Theory for Springs and Dampers on Joints
You can use the Spring and Damper feature to apply a spring or a dashpot to act on the relative motion in a joint. A dashpot can be used to account for various kinds of losses at the joint. The spring can be given a predeformation so that the spring equilibrium position is not the same as the initial state.
The Translational Spring and Damper, Rotational Spring and Damper, and Inclination Angle and Axial Rotation Spring and Damper versions of this feature are discussed in this section.
Translational Spring and Damper
The translational type of spring and damper is available for the Prismatic Joint, Cylindrical Joint, Screw Joint, Planar Joint, Slot Joint, and Reduced Slot Joint features.
The spring force, which is proportional to the relative displacement is
where ku is the spring constant and u0 is the predeformation.
You can also specify the spring force as function of extension explicitly as
where us is the spring extension.
The damping force, which is proportional to the relative velocity, is:
where cu is the damping coefficient.
The virtual work contribution is
In the case of a Planar Joint, springs and dashpots are attached along user-defined axes oriented in a plane perpendicular to the joint axis. The spring axis and dashpot axis can differ. The forces are proportional to the relative displacement along the spring axis (us) and relative velocity along the dashpot axis (vdp), respectively:
where es is the normalized spring axis and edp is the normalized damper axis.
Rotational Spring and Damper
The rotational type of spring and damper is available for the Hinge Joint, Cylindrical Joint, Screw Joint, Planar Joint, and Reduced Slot Joint features.
The theory for rotational spring and damper is similar to what is described above for Translational Spring and Damper. All translational quantities are replaced by corresponding angular quantities.
Inclination Angle and Axial Rotation Spring and Damper
The inclination angle and axial rotation types of spring and damper are available for the Ball Joint and Slot Joint features.
The spring and damper here can affect two types of relative motion — the inclination angle and the axial rotation of the destination attachment. The inclination angle and axial rotation are computed as:
The definitions of inclination angle and axial rotation, as well as the expressions of cosines, are discussed in the Inclination Angle Constraints and Axial Rotation Constraints sections.
As these rotations are computed using the inverse cosine function, they are unique as long as the magnitudes of the actual rotations (θ and Ψ) are less than 180°.
The spring and damping forces are for both cases similar to those used for a standard Rotational Spring and Damper.