Class Notes 5, Phyx 2110
Oscillators, Attractors, and Resonance
I. INTRODUCTION SHO DISPLACEMENT vs TIME 2 A summary of harmonic oscillators is presented, with special emphasis on the late-time motion of each type of oscillator. When the late time motion is independent of the initial conditions, which are the values of the dis- placement x and velocity vx at time t = 0, the oscillator (or any dynamical system) is said to have an attractor: 0 the attractor is the dynamical behavior of the system x(t) (m) after a sufficient waiting period, that is, at long times after the system has been moving. An attractor is an important concept in chaos theory, which we will study shortly. 2 The relationship of the driven damped oscillator to the 0246810 ubiquitous phenomenon of resonance is also discussed. TIME (s) Resonance, which is a large amplitude response to a si- ME = 4 Joules nusoidal driving force, is an important concept that is ME = 16 Joules observed is a wide variety of systems, often much more complex that a simple harmonic oscillator. Figure 1. Displacement vs time for two SHO's, one with 4 J of mechanical energy and one with 16 J of energy. Both have k = 8 N/m and m = 0.5 kg. II. SIMPLE HARMONIC OSCILLATOR
A simple harmonic oscillator (SHO) is a mass attached III. DAMPED HARMONIC MOTION to an ideal spring with no friction and no external driving force. The SHO equation of motion (i.e., the appropriate form of Newton’s second law for the mass) A damped harmonic oscillator (DHO) is a mass on an is ideal spring with friction, but no external driving force. Its equation of motion is
max = −kx, (1) max = −kx + friction. (3) where m is the oscillator mass, ax is the acceleration, and k is the spring constant. The friction force is usually represented by a drag force At late times the motion of the SHO is given by that is linear in the velocity: friction = −bvx. The late-time motion is rather trivial: friction converts the coherent mechanical energy of the oscillator into in- x (t) = A sin (ω0t + φ) , (2) coherent thermal energy. This process is known as dis- sipation. As a result a mass on a spring moving with p where ω0 = k/m, the natural frequency of the os- friction always “runs down” and ultimately stops; this cillator, is related to its period T via ω0 = 2π/T . For “dead” end state (x = 0, vx = 0) is the attractor of this oscillator the late time motion is the same as the be- the dynamics because all initial states ultimately end ginning motion: the amplitude A and phase φ are com- up there. In other words, the late-time amplitude of the pletely determined by the initial conditions: the more motion is always zero for a DHO. energy put in to start, the larger the amplitude of the For a DHO there are three different types of motion motion. The phase shift φ also depends upon exactly that can be observed before all of the mechanical en- how the oscillator is started in its motion (we won’t worry ergy is dissipated. The first, known as underdamped about that detail now). Figure 1 shows the displacement motion, occurs when the damping parameter b is small, vs time for two simple harmonic oscillators, both with k specifically when b < 2mω0. This is illustrated in Fig. 2 = 8 N/m and m = 0.5 kg. However, the large-amplitude for two values of the damping parameter b. As shown oscillation has a mechanical energy of 16 J; the other has there the motion oscillates back and forth about x = 0 4 J. Note also, the phase φ is different for the two oscil- as the mechanical energy is dissipated. For b << 2mω0 lators since their peaks in displacement do not occur at the oscillation frequency is very close to the natural fre- the same time. Because there is no damping in a SHO, quency ω0 of the corresponding SHO. When b is very it oscillates indefinitely with constant amplitude A. small only a small amount of energy is removed from
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large, specifically when b > 2mω . Overdamped mo- UNDERDAMPED DHO x(t) vs time 0 tion is illustrated in two of the curves (solid and dotted) in Fig. 3. When b is very large the oscillator quickly 0.2 loses much of the kinetic energy that it may have had, as can be seen in the rapid initial drop in the displacement curves in Fig. 2. However, the oscillator will likely not be at x = 0 (and thus still possess a significant amount of potential energy). It then takes a very long time for the 0 oscillator to return to x = 0 because the large value of b x(t) (m) keeps the velocity small. Note that for the overdamped oscillator the the larger value of b results in a longer time for the oscillator to come to rest. The third type of motion is known as critically 0.2 damped motion. In this case b = 2mω0. In this amount of damping the coherent mechanical energy is most effi- 0246810 ciently converted into incoherent thermal motion and the t (sec) oscillator comes to rest most quickly. Critically damped b = 0.1 N m/s motion is illustrated in the dashed curve of Fig. 3. No- b = 0.5 N m/s tice that the displacement for this oscillator goes to zero much more rapidly than for either the underdamped os- Figure 2. Displacement vs time for two underdamped DHO's, cillators of Fig. 2 or the overdamped oscillators of Fig. one with b = 0.1 N m/s and one with b = 0.5 N m/s Both have 3. k = 8 N/m and m = 0.5 kg. An excellent example of a critically damped oscillator is provided by the suspension system of an automobile. The amount of damping (provided by the shock absorber) is such that the wheel, when it encounters a bump, stops OVERDAMPED AND CRITICALLY DAMPED 2 moving as quickly as possible. When the shock absorber becomes worn out, the system becomes underdamped, and the wheel oscillates wildly after encountering a bump in the road. 1.5
