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Chapter 2 THE DAMPED HARMONIC OSCILLATOR Reference: George C. Kings, Vibrations and waves, A John Wiley and Sons, Ltd., Publication, 2009. In this lecture we will concentrate our attention to the damped harmonic oscillator. Before state the plan of the lecture let us briefly summarize preceeding topic, simple harmonic motion (SHM). In the previous chapter we have already discussed SHM. SHM is periodic and neither driven nor damped. We have stated that; SHM is that to a good approximation many real oscillating systems behave like simple harmonic oscillators when they undergo oscillations of small amplitude. Remember that we have investigated physical nature of the SHM on three fundamental examples: 1. Mass-Spring system The force acting on the mass is just due to spring. According to the Hook’s law we can write: No frictional force (damped force) No driven force. Therefore, according to the Newton’s second law In addition to displacement, velocity, acceleration, force, momentum, energy we define the quantities Frequency, (frequency is defined as a number of cycles per unit time.) Period (The period is the duration of one complete cycle in a repeating event) Angular frequency That describes sSHM. Note that force is proportional to the in a SHM then is defined as Return (restoring) force per unit displacement, per unit mass. Solution of the Newton’s equation yields The constants can be obtained from initial conditions. Solution of the equation include information about various physical quantities, such that Velocity Acceleration Force Momentum Energy 2. Simple Pendulum Again we can write the force acting on the mass m. For small oscillations , we can write the equation of motion: Its solution is given by We point out that, for the physical pendulum we consider inertia of the oscillating rod. In that case 3. The last example we have discussed is LC circuit: Using Kirchhoff’s law Its solution is again We also discussed correspondence between physical quantities of the oscillator take place of or take place of take place of or IMPORTANT!! The physical models do not include any damping or driving force! It is not realistic. We have ignored friction between mass and surface, air resistance, and resistance of the cables etc.. In order to make a more realistic physical model we have to take into consideration, resistive or frictional forces. I hope that at the end of this chapter you understand physical characteristic of damping oscillator and solution of the Newton’s equation including damping forces. Outline of the Chapter: 2. THE DAMPED HARMONIC OSCILLATOR 2.1 Physical Characteristics of the Damped Harmonic Oscillator 2.2 The Equation of Motion for a Damped Harmonic Oscillator 2.2.1 Light damping (weak damping) 2.2.2 Heavy damping (strong, over damping) 2.2.3 Critical damping 2.3 Rate of Energy Loss in a Damped Harmonic Oscillator 2.3.1 The quality factor Q of a damped harmonic oscillator 2.4 Damped Electrical Oscillations PROBLEMS The Damped Harmonic Oscillator We have described simple harmonic motion that; a mass i.e mass of the pendulum, swinging back and forth at the end of a string a mass oscillating up or down or back and forth attached to a sprinq charges oscillating over capacitor and inductor in a circuit. Note that this oscillating system is not ideal. In fact after we set the mass (or charge) in motion, the amplitude of oscillation steadily reduces and the apple eventually comes to rest. This is because there are dissipative forces acting and the system steadily loses energy.(remember energy is proportional to the square of the amplitude of the oscillation!) For example, the mass will experience a frictional force as it moves through the air. There exist between mass and surface when the mass oscillating horizontally attached to a spring. In a realistic model, there are resistive force acting on the charge in LC circuit, due to wires and internal resistance of the devices. The motion is damped and such damped oscillations are the subject of this chapter. We note that all real oscillating systems are subject to damping forces and will cease to oscillate if energy is not fed back into them. How can we model oscillating system with damping? How can we write the Newton’s equation including damping force? How can we relate the damping force with the physical quantities? Consider an object moving in a viscous fluids (or gas). Fluid (also gas) experience a force on the object is known as drug force. (result of interaction between molecules). Often these damping forces are linearly proportional to velocity. (In fact it is proportional to the nth power of the velocity but its simplified form for relatively small speeds is proportional to the first power of velocity). In an electric circuit this force (potential) is known as resistive force and it is proportional to the current. Example: In an electric circuit, power dissipated on a resistor of resistivity 100 Ohm, is 4.0 Watt. Calculate current through resistor and voltage across the resistor. Solution: Power of a resistor is defined as follows: Potential across the resistor is then power then For a mechanical system power can be expressed as ; Here F corresponds potential and corresponds current . Fortunately, this linear dependence leads to an equation of motion that can be readily solved to obtain solutions that describe the motion for various degrees of damping. Clearly the rate at which the oscillator loses energy will depend on the degree of damping and this is described by the quality of the oscillator. At first sight, damping in an oscillator may be thought undesirable (unwanted). However, there are many examples where a controlled amount of damping is used to quench unwanted oscillations. For instance quality of the suspension system of a car depend on damping! If damping is zero then the car’s bouncing up and down long after it has passed over a bump in the road. Think! What happens when a bridge does not have a damping? Additional damping was installed on London’s Millennium Bridge shortly after it opened because it suffered from undesirable oscillations. 