UNIT 2 | unit 2 simple harmonic motion damped and forced simple harmonic oscillator

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UNIT 2

SIMPLE HARMONIC MOTION, DAMPED AND FORCED SIMPLE HARMONIC OSCILLATOR

2.1 MECHANICAL AND ELECTRICAL SIMPLE HARMONIC OSCILLATORS

MECHANICAL SIMPLE HARMONIC OSCILLATORS

The figure-1 depicts simple mechanical oscillator having mass attached to the spring. This structure is similar to Pendulum. It is also known as harmonic oscillator.

Figure: 1

If a body attached to a spring is displaced from its equilibrium position, the spring exerts a restoring force on it, which tends to restore the object to the equilibrium position. This force causes oscillation of the system, or periodic motion.
When the restoring force is directly proportional to the displacement from equilibrium, the resulting motion is known as simple harmonic motion and the oscillator is known as Harmonic Oscillator.

In this type of system body itself changes its position.

For mechanical oscillation two things are especially responsible i.e. Inertia & Restoring force.

Differential Equation

A particle is said to be execute simple harmonic oscillation is the restoring force is directed towards the equilibrium position and its magnitude is directly proportional to the magnitude and displacement from the equilibrium position. If F is the restoring force on the oscillator when its displacement from the equilibrium position is x, then

F–x

Here, the negative sign implies that the direction of restoring force is opposite to that of displacement of body i.e. towards equilibrium position.

F= -kx ............. (1)

Where, k= proportionality constant called force constant.

An example of such a simple system is the mass m, attached to a spring of stiffness k.

Figure: 2

In the case of a mass on a spring, the restoring force for small oscillations obeys Hooke’s law:

F=−kx,

Where k is the stiffness of the spring. Here the coordinate x=0 corresponds to the point of equilibrium, in which the force of gravity is balanced by the initial tension of the spring.

Then, according to Newton’s second law, the movement of the mass will be described by the differential equation

Mx′′=−kx,

⇒x′′+ x=0.

Solution of differential equation is

x= Acosωt

Thus, the mass on the spring will perform undamped harmonic oscillations with the circular frequency

ω=

The period of oscillation, respectively, will be equal to

T= = 2π

A similar analysis of other oscillatory system – a simple (mathematical) pendulum – leads to the following formula for the oscillation period:

T=2π

Where L is the length of the pendulum, g is the acceleration of gravity.

In the case of a compound or physical pendulum, the period of oscillation is given by

T=2π

Where I is the moment of inertia of the pendulum about the pivot point, m is the mass of the pendulum, a is the distance between the pivot point and the centre of mass of the pendulum.

Example: oscillation of mass spring system, oscillation of fluid-column in a U-tube, oscillation of simple pendulum, rotation of earth around the sun, oscillation of body dropped in a tunnel along earth diameter, oscillation of floating cylinder, oscillation of a circular ring suspended on a nail, oscillation of atoms and ions of solids, vibration of swings etc.

2.2 DAMPED OSCILLATIONS

We have seen that the total energy of a harmonic oscillator remains constant. Once started, the oscillations continue forever with a constant amplitude (which is determined from the initial conditions) and a constant frequency (which is determined by the inertial and elastic properties of the system). Simple harmonic motions which persist indefinitely without loss of amplitude are called free or undamped.

However, observation of the free oscillations of a real physical system reveals that the energy of the oscillator gradually decreases with time and the oscillator eventually comes to rest. For example, the amplitude of a pendulum oscillating in the air decreases with time and it ultimately stops. The vibrations of a tuning fork die away with the passage of time. This happens because, in actual physical systems, friction (or damping) is always present. Friction resists motion.

The presence of resistance to motion implies that frictional or damping force acts on the system. The damping force acts in opposition to the motion, doing negative work on the system, leading to a dissipation of energy. When a body moves through a medium such as air, water, etc. its energy is dissipated due to friction and appears as heat either in the body itself or in the surrounding medium or both.

