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Topic: Damped pendulum

	Equations of motion of damped and driven pendula
		The derivation of the equations of motion of damped and driven pendula extends the derivation of the undamped and undriven case.
		Damping and driving are caused by two additional forces acting on the pendulum: The damping force and the driving force.
		Here, the main sources of damping are aerodynamical friction due to the motion of the mass through the air and friction caused by bending the rope at the suspension point.
	monet.physik.unibas.ch /~elmer/pendulum/eqm2.htm (609 words)

	Pendulum - Wikipedia, the free encyclopedia
		The pendulum was discovered by Ibn Yunus al-Masri during the 10th century, who was the first to study and document its oscillatory motion.
		Pendulums (these may be a crystal suspended on a chain, or a metal weight) are often used for divination and dowsing.
		A pendulum in which the rod is not vertical but almost horizontal was used in early seismometers for measuring earth tremors.
	en.wikipedia.org /wiki/Pendulum (563 words)

	Encyclopedia :: encyclopedia : Pendulum (Site not responding. Last check: 2007-10-03)
		A gravity pendulum (plural pendula) is a weight on the end of a rigid rod (or a string/rope), which, when given an initial push, will swing back and forth under the influence of gravity over its central (lowest) point.
		The blue arrow is the gravitational force acting on the bob, violet arrows are that same force resolved into components parallel and perpendicular to the bob's instantaneous motion, the motion along the red axis, which is always perpendicular to the cable/rod.
		It is the vertical distance the pendulum fell.
	www.hallencyclopedia.com /Pendulum (1260 words)

	Attractor - Wikipedia, the free encyclopedia
		A fixed point is a point that a system evolves towards, such as the final states of a falling pebble, a damped pendulum, or the water in a glass.
		The ideal pendulum is not an example because its orbits are not isolated.
		In phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit.
	en.wikipedia.org /wiki/Attractor (978 words)

	Pendulum
		A simple gravity pendulum (plural pendulums or pendula), also called a bob pendulum, is a weight on the end of a rigid rod (or a string/rope), which, when given an initial push, will swing back and forth under the influence of gravity over its central (lowest) point.
		is the semi-amplitude of the oscillation, that is the maximum angle between the rod of the pendulum and the vertical.
		For a swing of the bob is balanced over its pivot point and so (keep in mind the pendulum is made of a rigid rod).
	www.brainyencyclopedia.com /encyclopedia/p/pe/pendulum.html (1418 words)

	Chaotic Pendulum Physics Simulation
		The pendulum is modeled as a point mass at the end of a massless rod.
		The damping (friction) is proportional to the angular velocity of the pendulum.
		This is the equation of motion for the driven damped pendulum.
	www.myphysicslab.com /pendulum2.html (620 words)

	Cartoons and Movies
		Chaotic Pendulum Bifurcation Diagram: Plots the bifurcations of the chaotic pendulum as a function of g.
		In the chaotic regime, the Poincare plot of the damped driven pendulum is highly folded fractal.
		The basins of attraction of the damped driven pendulum have fractal boundaries.
	ist-socrates.berkeley.edu /~fajans/Teaching/cartoons (926 words)

	Pendulum (Site not responding. Last check: 2007-10-03)
		Pendulum is one of the topics in focus at Global Oneness.
		Another variety of a torsion pendulum is a fixed elastic coil connected to a rod-like object; once moved off its resting position, the co...
		To begin, we shall make three assumptions about the simple pendulum The rod/string/cable on which the bob is swinging is massless and always remains taut; The bob is a point mass; Motion occurs in a vertical plane, i.e.
	www.experiencefestival.com /pendulum (807 words)

	Simple Pendulum
		The pendulum should be thought of as a weight hung on a rigid rod of negligible mass from a pivot without friction in a medium which offers no resistance to things moving through it.
		The position is measured as the angle from the vertical of the pendulum with displacement to the right of vertical being positive and that to the left being negative.
		In the case of the pendulum it would be one dimension measured in position and one dimension measured in velocity.
	www.mcasco.com /pend1.html (1732 words)

	[No title]
		The forced damped pendulum is a computer model that makes an image of the paths a pendulum would go with several forces acting on it.
		This virtual pendulum is a mathematical laboratory tool, one whose behavior is caused by two factors, values which can be graphed in two dimensions; when the outcomes (just two) were matched to their respective causal factors on the graph, a phase space map is created.
		Equations that describe the motions of a ¡°forced damped pendulum.¡± The conclusion was that it revealed that when the pendulum was on the boarder of two physical fates, it actually was on the border of all possible fates.
	www.colorado.edu /physics/phys2900/replies8.html (3014 words)

	PIRA 5K20.00 EDDY CURRENTS (Site not responding. Last check: 2007-10-03)
		A heavy copper disk swings as a pendulum between the poles of an electromagnet.
		A pendulum with a copper plate bob is swung through a big electromagnet.
		A bar magnet suspended as a pendulum is damped as it swings over a copper plate.
	www.physics.ncsu.edu /pira/5eandm/5K20.html (748 words)

