Tuning Fork Resonance (3D)

The 3D bench

The forks, resonance boxes, table and mallet are Blender models. The prongs bend by the first cantilever mode shape, driven live by the coupled-oscillator model, so the amplitudes and beats match the 2D page and the checks. At slow time scales you see the real vibration; at fast time scales the prongs show the envelope.

What you are looking at

Two metal tuning forks stand on a table. Fork A is struck (or driven by a tone generator); fork B is its neighbour. Each fork is modelled by its fundamental bending mode as one damped oscillator, and the two oscillators are coupled weakly through the table. A resonance box under a fork makes it louder but makes it ring for a shorter time.

Why a fork has a pitch: the cantilever

Each prong is a beam clamped at the stem. For a uniform beam (Euler-Bernoulli theory) the fundamental frequency is f = (beta^2 / 2 pi) (a / L^2) sqrt(E / 12 rho) with beta L = 1.8751 the first root of cos(beta L) cosh(beta L) = -1, prong length L, thickness a in the bending direction, Young's modulus E and density rho. The default (steel, L = 100 mm, a = 5.4 mm) gives 440 Hz, concert A. Doubling the length lowers the pitch by a factor of four. The overtone is 6.27 times higher, which is why a fork sounds almost pure: the first overtone dies out quickly and the plain fundamental is what you hear.

Resonance and ring-down

The oscillator is x'' + (w/Q) x' + w^2 x = w^2 x_s cos(w_d t). Struck, it decays as exp(-t / tau) with tau = Q / (pi f). Driven at its own frequency, the steady amplitude is Q x_s: a push of one micrometre becomes several millimetres. The width of the resonance peak is f / Q, so a high-Q fork responds only to an almost exact match.

Sympathetic resonance and beats

The table adds a coupling stiffness k_c = c k between the two modal coordinates (the distance dependence of c here is a stylised law, not a measurement). Two identical forks then have two normal modes: both swinging together (frequency f) and in opposition (f sqrt(1 + 2c)). Striking A excites both, and the energy slowly moves between the forks with first full hand-over after pi / (w_a - w_s). If B is slightly mistuned by a clip, only part of the energy transfers and the pitches beat at the difference frequency. The clip changes the frequency as f' = f / sqrt(1 + M / (0.2357 m)) where m is the prong mass.

Numerical checks on this page

• A finite-element cantilever (30 Hermite beam elements, inverse iteration) gives (beta L)^2 = 3.5160 and the overtone ratio 6.267, and the fork frequency computed from it is compared with the closed-form formula.
• The ring-down time of an isolated simulated fork is measured from two amplitudes and compared with Q / (pi f).
• For two identical forks the simulated time of the first complete energy transfer is compared with pi / (w_a - w_s).
• The driven steady-state amplitudes from the time integration are compared with the closed-form complex solution of the 2 by 2 linear system.

What is simplified

Each fork is reduced to its fundamental mode; the real prongs also have the stem motion and a small second-harmonic content when struck. Both prongs move exactly in opposition (no net force on the stem), so the coupling is a lumped, symmetric effect; the two forks have the same modal mass. The resonance-box factors (Q divided by five, louder sound) are illustrative, not acoustic measurements. Amplitude is the tip displacement; the dB levels are relative to 1 mm, not sound-pressure levels. Sound output (Web Audio) plays the two fundamental frequencies with gains following the simulated amplitudes, at real time speed.

What to try

• Strike A with the default settings: B starts to ring a little later, then the energy returns.
• Put a 0.12 g clip on B: the forks now beat; turn on Sound to hear it.
• Choose the sweep preset and watch the measured dots follow the closed-form resonance curve.
• Lower Q to 500 or add boxes and see the wider peak and shorter ring.
• Switch the material: steel and aluminium nearly share a pitch because the speed of sound in them is about the same.