# Py231 Notes On Linear And Nonlinear Oscillators And-Books Pdf

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2 1 The Simple Oscillator, all it does is shift the equilibrium point We have seen in class that when pulled down the. mass will oscillate about the point where it was originally at rest which is often called the. equilibrium position, We say that the ideal spring provides a linear restoring force that means that it pulls the. mass back towards its equilibrium point with a force which is proportional to how far you. stretch it If we choose y 0 to be the equilibrium point then the force the spring exerts. on the mass is given by, This is a linear restoring force the sign tells us the force pushes or pulls the mass back. towards the equilibrium point K is a constant which tells how stiff the spring is. aside If the force included a term like y 2 or y 3 then it would be a much more difficult. problem to solve We will briefly discuss such nonlinear forces at the end of these notes. Because all of the physical systems which appear in musical acoustics are nonlinear we will. have to address this issue, Fig 2 The motion of a simple oscillator We show the vertical position as a function of time. assuming that at the beginning the mass was released from the maximum position. in the positive direction The time when the mass has finished one cycle period is. indicated by P, If we used the laws of physics to obtain the equation which described the motion of the.
mass on a spring and its solution we would find several simple facts about the motion of. 1 The mass bounces up and down We say that the motion is sinusoidal in time which. is sketched in Fig 2 This means that if we drew a graph of the position as a function. of time we would obtain a graph, y A cos 2 f0 t 2, where the frequency is given by s. and the amplitude A is just the maximum vertical distance the mass travels. 1 1 The Undamped Oscillator and Energy 3, 2 There is only one frequency with which this system can oscillate and that frequency. is f0 The period of the motion is just the inverse of the frequency i e P 1 f0. 3 The frequency does not depend on the amplitude or how the system is set into motion. but only on how stiff the spring is and how large the mass is. 4 If we increase the mass inertia of the system we lower the frequency it will vibrate. 5 If we increase the stiffness of the spring we will increase the frequency. 1 1 The Undamped Oscillator and Energy, If we stretch the spring we have to do work against the spring force. The amount of work depends on the distance squared and is given by. If we pull the spring out to a distance y A then the work which we do is W 1 2 KA2. This is the amount of energy that we put into the oscillator and we conclude that the total. energy of the oscillator is proportional to the square of the amplitude of the oscillation. After we have pulled the spring back we have given the mass spring system some potential. energy which is energy which exists by virtue of the configuration shape of the system. If we release the mass after pulling it aside the spring will accelerate the mass and it. will return to the equilibrium position but at that moment where the potential energy is. zero the potential energy will have been transformed into kinetic energy energy of motion. which is given by 1 2 mv 2 where v is the velocity. Since the mass has inertia associated with it it will keep moving in the same direction. past the equilibrium point After the mass passes the equilibrium point the spring starts. to decelerate the mass and it will stop when it reaches a distance A on that side of the. equilibrium point During this part of the cycle the spring is doing work on the mass. turning the kinetic energy into potential energy again. The spring then accelerates the mass back in the direction it came from changing the. potential energy into kinetic energy, Note that at any point except y 0 which is the equilibrium position or y A which. is the very top or bottom of the motion the energy of the system is a mixture of kinetic. and potential energy Note also that the total mechanical energy potential plus kinetic is. a constant at any time during the motion This is often called conservation of energy. In Fig 3 the total energy the kinetic energy and the potential energy are shown as a. function of time for the oscillator which starts its motion from rest at y A. The total mechanical energy is a constant which only depends on the maximum displace. ment and how stiff the spring is It is a constant because there is no mechanism in the simple. oscillator to dissipate the energy The real oscillator will of course have damping and we. will study this in the next section, 4 2 The Damped Oscillator.
Total Energy, Fig 3 The total kinetic and potential energy of a simple oscillator through several cycles. The total energy is constant 12 KA2 The kinetic energy is shown with a dashed. line and the potential energy is shown with the solid line The time for one and two. periods of oscillation are indicated by P and 2P Initially the mass was at rest at. 1 2 Summary of the Simple Oscillator, A linear restoring force leads to simple harmonic motion which occurs at a frequency de. termined by the square root of the spring stiffness divided by the mass The period of the. motion is given by the inverse of the frequency The amplitude of the oscillation is con. stant since there is no way for energy to leave the system since we said that for the simple. oscillator there are no dissipative forces, The example of the simple oscillator which we used was the mass on a spring Another. almost simple example which we have all seen is a swing like those found on all playgrounds. Just as a clock pendulum the rider swings back and forth with a frequency which for small. amplitudes only depends on the acceleration of gravity and the length of the swing We. return to this example below when we discuss the driven oscillator. Up to this point we have only discussed simple oscillators with one way to vibrate i e. they have only one frequency However there are many examples of oscillators which have. more than one way to vibrate All musical systems e g bars drum heads strings air. columns in tubes even the coffee in your cup have more than one way to vibrate Neverthe. less the simple ideas developed above are still applicable. 2 The Damped Oscillator, The ideal oscillator discussed above does not really exist on the macroscopic scale Real. oscillators always experience a damping force often one proportional to velocity The pro. totypical system is a mass suspended on a spring but with a damper which is suspended in. a viscous fluid 1 Such a system is sketched in Fig 4. This sort of velocity dependent force is familiar to us all At 10 mph our hand sticking. out of the car window does not feel too much resistance At 60 or 70 mph the effect is. The viscosity of a fluid tells us how much it resists something moving through it Air is much less viscous. than water Water is much less viscous than molasses If you have ever bought oil for your car engine you. may have seen the label 10W40 or 5W30 which indicate values of the viscosity both cold and hot. Viscous Fluid, Fig 4 The Damped Oscillator, much more dramatic The oscillator experiences an additional force which in the simplest.
