Iterative Prediction of Spring-Mass System - Revision history

Richard.Udall: /* External links */

2019-07-25T20:00:13Z

External links

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	*[https://phet.colorado.edu/sims/mass-spring-lab/mass-spring-lab_en.html Interactive Spring-Mass Simulator]		*[https://phet.colorado.edu/sims/mass-spring-lab/mass-spring-lab_en.html Interactive Spring-Mass Simulator]
	*[http://www.feynmanlectures.caltech.edu/I_23.html]		*[http://www.feynmanlectures.caltech.edu/I_23.html The Feynman Lectures]

	===Further Reading===		===Further Reading===

Richard.Udall: /* See also */

2019-07-25T20:00:00Z

Richard.Udall: /* Damping */

2019-07-25T19:51:45Z

Damping

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	====Damping====		====Damping====

	We know from our observations of ~~say~~ a pendulum swinging or a spring bobbing that it's periodic motion eventually dies out. This ~~is because~~ dissipative forces ~~prevalent~~ such as air resistance, ~~friction, fluid drag, etc.~~ reduce the ~~net energy~~ of the system ~~by dissipating it into the surroundings~~. A formal derivation of the equations of motion for a damped system will not be considered here, but it is very straightforward to add damping to our computational model. Whenever we calculate the net force on the mass during the iterative process, all we have to do now is add a dissipative force. For air resistance, the standard formula is		We know from our observations of a pendulum swinging or a spring bobbing that it's periodic motion eventually dies out. This results from dissipative forces present in our environment, such as friction or air resistance. The question of why these are called dissipative will become clear when we learn about [[Work/Energy]], but sufficed to say that they will reduce the amount of motion in the system. A formal derivation of the equations of motion for a damped system will not be considered here, but it is very straightforward to add damping to our computational model. Whenever we calculate the net force on the mass during the iterative process, all we have to do now is add a dissipative force. For air resistance, the standard formula is

	<math>\vec{F} = -~~0.5CAρv~~^2\hat{v}</math>		<math>\vec{F} = -\frac{CAρv^2\hat{v}}{2}</math>

	More frequently though, when we talk about the damped harmonic oscillator, we will discuss this force<ref> https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-ii-second-order-constant-coefficient-linear-equations/damped-harmonic-oscillators/MIT18_03SCF11_s13_2text.pdf </ref> <ref> http://www.feynmanlectures.caltech.edu/I_24.html </ref> :		More frequently though, when we talk about the damped harmonic oscillator, we will discuss this force<ref> https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-ii-second-order-constant-coefficient-linear-equations/damped-harmonic-oscillators/MIT18_03SCF11_s13_2text.pdf </ref> <ref> http://www.feynmanlectures.caltech.edu/I_24.html </ref> :
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	<math>\vec{F} = -b\vec{v}</math>		<math>\vec{F} = -b\vec{v}</math>

	where b is the damping constant. The important difference between the two is that in the latter the force has a linear relationship to <math> \vec{v} </math>. There are three types of damping, depending on the coefficient, and labelled by how quickly the system damps out (reaches equilibrium). If the coefficient is small relative to the mass and spring constant, then the oscillator will be under damped, and still have some oscillatory behavior. If it is large relative to the mass and spring constant, the oscillator will be over damped, meaning that there will be no oscillations, but the system will take longer to arrive at an equilibrium because it has been slowed down by the dissipation. The fastest time of decay is when the system is critically damped, which occurs when <math> b^2 = 4mk </math>. The figure below illustrates these three possibilities, and is generated using the numpy based code from the Computational Methods section below.		where b is the damping constant. The important difference between the two is that in the latter the force has a linear relationship to <math> \vec{v} </math>. There are three types of damping, depending on the coefficient, and they are labelled by how quickly the system damps out (reaches equilibrium). If the coefficient is small relative to the mass and spring constant, then the oscillator will be under damped, and still have some oscillatory behavior. If it is large relative to the mass and spring constant, the oscillator will be over damped, meaning that there will be no oscillations, but the system will take longer to arrive at an equilibrium because it has been slowed down by the dissipation. The fastest time of decay is when the system is critically damped, which occurs when <math> b^2 = 4mk </math>. The figure below illustrates these three possibilities, and is generated using the numpy based code from the Computational Methods section below.

