Inertial Loads
Deformable bodies undergoing rigid body motions are subject to inertial loads. To model these phenomena correctly it is convenient to separate rigid displacements due to rigid motion from non-rigid displacements due to the deformation of the solid.
In Solid Mechanics, the total motion of the body can be described in a moving reference frame.
Kinematics of Deformable Bodies Subject to Rigid Motions
where
To determine the material velocity and acceleration in the presence of a rigid body motion, the current position vector is expressed in the time-dependent form:
From this, the total derivative with
respect to time provides the material velocity at point
The time derivative of the rotation matrix is often expressed as:
Inserting Eqn. (4470) into Eqn. (4468), and removing the explicit time dependence of the different variables, leads to an alternative expression for the velocity:
Virtual work due to Inertial Forces
The fourth term is the inertial load
due to local acceleration. This term leads to the mass matrix contribution in the
finite element discretization. The fifth term is the translational rigid body motion
acting as an external load, independent of the unknowns
The centrifugal force (second term)
and the Euler acceleration (first term), are loads which are linearly dependent on
Linearization of the Inertial Force Virtual Work
The linearization of the centrifugal force introduces an apparent stiffness in the system (centrifugal stiffness or spin softening effect). This stiffness is important for rotor applications and also influences the modal analysis of rotating structures. From Eqn. (4478) the importance of this term increases quadratically as the rotational speed increases.
To clarify the anti-symmetry of the
linearizations the integrand terms have been rearranged. Eqn. (4477) shows the symmetric bilinear form of the
centrifugal stiffness term, whilst the linearization of the Euler acceleration term
is in a skew-symmetric bilinear form due to the skew-symmetric nature of the angular
acceleration tensor (