IV. DRIVEN DAMPED HARMONIC 1 OSCILLATOR x(t) (m) Often one is interested in a damped harmonic oscil- 0.5 lator that also has a time dependent external driving force, in addition to the spring and damping forces. The oscillator is then known as a driven, damped harmonic oscillator (DDHO) and its equation of motion is given by 0 012345 t (sec) b = 20 N m/s max = −kx − bvx + driving. (4) b = 0 N m/s 1 In principle, the driving force can have any time depen- b = 4 N m/s dence. For example, for the suspension on a car the Figure 3. Displacement vs time for overdamped and critically driving force would be the variations in normal force on damped DHO's. The two over damped oscillators have b = the tire from the road. However, to keep things rela- 10 N m/s and b = 20 N m/s. The critically damped oscillator tively simple, we often are interested in a driving force has b = 4 N m/s. All oscillators have k = 8 N/m and m = 0.5 kg. with a sinusoidal time dependence. That is, driving = F0 sin(ωdt). Note that because the driving force is from some external source the driving frequency ωd does not necessarily equal the natural frequency ω0 of the oscilla- the oscillator during each oscillation; it thus takes a long tor. time for the oscillator to come to rest. Note that the So what is the motion of a DDHO? At late times fric- oscillator with the smaller value of b takes longer to stop tion causes the coherent motion associated with the ini- oscillating. tial conditions [x (t = 0) and vx (t = 0)] to be converted The second type of motion is known as overdamped into incoherent thermal motions (as in the case of the motion: this occurs when the damping parameter b is DHO). At the same time the driving force continually
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6 x(t) vs t for a DDHO 3
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x(t) (m) 2 0 ENERGY DISSIPATION (J/s)
1 0 02468 2 FREQUENCY (radians/s) 0 5 10 15 b = 0.1 N s/m t (s) b = 0.5 N s/m Initial ME = 4 J Initial ME = 16 J Figure 6. Energy dissipated (in J/s) vs frequency ω. The solid line is for an oscillator with b = 0.1 N s/m. Figure 4. Displacement vs time for a DDHO. Both oscillators The dotted curve is for an oscillator with b = 0.5 N s/m. are identical, except for the initial conditions. The solid line is for an oscillator with an initial energy of 4 J while the dotted line corresponds to an initial energy of 16 J.
provides coherent ME which is also dissipated; however, since the driving force is continually providing a source of coherent motion, there ends up being a balance be- 1.5 tween the driving force and friction, resulting in a certain amount of late-time coherent motion. This late-time mo- tion can be written as
1 xlate (t) = Alate sin (ωdt + φlate) , (5)
where Alate is the amplitude of the late time motion, ωd is the frequency of the driving force – not the natural frequency ω of the oscillator, and φ is a phase shift. 0.5 0 late Note that these three quantities do not depend upon the initial conditions (x0 and vx0) of the oscillator. That is,
ATTRACTOR AMPLITUDE (m) this late-time motion is the same no matter what the initial conditions of the motion. In other words, this os- cillator also has an attractor, but instead of being “dead” 0 0246810as in the previous case, it is “alive” and wiggling with the FREQUENCY (radians/s) same frequency as the driving force. The phase shift φlate b = 0.1 N s/m in the argument of the sin function indicates that the mo- b = 0.5 N s/m tion is not necessarily in phase with the driver. That is, the oscillator position x need not be a maximum when the driving-force is a maximum. Figure 4 shows two ex- Figure . Attractor amplitude A (ω) vs driving 5 late amples of damped, driven oscillations, both with k = 8 frequency ω. The solid line is for an oscillator with b N/m, m = 0.5 kg, b = 0.5 Ns/m, and F = 5 N, but one = 0.1 N s/m. The dotted curve is for an oscillator 0 with a starting mechanical energy of 4 J and the other with b = 0.5 N s/m. with a starting mechanical energy of 16 J. Notice that after about 8 seconds both motions are identical – each system is now on the attractor.
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V. RESONANCE energy flow and resonance: the rate of energy flow is a maximum when the oscillator is driven at resonance. This is shown in Fig. 6, which plots the energy dissipa- The late time amplitude, Alate, (see Eq. (5)) is a func- tion (in J/s = Watts) vs the driving frequency. For all tion of the driving-force amplitude (F0), the amount of values of b the energy dissipation is a maximum when friction (b), and the driving frequency (ωd). Figure 5 the driving frequency equals the oscillator’s natural fre- shows graphs of the attractor amplitudes Alate as a func- tion of driving frequencies for two nearly identical oscil- quency. As with the amplitude plot (Fig. 6), smaller lators, the difference being the sharper curve corresponds values of b result in larger values of dissipation. This to an oscillator with less friction (smaller value of b) than may be a bit surprising, but one must keep in mind that the other. The maxima of the amplitudes occur when the smaller value of b result in larger oscillator velocities, so that the damping force magnitude bv is larger. driving frequency ωd is essentially equal to the natural frequency ω0 of the oscillator. This phenomenon of max- imum amplitude of vibration at the natural frequency of the oscillator is called resonance. In general, resonant VI. THE BIG PICTURE response becomes greater as the friction in the system becomes smaller. As Fig. 5 shows, when damping (b) Generally, attractors are dynamical behaviors that are is increased, the response is flatter and broader and the robust against environmental changes. In the absence of system (attractor) amplitude is not as large at resonance. dissipation every starting condition produces a different Another manifestation of resonance concerns energy behavior and no attractor can exist. However, when dis- dissipation. For late-time motion the driving force puts sipation is present attractors can appear. When there’s energy into the oscillator and the damping force takes no energy input (via a time dependent driving force) the energy out of the oscillator. However, because the am- attractor is “dead;” when there’s energy input as well plitude of the oscillator is constant, it mechanical energy as dissipation, the attractors are more interesting, reso- is also constant (remember, ME = 1/2kx2). Thus, for nance being the most interesting of all. This most inter- late-time motion energy simply flows from the external esting attractor occurs when friction is dissipating energy body (that is driving the oscillator) and into microscopic at its maximum rate. When we study thermodynamics, (thermal) energy, with the oscillator facilitating this en- we will see that this situation corresponds to the maxi- ergy flow. Here is the important thing regarding this mum rate of entropy production.