How To Find Frequency Of Damped Oscillation

We take seen that the total free energy of a harmonic oscillator remains constant. Once started, the oscillations go on forever with a constant amplitude (which is adamant from the initial conditions) and a abiding frequency (which is adamant by the inertial and rubberband backdrop 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 free 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 abroad with the passage of time. This happens because, in actual concrete systems, friction (or damping) is ever nowadays. Friction resists move.

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

In that location is another mechanism by which an oscillator loses energy. The energy of an oscillator may decrease non only due to friction in the system just likewise due to radiations. The aquiver trunk imparts periodic move 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 discipline to dissipative forces, or otherwise, there would be no loss of free energy by the trunk and consequently, no emission of sound free energy could occur. Thus, sound waves are produced by radiation from mechanical oscillatory systems. We shall learn after that the electromagnetic waves are produced by radiations from aquiver electric and magnetic fields.

The upshot of radiations by an oscillating organisation and of the friction present in the system is that the aamplitude of oscillations gradually diminishes with time. The reduction in amplitude (or free energy) of an oscillator is chosen damping and the oscillation are said to be damped.

Damping Forces

The damping of a real system is a complex phenomenon involving several kinds of damping forces.

• The damping strength opposes the movement of the body
• The magnitude of the damping force is direct proportional to the velocity of the body
• The management of the damping forcefulness is reverse to the velocity.
• Damping force is denoted past F_d.F_d = – pvWhere,v is the magnitude of the velocity of the object and p, the viscous damping coefficient, represents the damping force per unit of measurement velocity. The negative sign indicates that the force opposes the move, tending to reduce velocity. In other words, the glutinous damping strength is a retarding force.

The damping strength that depends on velocity is referred to as viscous damping force. Since the velocity of near aquiver systems is normally modest, the damping force exerted by the fluid in contact with the organization is likely to exist viscous. Gummy forces are generally much smaller than inertial and rubberband forces in a system.

Notwithstanding, damping devices called dampers are sometimes deliberately introduced in a system for vibration control. The damping strength exerted by such devices may exist comparable in magnitude to the inertial and elastic forces.

In real systems, information technology is likely that the moving part is in contact with an unlubricated surface, as in the case of horizontal oscillations of a trunk fastened to a spring (run across figure). The oscillating body is always in contact with the horizontal surface. The resulting frictional force opposes the movement and tin often be idealized equally a force of constant magnitude. Such a force is usually referred to every bit a Coulomb friction force.

In a solid, some part of energy may be lost due to imperfect elasticity or internal friction of the material. It is very difficult to gauge this type of damping. Experiments suggest that a resistive force proportional to the amplitude and contained of the frequency may serve as a satisfactory approximation. This kind of damping in solids is referred to equally structural damping.

Thus, the damping of a real system is a complex phenomenon involving several kinds of damping forces such equally gluey damping, Coulomb friction and structural damping. Because information technology is by and large very difficult to predict the magnitude of the damping forces, ane ordinarily has to rely on feel and experiment and so as to brand a reasonably good estimate. Information technology is a mutual do to guess the damping of a system by equivalent viscid damping, for the simple reason that viscous damping is the virtually convenient to handle mathematically. Thus, according to this approximation, the magnitude of the gluey force to be used in a particular trouble is chosen to be the ane that would produce the same charge per unit of energy dissipation as the bodily damping forces. This unremarkably provides a skilful estimate.

The inclusion of damping forces complicates the analysis considerably. Fortunately, in actual systems, the damping forces are usually small and tin oft be ignored. In situations, where they are not negligibly small, the sticky damping model is the near convenient mathematically. We shall use this model, under the simplifying supposition that the velocity of the moving part of the organisation is minor so that the damping force is linear in velocity.

If the velocity is not small, the damping forcefulness exerted on the system may exist represented more closely past a force proportional to the square of the velocity. Nosotros shall not deal with such forces. The effect of the linear gummy damping strength on the complimentary oscillations of unproblematic systems, with one caste of freedom, is considered in the next section.

Damped Oscillations Of A System Having One Degree Of Liberty

We shall investigate the effect of damping on the harmonic oscillations of a uncomplicated system having one caste of freedom. One such system is shown in the figure. When the arrangement is displaced from its equilibrium state and released, it begins to movement. The forces acting on the organisation are:

(i) a restoring strength –Kx, where K is the coefficient of the restoring forcefulness and x is the displacement, and

(2) a damping force

\(\begin{array}{l}-p\frac{dx}{dt},\end{array} \)

where p is the coefficient of the damping force and

\(\brainstorm{assortment}{50}\frac{dx}{dt}\stop{assortment} \)

is the velocity of the moving part of the system. From Newton's law for a rigid torso in translation, these forces must residual with Newton's force

\(\brainstorm{array}{fifty}yard\frac{{{d}^{ii}}x}{d{{t}^{two}}},\end{array} \)

where m is the mass of the oscillator and

\(\brainstorm{array}{50}\frac{{{d}^{ii}}ten}{dt^{two}},\end{assortment} \)

its accelerations. Since, the restoring force and the damping strength acts in a direction contrary to Newton'south strength, we have

\(\brainstorm{array}{l}m\frac{{{d}^{2}}x}{d{{t}^{2}}}=-Kx-p\frac{dx}{dt}\terminate{array} \)