There is another mechanism by which an oscillator loses energy. The energy of an oscillator may decrease not only due to friction in the system but also due to radiation. The oscillating body imparts periodic motion to the particles of the medium in which it oscillates, thus producing waves. For example, a tuning fork produces sound waves in the medium which results in a decrease in its energy.

All sounding bodies are subject to dissipative forces, or otherwise, there would be no loss of energy by the body and consequently, no emission of sound energy could occur. Thus, sound waves are produced by radiation from mechanical oscillatory systems. The electromagnetic waves are produced by radiations from oscillating electric and magnetic fields.

The effect of radiation by an oscillating system and of the friction present in the system is that the amplitude of oscillations gradually diminishes with time. The reduction in amplitude (or energy) of an oscillator is called damping and the oscillation are said to be damped.

When the energy of a oscillating system is gradually dissipated by friction and other resistances, the oscillations are said to be damped. The oscillations gradually reduce or change in frequency or intensity or cease and the system rests in its equilibrium position.

Damped oscillations refer to oscillations where the oscillating object loses its energy to the surroundings.

Damping, in physics, restraining of vibratory motion, such as mechanical oscillations, noise, and alternating electric currents, by dissipation of energy. Unless a child keeps pumping a swing, its motion dies down because of damping. Shock absorbers in automobiles and carpet pads are examples of damping devices.

A system may be so damped that it cannot oscillate. Critical damping just prevents oscillation or is just sufficient to allow the object to return to its rest position in the shortest period of time.

Calculations for Damped Systems Undergoing Free Vibration | Engineerin

Figure: 3

There are many types of mechanical damping. Friction, also called in this context dry, or Coulomb, damping, arises chiefly from the electrostatic forces of attraction between the sliding surfaces and converts mechanical energy of motion, or kinetic energy, into heat. The motion of a oscillating body is also checked by its friction with the gas or liquid through which it moves.

2.3 DAMPED HARMONIC OSCILLATOR – HEAVY, CRITICAL AND LIGHT DAMPING

Damped Oscillations

In real systems, there is always a resistance or friction, which leads to a gradual damping of the oscillations. In many cases, the resistance force (denoted by FC) is proportional to the velocity of the body, that is

FC=−cx′.

Then, taking into account the force of resistance, the differential equation for the “mass-spring” system is written as

Mx′′+cx′+kx=0, ⇒ x′′+ x′+x=0

We introduce the following notations: =2β, = . Here ω0 is the natural frequency of the undamped oscillator (previously, we denoted it as ω), β is the damping coefficient. In the new notations, the differential equation looks like

x′′+2βx′+x=0.

We will seek the solution of this equation as a function

x(t)=Aeλt.

The derivatives are given by

x′(t)=Aλeλt, x′′(t)=Aλ2eλt.

Substituting this into the differential equation, we obtain the algebraic characteristic equation:

Aλ2eλt+2βAλeλt+Aeλt=0,

⇒λ2+2βλ+ =0.

The roots of this equation are

D=4β2−4,

It can be seen that depending on the sign of the radicand β2−, there may be three different types of solutions.

Case 1. Overdamping or Heavy Damping: β>ω0

When resistance to motion is very strong, the system is said to be heavily damped. Can you name a heavily damped system of practical interest? Springs joining wagons of a train constitute the most important heavily damped system. In your physics laboratory, vibrations of a pendulum in a viscous medium such as thick oil and motion of the coil of a dead beat galvanometer are heavily damped systems.

In this case (the case of strong damping), the radicand is positive: β2>. The roots of the characteristic equation are real and negative. The general solution of the differential equation has the form

x(t)=C1eλ1t+C2eλ2t,

Where the coefficients C1, C2, as usual, depend on the initial conditions.

It follows from this expression that there are no oscillations and the system returns to equilibrium exponentially, i.e. a periodically (Figure 4).

