modalpotential

Modalpotential is a theoretical construct used in physics and applied mathematics to describe how potential energy of a system is distributed when expressed in modal coordinates. The term combines modal, referring to independent oscillation patterns or normal modes, with potential, the energy associated with configuration in a conservative system.

In a linear elastic system with displacement vector q and potential energy V(q) = 1/2 q^T K q, where K is the stiffness matrix, suppose q = Φ η, with Φ the matrix of normalized mode shapes and η the modal coordinates. The modal potential is defined as V_modal(η) = 1/2 η^T Λ η, where Λ = diag(λ1,...,λn) contains the eigenvalues of K in the modal basis. Equivalently, V_modal can be written as the sum V_modal = sum_i 1/2 λ_i η_i^2, separating energy contributions from each normal mode.

This decomposition clarifies how energy is partitioned among modes and is central to modal analysis, reduced-order modeling, and control design for mechanical systems. In nonlinear or strongly coupled systems, cross terms or a nonlinear modal potential may arise, complicating the interpretation but still guiding energy-based analysis.

Applications include structural vibration, acoustics, automobile or aerospace engineering, and certain areas of electrical circuits and fluid-structure interaction. The concept also features in numerical methods that rely on modal decomposition to simulate high-dimensional systems efficiently.

See also: modal analysis, normal modes, eigenvalue decomposition, potential energy, reduced-order modeling.