——-(3.ane)

Recollect, this equation holds only for small displacements and pocket-size velocities. This equation can exist rewritten every bit:

\(\begin{array}{50}\frac{{{d}^{two}}x}{d{{t}^{ii}}}+\gamma \frac{dx}{dt}+\omega _{0}^{two}ten=0\end{assortment} \)

… (iii.ii)

With

\(\begin{array}{l}\gamma =p/thou\cease{assortment} \)

… (3.3)

And

\(\begin{array}{l}\omega _{0}^{2}=K/g\end{array} \)

… (3.iv)

Notice that dimensionally

\(\begin{array}{fifty}\gamma =\frac{p}{g}=\frac{\text{force}}{\text{velocity }\!\!\times\!\!\text{ mass}}=\frac{\text{ML}{{\text{T}}^{-2}}}{\text{Fifty}{{\text{T}}^{-one}}M}={{T}^{-1}}.\terminate{array} \)

the same as the dimension of frequency.

It is like shooting fish in a barrel to see that in Eq. (3.2) the damping is characterized by the quantity γ, having the dimension of frequency, and the constant ω₀ represents the angular frequency of the system in the absence of damping and is called the natural frequency of the oscillator. Equation (3.2) is the differential equation of the damped oscillator. To find out how the displacement varies with fourth dimension, we need to solve Eq. (3.2) with constants γ and ω₀ given respectively past Eqs. (3.three) and (3.4).

The General Solution

To solve Eq. (three.2) we brand apply of the exponential function again. Permit usa presume that the solution is

\(\begin{assortment}{l}10=A{{e}^{\alpha t}}\cease{array} \)

And solve for α. Constants A and α are arbitrary and as yet undetermined. Differentiating, we accept

\(\brainstorm{assortment}{l}\frac{dx}{dt}=\alpha A\,\,{{e}^{\alpha t}}\end{array} \)

\(\begin{assortment}{fifty}\frac{{{d}^{2}}x}{d{{t}^{2}}}={{\alpha }^{2}}A\,\,{{e}^{\alpha t}}\cease{array} \)

Substitution in Eq. (three.2) yields

\(\begin{array}{fifty}\left( {{\alpha }^{ii}}+\gamma \alpha +\omega _{0}^{2} \right)A\,{{e}^{\blastoff t}}=0\end{array} \)

For this equation to hold for all values of t, the term in the brackets must vanish, i.eastward.

\(\begin{array}{l}{{\alpha }^{2}}+\gamma \alpha +\omega _{0}^{two}=0\end{assortment} \)

The two roots of this quadratic equation are

\(\brainstorm{assortment}{50}{{\alpha }_{1}}=-\frac{\gamma }{2}+\frac{i}{ii}{{\left( {{\gamma }^{2}}-four\omega _{0}^{2} \correct)}^{i/ii}}\end{array} \)

And

\(\begin{assortment}{l}{{\alpha }_{2}}=-\frac{\gamma }{2}-\frac{1}{ii}{{\left( {{\gamma }^{2}}-iv\omega _{0}^{2} \right)}^{ane/two}}\end{array} \)

Thus the two possible solutions of Eq. (3.ii) are

\(\begin{assortment}{l}{{x}_{1}}={{A}_{ane}}\,\,{{eastward}^{\blastoff }}{{1}^{t}}={{A}_{i}}\,\exp \left[ -\frac{\gamma }{2}+\frac{1}{2}{{\left( {{\gamma }^{2}}-4\omega _{0}^{2} \right)}^{ane/2}} \right]t\finish{array} \)

And

\(\brainstorm{array}{l}{{ten}_{2}}={{A}_{two}}\,\,{{due east}^{\alpha }}{{2}^{t}}={{A}_{2}}\,\exp \left[ -\frac{\gamma }{ii}-\frac{i}{2}{{\left( {{\gamma }^{2}}-4\omega _{0}^{two} \right)}^{ane/2}} \right]t\end{array} \)

Since Eq. (three.2) is linear, the superposition principle is applicable. Hence, the general solution is given past the superposition of the two solutions, i.e.

\(\begin{array}{fifty}ten={{x}_{one}}+{{x}_{2}}\end{assortment} \)

Or

\(\brainstorm{array}{l}x={{A}_{one}}\,\exp \left[ -\frac{\gamma }{2}+{{\left( \frac{{{\gamma }^{2}}}{4}-\omega _{0}^{2} \correct)}^{i/2}} \right]t\end{assortment} \)

\(\begin{array}{l}+{{A}_{2}}\,\exp \left[ -\frac{\gamma }{two}-{{\left( \frac{{{\gamma }^{2}}}{four}-\omega _{0}^{2} \correct)}^{1/2}} \correct]t\end{array} \)

…..(three.five)

Hither A_i and A₂ are arbitrary constants to be determined from the initial conditions, namely, the initial deportation and the initial velocity.

The nature of the motility depends on the character of the roots α_i and α_ii. The roots may be real or circuitous depending on whether γ > 2ω₀ or γ < 2ω₀ and γ < 2 ω₀. Each condition describes a detail kind of behaviour of the system. We shall now treat each case separately.

Example I:

γ > 2 ω₀ (Large Damping) In this case, the damping term γ/2 dominates the stiffness term ω₀ and the term

\(\begin{assortment}{l}{{\left( {{\gamma }^{2}}/4-\omega _{0}^{2} \correct)}^{1/ii}}\end{assortment} \)

in Eq. (3.5) is a real quantity with a positive value, say, q, i.e.