	Chaotic breakdown of a periodically forced, weakly damped pendulum (Site not responding. Last check: 2007-10-03)
		An investigation is made of the transition from periodic solutions through nearlyperiodic solutions to chaotic solutions of the differential equation governing forced coplanar motion of a weakly damped pendulum.
		The pendulum is driven by horizontal, periodic forcing of the pivot with maximum acceleration
		Further decrease of the forcing frequency leads to time intervals in which the motion is strongly unstable, with the pendulum passing intermittently over the pivot, interspersed with time intervals when the motion is nearly-periodic and only weakly unstable.
	anziamj.austms.org.au /V34/part2/Bryant.html (315 words)

	Control Experiment: Damped Compound Pendulum with Motor
		When voltage is applied to the motorized propeller, the pendulum oscillates until dynamic equilibrium is achieved; the pendulum settles at an angle where propeller lift balances pendulum weight.
		The step response graph shown in Figure 4 plots the pendulum angle after 2 Volts is applied to the motorized propeller.
		The angle was captured using an optical encoder mounted on pendulum pivot.
	prism2.mem.drexel.edu /~paul/thrustTester/thrustTester.html (2194 words)

	Pendulum Dynamics
		If you were to set the damping coefficient k = 0, the pendulum's motion would become simple again.
		It seems that the motion of the damped pendulum isn't terribly interesting; energy is always lost due to friction, and the pendulum is doomed to hang stationary however we start it off.
		The way to overcome this is to add energy to the pendulum; one of the many possible ways is to oscillate the pivot point.
	www.enm.bris.ac.uk /teaching/pendulum/maple2.htm (368 words)

	[No title]
		A viscous damping force, modeling for example the viscous damping of the oil in the bearing at the pendulum hinge, would to a good approximation be proportional to the angular velocity of the pendulum, with a coefficient we'll call alpha.
		Putting the damping just large enough to kill the oscillations also removes the energy as fast as possible: making the damping even larger is counterproductive.
		Sound is damped in air, light is absorbed as it passes through water, guitar strings don't vibrate forever.
	www.physics.cornell.edu /sethna/teaching/sss/galileo/galil02.htm (730 words)

	On the Dynamics of a Vertically Driven Damped Planar Pendulum (ResearchIndex) (Site not responding. Last check: 2007-10-03)
		Abstract: The dynamics of the planar pendulum with parametric vertical time-periodic forcing is considered.
		1 The inverted pendulum (context) - Kalmus - 1970
		1 The inverted pendulum (context) - Pippard - 1987
	citeseer.ist.psu.edu /432915.html (667 words)

	Pendulum 2D Phase Space, Damped (Site not responding. Last check: 2007-10-03)
		You will have noticed that on this display, the orbit is a spiral winding in toward the point where both position and velocity are zero.
		The phase space portrait of this pendulum would be a set of spirals starting adjacent to each other and ending at the equilibrium point.
		From the nature of the display we just observed you might conclude that whatever the initial conditions of position and velocity, the damped pendulum would eventually come to rest at the same phase space point, that of zero position and zero velocity.
	mcasco.com /oap2dpsd.html (179 words)

	Damped pendulum model (Site not responding. Last check: 2007-10-03)
		Formulate a linear pendulum model that includes damping and estimate the damping parameter experimentally.
		Arrange a pendulum so that you can measure the horizontal displacement of its bob.
		Couple that data with information obtained from the damped model (given in class) to determine the damping parameter p.
	users.wpi.edu /~heinrich/odeproj/node3.html (157 words)

	Jeremy Osinski's MATH 447 home page (Site not responding. Last check: 2007-10-03)
		The non-linear differential equations modeling the pendulum are the ones being used to test accuracy.
		The minimum d for which the pendulum is overdamped was found to be d=6.25 with initial conditions: pi/2 (position), 0 (velocity)
		With initial condition theta=3.1, and damping parameter d=1.0, parametric resonance occured within the range of aforce: 100-270.
	mason.gmu.edu /~josinski/p2/proj2.html (769 words)

	Vibe--Simple Pendulum with Damping (Site not responding. Last check: 2007-10-03)
		The damping function used is a velocity-squared damping which simulates air drag.
		Also, like the regular pendulum this oscillator accounts for the nonlinear performance inherent in large swings.
		The damped pendulum adds a damping factor which is multiplied by (velocity *
	www.iro.umontreal.ca /~eckdoug/vibe/Harmonic/PendulumDamped.html (96 words)

	CDS 280 - damped pendulum with torque
		Equation (1) describes the motion of a pendulum solely influenced by gravitation.
		The behavior of this pendulum is rather boring: eventually all oscillations die out.
		The position where the pendulum is hanging down vertically is a global attractor.
	www.cds.caltech.edu /~hinke/courses/CDS280/ddp.html (200 words)

	On the dynamics of a vertically driven damped planar pendulum
		On the dynamics of a vertically driven damped planar pendulum
		Results on the dynamics of the planar pendulum with parametric vertical timeperiodic forcing are reviewed and extended.
		We also calculate the Lyapunov exponents to show that for some parameter values the dynamics of the pendulum shows sensitivity to initial conditions.
	epubs.surrey.ac.uk /mathspubl/61 (142 words)

