Category: 08 Oscillations

# 137 Lissajous Figures on CRO

When the CRO is set to the *x*–*y* mode, it uses the Channel 1 input as the time-base. This means the *x* and *y* position of the CRO trace is controlled by inputs from Channel 1 and 2 respectively.

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If both channels are fed with sinusoids with the same frequency, then the following traces wil result, depending on the phase relationship between the two sinusoids.

In-phase. Much like the relationship between force and acceleration.

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Anti-phase. Much like the relationship between acceleration and displacement for a SHM.

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A quarter-cycle out of phase. Much like velocity leading displacement for a SHM.

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In this demo, because I was using analogue signal generators, I was unable to set the inputs to the exact frequencies I wanted. So even though I was aiming for 100 Hz for both channels, I could only get them to be close to but not exactly 100 Hz. This means there was a slight difference in the periods of the two signals. This causes their phase relationship to keep shifting, alternating between in-phase and anti-phase. This results in the lovely outcome of the tracing alternating between diagonals and flipped diagonals, including the ellipses between.

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When Channels 1 and 2’s frequencies are a nice integer ratio of each other, very intricate patterns are formed. By counting the number of times the trace cuts each axis, one can figure out the exact ratios. If you are interested and want to read up more, just google “Lissajous Figures”.

# 135 Natural Frequency

When an oscillatory system is displaced from its equilibrium position, and then released, it goes into oscillation. Basically, if an oscillating mass is subjected to only the restoring force (i.e. no damping force and no external driving force), the resulting oscillation is called a free oscillation.

It turns out that for a SHM, *a*=-*ω*^{2}*x*. This means that the *a*/*x* ratio determines the frequency of oscillation.

With this in mind, we can understand why for a spring-mass system, a stiffer spring (increases the restoring force per unit displacement) and a smaller mass (increases the acceleration for the same restoring force) results in a higher frequency of oscillation. In fact, it can be derived quite easily that for a spring-mass system *ω*^{2}=*k*/*m*.

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A pendulum is another famous SHM. The derivation is more complicated but it can be shown that for a pendulum *ω*^{2}=*g*/*L*. (Notice the amplitude has no effect on the period of oscillation?)

# 134 Restoring Force

An oscillation is any periodic repetitive to-and-fro motion.

Just like circular motion requires a centripetal force, an oscillation requires a restoring force. The restoring force serves to accelerate the oscillating mass towards the equilibrium position. But upon arriving at the equilibrium position, inertia causes the mass to overshoot the equiblirium position. This repeats again and again, hence oscillations.

A simple harmonic motion is a special class of oscillation whose displacement variation with time can be described by one single sinusoidal function (hence simple harmonic). It can also be shown that for an oscillation to be simple harmonic, the restoring force must be proportional to displacement.

The spring-mass system and the pendulum are probably the two most famous SHMs. For the spring-mass system, it is clear that at the equilibrium position, weight and tension forces are balanced. But at any other displacement *x*, the mass experiences a net force of *F*_{net}=*kx*. The restoring force is clearly proportional to displacement, resulting in a simple harmonic oscillation.

# 133 Pendulum Wave

In this simulation, 17 balls are programmed to oscillate at angular frequencies of *ω*, 17*ω*/16, 18*ω*/16, 19*ω*/16,… 2*ω*. Notice that the angular frequencies are evenly spaced out, which implies the periods are not.

Since angular frequency is the rate of change of phase angle, the leftmost ball lags behind the most in terms of phase. For example, after one period of the leftmost ball, the leftmost ball would have traversed only one cycle (2π), while the rightmost would have already traversed two cycles (4π), and the other balls evenly spaced out in between. This makes the balls line up along one cycle of cosine function.

After two periods, the positions of the balls would be spaced out evenly along two cosine functions. (Because the rightmost ball would have completed 2 cycles more than the leftmost).

# 131 Fortune Telling Sticks

The NO stick has a much lower natural frequency than the YES sticks.

When the base is moved to-and-fro subtlely but periodically, both sticks experience a periodic driving force.