approximation depends linearly on the velocity v which we write as bv where b is called. the damping coefficient 2 The negative sign tells us that the damping force opposes the. The resulting motion of a system depends on how large the damping force is Consider. what will happen when you pull the mass aside and let it go as we described above You. can imagine that the damping force could be so large that shortly after you release the mass. the damping force just balances the spring force and the mass slowly moves back to its. equilibrium position This situation is called overdamped. On the other hand the damping could be light enough to permit the mass to oscillate. a few cycles or many cycles for that matter Physicists call this underdamping or light. damping However each successive oscillation will have a reduced amplitude since part of. the system s energy will be lost to work done against the damping force. A good example of this is an automobile When a car hits a bump it may bounce up. and down once or twice but unless the shock absorbers are bad it will very quickly stop. oscillating up and down, We are all familiar with energy loss due to friction While this viscous damping force. is not exactly like friction some ideas do transfer over For example think about what. happens when you push a heavy box across the floor We have to do work against friction. to get the box to slide across the floor To get it started sliding we have to overcome the. static frictional force and the box s inertia If the box were on wheels then once you got. it rolling it would continue to roll However eventually it would stop because the wheels. also have some friction which will remove the kinetic energy which resulted from your doing. work on the box, If the velocity is sufficient to cause turbulence then a damping force quadratic in velocity becomes. important i e Fdamping b1 v b2 v 2 and the equation which describes the motion of the system becomes. nonlinear We will ignore this quadratic term which implies that the velocities under consideration are not. too large and that the coefficient b2 is small, 6 2 The Damped Oscillator. Fig 5 The motion of an under damped oscillator, The motion is sketched in Fig 5 where we see that the amplitude decreases with time. We can define the halving time T1 2 which is the time it takes for the amplitude to be. reduced by one half Note that this time T1 2 is independent of the time when you start. to measure T1 2 The important measure of the damping is the quantity b m the ratio. of the strength of the damping force to the mass, aside For the curious we give the the equation for the position as a function of time.
y t Ae 2 t cos 2 fd t 6, There are two important features to this solution which arise from damping The am. plitude of the oscillations dies away the heavier the damping the faster the oscillations die. out and the frequency of oscillation is lowered from f0 to some new value fd The size of. the frequency change is small and we will not worry about it here. The decrease in amplitude should come as no surprise since the damping force does work. on the system and thus takes energy away from it Each time the mass comes to rest let s. call the displacement where it comes to rest Ymax the total energy remaining in the the. system is given by 1 2 KYmax however since energy is lost Ymax is less each time. Where does the energy go The ultimate answer is into heat The damper moving. through the liquid stirs the molecules and heats them up Even if you just have a block of. wood on a spring in air there will be some loss of energy due to the viscosity of the air. In a real spring there is a second source of energy loss If you have ever bent a piece. of metal back and forth until it breaks you have noticed that it heats up A real spring. supporting the mass will dissipate some energy, 2 1 The Damped Oscillator in Music. A number of musical instruments are damped non simple oscillators which have several. frequencies with which they can vibrate The simplest example is a tuned bar or tuning fork. which is designed to give a single pitch for tuning More complicated examples are the string. of a guitar or violin the air in a trumpet and the head of a drum There are two ways in. 2 2 Radiation Damping 7, which energy is put into these systems For percussion instruments the piano and the guitar. we strike the instrument or string with a mallet or hammer or pluck it with a plectrum. Then the system is allowed to vibrate freely The amplitude will decrease with time due to. damping forces and eventually the oscillations become so low in amplitude that the sound. is no longer audible, For the bowed violin the trumpet or a number of other instruments the player provides. a constant source of energy and a steady sound is produced We discuss this further in the. section below on driven oscillators, 2 2 Radiation Damping.
The tuned bar used for demonstrations in this course has a resonator box below it which. makes the oscillations of the bar at 440 Hz sound louder In addition to the dissipative. forces which remove energy from the bar after it is struck some of the energy goes into the. sound waves which are radiated into the air, What happens to the air in the box below the bar also leads us to the topic of resonance. and the driven oscillator, 3 The Driven Oscillator. When you were small chances are you went to a playground and rode in a swing At first. you had to be pushed by someone bigger but soon you learned that you could pump the. swing to get it going and if you kept pumping the amplitude of your swinging increased to. a maximum value What you didn t realize was that you were providing a periodic driving. force and the period was that of the swing If you pumped too fast or too slow it didn t. work very well, This was your introduction to the phenomenon of resonance which is just the response. of a system which can oscillate to a driving force with a frequency equal or close to the. natural frequency of the oscillator, Although the equations are not that complicated our point here is to describe this. behavior in a qualitative way We can separate the response of an oscillatory system to a. PY231 Notes on Linear and Nonlinear Oscillators and Periodic Waves B Lee Roberts Department of Physics Boston University DRAFT January 2011 1 The Simple Oscillator In many places in music we encounter systems which can oscillate If we understand such a system once then we know all about any other situation where we encounter such a system

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