	Damped Pendulum Example
		The motion of a damped pendulum (i.e., a pendulum with friction slowing it) is described by a periodic function (cosine) whose amplitude decreases as time increases in proportion to a negative exponential function.
		The initial trial values for the parameters are specified on the Parameter statements.
		Title "Damped pendulum motion"; Variables Time,X; Parameter A = 90; Parameter alpha =.0008; Parameter w =.005; Parameter beta = 125; Parameter Phase = 0; Function X = Aexp(-alphaTime)cos(w(Time-Phase))+beta; Plot; Data; [ data goes here ]
	www.nlreg.com /pendulum.htm (173 words)

	[No title]
		A detailed walk-through of the concept underlying the method of control, and examples of the implementation of the control algorithm, are presented at a level accessible to the non-expert in chaotic dynamics.
		In his 1994 paper entitled “Control of the chaotic driven pendulum” Baker presented an excellent description of how the control algorithm is implemented for the situation of the driven, damped pendulum.
		The equation of motion for the system may be written (in dimensionless form) as EMBED Equation.3 (1) where (is the angular displacement of the pendulum from the vertical, q is a friction parameter, g is the forcing amplitude, and (d is the forcing frequency.
	www.math.mcgill.ca /~wilds/chaos/ChaosReport.doc (3248 words)

	Exercises (Site not responding. Last check: 2007-10-03)
		Describe and explain the motion of the pendulum.
		Zoom right in to the centre of the phase portrait to see the final state, as calculated by the computer.
		Warning: you will need to zoom in a very long way to see whether the pendulum is oscillating or not at the centre of the phase portrait.
	oldsite.vislab.usyd.edu.au /education/chaos/idjChaosLAB/node23.html (134 words)

	Citebase - Power fluctuations in a driven damped chaotic pendulum (Site not responding. Last check: 2007-10-03)
		Authors: Goldschmidt, Yadin Y. In this paper we investigate the power fluctuations in a driven, dampted pendulum.
		When the motion of the pendulum is chaotic, the average power supplied by the driving force is equal to the average dissipated power only for an infinite long time period.
		We measure the fluctuations of the supplied power during a time equal to the period of the driving force.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:nlin/0106016 (187 words)

	The Damped Pendulum: Linear Regime (Site not responding. Last check: 2007-10-03)
		There is a second effect which can determine how the pendulum moves: friction.
		The total torque on the pendulum is then
		Note: The frictional constant c needs to be specified from now on when running the program; there is a menu entry to do this.
	oldsite.vislab.usyd.edu.au /education/chaos/idjChaosLAB/node22.html (110 words)

	The Pendulum Driven by a Periodic Force
		Related topics in the lecture room: The harmonic oscillator with damping.
		Depending on the initial condition, the pendulum oscillates either with a large amplitude or with a small one.
		Related topics in the lecture room: Foldover and bistability.
	monet.physik.unibas.ch /~elmer/pendulum/spend.htm (210 words)

	Damped Pendulum
		Next, genetic programming tackled the damped pendulum system, given by Eqs.
		For this system, the tests produced direct hits quite quickly (not more than eight generations) in almost all cases.
		Figure 9: False result of a genetic programming Lyapunov search for a damped pendulum system.
	www.aerojockey.com /papers/lyapunovgp/node8.html (127 words)

	Lab on the Behavior of a Damped Pendulum
		Lab on the Behavior of a Damped Pendulum
		In this problem, there is a person on a swing.
		Several questions proposed are: How does the length of a swing affect the behavior; How does the weight and height of the rider affect the pendulum; For a given rider, what is the longest swing that allows you to "go over the top".
	www.sci.wsu.edu /idea/links/134.htm (69 words)

	NLREG -- Nonlinear Regression Analysis Program
		Since the NLREG language includes arrays, you can even use tabular look-up methods to define the function.
		Here is an example of an NLREG program for fitting a damped sine wave to some data:
		NLREG performs true nonlinear regression analysis and curve fitting, it does not transform the function into a linear form.
	www.nlreg.com (613 words)

	Forced Damped Pendulum (Site not responding. Last check: 2007-10-03)
		How to use the Forced Damped Pendulum Tool
		Use the sliders to set the values of the damping constant b, the forcing amplitude A, and the forcing frequency w.
		Press the mouse down on the slider knob for the parameter you want to change and drag the mouse back and forth, or click the mouse in the slider channel at the desired value for the parameter.
	www.maths.tcd.ie /pub/coursework/141/Applets/pndfrcph.html (114 words)

	Research Output: Department of Mathematics (Site not responding. Last check: 2007-10-03)
		Georgiou [2003] KAM Theorem and Stability of the Upside-Down Pendulum.
		Georgiou [2002] On the stability of the upside-down pendulum with damping.
		[2001] On the dynamics of a vertically-driven damped planar pendulum.
	www.maths.surrey.ac.uk /research/research-output.php (6822 words)

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