Figure 4

Case 2. Critical Damping: β=ω0

You may have observed that on hitting an isolated road bump, a car bounces up and down and the occupants feel uncomfortable. To minimise this discomfort, the bouncing caused by the road bumps must be damped very rapidly and the automobile is restored to equilibrium quickly. For this we use critically damped shock absorbers. Critical damping is also useful in recording instruments such as a galvanometer (pointer type as well as suspended coil type) which experience sudden impulses. We require the pointer to move to the correct position in minimum time and stay there without executing oscillations. Similarly, a ballistic galvanometer coil is required to return to zero displacement immediately.

In the limiting case when β=ω0, the roots of the characteristic equation are real and coincide:

λ1=λ2=−β=−ω0.

Here the solution is given by the formula

x(t)=(C1t+C2)

In this mode, the value of x(t) may even increase at the beginning of the process because of the linear factor C1t+C2. But in the end the deflection x(t) decreases rapidly due to the exponential decay with a characteristic time τ=2π/ω0.

Note that in this critical mode the relaxation occurs faster than in the case of the aperiodic damping (Case 1). Indeed, in this mode the relaxation time will be determined by the smaller (in absolute value) root λ1, and will be given by the formula

The function Φ(β/ω0) included in this expression is monotonically increasing. It is always greater than or equal to 1, as shown in Figure 5

Figure 5

In the critical case (Case 2) the ratio βω0 is 1, and βω0>1 in the case of the aperiodic damping (Case 1). Therefore for the aperiodic damping mode, we can write

τ= Φ() >

Thus, the critical damping mode provides the fastest possible return of the system to equilibrium. This is often used, for example, in door closing mechanisms.

Case 3. Underdamping or Light Damping β<ω0

Here the roots of the characteristic equation are complex conjugate:

λ1,2 =−β±i√

The general solution of the differential equation is oscillatory in nature and can be written as

x(t)=e−βt[C1cos(ω1t)+C2sin(ω1t)]

Where the oscillation frequency ω1 is equal to

ω1=

The resulting formula can be written in a somewhat different form:

x(t)=Ae−βtcos(ω1t+φ0),

Where φ0 is the initial phase of the oscillations and Acosφ0 is the initial amplitude of the oscillations. We see that classical damped oscillations occur in this mode. Here the oscillation frequency ω1 is less than the harmonic frequency ω0, and the oscillation amplitude decreases exponentially with e−βt.

2.4 ENERGY DECAY IN A DAMPED HARMONIC OSCILLATOR

We know that in the presence of damping the amplitude of oscillation decreases with the passage of time. This means that energy is dissipated in overcoming resistance to motion. We know that, the total energy of a harmonic oscillator is made up of kinetic and potential components. We can still use the same definition and write

E(t) = K.E. + P.E.

= m ()2+ kx2

Where () denotes instantaneous velocity. For a weakly damped harmonic oscillator, the instantaneous displacement is given by

x(t) = a0 exp(-bt) cos (ωdt +ϕ)

By differentiating it with respect to time, we get instantaneous velocity:

= v = a0 exp(-bt) [bcos (ωdt +ϕ) +ωd sin (ωdt +ϕ)]

Hence, kinetic energy of the oscillator is

K.E. = m ()2 = m a02 exp(-2bt) [bcos (ωdt +ϕ) +ωd sin (ωdt +ϕ)]2

= m a02 exp(-2bt) [b2cos2 (ωdt +ϕ) +ω2d sin2 (ωdt +ϕ) + b ωd 2sin (ωdt +ϕ)]

Similarly, the potential energy of the oscillator is

U =P.E.= kx2 =

Since k =

On substituting for x, we get

U = = a02 exp(-2bt) [cos (ωdt +ϕ)]2

Hence, the total energy of the oscillator at any time t is given by

E(t) = m a02 exp(-2bt) [(b2+ ω2o) cos2 (ωdt +ϕ) +ω2d sin2 (ωdt +ϕ) + b ωd 2sin (ωdt +ϕ)]

When damping is small, the amplitude of oscillation does not change much over one oscillation. So we may take the factor exp (-2bt) as essentially constant. Further, since < sin2 (ωdt +ϕ) > = < cos2 (ωdt +ϕ) > = and <sin (ωdt +ϕ)> = 0, the energy of a weakly damped oscillator when averaged-over one cycle is given by

<E> = m a02 exp(-2bt) [ + ]

<E> = m a02 exp(-2bt)

E0 =m a02 is the total energy of an undamped oscillator. Hence, we can write <E> = Eo exp (- 2bt)

This shows that the average energy of a weakly damped oscillator decreases exponentially with time. This is illustrated in Figure 6. You will also observe that the rate of decay of energy depends on the value of b; larger the value of b, faster will be the decay.