2.1 PHYSICAL CHARACTERISTICS OF THE DAMPED HARMONIC OSCILLATOR A tuning fork is an example of a damped harmonic oscillator. Indeed we hear the note because some of the energy of oscillation is converted into sound. After it is struck the intensity of the sound, which is proportional to the energy of the tuning fork, steadily decreases. However, the frequency of the note does not change!!(Be careful). The ends of the tuning fork make thousands of oscillations before the sound disappears and so we can reasonably assume that the degree of damping is small. We may suspect, therefore, that the frequency of oscillation would not be very different if there were no damping. Thus we infer that the displacement of an end of the tuning fork is described by a relationship of the form where the angular frequency is about but not necessarily the same as would be obtained if there were no damping. 2.2 THE EQUATION OF MOTION FOR A DAMPED HARMONIC OSCILLATOR An example of a damped harmonic oscillator is shown in Figure. It is similar to the simple harmonic oscillator described in previous section but now the mass is immersed in a viscous fluid. When an object moves through a viscous fluid it experiences a frictional force. This force dampens the motion: the higher the velocity the greater the frictional force. So as a car travels faster the frictional force increases thereby reducing the fuel economy, while the velocity of a falling raindrop reaches a limiting value because of the frictional force. (Talk about limiting-terminal velocity) The damping force acting on the mass in Figure is proportional to its velocity so long as is not too large (for large the force is proportional to the higher power of , i.e for an airplane frictional force is proportional to the square of the velocity). The resulting equation of motion is where the minus sign indicates that the force always acts in the opposite direction to the motion. The constant depends on the shape of the mass and the viscosity of the fluid and has the units of force per unit velocity. We introduce the parameters The equation takes the form: This is the equation of a damped harmonic oscillator. Now we designate this angular frequency and describe it as the natural frequency of oscillation, i.e. the oscillation frequency if there were no damping. This allows the possibility that the damping does change the frequency of oscillation. In the present example the damping force is linearly proportional to velocity. This linear dependence is very convenient as it has led to an equation that we can readily solve. As we mentioned before, this linear dependence is a good approximation for many other oscillating systems when the velocity is small. The damped oscillator equation can be solved by using various methods. Suppose that the damping is exponential. The general solution of the equation will be in the form Substituting into the equation we obtain: Then we obtain Then the general solution takes the form Depending on the relation between damping constant and natural frequency the degree of damping involved, corresponding to the cases of (i) light damping, (ii) heavy or over damping and (iii) critical damping. Light damping is the most important case for us because it involves oscillatory motion whereas the other two cases do not. Let us break here. We will continue to explore our analysis next lecture. Briefly review previous lecture notes! Remember, we have obtained a general solution: Where , describe type of the oscillation as we stated before. 2.2.1 Light damping Consider a mass m immersed in a fluid as in the figure. We assume that the viscosity of the fluid is low like thin oil or even just air. The parameters and are determined solely by the properties of the oscillator while the constants are determined by the initial conditions. If we let we obtain, as expected, our previous results for the simple harmonic oscillator. A graph of is shown in Figure for different values of Here we are dealing weak damping therefore for simplicity we set We see that successive maxima decrease by the same fractional amount. This is logarithmic decrement 2.2.2 Heavy damping(Over damped oscillator, strong damping) Heavy damping occurs when the degree of damping is sufficiently large that the system returns sluggishly to its equilibrium position without making any oscillations at all. This corresponds to the mass being immersed in a fluid of large viscosity like syrup. Consider frequency of the oscillation: If is larger than , then the system is over damped. In this case becomes imaginary and Hyperbolic function is not periodic, it is exponential. Therefore the system is over damped. Figure shows over damped motion of the mass. 1, 0 12 1 3 2 2 2 1, 0 6 1 3 2 2 2 4, 0 1 1 2 3 2 2 6, 0 1 1 2 3 2 2 2.2.3 Critical damping An interesting situation occurs when equation becomes . In this case the solution of the damping oscillator This is the case of critical damping. Here the mass returns to its equilibrium position in the shortest possible time without oscillating. Critical damping has many important practical applications. For example, a spring may be fitted to a door to return it to its closed position after it has been opened. In practice, however, critical damping is applied to the spring mechanism so that the door returns quickly to its closed position without oscillating. Similarly, critical damping is applied to analogue meters for electrical measurements. This ensures that the needle of the meter moves smoothly to its final position without oscillating or overshooting so that a rapid reading can be