Distributed Loads
The direction of the explicitly applied distributed load should be given with reference to a global or local coordinate system in the spatial (fixed) or corotating frame. The magnitude of the load must be with reference to the undeformed reference (or material) area. Since the reference area does not depend on the material or corotating frame, the magnitude of the load remains unaffected with respect to the frame in which it is applied. However, the direction of the load will depend strongly on the frame in which it is applied. The relation between the true force f acting on the current area da and the specified distributed load F acting on the material area dA is fda = FfdA. If the load is specified in the corotating frame, then the relation changes to fda = RFdA. Therefore, the contribution to the virtual work for a distributed force specified on a surface is
In a stationary analysis, irrespective of the choice of the frame, a load is always considered to act in the corotating frame.
When a pressure load, p, acts on the rotor, the true force on the surface element acts with the magnitude p over the current area da in the normal direction n:
Therefore, the pressure load type specifies the distributed load as
where both normal n and area element da are functions of the current displacement field. Another view of how to interpret the load is to express it in terms of the first Piola–Kirchhoff stress tensor P via the formula
where the normal n0 corresponds to the undeformed surface element. Such a force vector is often referred to as nominal traction.
Rotating Load in Frequency Domain
In rotordynamics, we often encounter loads that are rotating with respect to the frame in which the rotor is modeled. For example, an eccentric load observed in a space-fixed frame is rotating with a speed equal to the rotor’s angular speed. Similarly, a gravitational load will appear to be rotating with a speed equal to the rotor’s angular speed and opposite to the rotational direction when observed in a corotating frame. In frequency-domain analyses, such loads require special treatment to correctly model their rotating behavior.
Consider a load of magnitude F0 rotating about the rotor axis with angular speed Ω with an initial phase angle ϕ. The local components of this load as a function of time are
In vector form
With
a load rotating in positive Ω direction can in frequency-domain be represented as
Similarly, a load rotating in negative Ω direction is