Figure 6

2.5 QUALITY FACTOR

The energy loss rate of a weakly damped (i.e., ν) harmonic oscillator is conveniently characterized in terms of a parameter, Qf, which is known as the quality factor. This quantity is defined to be 2π times the energy stored in the oscillator, divided by the energy lost in a single oscillation period. If the oscillator is weakly damped then the energy lost per period is relatively small, and Qf is therefore much larger than unity. Roughly speaking, Qf is the number of oscillations that the oscillator typically completes, after being set in motion, before its amplitude decays to a negligible value. Let us find an expression for Qf.

As we have seen, the motion of a weakly damped harmonic oscillator is specified by

$\displaystyle x = a {\rm e}^{-\nu t/2} \cos(\omega_1 t-\phi).$ (1)

It follows that

$\displaystyle \dot{x} =- \frac{a\,\nu}{2}\,{\rm e}^{-\nu\,t/2}\,\cos(\omega_1\,t-\phi)- a\,\omega_1\,{\rm e}^{-\nu\,t/2}\,\sin(\omega_1\,t-\phi).$

(2)

The energy lost during a single oscillation period is

$\displaystyle \Delta E$ $\displaystyle = -\int_{\phi/\omega_1}^{(2\pi+\phi)/\omega_1} \frac{dE}{dt} dt$

$\displaystyle = m \nu a^{2}\int_{\phi/\omega_1}^{(2\pi+\phi)/\omega_1}{\rm e}... ...{2} \cos(\omega_1 t-\phi) + \omega_1 \sin(\omega_1 t-\phi)\right]^{ 2} dt.$ (3)

In the weakly damped limit, ν, the exponential factor is approximately unity in the interval t= ϕ/ω1 to (2π+ ϕ)/ ω1 , so that

$\displaystyle \Delta E \simeq \frac{m \nu a^{2}}{\omega_1}\int_0^{2\pi} \left... ...,\omega_1 \cos\theta \sin\theta + \omega_1^{ 2} \sin^2\theta\right)d\theta,$ (4)

Where θ = ω1t- ϕ . Thus,

$\displaystyle \Delta E \simeq \frac{\pi\,m\,\nu\,a^{2}}{\omega_1}\,(\nu^2/4+\omega_1^{\,2}) = \pi\,m\,\omega_0^{\,2}\,a^{2}\left(\frac{\nu}{\omega_1}\right),$ (5)

because, as is readily demonstrated,

$\displaystyle \int_0^{2\pi}\cos^2\theta d\theta = \int_0^{2\pi}\sin^2\theta d\theta$

$\displaystyle =\pi,$

(6)

$\displaystyle \int_0^{2\pi}\cos\theta \sin\theta d\theta$

$\displaystyle =0.$

(7)

The energy stored in the oscillator (at t=0 ) is [cf., Equation (16)]

$\displaystyle E = \frac{1}{2} m \omega_0^{ 2} a^{2}.$ (8)

Hence, we obtain

$\displaystyle Q_f = 2\pi\,\frac{E}{\Delta E} = \frac{\omega_1}{\nu}\simeq \frac{\omega_0}{\nu}.$ (9)

Yet another way of expressing the damping effect is by means of the rate of decay of energy.

We note that the average energy of a weakly damped oscillator decays to Eoe-1 in time t = = seconds. If ωd is its angular frequency, then in this time the oscillator will vibrate through ωd m/ radians. The number of radians through which a weakly damped system oscillates as its average energy decays to Eoe-1 is a measure of the quality factor Q.

