Normal coordinates and mode excitation

Collect mass-normalized eigenvectors as columns of a matrix E. Write the physical displacement η as E Q, where Q lists modal coordinates. Since Eᵀ M E is the identity, the inverse projection is Q=Eᵀ Mη. It is not generally E transpose alone. This distinction is easy to miss if all demonstration masses happen to be one. A displacement that looks perpendicular to a mode in a drawing can still have a nonzero mass-weighted projection onto that mode.

Substituting this transformation into the quadratic Lagrangian removes kinetic and potential cross terms. Each modal coordinate obeys its own oscillator equation with the corresponding natural frequency. For positive nonzero frequencies, its solution combines a cosine fixed by initial displacement and a sine fixed by initial velocity. Both projections matter. A mass released at equilibrium with a nonzero velocity can excite a mode whose displacement coefficient is initially zero; looking only at the initial shape would miss it.

If the initial displacement is proportional to one mode and initial velocity is proportional to the same mode, no other modes are excited in this linear autonomous model. An applied generalized force similarly projects as E transpose times the force vector. Spatial alignment can therefore suppress a resonance even when its frequency is present in the spectrum. This is a selection rule from the forcing pattern, not proof that the unexcited mode does not exist. Small nonlinearities can later couple modes that are independent in the approximation.

The quadratic energy is the sum of one half of modal velocity squared plus one half of frequency squared times modal coordinate squared. In an undamped autonomous linear system, each of those modal energies is constant. The energies assigned to individual physical masses and springs can vary as cross-coordinate contributions rearrange. Calling that behavior energy transfer is useful only after specifying the subsystem definition. It does not mean the independent normal modes are exchanging energy in the exact linear model.

Two frequencies can produce an envelope through trigonometric addition. For nearly equal frequencies, a fast oscillation is modulated by a slow difference-frequency factor, leading to familiar beats. A finite frequency separation still allows the identity but may not look like a slowly varying envelope. Equal frequencies with opposite phase instead give fixed cancellation. These are different statements: changing relative phase at one common frequency cannot reproduce the time-dependent relative phase of unequal-frequency modes.

The waves lab deliberately shows the equal-frequency case: it adds two waves with the same spatial and temporal frequency, controlled by phase difference and amplitude ratio. Predict cancellation at a phase difference of 180 degrees and amplitude ratio one, then test it. It does not have two mode-frequency controls, a mass matrix, or coupled spring energies, so it cannot demonstrate the worked system's beat dynamics. Use its cancellation example to understand superposition, while reconstructing unequal-frequency mechanical motion with the normal-coordinate equations.