Oscillations beyond SHM (Topic 13B)

Oscillators have a natural frequency.

An object oscillates at its natural frequency when performing a free oscillation, where no energy is being transferred to or from the surroundings.

For example, if a guitar string is plucked once, it will vibrate at its natural frequency. The notes from the different strings on a guitar correspond to their natural frequencies, which depend on their thickness and tension.

The amplitude of an object oscillating at its natural frequency varies with driving frequency.

When the driving frequency is below the natural frequency, the amplitude of oscillations is small.

As the driving frequency approaches the natural frequency of the oscillator, the amplitude increases rapidly, reaching a maximum when the driving frequency equals the natural frequency; this is known as resonance.

When the driving frequency exceeds the natural frequency, the amplitude begins to decrease again.

A Barton’s pendulum is a system used to display the effect of resonance.

A series of pendulums of different lengths is suspended from the same horizontal, flexible string.

One of the pendulums (X) is much heavier than the others. Pendulum X is displaced and begins moving with simple harmonic motion, with a frequency determined by its length.

The flexible string feels a force from the motion of pendulum X, and it vibrates at the same frequency. The other pendulums start to oscillate in response.

Pendulum D will oscillate with the greatest amplitude because it is the same length as pendulum X, so they will both have the same natural frequency.

Pendulum X provides an external periodic driving frequency which matches the natural frequency of pendulum D, causing it to resonate. The other pendulums will oscillate, but with a smaller amplitude than pendulum D.

Stringed instruments display the effect of resonance. When a string is plucked, it vibrates and stationary waves are formed. Their frequencies are the resonant frequencies of the string.

The stationary waves consist of a series of nodes and antinodes. Their wavelengths are determined by the length of the string.

The string vibrates at several of these resonance frequencies simultaneously. The higher frequencies, known as harmonics, determine the sound of the instrument.

However, these vibrations move very little air and produce almost no sound on their own. The body of the guitar is designed to have a natural frequency similar to the frequency of the stationary waves.

The body of the guitar resonates when the strings vibrate at the same frequency, causing the larger body to vibrate and displace a significantly greater amount of air, thereby creating louder sound waves.

Magnetic resonance imaging (MRI) exploits the resonance effect of hydrogen nuclei within water molecules in living tissue.

A magnetic field causes the hydrogen nuclei, which behave like small magnets, to precess (rotate). The natural frequency of precession depends on the tissue in which the nuclei are located and the strength of the magnetic field.

The scanner transmits radio waves with a frequency equal to the natural frequency of the nuclei, causing them to gain energy and resonate.

When the radio transmission is turned off, the nuclei relax and emit the energy gained in the form of radio photons. These emissions are detected by the machine, enabling the scanner to construct a detailed image of the patient’s tissue structure.

The kinetic energy of an object is given as:

where:

• is the kinetic energy measured in joules ()
• is the mass measured in kilograms ()
• is the velocity measured in metres per second ().

In simple harmonic motion:

• At the equilibrium position, the speed of the oscillator is maximum and, therefore, this is where the kinetic energy is also greatest.
• At the maximum displacement (equal to the amplitude), the velocity is zero; therefore, so is the kinetic energy.

The graph below illustrates the variation in kinetic energy of a pendulum bob over two distinct time periods.

The maximum kinetic energy of the system can be found using the equation:

The maximum velocity of an object moving with simple harmonic motion in is:

Where:

• is the angular velocity measured in radians per second ()
• is the amplitude measured in meters ().

The potential energy of an object in simple harmonic motion depends on the displacement from the equilibrium position:

• At equilibrium, the potential energy of the oscillator is zero.
• At maximum displacement, the potential energy is maximum.

The graph below illustrates the variation in potential energy of the oscillator over two time periods

If the potential energy of the oscillator is in the form of gravitational potential energy, then we can use the following equation:

where:

• is the height above the equilibrium and is measured in metres ()
• is the mass measured in kilograms ()
• is acceleration due to gravity measured in metres per second squared ().

If the potential energy is in the form of elastic potential energy, then we can use the following equation:

where:

• is the stiffness constant of the object and is measured in Newtons per metre ()
• is the change in length of the object and is measured in metres ().

The maximum elastic potential energy will be when the change in length is equal to the maximum displacement (amplitude), so the equation becomes:

where:

• is the amplitude measured in meters ().