You will note that Q is only a number and has no dimensions. In general, is small so that Q is very large. A tuning fork has Q of a thousand or so, whereas a rubber band exhibits a much lower (-10) Q. This is due to the internal friction generated by the coiling of the long chain of molecules in a rubber band. An undamped oscillator ' ( = 0) has an infinite quality factor.

For a weakly damped mechanical oscillator, the quality factor can be expressed in term; of the spring factor and damping constant. For weak damping,

ωd -

Q =

That is, the quality factor of a weakly damped oscillator is directly proportional to the square root of k and inversely proportional to.

In a more physically meaningful form

Q = =

2.6 FORCED MECHANICAL OSCILLATORS

FORCED OSCILLATION

The oscillation of a oscillator is said to be forced oscillator or driven oscillation if the oscillator is subjected to external periodic force.

If an external periodic sinusoidal force Fcosωt acts on a damped oscillator, its equation of motion is written as,

…………(1)

Where β= b/2m, = k/m and f0 = F/m , and β and respectively called as damping coefficient, natural frequency.

Equation (1) is also represented as

= f0cosωt

Equation (1) represents the general equation of forced oscillation.

Equation (1) is a non-homogenous differential equation with constant co-efficient. For weak damping ( >β 2) , the general equation contains,

x(t) = xc(t) + xp(t)

Where xc(t) is called complementary solution and its value is

…………(2)

Now xp(t) is called the particular integral part.

Let us choose

xp(t) = P cos (ωt-δ)

(t)= -Pωsin(ωt-δ)

(t)=-Pω2cos(ωt-δ) ……………(3)

Putting xp(t) , (t), (t) in equation (1) we get

Now, comparing the coefficient of cos(ωt-δ) and sin (ωt-δ) on both sides,

(-ω2)P = f0cosδ ……………(4)

2βPω = f0sinδ ……………(5)

Squaring and adding eqn (4) & (5)

……….(6)

Now dividing equation (5) by (6)

This is steady state solution. Now

x(t) = xc(t) + xp(t)

STEADY STATE BEHAVIOUR

Frequency:-The Oscillator oscillates with the same frequency as that of the periodic force.

ω0 and ω are very close to each other than beats will be produced and these beats are transient as it lasts as long as the steady state lasts. The duration between transient beats is determined by the damping coefficient β.

PHASE: The phase difference δ between the oscillator and the driving force or between the displacement and driving is

This shows that there is a delay between the action of the driving force and response of the oscillator.

Figure: 7

(In the above figure fQ= ω0 and fp= ω)

At ω=ω0, φ =π/2, the displacement of the oscillator lags behind the driving force by π/2.

At ω<<ω0 then δ=0→ δ=0

For ω>> ω0 then δ =-2p/ ω → -0= π

AMPLITUDE: The amplitude of driven oscillator, in the steady state, is given by

It depends upon (-ω2). If it is very small, then the amplitude of forced oscillation increases.

Case-1: At very high driving force i.e ω>>ω0 and damping is small (β is small) or (β→0)

A =

A = =

Amplitude is inversely proportional to the mass of the oscillator & hence the motion is mass controlled motion.

Case-2: At very low driving force (ω<<ω0) and damping is small ( β→0),

i.e. -ω2 =

A =

A = =

So, when the low driving force is applied to oscillator, the amplitude remains almost constant for low damping. The amplitude of the forced oscillator in the region ω<<ω0 and β< ω0 is inversely proportional to the stiffness constant (k) and hence motion is called the stiffness controlled motion.

Case:-3 (Resistance controlled motion)

When angular frequency of driving force=natural frequency of oscillator i.e.(ω=ω0)

A = =

A=f/bω=f/bω0

2.7 RESONANCE

Resonance occurs when:-

Frequency of the external periodic force = Natural frequency of the system.

Definition:- The phenomena of producing oscillatory motion in a system by the influence of an external periodic force having the same frequency as that of natural frequency of the system is called Resonance.

