Beam vibration and cavity acoustics: two interactive simulations

A beam with a mass and spring

A pinned beam carries a point mass, with a spring from the same point to the ground. The mass lowers the frequencies and the spring raises them, and the amount depends on where the attachment sits relative to each mode's shape.

Spring stiffness, K

100

Attached mass, M

1.00

Attachment position, ξ

0.50

 

Strike the beam

Try three things. At ξ = 0.5, the flat curve at 39.5 is a mode with a node at midspan, so the attachment never moves and K has no effect on it. Raise K, and the first mode climbs through that flat curve near K = 2,550, heading for about 61.7: the stiff spring acts as a pin in the middle of the beam, and each half then behaves as a pinned-clamped span. Then move ξ away from the center and sweep K again, and the curves approach each other and veer apart instead of crossing.

Equations

With beam modes sin(nπξ), generalized mass M_nn = ½ and bare-beam frequencies ω_n = (nπ)², the energies are:

T = ½ Σ M_nn q̇_n² + ½ M ż²,   V = ½ Σ M_nn ω_n² q_n² + ½ K z²,   z = Σ q_n sin(nπξ_s)

The constraint ties the mass and spring to the beam. Lagrange's equations then give a symmetric eigenproblem, which the page solves with sixteen modes each time a slider moves. Eliminating the multiplier gives the characteristic equation:

1 / (K − Mω²) + Σ sin²(nπξ_s) / [M_nn(ω_n² − ω²)] = 0

A flexible panel on an air cavity

A panel on an aircraft or rail car often closes a compartment. The trapped air and the panel affect each other: the panel's motion drives sound in the cavity, and the cavity's pressure pushes back on the panel.

Cavity depth

30 cm

Aluminum panel thickness

1.0 mm

 

Strike the panel

At the default setting, the panel alone would ring at about 7 Hz. With the cavity behind it, that first bending shape moves to about 51 Hz, because it changes the enclosed volume and has to compress the air. The second bending shape doesn't change the volume, so it barely moves, and it becomes the lowest mode of the system. Make the cavity shallower and the air spring gets stiffer. Make the panel thicker and the panel takes over.

Equations

In the cavity, the velocity potential obeys the wave equation with hard walls. Separating variables gives the cavity's modes and tones:

∇²φ − (1/c²) φ_tt = 0,   Φ_nm = cos(nπx/a) cos(mπz/d),   ω_nm = cπ √[(n/a)² + (m/d)²]

My homework stopped at those modes. This page adds the flexible panel at z = d, where the air's motion has to match the panel's. Expanding the pressure in Φ_nm and the panel in sine modes gives:

m_r (q̈_r + Ω_r² q_r) = Σ C_ir P_i,   P̈_i + ω_i² P_i = −(ρ₀c²/Λ_i) Σ C_ir q̈_r

These equations aren't symmetric as written. Rewriting the air in terms of a displacement potential makes them symmetric, so frequencies stay real and the page can solve them with a standard symmetric routine. The uniform pressure mode has zero frequency, and it survives only as the air spring on the panel.

Verification

These checks run the same code as this page, in checks/verify_vibration.js, along with a separate finite-element model.

The models are linear and two-dimensional, with simple support and rigid walls, so they're for building intuition rather than predicting a real panel. Light damping is added only so that a strike fades out.