	[[File:Damped Oscillators.png]]		[[File:Damped Oscillators.png]]

Richard.Udall: /* A Mathematical Model */

2019-07-25T19:48:04Z

A Mathematical Model

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	<math>\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}\Delta t</math>		<math>\vec{p}_{f} = \vec{p}_{i} + \vec{F}_{net}\Delta t</math>


	<div style="text-align: left;"> Kinematics, expressed in a similarly iterative manner, allows for the determination of the next iteration's position and velocity		<div style="text-align: left;"> Kinematics, expressed in a similarly iterative manner, allows for the determination of the next iteration's position and velocity

Richard.Udall: /* Connectedness */

2019-07-24T20:30:00Z

Connectedness

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	==Connectedness==		==Connectedness==
			[A student should expand upon this section, following the [[Template]]]

	While you may not need to precisely calculate the motion of spring-mass systems on a daily basis, understanding their behavior has led to the invention of many useful items. Among these are mattresses, clocks, and the suspension on cars.		While you may not need to precisely calculate the motion of spring-mass systems on a daily basis, understanding their behavior has led to the invention of many useful items. Among these are mattresses, clocks, and the suspension on cars.

Richard.Udall at 20:28, 24 July 2019

2019-07-24T20:28:44Z

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	'''~~claimed~~ by kgiles7 2015'''		'''Claimed by kgiles7 2015'''

	'''Edited by Richard Udall June 2019'''		'''Edited by Richard Udall June 2019'''
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	==The Main Idea==		==The Main Idea==

	~~A simple~~ spring-mass system is a ~~basic illustration of~~ the ~~momentum principle~~. ~~The principle~~ of ~~conservation~~ of ~~momentum~~ can be ~~repeatedly applied to predict~~ the ~~system's future motion~~.		Added complications bring the spring mass system much closer to a real system than the idealized system we have included before. Since we had to deal with the idealized system using iterative prediction, we will need to deal with this more complex situation using iterative prediction as well. Since our model presented in [[Fundamentals of Iterative Prediction with Varying Force]] is fairly general, adding these complications consists of adding new forces, and observing the resultant behaviors. These cases can also be explored analytically using differential equations, but doing so is outside of the scope of this class.



	===A Mathematical Model===		===A Mathematical Model===

Richard.Udall at 20:23, 24 July 2019

2019-07-24T20:23:21Z

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	claimed by kgiles7		'''claimed by kgiles7 2015'''

			'''Edited by Richard Udall June 2019'''

			The spring mass system is one of the classical examples of harmonic motion. The idealized case, where no forces except the spring force are acting upon the system, is the prototypical example of [[Simple Harmonic Motion]]. This page will explore less idealized cases, introducing damping, driving, and added gravitational force.

	==The Main Idea==		==The Main Idea==

Richard.Udall: /* Difficult Example */

2019-06-25T19:54:21Z

Difficult Example

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	# Set <math> b = 0 </math>, <math> F_0 = 1 </math>, <math> x_0 = 0 </math>, and <math> \omega_1 = 5 Hz </math>		# Set <math> b = 0 </math>, <math> F_0 = 1 </math>, <math> x_0 = 0 </math>, and <math> \omega_1 = 5 Hz </math>
	# Now, using the same parameters as above, set <math> x_0 = 1 </math>		# Now, using the same parameters as above, set <math> x_0 = 1 </math>
	# Using the same parameters from (2), now set <math> b = ~~0.5~~ </math>		# Using the same parameters from (2), now set <math> b = 2 </math>
	# Using the same parameters from (3), set <math> F_0 = 2 </math>		# Using the same parameters from (3), set <math> F_0 = 2 </math>
	# Using the same parameters from (4), set <math> \omega_1 = 4 Hz </math>		# Using the same parameters from (4), set <math> \omega_1 = 4 Hz </math>

