that when a car goes over a bump, the passenger in the car quickly and smoothly regains equilibrium.

However, car suspensions are often adjusted to slightly undercritically damped condition to give a more comfortable ride. Critical damping also leaves the car ready to respond to further bumps in the road quickly.

Moving-coil meters

Critical damping is an important feature of moving-coil meters which are used to measure current and voltage. When the reading changes, it is of little use if the pointer oscillates for a while or takes too long to settle down to the new reading. The new reading must be taken quickly in case it changes again.

Thus, a pointer is critically damped to allow it to move to the correct position immediately whenever a current flows through it or a voltage is applied across it.

Door closers

Devices fitted above doors to prevent slamming.

critical damping, b =2.0 SHM starts at position $ x_{o} = 2 $ with $ v_{o} = 0 $ |

critical damping, b =2.0 SHM starts at equilibrium position $ x_{o} = 0 $ with $ v_{o} = 2 $ |

critical damping, b =2.0 SHM starts at position $ x_{o} = -1 $ with $ v_{o} = 0 $ |

### Heavy Damping

System returns to equilibrium very slowly without any oscillation.

heavy damping, b =5.0 SHM starts at equilibrium position $ x_{o} = 0 $ with $ v_{o} = 2 $ |

heavy damping, b =5.0 SHM starts at position $ x_{o} = 2 $ with $ v_{o} = 0 $ |

heavy damping, b =5.0 SHM starts at equilibrium position $ x_{o} = -2 $ with $ v_{o} = 0 $ |