By applying virtual work from the d'Alembert force, the contribution of the first term {a _r} (Equation 14–54) to the virtual work introduces the mass matrix of the element (Guo et al.([361])).

(14–55)

where:

The Coriolis matrix is deduced from the last term of Equation 14–54 (Guo et al.([361])):

(14–56)

where:

[Ge] = element Coriolis damping matrix

and thus [G_e] are skew symmetric matrices.

The governing dynamics equation of motion in a rotating reference frame can be written as,

(14–57)

where:

n = number of elements

[K] = global stiffness matrix

[KSP] = global stiffness matrix due to spin softening

{F} = load vector (sum of external and centrifugal loads)

In a quasi-static analysis, the Coriolis effect is introduced as a load vector:

(14–58)

where:

{Fc} = Coriolis force

= nodal velocity vector (input using the IC or ICROTATE command).

The gyroscopic finite element matrix is calculated from the kinetic energy due to the inertia forces.

The kinetic energy for lumped mass and beam element (Nelson and McVaugh([359])) is detailed in Kinetic Energy for the Gyroscopic Matrix Calculation of Lumped Mass and Beam Element below.

The general expression of the kinetic energy used for the development of the gyroscopic matrices for all other elements (Geradin and Kill [377]) is presented in General Expression of the Kinetic Energy for the Gyroscopic Matrix Calculation

 

14.4.2.1.1. Kinetic Energy for the Gyroscopic Matrix Calculation of Lumped Mass and Beam Element

Both mass and beam should be axisymmetric around the spinning axis. The spinning axis is along one of the principal axis of inertia for lumped mass. For the beam, it is along the length.

Two reference frames are used (see Figure 14.4: Reference Frames) (OXYZ) which is stationary and which is attached to the cross-section with axis normal to it.

is defined using 3 successive rotations:

• θ_z around Z axis to give

• θ_y around axis to give

• θ_x around axis to give

Hence for small rotations θ_y and θ_z, the instantaneous angular velocity is:

(14–59)

1. For a lumped mass, considering only second order terms, kinetic energy is obtained using the instantaneous angular velocity vector in Equation 14–59.

   (14–60)

   where:

   m = mass

   Id = diametral inertia

   Ip = polar inertia

   The first two terms contribute to the mass matrix of the element and the last term gives the gyroscopic matrix.

2. The beam element is considered as an infinite number of lumped masses. The gyroscopic kinetic energy of the element is obtained by integrating the last term of Equation 14–60 along the length of the beam:

   (14–61)

   where:

   ρ = density

   Ix = moment of inertia normal to x

   L = length of the beam element

   Gyroscopic matrix is deduced based on the element shape functions. (See PIPE288).

 

14.4.2.1.2. General Expression of the Kinetic Energy for the Gyroscopic Matrix Calculation

A point, in element i, with coordinates (x,y,z) in the stationary reference frame is considered. The kinetic energy is

(14–62)

where:

EGki = gyroscopic kinetic energy of element i

Vi = volume of element i

dm = elementary mass

The gyroscopic matrix is then calculated using the element shape functions.

In a linear approach, the relation between displacements in the stationary reference frame (0XYZ) and displacements in the rotating reference frame (0X’Y’Z’) can be written as:

(14–63)

where:

Differentiating Equation 14–63 with respect to time, one obtains the expression for the velocity vector:

(14–64)

where:

= the velocity vector in the rotating reference frame

= the velocity vector in the stationary reference frame

= the rotational matrix, as defined in Equation 14–56

If structural damping is present in the rotating structure (proportional damping for example) or if there is localized viscous damping (as in a damper), damping forces in the rotating reference frame may be expressed as:

(14–65)

where:

= the damping forces in the rotating reference frame

[C] = the damping matrix

To obtain the damping forces in the stationary reference frame, first apply the transformation of Equation 14–63:

(14–66)

where:

Then replace Equation 14–64 in Equation 14–65, the resulting expression in Equation 14–66 yields:

(14–67)

If the damping is isotropic (implementation assumption):

(14–68)

Where [B] is the rotating damping matrix:

(14–69)

In the particular case of spring-damper element (COMBIN14), general joint (MPC184), and stiffness matrix (MATRIX27), the following average equation is used:

(14–70)

[B] from Equation 14–69 and Equation 14–70 is a non-symmetric matrix which will modify the apparent stiffness of the structure.

In a quasi-static analysis, the gyroscopic effect is introduced as a load vector:

(14–71)

where

= gyroscopic force

= nodal velocity vector (input using the IC or ICROTATE command)

In this case, an additional centrifugal load is introduced based on the rotational velocity and rotational velocity axis defined with ICROTATE. The translational acceleration used is similar to the third term of Equation 14–54.

The transient forces of a small mass unbalance are expressed as described below.

A body is rotating at 𝛚 around the Z axis. Point A is the center of the body cross-section in plane (X,Y). m is the mass unbalance located at point B. The distance AB is the eccentricity e, which is constant. The instantaneous location of A is (u_x,u_y).

Figure 14.5: Mass Unbalance

In the (X,Y) plane, the mass unbalance coordinates can be expressed as:

(14–72)

Hence, its velocity is:

(14–73)

The kinetic energy of the mass unbalance can be written as:

(14–74)

The mass unbalance is very small with respect to the mass of the rotating body, hence the first two terms are neglected. The last term is constant and has no contribution to the dynamics equations.

The kinetic energy reduces to:

(14–75)

Application of Lagrange’s equations leads to the following forces in the (X,Y) plane:

(14–76)

It is the sum of a centrifugal force due to the rotational velocity and a tangential force due to the rotational acceleration.