Richard.Udall: /* See also */

2019-06-25T18:34:16Z

Richard.Udall: /* History */

2019-06-25T18:33:03Z

History

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	==History==		==History==

	The history of the spring mass system is essentially that of Hooke's law, as described in the article [[Simple Harmonic Motion]], and repeated in part here. Thomas Hooke, an English scientist, discovered what is now known as Hooke's Law in 1660 while working on the springs of watches.<ref> https://www.britannica.com/science/Hookes-law </ref> A remarkable physicist, Hooke was also a pioneer in optics, astronomy, and fluid mechanics. He supported a theory of evolution nearly two hundred years before Darwin (although he did not know about the principle of natural selection), devised the inverse square law which Newton adapted, and accurately described air as individual particles separated by large distances.<ref> https://www.britannica.com/biography/Robert-Hooke </ref> Solutions to the more complicated differential equations that describe the systems we looked at here are attributed to Leonhard Euler, Joseph Lagrange, and Jean D'Alembert, each a very influential mathematician.		The history of the spring mass system is essentially that of Hooke's law, as described in the article [[Simple Harmonic Motion]], and repeated in part here. Thomas Hooke, an English scientist, discovered what is now known as Hooke's Law in 1660 while working on the springs of watches.<ref> https://www.britannica.com/science/Hookes-law </ref> A remarkable physicist, Hooke was also a pioneer in optics, astronomy, and fluid mechanics. He supported a theory of evolution nearly two hundred years before Darwin (although he did not know about the principle of natural selection), devised the inverse square law which Newton adapted, and accurately described air as individual particles separated by large distances.<ref> https://www.britannica.com/biography/Robert-Hooke </ref> Solutions to the more complicated differential equations that describe the systems we looked at here are attributed to Leonhard Euler, Joseph Lagrange, and Jean D'Alembert, each a very influential mathematician <ref> http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.112.3646&rep=rep1&type=pdf </ref>.



	~~Harmonic motion is thus named because of its connection to music: instruments vibrate, and in doing so produce the sounds we perceive as music.~~<ref> ~~https~~://~~www~~.~~britannica~~.~~com~~/~~science~~/simple-harmonic-motion </ref> The Fourier series allows for the expression of ''any'' periodic function in terms of sinusoids, meaning that a huge variety of phenomena in physics may be decomposed into simple harmonic motion. Furthermore, for reasons discussed in reference 8, harmonic oscillators are a strong approximation of many systems near equilibrium. The corresponding quantum harmonic oscillator is similarly useful in quantum mechanics. <ref> https://phys.~~libretexts~~.~~org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/7%3A_Quantum_Mechanics/7~~.~~5%3A_The_Quantum_Harmonic_Oscillator~~ </ref>.

	== See also ==		== See also ==

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 == See also ==
-[[Conservation of Momentum]]
+*[[Conservation of Momentum]]
-[[Predicting Change in multiple dimensions]]
+===External links===
-[[Tension]]
+*[https://phet.colorado.edu/sims/mass-spring-lab/mass-spring-lab_en.html Interactive Spring-Mass Simulator]
-[[Fundamentals of Iterative Prediction with Varying Force]]
+===Further Reading===
-[https://phet.colorado.edu/sims/mass-spring-lab/mass-spring-lab_en.html Interactive Spring-Mass Simulator]
+*Matter and Interactions, 4th Edition
 ==References==

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	[https://phet.colorado.edu/sims/mass-spring-lab/mass-spring-lab_en.html Interactive Spring-Mass Simulator]		[https://phet.colorado.edu/sims/mass-spring-lab/mass-spring-lab_en.html Interactive Spring-Mass Simulator]
			[http://www.feynmanlectures.caltech.edu/I_23.html]

	==References==		==References==