OCR Specification focus:
‘Resonance occurs when driving frequency matches natural frequency, producing large amplitudes.’
Resonance and natural frequency describe how oscillating systems respond to external periodic forces, producing maximum amplitude when frequencies align. Understanding this concept is vital for physics and engineering applications.
Understanding Natural Frequency
Every oscillating system possesses a natural frequency, the rate at which it vibrates when disturbed and then left to oscillate freely without external influence. This frequency depends on the system’s physical properties such as mass, stiffness, and tension.
Natural Frequency: The frequency at which a system oscillates when no external force acts upon it, determined solely by its internal characteristics.
For example, a mass–spring system vibrates at a frequency determined by the spring constant and the mass attached, while a tuning fork’s natural frequency depends on its material and shape. Systems can possess multiple natural frequencies, particularly when they can oscillate in different modes, as in musical instruments or bridges.
When a system oscillates at its natural frequency, its energy transfers seamlessly between kinetic energy (movement) and potential energy (stored deformation energy). This interchange continues indefinitely in an ideal system without damping.
Driving Frequency and Forced Oscillations
When an external periodic force acts on a system, it is said to be driven. The driving frequency is the frequency of this external force. If the driving frequency differs from the system’s natural frequency, the response is modest. However, when the driving frequency approaches or equals the natural frequency, a dramatic increase in oscillation amplitude occurs — this phenomenon is known as resonance.
Resonance: A condition in which an oscillating system experiences maximum amplitude when the frequency of an external driving force equals its natural frequency.
The concept of resonance applies widely across physical systems: from mechanical oscillators like bridges and car suspensions, to electrical circuits and acoustic systems. The energy input at the driving frequency continually reinforces the natural oscillation, allowing amplitude to build up significantly.
The Mechanism of Resonance
Resonance occurs due to constructive interference between the oscillating motion of the system and the periodic driving force. When both oscillations are in phase, each cycle of the external force contributes energy at just the right moment to increase the system’s motion.
Key features of resonance include:
Large amplitude response: When resonance occurs, small driving forces can produce large displacements.
Energy transfer efficiency: The driving force does maximum work on the oscillator because the energy transfer is perfectly timed.
Phase relationship: At resonance, the velocity of the oscillator is in phase with the driving force, meaning energy is continuously added to the system.
The amplitude increase is theoretically infinite in an undamped system, but real systems always experience damping, which limits the amplitude and ensures stability.
Mathematical Relationships
For a simple harmonic oscillator subject to a driving force, the amplitude AAA of oscillation depends on the driving frequency fdf_dfd relative to the natural frequency f0f_0f0.
EQUATION
—-----------------------------------------------------------------
Resonance Condition: fd=f0f_d = f_0fd=f0
fdf_dfd = Driving frequency (Hz)
f0f_0f0 = Natural frequency (Hz)
—-----------------------------------------------------------------
When this equality holds, resonance occurs and the amplitude reaches its maximum possible value for that level of damping.
Real systems display a resonance curve, where amplitude is plotted against driving frequency. The peak of this curve marks the resonance point.
Amplitude versus driving frequency for the same oscillator with small, medium, and heavy damping. The peak at f=f0f=f_0f=f0 marks resonance, where amplitude is largest; greater damping lowers and broadens the peak. The curve visualises why undamped resonance is idealised, while real systems respond finitely. Source.
The sharper and taller the peak, the smaller the damping present in the system.
Phase Relationships and Energy Transfer
The phase difference between the driving force and the system’s displacement determines how efficiently energy is transferred.
Below resonance: The system’s displacement lags the driving force by a small phase angle. The motion follows the driving force sluggishly.
At resonance: The phase difference between displacement and driving force is exactly 90°, but the velocity is in phase with the driving force, allowing continuous energy addition.
Phase difference versus normalised frequency for a forced, damped oscillator. The curve passes through 90° at resonance, illustrating maximum energy transfer timing. This graphic mentions inertial sensors on its file page, but the plotted phase behaviour is the standard result for any forced, damped oscillator. Source.
Above resonance: The displacement leads the driving force, reducing energy transfer and amplitude.
This interplay explains why resonance leads to powerful oscillations and why tuning is crucial to control or exploit the effect.
Damping and Real-World Resonance
In practical systems, damping arises from friction, air resistance, or internal material losses. Damping reduces amplitude and broadens the resonance peak. Without damping, oscillations at resonance could grow uncontrollably, leading to structural failure or system instability.
