14.1. Acceleration Effect

Accelerations are applicable only to elements with displacement degrees of freedom (DOFs).

The acceleration vector {ac} which causes applied loads consists of a vector with a term for every degree of freedom in the model. In the description below, a typical node having a specific location and accelerations associated with the three translations and three rotations will be considered:

(14–1)

where:

where:

= accelerations in global Cartesian coordinates (input on ACEL command or CMACEL command for component base acceleration)
= translational acceleration vector due to inertia relief (see Inertia Relief)
= rotational acceleration vector due to inertia relief (see Inertia Relief)
= translational acceleration vector due to rotations (defined below)
= angular acceleration vector due to input rotational accelerations (defined below)

Mechanical APDL defines three types of rotations:

Rotation 1: The whole structure rotates about each of the global Cartesian axes (input on OMEGA and DOMEGA commands)
Rotation 2: The element component rotates about an axis defined by user (input on CMOMEGA and CMDOMEGA commands).
Rotation 3: The global origin rotates about the axis by user if Rotation 1 appears or the rotational axis rotates about the axis defined by user if Rotation 2 appears (input on CGOMGA, DCGOMG, and CGLOC commands)

Up to two out of the three types of rotations may be applied on a structure at the same time. These rotations induce accelerations and thus inertia forces when the equations are expressed in a rotating reference frame.

 

14.1.1. Acceleration Due to One Rotation

When one rotation is defined (Rotation 1, 2, or 3), the angular acceleration vector is:

(14–2)

The translational acceleration vector used to calculate the inertia force in the rotating reference frame is deduced from Equation 14–54:

(14–3)

where

= vector cross product

The translational acceleration vector is the difference between the translational acceleration expressed in a stationary reference frame and the translational acceleration expressed in the rotating reference frame, ignoring the Coriolis term.

The second term in Equation 14–3 leads to the Euler force and the last term to the centrifugal force.

 

14.1.2. Acceleration Due to Two Rotations

The angular acceleration vector due to the rotations is:

(14–4)

The translational acceleration vector used to calculate the inertia force in the rotating reference frame (second rotation) is:

(14–5)

All other quantities are defined below, depending on the types of rotations considered.

 

14.1.2.1. Rotation 1 and Rotation 3

{ω} = angular velocity vector defined about the global Cartesian origin (input on OMEGA command)
{Ω} = angular velocity vector of the overall structure about the point CG (input on CGOMGA command)
= angular acceleration vector defined about the global Cartesian origin (input on DOMEGA command)
= angular acceleration vector of the overall structure about the point CG (input on DCGOMG command)
{r} = position vector
{R} = vector from CG to the global Cartesian origin (computed from input on CGLOC command, with direction opposite).

Figure 14.1: Rotational Coordinate System (Rotations 1 and 3)

Rotational Coordinate System (Rotations 1 and 3)

 

14.1.2.2. Rotation 1 and Rotation 2

{ω} = angular velocity vector defined about the rotational axis of the element component (input on CMOMEGA command)
{Ω} = angular velocity vector defined about the global Cartesian origin (input on OMEGA command)
= angular acceleration vector defined about the rotational axis of the element component (input on CMDOMEGA command)
= angular acceleration vector defined about the global Cartesian origin (input on DOMEGA command)
{r} = position vector
{R} = vector from about the global Cartesian origin to the point on the rotational axis of the component.

Figure 14.2: Rotational Coordinate System (Rotations 1 and 2)

Rotational Coordinate System (Rotations 1 and 2)

 

14.1.2.3. Rotation 2 and Rotation 3

{ω} = angular velocity vector defined about the rotational axis of the element component (input on CMOMEGA command)
{Ω} = angular velocity vector of the overall structure about the point CG (input on CGOMGA command)
= angular acceleration vector defined about the rotational axis of the element component (input on CMDOMEGA command)
= angular acceleration vector of the overall structure about the point CG (input on DCGOMG command)
{r} = position vector
{R} = vector from CG to the point on the rotational axis of the component

Figure 14.3: Rotational Coordinate System (Rotations 2 and 3)

Rotational Coordinate System (Rotations 2 and 3)

 

14.1.2.4. Additional Term for Point Mass Element

For MASS21 with KEYOPT(3) = 0 and MATRIX27 with KEYOPT(3) = 2, additional Euler's equation terms are considered:

(14–6)

where:

{M} = additional moments generated by the angular velocity
[I] = matrix of input moments of inertia
{ωT} = total applied angular velocities: = {ω} + {Ω}