The equilibrium equations for rotordynamics are based on Newton’s second law. It is usually written using a spatial formulation in terms of the Cauchy stress tensor σ:

Here, fV is a body force per unit deformed volume, and ρ is the current mass density. For the material frame formulation used in COMSOL Multiphysics, it is more appropriate to use a Lagrangian version of the equation:

Here, the first Piola–Kirchhoff tensor, P, is used. The term FV is the body force with the components in the current configuration but given with respect to the undeformed volume, and ρ0 is initial mass density. Also note that the gradient operators are not the same. In the first case, the gradient is taken with respect to spatial coordinates, and in the second case it is taken with respect to material coordinates.

Considering that Equation 3-13 is valid in an inertial frame, the fixed-to-corotating frame transformation and Equation 3-4 are used to write Equation 3-13 in the corotating frame as:

All the variables in Equation 3-14 are interpreted in the corotating frame. Here, Sr is the second Piola–Kirchhoff stress tensor, and Fr is the deformation gradient tensor. The COMSOL Multiphysics implementation of the equations in the Solid Rotor interface is, however, not based on the equation of motion directly but rather on the principle of virtual work.

The principle of virtual work states that the sum of the internal virtual work and the external virtual work are equal. The internal virtual work is the work done by the current stress state on a kinematically admissible variation in strains. The external virtual work is the work done by all forces (acting on domains, boundaries, edges, or points) when multiplied with the variation in displacements corresponding to the variation in strains. The virtual displacements δur are in the finite element formulation represented by the test() operator in COMSOL Multiphysics. For a transient case, the virtual work δW is written as

The momentum balance equation (Equation 3-13) can be transformed by incorporating the quantities obtained by using rotated coordinates as a reference (see Equation 3-8 and Equation 3-12). The momentum balance then becomes

Using Equation 3-6 to transfer the derivative with respect to the material coordinate X to the derivative with respect to the rotating coordinate xR, the momentum balance equation changes to

For the transient case, virtual work can be obtained by multiplying the momentum balance equation by δu and integrating the resulting expression over the material volume. Since many of the quantities are expressed in terms of rotated coordinates xR rather than the material coordinate X, we can change the integral over the material volume V to an integral over the rotating volume VR. The resulting expression is