1 Driven Oscillations Driven Oscillators –Equation of motion –Algebraic solution –Complex algebra solution Phase and amplitude of particular solution –Algebraic –Complex phase methods Argand diagrams Resonance phenomena –Position of amplitude resonance –Phase behavior of x(t) with respect to driving force. Transient solution –Why transients are necessary? –Typical transient behavior Q as the FWHM of the kinetic energy resonance –Average kinetic energy of a driven oscillator over one cycle –Resonance approximation –Definition of FWHM

2 What will we do in this chapter? We discuss sinusoidally driven oscillators with velocity dependent damping. The equation of motion is an inhomogeneous differential equation. We show that the solution consists of a transient solution (which looks like a free oscillation solution) and a particular solution which depends on the driving force and frequency. We present an algebraic solution (text) and a complex algebra solution which I believe is better and introduces many useful “complex” techniques. We next discuss the amplitude and phase shift between the x motion and driving force as one passes through the resonant frequency which is close to the natural frequency. We show that the relative phase between the force and x(t) passes from 0 to as one sweeps through frequencies and 90 0 at the “natural” frequency of  o 2 = k/m. We also show that the width of resonance is proportional to 1/Q where Q ( for Quality) is a dimensionless variable proportional to  o divided by the damping coefficient. We conclude by discussing the role of transient solutions and by discussing the resonance behavior of the average kinetic energy as a function of driving frequency.

3 Driven Oscillations We need the transient piece to match the initial velocity and position, but it always dies out if there is damping. As a practical matter, it often suffices to know the particular solution. We will review the way that the text does this and show an alternative method using complex variable representations.

4 Algebraic Method

5 Contrast this with a “complex” solution

6 “Complex” solution continued We get the same result but I believe the technique is much easier and gives more insight. Extracting the phase and amplitude

7 Phase and amplitude of x p “complex” method

8 Polar forms Draw the D complex number in an Argand diagram where we plot the real part on the horizontal axis and the imaginary part on the vertical axis. The modulus is the length and the phase is the polar angle in such a diagram.

9 Resonance Q is a dimensionless variable which gives the “quality” of the resonance The amplitude resonates at the slightly different value of  R in a way that depends on Q or  It is remarkable that the x(t) response is out of phase with the driver force. At low , x(t) and F(t) are in phase. At    they are out of phase by At high , they are out of phase by !

10 Resonance phenomena We plot the amplitude and phase versus the driving frequency for 3 different Q values in the upper two plots. The larger the Q value -- the sharper the resonance both in phase and amplitude. The lower the plot shows the ratio of the resonant and “natural” frequency as a function of Q. For high quality resonances, the amplitude resonance occurs very nearly at the frequency of un-driven oscillations.

11 What’s the deal with transients? time You can get some pretty complicated motions when driving frequency is very different from the natural frequency. But the sum (upper) makes sense when you look at the 2 components (lower). Transient solutions are necessary since as shown above the “particular” solution is too specific. All aspects of x p are specified in terms of the driving frequency and driving force. Yet we still need some way of specifying arbitrary x(0) and v(0). We thus need to add a function w/ two coefficients. A and  give us the freedom to match x(0) and v(0)

12 The ubiquitous Q

13 More on Q FWHM Here is a plot to illustrate the FWHM concept on the average kinetic energy of the driven oscillator. I have selected a damping coefficient to create a fairly broad Q=4 oscillator. Even so the FWHM is close to 1/Q

14 How can we show this?

15 E loss (continued)

16 Electrical analogs There are many important analogs to the motion of the driven, damped harmonic oscillator. They occur in fluid mechanics, electrodynamics, quantum mechanics, etc and often appear in more practical problems than that of the mechanical oscillator. We will work out the electrical analogy in depth. We begin with a review of some basic elements of AC circuit theory that you presumably learned in physics 112.

17 Voltage drop across a capacitor A very crude analogy is that voltage is analogous to pressure, and charge is analogous to fluid mass. A capacitor stores charge like a a tank of a certain “capacity” stores water. Since the pressure difference between the top and bottom of the tank is proportional to the height of the water, a large capacity (area) tank develops little pressure drop when storing a certain amount of fluid. The electrical analogy is a capacitor with a large capacitance C requires a small voltage drop to store a given charge Q.

18 Voltage drop across a resistor A crude analogy for the voltage drop when current flow through a resistor is the “pressure” drop required to flow a viscous fluid through a narrow pipe. The physics behind Ohm’s law, a very simple and highly intuitive result, is actually fairly advanced and extremely important! In a perfect crystalline metal, an electron can flow from lattice site to lattice site unimpeded as is the case in a superconductor. This extremely unintuitive conclusion follows from a quantum mechanical treatment of electrons subjected to a perfect lattice of potentials due to the regularly spaced positive ion cores in the metal. The energy loss which requires a potential drop or electrical force to allow flow follows through through inelastic collisions of the electrons with imperfections in the lattice site pattern. This collision energy is transferred through quantum vibrations called “phonons”. The density of such phonons is roughly proportional to the temperature of the metal and hence ordinary ohmic resistance is roughly proportional to temperature. Of course there are conditions such that the resistance vanishes completely when certain materials are cooled below a critical temperature and the metal turns superconducting !

19 The voltage drop across an inductor I know of no simple fluid analogy to an inductor, so we will begin by discussing the real physics behind it. A solenoid is often used as an inductor. In a solenoid, one generates a magnetic field proportional to the current flowing through the coil.

20 The series RLC circuit The series RCL circuit forms a perfect analogy to the driven mechanical oscillator. An easy way to get the relevant differential equation in the charge is to set the total voltage drop around the loop along the indicated path to zero. Otherwise the voltage at a point is ill-defined This Kirchoff law is very similar in spirit to the work energy theorem.

21 Electrical/Mechanical analogies Q is analogous to x L is analogous to m R is analogous to b E is analogous to F The analogies follow directly by comparing the two differential equations:

22 Energy analogs Hence in the mechanical oscillator the energy oscillates between kinetic and potential energy. In the RLC circuit, the energy oscillates between electrical energy ( or the energy stored in the capacitor) and magnetic energy ( or the energy stored in the inductor). When driven at the natural frequency (    equal average amounts of kinetic and potential energy appears for the mechanical driven harmonic oscillator. When driven at the natural frequency, equal average amounts of electric and magnetic energy appear for the RLC circuit.