\(\begin{array}{fifty}{{\left( \frac{{{\gamma }^{2}}}{iv}-\omega _{0}^{2} \correct)}^{1/ii}}=q\cease{array} \)

And so that displacement ψ as a office of time is given past

\(\begin{array}{l}x={{A}_{1}}\,\exp \left( -\frac{\gamma }{2}+q \right)t+{{A}_{2}}\,\exp \left( -\frac{\gamma }{2}+q \right)t\end{array} \)

…..(iii.6)

The velocity is given past

\(\begin{array}{l}\frac{dx}{dt}=\left( -\frac{\gamma }{two}+q \right){{A}_{1}}\,\exp \left( -\frac{\gamma }{2}+q \right)t\end{assortment} \)

\(\brainstorm{array}{l}-\left( \frac{\gamma }{2}+q \right){{A}_{2}}\,\exp \left( -\frac{\gamma }{ii}+q \right)t\end{array} \)

……(three.7)

These equations describe the behaviour of a heavily damped oscillator, every bit for instance, a pendulum in a glutinous medium such equally a dense oil. As stated earlier, the constants A₁ and A_ii are determined from the initial conditions. Let us presume that the oscillator is at its equilibrium position) (ψ = 0) at time t = 0. At this instant information technology is given a kicking so that it has a finite velocity, say, V₀ at this fourth dimension, i.due east. at t = 0.

x = 0

Equations (3.six) and (3.7) then give (setting t = 0)

\(\brainstorm{assortment}{l}0={{A}_{i}}+{{A}_{2}}\end{assortment} \)

\(\begin{array}{l}{{V}_{0}}=\left( -\frac{\gamma }{2}+q \correct){{A}_{i}}-\left( \frac{\gamma }{2}+q \right){{A}_{2}}\end{array} \)

giving

\(\begin{array}{l}{{A}_{one}}=-{{A}_{ii}}=\frac{{{V}_{0}}}{2q}\end{assortment} \)

Thus, nether the above initial conditions, Eqs. (3.half dozen) and (3.7) go

\(\begin{assortment}{l}x=\frac{{{Five}_{0}}}{2q}{{e}^{-\gamma t/ii}}\left( {{e}^{qt}}-{{eastward}^{-qt}} \right)\end{array} \)

Or

\(\begin{array}{l}10=\frac{{{Five}_{0}}}{q}{{e}^{-\gamma t/2}}\sinh \left( qt \right)\end{array} \)

And

\(\begin{array}{l}\frac{dx}{dt}=\frac{{{V}_{0}}}{two}{{e}^{-\gamma t/ii}}\left\{ \left( {{due east}^{qt}}+{{e}^{-qt}} \right)-\frac{\gamma }{2q}\left( {{east}^{qt}}-{{eastward}^{-qt}} \right) \right\}\end{array} \)

Or

\(\begin{array}{l}\frac{dx}{dt}={{5}_{0}}{{e}^{-\gamma t/two}}\left\{ \cosh \left( qt \correct)-\frac{\gamma }{2q}\sinh \left( qt \correct) \correct\}\end{array} \)

……(3.9)

Figure 1 illustrates the behaviour of a heavily damped system when it is disturbed from equilibrium past a sudden impulse at t = 0. It is the displacement – time graph of Eq. (3.8). For modest values of fourth dimension t, the term

\(\begin{array}{l}{{e}^{-\gamma t/ii}}\end{array} \)

is very nearly unity, the displacement increases with time since sinh (qt) increases as t increases. Very shortly, however, the term

\(\brainstorm{array}{50}{{east}^{-\gamma t/2}}\end{array} \)

starts contributing and the displacement decays exponentially with time, eventually becoming zero. The turning signal occurs at a time

\(\begin{array}{l}t={{t}_{0}}\end{array} \)

when

\(\begin{array}{fifty}dx/dt=0.\end{array} \)

Equation (3.9) tells the states that this happens at a time t = t₀ satisfying

\(\brainstorm{array}{l}\tan \,h\left( q{{t}_{0}} \right)=\frac{2q}{\gamma }\end{array} \)

Thus, the deportation increases until time

\(\begin{array}{l}t={{t}_{0}},\end{array} \)

after which it slowly returns to zero. Since, displacement ψ never becomes negative, at that place is no oscillation at all. Such a motility is called dead vanquish. We come up across such a motion in the case of a dead vanquish galvanometer (encounter sec. iii.half dozen).