taken. Springs are used in motor cars to provide a smooth ride. However damping is also applied in the form of shock absorbers as illustrated schematically in Figure. Without these the car would continue to bounce up and down long after it went over a bump in the road. Schematic diagram of a car suspension system showing the spring and shock absorber. Shock absorber consists essentially of a piston that moves in a cylinder containing a viscous fluid. Holes in the piston allow it to move up and down in a damped manner and the damping constant is adjusted so that the suspension system is close to the condition of critical damping. You can see the effect of a shock absorber by pushing down on the front of a car, just above a wheel. The car quickly returns to equilibrium with little or no oscillation. You may also notice that the resistance is greater when you push down quickly than when you push down slowly. This reflects the dependence of the damping force on velocity. 2, 0 1 1 2 3 2 2 To appreciate the physical origin of these different types of motion, we recall that is the damping term while is proportional to the spring constant . When the damping term is small compared with the motion is governed by the restoring force of the spring and we have damped oscillatory motion. Conversely, when the damping term is large compared with the damping force dominates and there is no oscillation. At the point of critical damping the two forces balance. We finally note that the relative size of compared with also determines the response of the oscillator to an applied periodic driving force, as we shall see in next Chapter. Worked example (I will add an example) 2.3 RATE OF ENERGY LOSS IN A DAMPED HARMONIC OSCILLATOR The mechanical energy of a damped harmonic oscillator is eventually dissipated as heat to its surroundings. We can deduce the rate at which energy is lost by considering how the total energy of the oscillator changes with time. The total energy E is given by For undamped oscillator the energy is given by For damped oscillation the displacement can be written as: We can write In this case energy takes the form: where is the total energy of the oscillator at t = 0. We have the important and simple result that the energy of the oscillator decays exponentially with time as shown in Figure. We also have an additional physical meaning for . The reciprocal of is the time taken for the energy of the oscillator to reduce by a factor of . Defining , we obtain where has the dimensions of time and is called the decay time or time constant of the system. There are many examples of both classical and quantum-mechanical systems that give rise to exponential decay of their energy with time as described here and for some of these is called the lifetime. The energy of an oscillator is dissipated because it does work against the damping force at the rate (damping force × velocity). We can see this by differentiating . Power dissipated by resistive or damped or drag force is 2.3.1 The quality factor Q of a damped harmonic oscillator From the foregoing analysis, it is clear that the damped oscillator is characterized by two parameters, . The constant is the angular frequency of undamped oscillations and is the reciprocal of the time required for the energy to decrease to of its initial value. Thus are quantities of the same dimensions. For convenience in applying our results to diverse kinds of physical systems, we define a parameter called the value ( for quality) of the oscillatory system, given by the ratio of these two quantities: Note that meaning of the have the same dimensions therefore then we define -value is a pure number. Remember physical Angular frequency can be expressed as: It is quite evident that the higher the value, the greater the number of oscillations. We can define the quality factor in a different way by considering the rate at which the energy of the oscillator is dissipated. If we consider the energy of a very lightly damped oscillator one period apart we have Giving The series expansion of Thus, when If is , to a good approximation. And therefore where we have substituted . The fractional change in energy per cycle is equal to so the fractional change in energy per radian is equal to . 2.4 DAMPED ELECTRICAL OSCILLATIONS In our mechanical example of a mass moving through a fluid we saw that the fluid offered a resistance that damped the motion. In the case of an electrical oscillator it is the resistance in the circuit that impedes the flow of current. An electrical oscillator is shown in Figure. It consists of an inductor and capacitor as before but now there is also the resistor We charge the capacitor to voltage switch. Kirchoff’s law gives and and then close the This has the identical form to Equation, and we recognise the analogous quantities we met before: is equivalent to , to and to . However, we see that is analogous to the mechanical damping constant and so is the equivalent of . Since the above differential equations have identical forms, their solutions also have identical forms. The importance of this is that we can use our results for the mechanical oscillator to immediately write down the corresponding results for the electrical case. Thus solution of the equation is Where , describe type of the oscillation as we stated before. Here is initial charge on the capacitor and Similarly we find that the quality factor of the circuit is given by For example, with , which is a typical value for an electrical oscillator. Again we emphasize the exact correspondence between the equations and solutions that describe the mechanical and electrical systems, so that mechanical systems can be simulated by electrical circuits. Such analogue computers can greatly facilitate the design and development of mechanical systems. Now, we have completed chapter 2. Next lecture I will solve sample problems.