The vibration of a spinning body will cause relative circumferential motions, which will change the direction of the centrifugal load which, in turn, will tend to destabilize the structure. As a small deflection analysis cannot directly account for changes in geometry, the effect can be accounted for by an adjustment of the stiffness matrix, called spin softening. The spin softening contribution is included if any of the following criteria are met:
For the first two criteria, spin softening is an additional contribution to the tangent matrix (Equation 3–32). For the last criteria, it is part of the equations of a rotating structure when expressed in a rotating reference frame (Equation 14–12).
In the following sections, equations are first developed for a spring-mass system, and then a general system equation is formed:
Consider a simple spring-mass system, with the spring oriented radially with respect to the axis of rotation, as shown in Figure 3.7: Spinning Spring-Mass System. In the rotating reference frame, the equilibrium of the spring and centrifugal forces on the mass using small deflection logic requires:
(3–68) |
where:
u = radial displacement of the mass from the rest position along the rotating reference frame direction X' |
r = radial rest position of the mass along the rotating reference frame direction X' |
= angular velocity of rotation |
However, to account for large-deflection effects, Equation 3–68 must be expanded to:
(3–69) |
Rearranging terms,
(3–70) |
Defining:
(3–71) |
and
(3–72) |
Equation 3–70 becomes simply,
(3–73) |
is the stiffness needed in a small deflection solution to account for large-deflection effects. is the same as that derived from small deflection logic. This decrease in the effective stiffness matrix is called spin (or centrifugal) softening. See also Carnegie([104]) for additional development.
Based on the expression of the centrifugal acceleration (Equation 14–39), the spin softening element matrix is expressed as:
(3–74) |
where:
= spin softening element matrix |
= shape function matrix |
= element density |
= rotational matrix associated with the angular velocity vector (input with OMEGA or CMOMEGA) – see Equation 14–41 |
The tangent matrix can then be written as:
(3–75) |
with
(3–76) |
where:
ω_{x}, ω_{y}, ω_{z} = x, y, and z components of the angular velocity |
If there are more than one non-zero component of angular velocity of rotation, the stiffness matrix may become unsymmetric. For example, for a diagonal mass matrix with a different mass in each direction, the matrix becomes nonsymmetric with the expression in Equation 3–75 expanded as:
(3–77) |
(3–78) |
(3–79) |
(3–80) |
(3–81) |
(3–82) |
(3–83) |
(3–84) |
(3–85) |
where:
K_{xx}, K_{yy}, K_{zz} = x, y, and z components of stiffness matrix as computed by the element |
Kxy, Kyx, Kxz, Kzx, Kyz, Kzy = off-diagonal components of stiffness matrix as computed by the element |
M_{xx}, M_{yy}, M_{zz} = x, y, and z components of mass matrix |
From Equation 3–77 thru Equation 3–85, it may be seen that there are spin softening effects only in the plane of rotation, not normal to the plane of rotation. Using the example of a modal analysis, Equation 3–71 can be combined with Equation 15–49 to give:
(3–86) |
or
(3–87) |
where:
ω = the natural circular frequencies of the rotating body. |
If stress stiffening is added to Equation 3–87, the resulting equation is:
(3–88) |
Stress stiffening is normally applied whenever spin softening is activated, even though they are independent theoretically. The modal analysis of a thin fan blade is shown in Figure 3.8: Effects of Spin Softening and Stress Stiffening.