Case II:

\(\begin{array}{50}\gamma =2\,{{\omega }_{0}}\stop{array} \)

(Critical Damping). This is a special case of a heavily damped motion. Using the notation

\(\begin{array}{fifty}q={{\left( {{\gamma }^{2}}/four-\omega _{0}^{2} \correct)}^{t/2}}\stop{assortment} \)

of instance I, we see that, in this case, q = 0 and Eq. (3.six) becomes

\(\begin{array}{fifty}x=\left( {{A}_{1}}+{{A}_{2}} \right){{e}^{-\gamma t/2}}\stop{assortment} \)

Or

\(\brainstorm{array}{l}ten=B\,{{e}^{-\gamma t/2}}\cease{array} \)

……(3.10)

Where B = A_one + A₂, is a abiding. In other words, Eq. (3.10) is the solution of Eq (three.2) for γ = 2 ω₀. In this instance, the 2 roots α₁ and α_ii become identical. Notice that the solution (three.10) contains only i adaptable constant B. This solution is only a fractional solution, since the solution of any second – order differential equation must contain two adjustable constants. This tin be understood as follows. If Eq. (3.10) were a complete solution of Eq. (3.2), then the velocity of the oscillator would be given past

\(\begin{array}{l}\frac{dx}{dt}=-B\frac{\gamma }{2}{{east}^{-\gamma t/2}}\end{array} \)

When the system is disturbed from equilibrium (x = 0) by giving an impulse (i.e. past imparting a velocity V₀) at t = 0, nosotros take, from the above two equations

B = 0

\(\begin{array}{l}{{V}_{0}}=-\frac{\gamma }{2}B\end{assortment} \)

implying, thereby, that Five₀ is also zero, which is non our initial status. Hence our trial solution yields only a fractional solution in the case when q = 0.

Nosotros can verify that a 2nd solution is represented by the trial solution

\(\begin{assortment}{50}x=Ct\,{{east}^{-\gamma t/2}}\end{array} \)

Giving

\(\begin{array}{l}\frac{dx}{dt}=C\,{{due east}^{-\gamma t/2}}\left( 1-\frac{\gamma t}{ii} \right)\end{array} \)

And

\(\begin{array}{l}\frac{{{d}^{ii}}x}{d{{t}^{2}}}=C\,\frac{\gamma }{2}{{eastward}^{-\gamma t/ii}}\left( -two+\frac{\gamma t}{2} \right)\end{array} \)

Substituting for x,

\(\begin{assortment}{l}\frac{dx}{dt}\,and\,\,\frac{d{{x}^{2}}}{d{{t}^{2}}}\end{array} \)

in Eq. (3.two) with ω_o^two  replaced by

\(\begin{array}{l}\frac{{{\gamma }^{ii}}}{4}\finish{array} \)

i.e.

\(\begin{array}{l}\frac{{{d}^{ii}}x}{d{{t}^{two}}}+\gamma \frac{dx}{dt}+\frac{{{\gamma }^{2}}}{4}10=0\end{assortment} \)

We accept,

\(\brainstorm{array}{l}C\frac{\gamma }{2}{{due east}^{-\gamma t/2}}\left( -two+\frac{\gamma t}{2} \correct)+\gamma C\,{{e}^{-\gamma t/2}}\left( 1-\frac{\gamma t}{2} \correct)+\frac{{{\gamma }^{2}}}{4}Ct\,{{e}^{-\gamma t/2}}=0\end{array} \)

Or

\(\begin{array}{l}\frac{\gamma }{2}C\,{{e}^{-\gamma t/2}}\left( -2+\frac{\gamma t}{2}+2-\gamma t+\frac{\gamma t}{2} \right)=0\cease{array} \)

Or 0 = 0

Thus, Eqs. (3.x) and (3.eleven) are both possible solutions of Eq. (3.two) in the special case when

\(\begin{assortment}{l}\gamma =ii\,{{\omega }_{0}}.\finish{array} \)

From the superposition principle, the general solution is given by

\(\brainstorm{array}{fifty}x=B\,{{due east}^{-\gamma t/2}}+Ct\,{{e}^{-\gamma t/2}}=\left( B+Ct \right){{e}^{-\gamma t/two}}\end{array} \)

…..(three.12)

And

\(\begin{array}{fifty}\frac{dx}{dt}=\left\{ C-\frac{\gamma }{two}\left( B+Ct \right) \right\}{{e}^{-\gamma t/2}}\end{array} \)

The constants B and C tin can exist determined from the initial conditions. If at t = 0, x = 0 and

\(\begin{array}{l}\frac{dx}{dt}={{V}_{0}},\terminate{assortment} \)

we have, from the above equations,

B = 0

C = V₀

Thus, under these initial weather condition, the deportation 10 in Eq, (three.12) is given by

\(\brainstorm{assortment}{l}10={{V}_{0}}t\,{{e}^{-\gamma t/two}}\finish{array} \)

…..(3.13)

And

\(\begin{assortment}{l}\frac{dx}{dt}=C\left( i-\frac{\gamma t}{two} \correct){{due east}^{-\gamma t/2}}\cease{assortment} \)

….(3.14)

Figure ii is a graph of x confronting t in Eq. (3.13). It illustrates the displacement – time behaviour of a damped system with

\(\begin{array}{50}\gamma =2{{\omega }_{0}},\terminate{array} \)

when it is disturbed from equilibrium past a sudden impulse For small values of t, the term

\(\brainstorm{array}{l}{{e}^{-\gamma t/2}}\end{assortment} \)

is very virtually unity and displacement [Eq. (iii.13)] increases linearly with fourth dimension t. Afterward onetime

\(\begin{array}{fifty}{{e}^{-\gamma t/2}}\end{assortment} \)

starts changing and the displacement decays exponentially with time, eventually get zero. The turning point occurs at a fourth dimension t₀, when

\(\begin{array}{l}\frac{dx}{dt}=0.\finish{array} \)

From Eq. (iii.xiv) this happens at t = t₀ given past

\(\begin{array}{fifty}1-\frac{\gamma {{t}_{0}}}{two}=0\terminate{assortment} \)

Or

\(\begin{array}{l}{{t}_{0}}=\frac{two}{\gamma }=\frac{1}{{{\omega }_{0}}}\end{assortment} \)

The deportation increases until time t = t₀, afterwards which it decays to naught. A comparison of Eqs. (3.viii) and (3.xiii) reveals that the decay charge per unit is much faster when

\(\brainstorm{array}{fifty}\gamma =2\,{{\omega }_{0}}\end{assortment} \)

and then when

\(\begin{array}{l}\gamma >two\,{{\omega }_{0}}.\stop{array} \)

In both cases, there is no oscillation at all, since ψ never becomes negative.

