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Oscillations

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An oscillation is a repetitive to-and-fro movement. There are two types of vibration: free and forced. . Oscillations .

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An oscillation is a repetitive to-and-fro movement. There are two types of vibration: free and forced.

Oscillations

A forced vibration is produced when a system is made to oscillate by another system at the other system’s frequency. The frequency of the oscillation is not the object’s natural frequency.

A free vibration is produced when a system is displaced from its equilibrium position and then released. It then oscillates at its natural frequency.

A forced vibration is something like a loudspeaker cone; the cone is forced to vibrate at the frequency of the current that passes in the coil. The amplitude of the vibration depends on the size of the current.

A pendulum is an example of a free vibration. Other examples include a swing at a playground, or the string of a guitar when plucked. A free vibration has no driving mechanism and zero friction. Once started, theoretically, it would continue to oscillate for ever. In practice, no oscillation is ever truly ‘free’.

For a pendulum swinging, the time period does not depend on the amplitude of the oscillations, for small angles of θ where sin θ approximately equals θ in radians.

Amplitude: maximum displacement from the mean equilibrium position

Period: time to complete one full oscillation

Frequency: number of oscillations per unit time

Angular frequency: 2πf – one oscillation corresponds to one rotation, this is an angle of 2π. If there are f oscillations per unit time, there will be a corresponding angle of 2πf radians per unit time.

Phase Difference : angle in radians between two oscillations.

Simple Harmonic Motion: A body executes simple harmonic motion if its acceleration is directly proportional to its displacement from its equilibrium position and the acceleration is in the opposite direction to the displacement, always towards the equilibrium position.

A body has maximum acceleration to the right when at –A, and therefore maximum resultant force is exerted. It’s velocity is zero.

A body has maximum acceleration to the left when at A, and therefore maximum resultant force is exerted. It’s velocity is zero.

A body has maximum velocity and zero acceleration when at equilibrium (displacement is zero)

-Xmax

Xmax

Equilibrium

Acceleration is greatest at maximum displacement; velocity is greatest at zero displacement; resultant force greatest at maximum displacement.

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Plotting a displacement against time graph for a body freely vibrating, we would get something that looked like this;

We can use this graph to a plot velocity-time graph…

The gradient of this graph gives us rate of change of displacement – which is velocity. So, where the gradient is zero, the velocity is zero; where the gradient is steepest, velocity is at its maximum.

V

T

We can then use this graph to a plot an acceleration-time graph…

The gradient of this graph gives us rate of change of velocity – which is acceleration. So, where the gradient is zero, the acceleration is zero; where the gradient is steepest, acceleration is at its maximum.

From these 3 graphs, we can see that velocity is radians out of phase with displacement, and acceleration is π radians out of phase with displacement (it is in antiphase).

The conclusion we can draw is that acceleration is proportional to – displacement, as the acceleration and displacement are in opposite directions.

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The displacement of an object in simple harmonic motion can be calculated using either of these equations, depending on the nature of the wave.

The maximum speed of an object performing simple harmonic motion is given by:

Note: 2πf appears in all of these equations. It is a quantity known as the angular frequency, denoted by the letter ω omega. Its units are rads-1.

Energy Changes in Simple Harmonic Motion

The energy of the oscillating mass is transformed back and forth between kinetic and potential forms as it oscillates. The total amount of energy stays constant.

Total Energy = Kinetic Energy + Potential Energy

At the endpoints, there is maximum gravitational potential energy and zero kinetic energy.

At the midpoints, there is zero gravitational potential energy and maximum kinetic energy.

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Damping

Damping is the effect of resistive forces removing energy from a vibrating object.

Light damping gradually reduces the energy and therefore the amplitude of the vibrating object. An example is a swing in playground; it will gradually come to rest.

Critical damping returns the object to the equilibrium position in the shortest time possible. An example is shock absorbers in a car; they increase the resistive force so that after being displaced when going over a bump, the vehicle returns to its original position without oscillating.

Heavy damping is the result of a very large resistive force. An example could be a mass on a spring in a thick, viscous liquid. Heavy damping results in a long time before the object comes to rest.

A system can be made to oscillate by means of a periodic external driving force. Resonance occurs when the driving frequency most efficiently transfers its energy to the driven system. The amplitude therefore increases as more of the driving force's energy is taken into the system.

Resonance is a maximum amplitude oscillation of a system when it is forced to vibrate at its natural frequency.

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Resonance

As the degree of damping increases, the amplitude of the resonant vibrations decreases. The resonant peaks become broader.

The frequency at which resonance occurs becomes lower for a damped system.

Maximum amplitude when the driving frequency is at the natural resonant frequency of the object.

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