Mass-weighted elastic normal modes ($\hat{H}{\mathbf{u}}_{\alpha }=\rho {\omega }_{\alpha }^2{\mathbf{u}}_{\alpha }$)

For a conservative elastic body, the stiffness operator is a self-adjoint operator under compatible boundary conditions. The dynamical operator ${\rho }^{-1}\hat{H}$ is self-adjoint in the mass inner product $⟨\mathbf{u},\mathbf{v}{⟩}_{\rho }=\int \rho {\mathbf{u}}^{\ast }\cdot \mathbf{v}dV$. Choosing mass-normalized normal modes converts the field equation into independent forced oscillators. If the normal modes have mass norm $N_{\alpha }$ instead, each projected force must be divided by $N_{\alpha }$.

For an impulse vector $\mathbf{J}=M\Delta \dot{\mathbf{q}}$ delivered at an unstrained equilibrium, project the velocity jump onto mass-orthogonal normal modes. Multiplying $M\Delta \dot{\mathbf{q}}$ by ${\mathbf{u}}_j^T$ gives modal velocity coefficient ${\mathbf{u}}_j^T\mathbf{J}/({\mathbf{u}}_j^{T}M{\mathbf{u}}_j)$. Its kinetic energy is the displayed expression, including a translational zero mode if one is present. A single endpoint kick in an equal-mass free three-mass spring chain puts fractions $1/3,1/2,1/6$ into translation and the two internal modes.