The motion described by Eq. (3.13) is called critically damped. The necessary condition for critical damping is

\(\begin{assortment}{l}\gamma =2\,{{\omega }_{0}}.\end{assortment} \)

Suppose we are faced with a trouble in which we desire a high rate of disuse without oscillation. Patently, the optimum choice is critical damping. We come beyond such a problem in pointer-type galvanometers, where nosotros would want the pointer to move immediately to the correct position and stay in that location without annoying oscillation (see sec. 3.6).

Instance III

\(\begin{array}{fifty}\gamma <2\,{{\omega }_{0}}\cease{array} \)

(Small Damping).

When

\(\begin{array}{l}\gamma <ii\,{{\omega }_{0}},\cease{array} \)

the damping is modest and this gives the near of import kind of behaviour, namely, oscillatory damped harmonic move, for and then, the expression

\(\brainstorm{array}{l}{{\left( \frac{{{\gamma }^{2}}}{iv}-\omega _{0}^{2} \right)}^{1/2}}\finish{array} \)

in the exponentials in Eq. (iii.5) is an imaginary quantity. Writing this as

\(\brainstorm{array}{50}{{\left( \frac{{{\gamma }^{ii}}}{four}-\omega _{^{0}}^{2} \right)}^{1/2}}=\sqrt{-1}\left( \omega _{0}^{2}-\frac{{{\gamma }^{ii}}}{iv} \right)^{1/2}\end{array} \)

\(\brainstorm{array}{fifty}=i\omega *\end{array} \)

Where,

\(\brainstorm{assortment}{l}\omega *={{\left( \omega _{0}^{2}-\frac{{{\gamma }^{2}}}{four} \correct)}^{one/ii}}\end{array} \)

is a real positive quantity, the displacement Eq. (3.5) may be rewritten as

\(\begin{array}{l}x={{east}^{-\gamma t/two}}\left\{ \left( {{A}_{1}}\exp \left( i\omega *t \correct)+{{A}_{2}}\exp \left( -i\omega *t \right) \correct) \right\}\cease{array} \)

……(3.15)

To compare the behaviour of a damped oscillator with the ideal case in which damping is ignored, we volition recast Eq. (3.15) into a more than familiar form. We tin can practice this by using the identities,

\(\begin{assortment}{l}{{e}^{i\theta }}=\cos \theta +i\sin \theta\end{array} \)

\(\begin{array}{l}{{e}^{-i\theta }}=\cos \theta -i\sin \theta\end{array} \)

And then that Eq. (3.15) can be written as

\(\brainstorm{array}{l}ten={{e}^{-\gamma t/2}}\left\{ \left( {{A}_{ane}}+{{A}_{2}} \correct)\cos \omega *t+i\left( {{A}_{1}}-{{A}_{ii}} \right)\sin \omega *t \right\}\finish{array} \)

If nosotros choose

\(\brainstorm{array}{l}{{A}_{1}}+{{A}_{ii}}=A\cos \delta\terminate{assortment} \)

\(\begin{array}{fifty}i\left( {{A}_{1}}-{{A}_{2}} \correct)=A\sin \delta\cease{array} \)

Where A and δ are constants which depend upon the initial conditions, we find, after substitution,

\(\brainstorm{array}{l}ten=A\,{{e}^{-\gamma t/2}}\cos \left( \omega *t-\delta \correct)\end{array} \)

……(3.16)

With

\(\begin{array}{l}\omega *={{\omega }_{0}}{{\left( 1-\frac{{{\gamma }^{2}}}{iv\omega _{0}^{two}} \correct)}^{1/ii}}\stop{array} \)

…….(3.17)

Differentiating Eq. (3.16), we obtain an expression for the velocity of the oscillator, which reads

\(\begin{assortment}{50}\frac{dx}{dt}=-A\,\,{{e}^{-\gamma t/2}}\left\{ \omega *\sin \left( \omega *t-\delta \right)+\frac{\gamma }{2}\cos \left( \omega *t-\delta \correct) \correct\}\stop{assortment} \)

……(3.18)

Equation (3.xvi) shows that the movement is oscillatory. The oscillation is not elementary harmonic, since its, 'aamplitude'

\(\begin{array}{l}A\,{{e}^{-\gamma t/2}}\end{array} \)

is not constant just decreases with time. The motion is non periodic, since it never repeats itself, each swing being of smaller amplitude than the preceding one. However, if

\(\begin{array}{l}\gamma\end{array} \)

is very small compared to

\(\begin{array}{l}{{\omega }_{0}},\end{array} \)

the amplitude will remain sensibly abiding over a large number of cycles of the harmonic term

\(\brainstorm{assortment}{50}\cos \left( \omega *t-\delta \right)\terminate{array} \)

in which case, the motion is nearly periodic and uncomplicated harmonic.

