FLEXION-TORSION

Connection type FLEXION-TORSION provides a rotational connection between two nodes. It models the bending and twisting of a cylindrical coupling between two shafts. In this case the response to twist rotations about the shafts may differ from the response to bending of the shafts. Connection type FLEXION-TORSION cannot be used in two-dimensional or axisymmetric analysis.

The flexural part of the connection resists angular misalignment of the two shafts, whereas the torsional part of the connection resists relative rotations about the shafts. Connection type FLEXION-TORSION can be used in conjunction with connection type RADIAL-THRUST when resistance to relative radial and thrust displacements is modeled.

Related Topics
Connector elements
Connector element library
In Other Guides
*CONNECTOR BEHAVIOR
*CONNECTOR SECTION

ProductsAbaqus/StandardAbaqus/ExplicitAbaqus/CAE

Description

Figure 1. Connection type FLEXION-TORSION.

The FLEXION-TORSION connection does not impose kinematic constraints. The FLEXION-TORSION connection describes a finite rotation by three angles: flexion, torsion, and sweep (α, β, and θ). However, the flexion, torsion, and sweep angles do not represent three successive rotations. The flexion angle between two shafts measures the angle of misalignment of the two shafts and is always reported as a positive angle. The torsion angle measures the twist of one shaft relative to the other.

The sweep angle orients the rotation vector, in the e1ae2a plane, for the flexion motion. See Figure 1. Since the flexion angle is never negative, the sweep angle may undergo discontinuous jumps by up to π radians when the flexion angle passes through zero. An analysis may give inaccurate results or may not converge if any jump occurs in the sweep angle. In general, the sweep angle is not used as an available component of relative motion for which connector behavior is defined. Rather, it is used to define angular dependence for the elastic constitutive response in flexion deformations (as an independent component in the connector elastic behavior definition). Since the sweep angle is restricted to the interval -π to π radians, any dependence on the sweep angle should be periodic, such that the behavior for θ=-π is the same as θ=π. Since α=0 is a singular point for which the sweep angle is not uniquely defined, it is strongly recommended that any connector behavior that defines flexural moment versus flexion angle gives zero moment at zero flexion angle. If connector behavior is defined in the sweep available component, the sweep moment must be zero at flexion angles α=0 and α=π.

The FLEXION-TORSION connection is similar to a finite successive rotation parameterization 3–2–3. However, in terms of the 3–2–3 parameterization, the sweep angle is the first rotation angle, the flexion angle is the second rotation angle, and the torsion angle is the sum of the first and third rotation angles.

The first shaft direction at node a is e3a, and the second shaft direction at node b is e3b. Let the two shafts form an angle α, called the flexion angle. Then,

α=cos-1(e3ae3b),    where    0απ.

The flexion angle is a rotation by α about the (unit) rotation vector

q=1sinαe3a×e3b,    where    sinα=e3a×e3b.

The torsion angle β between the two shafts is defined as

β=tan-1(e2ae1b-e1ae2be1ae1b+e2ae2b)+mπ,

where positive torsion angles are rotations about the positive e3b-direction, and m is an integer.

The sweep angle θ measures the angle from e1a to the projection of e3b onto the e1ae2a plane. With this definition

θ=tan-1(e2ae3be1ae3b),    where    -πθπ.

It follows that the flexion rotation vector, q, can be written

q=-sinθe1a+cosθe2a.

A singularity in the definition of the sweep angles occurs when the flexion angle α vanishes. In this case e3b=e3a; that is, the torsion and sweep angle axes are coincident, and the two angles are no longer independent. When α=0, the sweep angle is assumed zero, θ=0.

The available components of relative motion ur1, ur2, and ur3 are the changes in the flexion, torsion, and sweep angles and are defined as

ur1=α-α0,    ur2=β-β0,    and    ur3=θ-θ0,

where α0 and β0 are the initial flexion and torsion angles, respectively. The initial value of the sweep angle θ0 is chosen to be zero if the shafts align initially. The connector constitutive rotations are

ur1mat=α-θ1ref,    ur2mat=β-θ2ref,    and    ur3mat=θ-θ3ref.

The kinetic moment in a FLEXION-TORSION connection is determined from the three component relationships:

m1=mflex-torq;    m2=mflex-tore3b;    and    m3=mflex-tore3a-mflex-tore3b.

Summary

FLEXION-TORSION
Basic, assembled, or complex: Basic
Kinematic constraints: None
Constraint moment output: None
Available components: ur1,ur2,ur3
Kinetic moment output: m1,m2,m3
Orientation at a: Required
Orientation at b: Optional
Connector stops: θ1minαθ1max,
  θ2minβθ2max,
  θ3minθθ3max
Constitutive reference angles: θ1ref,θ2ref,θ3ref
Predefined friction parameters: None
Contact force for predefined friction: None