 Nonclassical damping

Nonclassical damping occurs when a linear mechanical system's damping matrix cannot be diagonalized together with its mass matrix and stiffness matrix by an undamped normal mode transformation. Consequently the damping terms couple the modal equations even though their inertia and stiffness terms are separated. This can matter when a localized dissipative element is attached to a structure whose elastic and inertial responses are distributed differently.^[ref]

For $M\ddot{u}+C\dot{u}+Ku=0$, take a complete mass-normalized undamped mode matrix $\Phi$ with ${\Phi }^{T}M\Phi =I$ and ${\Phi }^{T}K\Phi ={\Omega }^2$. Then $u=\Phi q$ givesWith distinct undamped natural frequencies, a nonzero off-diagonal $D_{ij}$ shows modal damping coupling. If a frequency is repeated, first allow a rotation of the modes within its repeated-frequency subspace: an off-diagonal entry in one arbitrary basis is not by itself proof of nonclassical damping. For symmetric $M,K,C$ with $M$ positive definite, the basis-independent condition for classical decoupling isFailure of this condition establishes nonclassical damping under those assumptions.^[ref]

For an original example in consistent normalized units, letThe damping acts only on the first physical coordinate. The undamped mode vectors $(1,1)/\sqrt{2}$ and $(1,-1)/\sqrt{2}$ giveThus an initial modal state $q=(1,0{)}^T$, $\dot{q}=(1,0{)}^T$ has ${\ddot{q}}_2=-0.2$. Dropping the off-diagonal damping terms would incorrectly predict zero initial acceleration in the second modal coordinate. Nevertheless ${\dot{u}}^{T}C\dot{u}=0.4{\dot{u}}_1^2\ge 0$: passive dissipation does not require modal decoupling.

The damped quadratic eigenvalue problem follows from $u(t)=r{e}^{\lambda t}$:In oscillatory cases its complex vectors can encode different phases at different physical coordinates. One can solve the coupled equations directly or use an appropriate damped-mode method; assigning separate scalar decay equations to the undamped modes is an additional approximation, not a consequence of positive dissipation.