The angular frequency of the oscillation is ω* given Eq. (3.17) which is less than the natural angular frequency of free undamped oscillations. Strictly speaking, nosotros are really non justified in using the terms 'amplitude' and 'frequency' for a motion which is not periodic. But, when damping is small, the motion is most periodic, we may use these terms with some reservations.

To illustrate the behaviour of a weakly damped oscillator, let us choose the initial weather, namely, that at t = 0, x = 0 and

\(\begin{array}{l}\frac{dx}{dt}={{5}_{0}}.\terminate{array} \)

Using these conditions in Eqs. (iii.16) and (3.18) nosotros get

\(\brainstorm{array}{50}0=A\cos \delta\stop{array} \)

And

\(\begin{array}{50}{{V}_{0}}=-A\left( \frac{\gamma }{2}\cos \delta -\omega *\sin \delta \correct)\end{array} \)

yielding

\(\begin{array}{l}\delta =\frac{\pi }{2}\end{array} \)

(A = 0; being a trivial case)

And

\(\begin{assortment}{fifty}A=\frac{{{V}_{0}}}{\omega *}\terminate{array} \)

Using these values of A and δ in Eqs (3.16) and (3.18) we find that, nether the above initial conditions, the displacement and velocity of the oscillator are, respectively, given by

\(\begin{assortment}{fifty}ten=\frac{{{Five}_{0}}}{\omega *}{{east}^{-\gamma t/two}}\sin \omega *t=A\left( t \right)\sin \omega *t\end{array} \)

With

\(\brainstorm{array}{l}A\left( t \right)=\frac{{{Five}_{0}}}{\omega *}{{e}^{-\gamma t/2}}={{A}_{0}}{{e}^{-\gamma t/2}}\end{array} \)

A₀ existence the value of A(t) when γ = 0

And

\(\begin{array}{l}\frac{dx}{dt}={{Five}_{0}}\,{{eastward}^{-\gamma t/2}}\left( \cos \omega *t-\frac{\gamma }{ii\omega *}\sin \omega *t \right)\terminate{assortment} \)

….(iii.twenty)

Figure three.four depicts the behaviour of a weakly damped oscillator. Information technology is a graph of ψ against t of the move described past Eq. (3.19). The constant A₀ is the value of

\(\begin{array}{l}A\left( t \right)=\frac{{{V}_{0}}}{\omega *}{{e}^{-\gamma t/2}}\end{array} \)

in the absence of damping (γ = 0), i.east.

\(\begin{assortment}{50}{{A}_{0}}={{V}_{0}}/\omega *.\end{array} \)

Since the maximum values of

\(\brainstorm{array}{50}\sin \left( \omega *t \right)\terminate{assortment} \)

are + 1 and -i alternately, the displacement-time graph of oscillation is bounded past the dotted curves

\(\begin{array}{l}{{A}_{0}}{{e}^{-\gamma t/two}}\terminate{assortment} \)

and

\(\begin{array}{l}-{{A}_{0}}{{eastward}^{-\gamma t/ii}}\end{array} \)

.

Thus, although, the amplitude decreases exponentially with time, the weekly damped oscillator executes some sort of oscillatory motion. The motion does not repeat itself and is, therefore, not periodic in the usual sense of the term. Even so, it still has a time flow

\(\begin{array}{l}T*=2\pi /\omega *,\stop{assortment} \)

which is the time interval betwixt ii alternate zeros of deportation. The time menses betwixt two successive zeros of displacement is T*/2. This is also the time interval between a maximum and minimum value of the displacement, but the maxima and minima are not exactly halfway between the zeros. This is obvious from Eq. (iii.20), because at a maximum or a minimum of displacement, the velocity is nada, giving

\(\begin{array}{50}\cos \omega *t-\frac{\gamma }{2\omega *}\sin \omega *t=0\cease{array} \)

Or

\(\begin{array}{l}\tan \omega *t=\frac{2\omega *}{\gamma }\end{assortment} \)

Displacement-time behaviour of a weakly damped oscillator

The values of t satisfying this equation are the instants at which ψ is either a positive maximum or a negative maximum. In the case when

\(\brainstorm{array}{l}\gamma <2{{\omega }_{0}},\frac{2{{\omega }_{0}}}{\gamma }\to \infty ,\end{array} \)

and so that

\(\brainstorm{array}{50}\omega *t\to \frac{\pi }{2},\frac{3\pi }{2},\frac{five\pi }{2},…,\stop{array} \)

The first maximum of ψ occurs at a time

\(\brainstorm{assortment}{l}t={{t}_{1}}\finish{array} \)

given past

\(\brainstorm{assortment}{fifty}\omega *{{t}_{1}}=\frac{\pi }{2}\end{array} \)

Or

\(\begin{array}{l}{{t}_{i}}=\frac{\pi }{2\omega *}=\frac{T*}{4}\end{array} \)

i.e. the maximum is exactly midway betwixt the 2 zeros of ψ. Thus, only in the case of negligibly small damping, are the maxima and minima halfway between the zeros of deportation every bit in the case of simple harmonic motion.