EQUATION
—-----------------------------------------------------------------
Amplitude at Resonance (A₍res₎) ∝ 1 / Damping Coefficient (c)
A₍res₎ = Maximum amplitude (m)
c = Damping coefficient (kg s⁻¹)
—-----------------------------------------------------------------
Thus, increasing damping reduces the resonant amplitude and helps stabilise the system. Engineers often introduce controlled damping to prevent resonance-induced damage. For example:
Car suspension systems use shock absorbers to reduce resonant bouncing.
Buildings and bridges employ tuned mass dampers to prevent destructive resonances during earthquakes or strong winds.
Practical Significance of Resonance
Resonance is both beneficial and hazardous depending on context:
Constructive uses: In musical instruments, resonance amplifies sound; in microwave ovens, electromagnetic resonance efficiently transfers energy to food molecules.
Destructive consequences: In structures or machinery, uncontrolled resonance can lead to catastrophic failure, such as the collapse of the Tacoma Narrows Bridge in 1940.
Designers and engineers must therefore calculate natural frequencies and ensure that external forces never coincide with these frequencies unless resonance is intentionally harnessed.
Summary of Key Characteristics
To reinforce understanding, resonance and natural frequency can be characterised by the following principles:
Every oscillating system has a natural frequency determined by its physical parameters.
A driving force introduces forced oscillations; when its frequency matches the natural frequency, resonance occurs.
Amplitude increases sharply at resonance due to maximum energy transfer efficiency.
Damping limits amplitude and determines the sharpness of the resonance peak.
Awareness of resonance is crucial in safe and efficient design of physical systems.
FAQ
The natural frequency depends on the physical properties of the system, particularly its mass and stiffness.
For a mass–spring system, increasing the mass decreases the natural frequency, while increasing the spring constant raises it.
In pendulums, the natural frequency depends on the gravitational field strength and the effective length of the pendulum.
Changes in shape, material properties, or tension can also alter the natural frequency in structures such as bridges or guitar strings.
At resonance, the driving force and the oscillator’s motion are synchronised in phase so that energy is added to the system at just the right moment each cycle.
Because the energy input and the oscillation are perfectly aligned, each successive wave of the driving force reinforces the motion rather than opposing it.
This continuous reinforcement allows amplitude to grow rapidly, limited only by the damping present in the system.
Resonance is prevented by designing systems so that their natural frequencies do not coincide with common external driving frequencies.
Methods include:
Adding damping materials to absorb energy.
Changing the mass or stiffness to shift natural frequency.
Installing tuned mass dampers that counteract oscillations.
Engineers use vibration analysis to identify and modify risky resonant modes during the design stage.
Below resonance, displacement lags slightly behind the driving force.
At resonance, the displacement is 90° out of phase, while velocity is in phase with the driving force.
Beyond resonance, displacement leads the driving force by almost 180°.
This shifting phase relationship affects how efficiently energy is transferred to the oscillator — it’s greatest at resonance and falls away at other frequencies.
Electrical resonance occurs in RLC circuits when the inductive and capacitive reactances are equal and cancel each other out.
This condition is analogous to mechanical resonance, as both involve maximum energy transfer at the system’s natural frequency.
At resonance, current in the circuit reaches a maximum (for series resonance) or voltage across components peaks (for parallel resonance).
Both types demonstrate the same physical principle: efficient energy exchange when the driving and natural frequencies are matched.
Practice Questions
Question 1 (2 marks)
Define resonance and state the condition required for it to occur in an oscillating system.
Mark scheme:
1 mark for correctly defining resonance as the condition when a system oscillates with maximum amplitude due to a periodic driving force.
1 mark for stating that resonance occurs when the driving frequency equals the natural frequency of the system.
Question 2 (5 marks)
A student investigates the behaviour of a mechanical oscillator driven by a variable-frequency motor. The amplitude of oscillation increases sharply as the driving frequency approaches a certain value, then decreases again at higher frequencies.
(a) Explain why the amplitude increases rapidly at a particular frequency. (2 marks)
(b) Describe the role of damping in this system and how it affects the resonance curve. (3 marks)
Mark scheme:
(a)
1 mark for identifying that maximum amplitude occurs when the driving frequency equals the natural frequency (resonance).
1 mark for explaining that energy transfer from the driver to the oscillator is most efficient when they oscillate in phase.
(b)
1 mark for stating that damping reduces the amplitude of oscillation at all frequencies.
1 mark for stating that damping lowers and broadens the resonance peak on the amplitude–frequency graph.
1 mark for explaining that damping dissipates energy as heat or friction, preventing excessively large amplitudes and potential damage.