Example:- Resonance causes disaster during earthquake. When the natural frequency of the building becomes equal to the frequency of the periodic vibration present in the earth during earthquake, then the building starts vibrating with large amplitude and hence it collapses.

Sit in front of a piano sometime and sing a loud brief note at it with the dampers off its strings. It will sing the same note back at you—the strings, having the same frequencies as your voice, are resonating in response to the forces from the sound waves that you sent to them. Your voice and a piano’s strings is a good example of the fact that objects—in this case, piano strings—can be forced to oscillate but oscillate best at their natural frequency. In this section, we shall briefly explore applying a periodic driving force acting on a simple harmonic oscillator. The driving force puts energy into the system at a certain frequency, not necessarily the same as the natural frequency of the system. The natural frequency is the frequency at which a system would oscillate if there were no driving and no damping force.

Most of us have played with toys involving an object supported on an elastic band, something like the paddle ball suspended from a finger in Figure 8. Imagine the finger in the figure is your finger. At first you hold your finger steady, and the ball bounces up and down with a small amount of damping. If you move your finger up and down slowly, the ball will follow along without bouncing much on its own. As you increase the frequency at which you move your finger up and down, the ball will respond by oscillating with increasing amplitude. When you drive the ball at its natural frequency, the ball’s oscillations increase in amplitude with each oscillation for as long as you drive it.

The phenomenon of driving a system with a frequency equal to its natural frequency is called resonance. A system being driven at its natural frequency is said to resonate. As the driving frequency gets progressively higher than the resonant or natural frequency, the amplitude of the oscillations becomes smaller, until the oscillations nearly disappear and your finger simply moves up and down with little effect on the ball.

The given figure shows three pictures of a horizontal viewed single finger containing a string, suspended downward vertically, being tied to a paddle ball at its downward end. In the first figure the ball is stretching up and down very slowly having less displacement, the displacement shown in the figures as faded shades of the ball and is depicted as 2X. Whereas in the second figure the movement of the ball is highest, while in the third the movement is least. In all the three figures the ball is at its equilibrium with respect to its movement. The frequency, f, for the first figure is very low, for the second figure as f not, while for the third figure it is highest.

Figure 8: The paddle ball on its rubber band moves in response to the finger supporting it. If the finger moves with the natural frequency f0 of the ball on the rubber band, then a resonance is achieved, and the amplitude of the ball’s oscillations increases dramatically. At higher and lower driving frequencies, energy is transferred to the ball less efficiently, and it responds with lower-amplitude oscillations.

Figure 9 shows a graph of the amplitude of a damped harmonic oscillator as a function of the frequency of the periodic force driving it. There are three curves on the graph, each representing a different amount of damping. All three curves peak at the point where the frequency of the driving force equals the natural frequency of the harmonic oscillator. The highest peak, or greatest response, is for the least amount of damping, because less energy is removed by the damping force.

The given graph is of amplitude, X, along y axis versus driving frequency f, along x axis. There are three points on the x axis as f not divided by two, f not, three multiply f not divided by two. There are three curves along the x axis, in a one crest oscillation way, which are one over each other in correspondence. The curves start at a point just over the origin point and ends up at a same level along the x axis on the far right. The crests of the three curves are exactly over the f not point. The uppermost crest shows the small damping, whereas the middle one shows the medium damping, and the last one below shows the heavy damping.

Figure 9: Amplitude of a harmonic oscillator as a function of the frequency of the driving force. The curves represent the same oscillator with the same natural frequency but with different amounts of damping. Resonance occurs when the driving frequency equals the natural frequency, and the greatest response is for the least amount of damping. The narrowest response is also for the least damping.

It is interesting that the widths of the resonance curves shown in Figure 9 depend on damping: the less the damping, the narrower the resonance. The message is that if you want a driven oscillator to resonate at a very specific frequency, you need as little damping as possible. Little damping is the case for piano strings and many other musical instruments. Conversely, if you want small-amplitude oscillations, such as in a car’s suspension system, then you want heavy damping. Heavy damping reduces the amplitude, but the tradeoff is that the system responds at more frequencies.

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