Harmonic Oscillator

Experiment: Spring pendulum

A weight (orange box) is attached to a linear spring. If the weight is pulled down and then released, it starts swinging up and down.

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Left: Oscillation with friction
The oscillation loses energy due to friction, thereby the weight oscillates ever closer around the equilibrium position and eventually stops oscillating.

Right: Oscillation without friction
The weight oscillates smoothly around equilibrium position.

We first deal with the oscillation without friction. For more information about oscillation with friction see damped oscillator.

General definition of oscillation

Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. If the system is displaced from the equilibrium, there is a net restoring force on the mass, tending to bring it back to equilibrium.

Application on the spring pendulum

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Left: Stable equilibrium
The tension of the spring (upwards) and the acceleration due to gravity (downwards) cancel each other. The box does not move.

Right: Displacement and restoring force
If the weight is displaced from the equilibrium position (e.g. pulled by a hand), the result is a force imbalance between the tension of the spring and the gravitational acceleration.
The resulting total force acting on the weight, is called restoring force, because it "tries" to "restore" the initial position of the weight.

General definition of oscillation (continued)

[...] Oscillations are based on the periodic energy conversion between two forms of energy. In this case, the system will be in the initial state after after a fixed interval of time.

Application on the spring pendulum

To explain the oscillation of the spring pendulum a description of the velocity of the weight is needed.

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We observe:

At maximum displacement
The velocity of the weight is minimal (\(0 \,\, m/s\)). The restoring force is maximal.

When passing the equilibrium position
The restoring force is minimal (\(0 \,\, N\) because the restoring force and the gravitational force cancel. The velocity is maximal.
The weight continues to move due to inertia.

There is an energy conversion between the potential energy of the spring, and the kinetic energy of the weight.

The restoring force

The force that is occurring when a spring is deformed is given by:
$$ F = - D \cdot s $$

\(D\) = Spring constant, \(s\) = Displacement

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\( F_{Restore} = F_G + F_{Tension} = F_G - D \cdot s_1 = 0 \)

\( F_{Restore} = F_G + F_{Tension} = \underset{0}{\underbrace{F_G - D \cdot s_1}} - D \cdot s_2 = - D \cdot s_2 \)

Differential equation for oscillations

Using the formulas \(F = m \cdot a \) and \(a = \ddot{s} \) (Acceleration is the second derivative of the distance) we obtain the following Differentialgleichung:
\begin{aligned} F_{Rück} & = - D \cdot s \\ m \cdot a & = - D \cdot s \\ m \cdot \ddot{s} & = - D \cdot s \end{aligned} How this equation can be solved, is not described here in greater detail.

Solution of differential equation for oscillations

Solving the differential equation results in the following equation that describes an oscillation: $$ s(t) = s_0 \cdot \sin (2 \pi f t + \phi_0) $$

\(s(t)\) = Displacement after time \(t\), \(s_0\) = Amplitude, \(f\) = Frequency, \(\phi_0\) = Phase angle

  • Amplitude
    The amplitude \( s_0 \) is the maximal displacement of an oscillation.
  • Period
    The period is the time that elapses while the oscillating system undergoes exactly one oscillation period, i.e. after which time it is located in the initial state. The reciprocal of the period \(T \) is the frequency \(f \), that is: \(f = \frac{1}{T} \).
  • Frequency
    The frequency \(f \) indicates the number of revolutions per unit of time and is measured after the German physicist Heinrich Hertz in Hertz (\(Hz = \dfrac{1}{s} \)).
  • Phase angle
    The phase angle \(\ phi_0 \) indicates at what angle of the oscillation starts. A phase angle of \(\phi_0 = 2 \cdot \pi \) corresponds to a shift by one period.
    At a phase angle of \(\phi_0 = \frac{1}{4} \cdot 2 \cdot \pi = \frac{1}{2} \cdot \pi \), the oscillation would shift by a quarter period. (That would mean the spring pendulum would start at maximum displacement).

Example 1:
\( s_0 = 2   m \),  \( f = \frac{1}{10}   Hz \) und \( \phi_0 = 0 \)

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The period is $$ T = \dfrac{1}{f} = \dfrac{1}{\frac{1}{10} Hz} = 10   s $$

Angular frequency

An oscillation can be understood as a projection of a circular motion.

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The angular velocity \(\ omega \) of such motion is already known from earlier grades: $$ \omega = 2 \pi f $$ It corresponds to the angle highlighted by the blue pointer per second. Overall In the animation on the left a weight oscillates with the frequency \(f = 0.25   Hz \), the angular velocity is therefore: $$ \omega = 2 \pi f = 2 \pi \cdot 0.25   Hz = \dfrac{1}{2} \pi   Hz $$ In oscillations \(\ omega \) is called angular frequency.

The equation for oscillation is now: $$ s(t) = s_0 \cdot \sin (\omega t + \phi_0) $$

\(s(t)\) = Displacement after time \(t\), \(s_0\) = Amplitude, \( \omega \) = Angular frequency, \(\phi_0\) = Phase angle

Example 2:
\( s_0 = 5   m \),  \( \omega = \frac{1}{2} \pi   Hz \) und \( \phi_0 = \frac{1}{4} \cdot 2 \cdot \pi = \frac{1}{2} \cdot \pi \)

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The frequency is $$ f = \dfrac{\omega}{2 \pi} = \dfrac{\frac{1}{2} \pi   Hz}{2 \pi} = \dfrac{1}{4}   Hz $$ The period is $$ T = \dfrac{1}{f} = \dfrac{1}{\frac{1}{4} Hz} = 4   s $$

Velocity and acceleration

The equations for the velocity and acceleration are obtained by deriving the equation for oscillations. \begin{aligned} s(t) & = s_0 \cdot \sin (\omega t + \phi_0) \\ & \\ v(t) & = \omega \cdot s_0 \cdot \cos (\omega t + \phi_0) \\ & \\ a(t) & = -\omega^2 \cdot s_0 \cdot \sin (\omega t + \phi_0) \end{aligned}
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The velocity function is shifted with respect to the oscillation function by \(\frac{1}{2} \pi \) to the left.
The acceleration function is shifted with respect to the oscillation function by \(1 \pi \) to the left.