When the base is shaken at the natural frequency of the NO stick, energy is transfered efficiently to the NO stick. The NO stick goes into resonance and attain a large amplitude of oscillation. The YES stick, being very quick, easily moves in line with the base and attains negligible amplitude of oscilation.

When the base is shaken at the natural frequency of the YES stick, the YES stick goes into resonance and attain a large amplitude of oscillation. The NO stick is too slow to react to the rapidly changing driving force, and remains motionless.

# 130 Damped Oscillation

This video demonstrates both EMI and Oscillations.

Electromagnetic Induction

The oscillating aluminium plate experiences a damping force which is EMI in nature. The aluminium plate cuts the magnetic flux (originating from the neodymium magnets). The induced emf results in induced eddy current in the plate. The induced current interacts with the magnetic field of the neodymium magnets, resulting in *F*=*BIL* magnetic forces. The direction of the magnetic forces are always opposite in direction to the (current) velocity of the oscillating plate. How do we know that? Lenz’s Law!

Oscillations

The G-clamp provides a convenient mechanism to gradually increase the amount of damping, thus showing the behaviour of underdamped, critically damped and overdamped oscillations.

An underdamped oscillation always overshoots the equilibrium position and comes to rest only after a number of oscillations.

Critical damping returns the pendulum to rest at the equilibrium position in the shortest amount of time possible, without overshooting the equilibrium position.

An overdamped oscillation does not overshoot the equilibrium position, but takes a longer time before coming to rest at the equilibrium position (compared to critical damping).

# 128 Tied Buoy Accelerator

You can think of this contraption as an acceleration meter. The length of the rubber is an indicator of the acceleration. When it is at the usual extension, acceleration is zero. When it stretches beyond this length, acceleration is upward. When it shrinks, acceleration is downward.

As in any oscillation, the acceleration is always opposite to the displacement. So in this vertical oscillation, the acceleration is upward (shown by the stretched rubber band) whenever it is below the equilibrium position. It does not matter whether it is going downward or upward.

Likewise, regardless of whether the velocity is upward or downward, whenever it is above the equilibrium position, the acceleration is downward (as shown by the shrunk rubber band)

# 068 Oscillations: Critical Damping

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Critical Damping

Damped oscillations must return to the equilibrium position eventually as the damping force saps energy continously from the oscillation. However, the manner in which the oscillations cease depends on the degree of damping.

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In the video, **critical damping** was achieved when the aluminium block was 2.0 cm from the magnets (1:46). Under critical damping, the pendulum returned immediately to the equilibrium position in the shortest amount of time possible, without overshooting the equilibrium position.

With less damping than critical (**underdamping** or light damping), the pendulum overshoots the equilibrium position and oscillates around it. The amplitude of oscillation decays exponentially as the damping force withdraws energy from the pendulum.

With more damping than critical (**overdamping** or heavy damping), the pendulum also does not overshoot the equilibrium position, but returns to the equilibrium position more slowly.

The **collage **at the end of the video shows very clearly that critical damping brings the oscillator to rest in the shortest time.

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Why is there a damping force?

As the magnet swings, the aluminium along side the magnet experiences first an increasing and then a decreasing flux linkage, resulting in induced emf. (*E*=*dΦ*/*dt*)

Since aluminium blocks are conductors, (eddy) currents formed in the blocks, producing their own magnetic field.

The polarity of the induced emf must be such as to produce a induced current and magnetic field that opposes the change that caused the induction in the first place. (Lenz’s Law). This predicts a retarding magnetic force on the pendulum that is always in opposite direction to the velocity of the pendulum, hence a damping force.

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How is the degree of damping varied?

By moving the block closer to the magnets, the rate of change of magnetic flux linkage is increased, resulting in a larger induced emf, current and thus damping force.

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# 059 Coupled Inverted Pendulum

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The oscillation of one pendulum sets the horizontal tube vibrating at the same frequency. The tube is thus shaking the other pendulum at the same frequency as the first ball. In other words, all the other pendulum experience a periodic driving force at the frequency of the first ball.

Resonance occurs only for the pendulum with the same length since its resonant frequency matches the frequency of the driving force. Which is why only one ball goes into large amplitude oscillation.