Effect of Damping:

The event of damping is two-fold: (a) The aamplitude of oscillation decreases exponentially with time as

\(\begin{array}{50}A\left( t \right)={{A}_{0}}{{due east}^{-\gamma t/two}}\terminate{assortment} \)

Where A₀ is the amplitude in the absence of damping and (b) The angular frequency ω* of the damped oscillator is less than ω₀, the frequency of the undamped oscillation. The relation between them is

\(\brainstorm{assortment}{50}\omega *={{\omega }_{0}}{{\left( 1-\frac{{{\gamma }^{2}}}{4\omega _{0}^{ii}} \correct)}^{ane/2}}\end{array} \)

Energy Of A Weakly Damped Oscillator

We shall now develop an expression for the boilerplate energy of a weakly damped oscillator at whatsoever instant of fourth dimension. We have seen that, in the case of weak damping

\(\begin{array}{l}\left( \gamma <2{{\omega }_{0}} \correct),\terminate{array} \)

the deportation and velocity of the oscillator are respectively given by Eqs. (3.16) and (3.18). If m is the mass of the oscillator, its instantaneous kinetic free energy is

\(\begin{array}{fifty}\frac{one}{ii}grand{{\left( \frac{dx}{dt} \right)}^{2}}\end{assortment} \)

Which with the help of Eq. (3.18) becomes

KE

\(\brainstorm{array}{l}=\frac{1}{2}m{{A}^{2}}{{due east}^{-\gamma t}}{{\left\{ \omega *\sin \left( \omega *t-\delta \right)+\frac{\gamma }{2}\cos \left( \omega *t-\delta \right) \right\}}^{ii}}\end{array} \)

\(\begin{assortment}{l}=\frac{1}{ii}chiliad{{A}^{2}}{{e}^{-\gamma t}}\left\{ \omega {{*}^{2}}{{\sin }^{2}}\left( \omega *t-\delta \right)+\omega *\gamma \sin \left( \omega *t-\delta \correct)\cos \left( \omega *t-\delta \right)+\frac{{{\gamma }^{ii}}}{4}{{\cos }^{ii}}\left( \omega *t-\delta \right) \correct\}\end{array} \)

The instantaneous potential energy of the oscillator is given by

\(\begin{array}{l}PE=\int\limits_{0}^{x}{Kx}dx=\frac{1}{2}K{{x}^{two}}\end{assortment} \)

Using Eq. (3.16) we have, since

\(\brainstorm{array}{l}Chiliad=thou\omega _{0}^{2},\end{array} \)

\(\begin{array}{50}PE=\frac{1}{2}m\omega _{0}^{ii}{{A}^{2}}{{eastward}^{-\gamma t}}{{\cos }^{two}}\left( \omega *t-\delta \right)\end{array} \)

The total energy of the oscillator at whatsoever instant of time is then given by

\(\begin{array}{50}Due east\left( t \right)=KE+PE\terminate{array} \)

\(\begin{array}{l}=\frac{1}{ii}one thousand\,{{A}^{2}}{{east}^{-\gamma t}}\left\{ \omega {{*}^{2}}{{\sin }^{ii}}\left( \omega *t-\delta \right)+\frac{\omega *\gamma }{ii}\sin 2\left( \omega *t-\delta \correct)+\left( \frac{{{\gamma }^{2}}}{iv}+\omega _{0}^{2} \right){{\cos }^{two}}\left( \omega *t-\delta \right) \right\}\end{array} \)

….(3.21)

If damping is very minor

\(\begin{assortment}{50}\left( \gamma <two{{\omega }_{0}} \right),\end{array} \)

as is unremarkably the case, the term

\(\begin{array}{l}{{due east}^{-\gamma t}}\end{array} \)

in Eq. (three.21) does not change appreciably during one time flow

\(\begin{array}{l}T*=2\pi /\omega *\end{assortment} \)

of the oscillation. Thus, bold that

\(\begin{array}{l}{{due east}^{-\gamma t}}\terminate{array} \)

is sensibly constant during catamenia T* of the oscillations, the time-averaged energy of the oscillator is given by

\(\begin{array}{l}<E\left( t \right)>=\frac{1}{2}g{{A}^{two}}\,{{e}^{-\gamma t}}\left\{ \omega {{*}^{2}}<{{\sin }^{2}}\left( \omega *t-\delta \right)>+\frac{\omega *\gamma }{two}<\sin ii\left( \omega *t-\delta \right)>+\left( \frac{{{\gamma }^{2}}}{4}+\omega _{0}^{two} \right)<{{\cos }^{2}}\left( \omega *t-\delta \right)> \right\}\stop{array} \)

……(3.22)

Where note < > implies averaging over one time period T*. A function f(t) averaged over T, is past definition, given past

\(\brainstorm{array}{fifty}<f\left( t \correct)>=\frac{\int\limits_{0}^{T}{f\left( t \right)dt}}{\int\limits_{0}^{T}{dt}}=\frac{1}{T}\int\limits_{0}^{T}{f\left( t \correct)dt}\end{array} \)

Thus,

\(\brainstorm{assortment}{l}<{{\sin }^{2}}\left( \omega *t-\delta \correct)>=\frac{1}{T*}\int\limits_{0}^{T*}{{{\sin }^{two}}}{{\left( \frac{2\pi t}{T*}-\delta \right)}^{2}}dt\terminate{array} \)