The total energy of a simple harmonic oscillator is the sum of its kinetic energy and potential energy.

The total energy remains constant unless an external force, such as friction, acts on the system which causes a damping effect.

An example of this is a pendulum bob undergoing simple harmonic motion:

1. At the top of the swing, the bob is stationary and all of its energy is in the form of potential energy.
2. As the bob descends, it gains speed and hence kinetic energy, but loses potential energy as its height decreases.
3. At the equilibrium position, all of the bob’s energy is in the form of kinetic energy.
4. As the bob rises again, it loses speed and, hence, kinetic energy, but gains potential energy again.

To calculate the total energy of the system, it is easier to calculate the maximum value of the kinetic energy or the potential energy than the sum of the kinetic energy and potential energy at some point between the equilibrium and the amplitude:

As an object moving with simple harmonic motion oscillates between two amplitudes, it constantly exchanges energy between kinetic and potential forms.

• The kinetic energy is greatest at the equilibrium, where the object is moving fastest and zero at the amplitude.

• The potential energy is greatest at the amplitudes and zero at the equilibrium.

• The total energy is the sum of the kinetic and potential energies at any particular point and is a constant as long as no external forces are acting, such as friction.

Shown below is a graph demonstrating how the different types of energy vary with displacement from the equilibrium position.

Energy changes for an ideal simple pendulum over time, where is the displacement from the equilibrium position:

Energy changes for an ideal mass spring system over time, where is the displacement from the equilibrium position:

Forced oscillations occur when a periodic external force acts on an object that is already oscillating.

The frequency of the external force is referred to as the driving frequency.

The external force provides energy to overcome the losses caused by damping.

An example of a forced oscillation is a child being pushed on a swing by another person.

The child is given a push at the start of each swing to maintain the swinging motion and counteract the damping forces.

In real-life situations, an oscillating object experiences resistive forces, which act in the opposite direction to the velocity of the object.

Examples of resistive forces:

• Air resistance is experienced by an object as it moves through air.
• Drag is experienced by an object moving through water (or any fluid).
• Friction is experienced by an object moving along a surface.

Resistive forces cause the amplitude and oscillation to decrease due to energy being transferred away from the oscillator. This is known as damping.

Although the amplitude of an oscillator decreases when it experiences damping, the frequency of the oscillations remains constant, as long as the damping force is not so large that it stops the oscillation completely.

For example, pendulum clocks still tell the time accurately as the swings of the pendulum decrease.

There are three types of damping that an oscillator can experience. One of these is light damping:

• The size of the resistive force is small.
• The amplitude size decays over time.
• The frequency of oscillation remains constant.

An example of light damping is a real pendulum’s bob as it swings through the air.

There are three types of damping that an oscillator can experience. One of these is heavy damping:

• The size of the resistive force is large.
• Similar to light damping, but the amplitude decays exponentially more quickly over time.
• The frequency of oscillation remains constant.

An example of heavy damping is the suspension on a rough terrain vehicle, which dissipates energy quickly so the oscillations die away rapidly and the vehicle does not continue to bounce after going over a bump.

There are three types of damping that an oscillator can experience. One of these is critical damping:

• A critically damped oscillator will return to the equilibrium position almost immediately after being initially displaced.
• The damping force is great enough to prevent oscillations, in contrast to both light and heavy damping.

An example of critical damping is a slow-closing, heavy door that incorporates a damping mechanism to prevent it from slamming shut.

In a forced oscillation, an external force is applied periodically. This is known as the driving frequency.

When the driving frequency of the external force matches the natural frequency of the oscillator, then a phenomenon called resonance occurs.

During resonance, the energy transfer from the driving force to the oscillator is most efficient. This causes the amplitude of the oscillator to increase quickly to its maximum.

An example of resonance is a child being pushed on a swing.

The child on the swing acts like a pendulum, with a natural frequency which depends on the length of the swing.

The adult applies a force at regular intervals. If the frequency of the driving force applied matches the natural frequency of the swing, then the amplitude of the oscillations will dramatically increase.

If a system undergoing a forced oscillation experiences damping, the amplitude of the oscillations decreases.

The greater the damping, the greater the decrease in amplitude at all frequencies.

For a resonating system, as the damping force increases, the resonance peak decreases and becomes broader. In addition, the peak shifts to a lower driving frequency.