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Natural and driven oscillations

Physical Background

Harmonic Oscillator

A body making an oscillatory harmonic motion is called the harmonic oscillator. At the first approximation it may be e.g. a body suspended on a spring (Fig. 1) or an atom or a molecule of a solid.

If the damping resistance of the environment may be neglected, we speak about the undamped harmonic oscillator. We can demonstrate that the oscillation is harmonic, if the acting force is proportional to the deflection from the equilibrium, and its direction is opposite to that of the deflection.

Let us have a spring, for the deflection of which from the equilibrium position by r is necessary to exert the force F = -kr, where k is the spring stiffness. An equation of the motion of the harmonic oscillator, consisting of the spring and the weight with the mass m, is −kr = ma (as the application of the second Newton΄s law Σ F = ma), i.e.

Introducing ω₀ = (k/m)^1/2 this equation may be expressed as follows A general solution of this equation in one dimensional case in the y axis (r = yj) where y is the deflection, A is the amplitude, [A] = m, ω_o is the natural angular frequency of the harmonic oscillator, [ω_o] = rad s⁻¹, and φ, [φ] = rad, is its initial phase (Fig. 2).

We can write for the natural angular frequency of the undamped harmonic oscillator

here f_o is its natural frequency and T_o is the natural period of the harmonic oscillator.

Damped oscillator

During the motion of a harmonic oscillator under real-world conditions friction forces are always acting decreasing the amplitude of the oscillating motion, leading to the standstill after some time. This kind of oscillations is then referred to as a damped oscillator.

The friction force at low velocities is directly proportional to the instantaneous velocity v of the weight

or

Let us denote the quantity b = k_b/(2m), referred to as the damping, [b] = s⁻¹. For a small damping (b < ω_o) equation (5) has a solution for r = yj

where ω_b = (ω_o² − b²)^1/2 is the angular frequency of the damped oscillator.

Driven oscillator

The important case of the ocillating motion is the driven oscillator, where an external force makes a material object to oscillate at a generally different frequency, than that of the natural frequency of the oscillator. If, however, both frequencies are approaching each other, the important phenomenon arises, known as the resonance.

Let us assume the weight in Fig. 3 is acted upon by a driving periodic force

where F_o is the amplitude and ω_v the angular frequency of the driving force F_v.

If all the forces that make the weight to move are considered, the following equation of the motion may be written

and after some adjustment Solution of this equation then is where A_v and φ_v are the amplitude and the initial phase of the driven oscillations, fulfilling

The graphs in Fig. 6 reveal some interesting features of the driven oscillations. First, it is obvious that the amplitude of the driven oscillations A_v (see Fig. 6 a) is the function of the angular frequency of the driving force ω_v. The maximum amplitude of the driven oscillations occurs for the so called resonance frequency, ω_{v, res}

with the value A_v,res = F_o / (2bm(ω_o² - b²)^1/2. Likewise the initial phase φ_v of the driven oscillations (or the initial phase shift between the driving force F_v and the deflection y) is dependent on the is frequency of the driving force ω_v and at the resonance equals φ_v,res = -π/2 ( see Fig. 6 b).

The dependencies in Fig. 6 will be used to determine the energy transferred to the oscillator. This phenomenon is very important in the Nature and for the technology in general as it represents a selective energy transfer between the transmitter of energy and the oscillator only within a certain interval of frequencies or wavelengths. This is e.g. the basis of selective transfer of telecommunication signals or also an absorption of some wavelength of radiation from the Sun by certain molecules, which otherwise would reach the Earth surface.

The energy transferred per a unit time to the oscillator (or the instantaneous time dependent power ) from the driving source

The medium energy transfered in a single period of oscillatory motion from the driving source into the oscillator is then dependent of driving angular frequency ω_v (mentioned here without derivation; for more information, see R. B. Lindsay, Physical Mechanics, 3rd ed., Van Nostrand 1962).

In Fig. 7 is the the medium energy transferred in one period of oscillatory motion to the oscillator from the driving source on the driving angular frequency ω_v for various values of damping b.

Author of study text: Prof. Dr. František Schauer, Univerzita Tomáše Bati ve Zlíně

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