To integrate, let us use the transformation

\(\begin{assortment}{l}\frac{two\pi t}{T*}-\delta =\alpha\stop{array} \)

So that

\(\brainstorm{array}{fifty}dt=\frac{T*}{two\pi }d\infty\end{array} \)

Then,

\(\begin{assortment}{l}<{{\sin }^{2}}\left( \omega *t-\delta \correct)>=\frac{i}{two\pi }\int\limits_{-\delta }^{2\pi -\delta }{{{\sin }^{ii}}\alpha dx}=\frac{ane}{iv\pi }\int\limits_{0}^{2\pi }{\left( i-\cos 2\alpha \right)dx=\frac{1}{ii}}\stop{array} \)

Similarly,

\(\brainstorm{assortment}{l}<{{\cos }^{2}}\left( \omega *t-\delta \right)>=\frac{i}{2}\end{array} \)

And

\(\begin{array}{l}<\sin two\left( \omega *t-\delta \right)>=0\cease{array} \)

Substituting for these time-averaged values in Eq. (3.22), we get

\(\begin{array}{50}<East\left( t \correct)>=\frac{ane}{4}m{{A}^{2}}{{eastward}^{-\gamma t}}\left( \omega {{*}^{2}}+\frac{{{\gamma }^{2}}}{4}+\omega _{0}^{2} \right)\end{array} \)

Now, since

\(\begin{assortment}{fifty}\omega {{*}^{two}}=\omega _{0}^{2}-\frac{{{\gamma }^{ii}}}{4},\end{array} \)

we take

\(\brainstorm{array}{50}<E\left( t \right)>=\frac{1}{2}m{{A}^{ii}}\omega _{0}^{ii}{{e}^{-\gamma t}}\stop{array} \)

Or

\(\begin{assortment}{fifty}<E\left( t \right)>={{Eastward}_{0}}{{e}^{-\gamma t}}\end{array} \)

Where

\(\begin{array}{l}{{E}_{0}}=\frac{1}{2}yard{{A}^{2}}\omega _{0}^{ii},\cease{assortment} \)

is the total energy of an undamped oscillator. Hence, the energy of a weakly damped oscillator diminishes exponentially with time. The decay of the total free energy is illustrated in Figure.

Figure: Exponential disuse of full energy during damping of harmonic oscillations

The average ability dissipation during one time period is given by

< P (t) > = charge per unit of loss of energy

\(\begin{array}{50}=\frac{d}{dt}<E\left( t \right)>\end{array} \)

\(\begin{array}{l}=\gamma {{E}_{0}}\,{{eastward}^{-\gamma t}}\end{assortment} \)

\(\brainstorm{array}{l}=\gamma <Due east\left( t \right)>\end{array} \)

This expression may besides be obtained equally follows: Since the loss of free energy is due to the work washed by the oscillator to overcome the force of friction

\(\begin{array}{l}F=-p\frac{dx}{dt},\end{array} \)

the instantaneous power P(t) is given past

\(\begin{array}{fifty}P\left( t \right)=\frac{\text{work}}{\text{time}}=\frac{F.\delta x}{\delta t}=F\frac{dx}{dt}\end{assortment} \)

Where δψ is the change in displacement in time δt. Thus

\(\begin{array}{l}P\left( t \right)=p{{\left( \frac{dx}{dt} \correct)}^{2}}=m\gamma {{\left( \frac{dx}{dt} \correct)}^{two}}\end{array} \)

….(three.24)

Now using Eq. (3.18) we have,

\(\begin{array}{l}P\left( t \right)=thou\gamma {{A}^{ii}}{{east}^{-\gamma t}}\left\{ \omega {{*}^{2}}{{\sin }^{ii}}\left( \omega *t-\delta \right)+\frac{\omega *\gamma }{two}\sin 2\left( \omega *t-\delta \right)+\frac{{{\gamma }^{ii}}}{four}{{\cos }^{2}}\left( \omega *t-\delta \right) \correct\}\end{array} \)

Hence, the ability dissipation during once period of oscillation is given by

\(\brainstorm{assortment}{fifty}<P\left( t \correct)>=m\gamma {{A}^{2}}{{e}^{-\gamma t}}\left\{ \omega {{*}^{2}}<{{\sin }^{two}}\left( \omega *t-\delta \right)>+\frac{\omega *\gamma }{2}<\sin two\left( \omega *t-\delta \right)>+\frac{{{\gamma }^{2}}}{4}<{{\cos }^{2}}\left( \omega *t-\delta \right)> \right\}\stop{array} \)

\(\begin{array}{fifty}=\frac{one}{ii}\gamma \,yard\,{{A}^{2}}\,{{eastward}^{-\gamma t}}\left( \omega {{*}^{2}}+\frac{{{\gamma }^{2}}}{4} \correct)\finish{array} \)

\(\begin{array}{l}=\frac{1}{2}\,\gamma m{{A}^{2}}\omega _{0}^{2}{{e}^{-\gamma t}}\end{array} \)

\(\brainstorm{assortment}{l}=\gamma <East\left( t \correct)>\end{array} \)

…..(3.25)

As mentioned earlier, this loss of energy is due to the friction in the system (leading to heating) and the emission of radiation from the organisation (resulting in waves).

Source: https://byjus.com/jee/damped-oscillation/

Posted by: slaytonopeashom.blogspot.com

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