Simple Harmonic Motion Concept Page - 13

Result

Free vs forced vibrations

Free vibrations	Forced vibrations
Vibrations of body in absence of any external force.	Vibrations of body in presence of external driving force.
Frequency of vibrations depends on the source of vibrations.	Frequency of vibrations depends on the source of vibrations and the frequency of the driving force.
Frequency of vibrations remains constant.	Frequency of vibrations can be changed by changing frequency of driving force.
Amplitude of vibration is constant.	Amplitude of vibration can be decreasing, increasing or constant.

Definition

Free oscillations

The amplitude remains constant as time passes, there is no damping. This type of oscillation will only occur in theory since in practice there will always be some damping. The displacement will follow the formula x=rsinwt$x=rsinwt$ where r$r$ is the amplitude. 

Definition

Forced Oscillations

If an oscillator is displaced and then released it will begin to vibrate. If a force is continually or repeatedly applied to keep the oscillation going, it is a forced oscillation.

Result

Write total force and write differential equation of motion for damped oscillations

Total force in damped oscillations is:
cdxdt+kx$c{\displaystyle \frac{dx}{dt}}+kx$
(Due to damper and spring.)

Final differential equation for the damper is:
md2xdt2+cdxdt+kx=0$m{\displaystyle \frac{d^{2}x}{d{t}^2}}+c{\displaystyle \frac{dx}{dt}}+kx=0$

Example

Write force equation and differential equation of motion in forced oscillation

Example: A weakly damped harmonic oscillator is executing resonant oscillations.  What is the phase difference between the oscillator and the external periodic force?

Solution:
The equation for forced oscillation in a damped system is given as-
md2xdt2+bdxdt+kx=F0cosωt$m{\displaystyle \frac{d^{2}x}{d{t}^2}}+b{\displaystyle \frac{dx}{dt}}+kx=F_{0}cos\omega t$⟹d2xdt2+2βdxdt+ω20x=Acosωt$⟹{\displaystyle \frac{d^{2}x}{d{t}^2}}+2\beta {\displaystyle \frac{dx}{dt}}+{\omega }_0^{2}x=Acos\omega t$The expected solution is of form x=Dcos(ωt−δ)$x=Dcos(\omega t-\delta )$Put this is in above equation gives,tanδ=2βωω20−ω2$tan\delta ={\displaystyle \frac{2\beta \omega }{{\omega }_0^2-{\omega }^2}}$For resonant oscillation, ω0=ω${\omega }_0=\omega$⟹δ=π/2$⟹\delta =\pi /2$ which is the phase difference between x$x$ and F$F$.

Definition

Damped vibrations

The periodic vibrations of a body of decreasing amplitude in presence of a resistive force are called damped vibrations. Some examples of damped vibrations are oscillations of branch of a tree, sound produced by tuning fork over longer distances, etc. In fact all vibrations on earth's surface in the absence of an external force are damped vibrations.

Result

Nature of damped vibrations

Usually in damped vibrations, frictional force is proportional to the velocity of the vibrating object and has tendency to resist the motion. The rate of energy lost to the medium depends on the nature of the medium. 

Definition

Damped Oscillations

Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems.

Definition

State the parameters of damped SHM

In damped oscillation:
1. Damping Force: −cdxdt$-c{\displaystyle \frac{dx}{dt}}$
where c is damping coefficient.

2. Restoring spring force: −kx$-kx$

Result

Displacement as a function of time in damped SHM

x(t)=Acos(w′t+ϕ)e−bt2m$x(t)=Acos(w^{\prime }t+\varphi )e^{{\displaystyle \frac{-bt}{2m}}}$

Diagram

Plot displacement as a function of time for different values of frequency in damped oscillations

Definition

Understand damping as loss due to resistance of medium

The decreasing amplitude of oscillation of the damping system is given by:
A=Aoe−bt2m$A=A_o{e}^{{\displaystyle \frac{-bt}{2m}}}$

As it's clear from the above equation amplitude falls with time. Fundamentally, this represents loss due to resistance of medium

BookMarks

Page 1  Page 2  Page 3  Page 4  Page 5  Page 6  Page 7  Page 8  Page 9  Page 10
Page 11  Page 12  Page 13  Page 14