Theory for Joint Locking
The Locking feature is used to lock the relative motion between the two components connected to a joint. You specify limits on the relative motion in the form of an upper and lower bound. This can be interpreted as an imaginary object (like a snap hook) that locks one of the free relative degrees of freedom of the destination attachment when it reaches the limiting value.
The Locking feature differs from Constraints in a sense that the value of the particular joint degrees of freedom becomes constant for the rest of the analysis after it attains its prescribed limiting value.
The Translational Locking, Rotational Locking, Inclination Angle Locking, and Axial Rotation Locking types are discussed in this section.
Translational Locking
Translational locking is available for the Prismatic Joint, Cylindrical Joint, Screw Joint, Planar Joint, Slot Joint, and Reduced Slot Joint features.
In this locking type, a lock is applied to the relative displacement (u). When the lock is active, its relative displacement becomes constant at the prescribed limiting value and the relative velocity goes to zero. If lock conditions are active for all the joint degrees of freedom, then the source and destination attachment move as a single object.
You prescribe the maximum (umax) and minimum (umin) limits of relative displacement. The activation condition for locking is:
Locking is implemented using a penalty method where the relative velocity is forced to become zero for the rest of the time. There is a loss of energy during locking, which can be interpreted as the energy given to the imaginary object responsible for locking the degrees of freedom. Due to the locking phenomenon, stress waves are generated in the flexible components. You can add damping to the problem to suppress unwanted stress waves.
The locking indicators (iumax and iumin) are ODE variables used to ensure that the lock is active for the rest of the simulation. The values of the indicators are switched from zero to a nonzero value when the lock is active, through the following equations:
The force required to lock the relative motion is
where pu1 and pu2 are the penalty factors. pu1 can be interpreted as a spring constant and its default value is a function of an equivalent Young’s modulus and geometric dimensions. pu2 is a damping coefficient and its default value is 10 ms times the value of pu1. these penalty factors are inputs and can you can modify them as required.
The value of the penalty factor (pu2) controls the rate of decay of the relative velocity. A large value gives a high rate of decay but can cause convergence problems.
To enforce the locking condition, a contribution is added to the virtual work:
In the case of a Planar Joint, the relative displacement is locked along a user-defined axis (the locking direction) oriented in a plane perpendicular to the joint axis. The relative displacement in the locking direction, ul, is:
where el is the normalized locking axis given by the user.
Rotational Locking
Rotational locking is available for the Hinge Joint, Cylindrical Joint, Screw Joint, Planar Joint, and Reduced Slot Joint features.
The theory for rotational constraints is similar to what is described above for Translational Locking.
Here, the penalty factor should be considered as a torsional spring.
Inclination Angle Locking
Inclination angle locking is available for the Ball Joint and Slot Joint features.
Here, the lock is applied on the inclination of the destination axis relative to the reference axis (the inclination angle or polar angle). The ODE variable used as locking indicator is computed from
The moment required to apply an inclination angle lock is written as
To apply the lock, virtual work contributions are added for the cosine of the inclination angle:
Axial Rotation Locking
Axial rotation locking is available for the Ball Joint and Slot Joint features.
Here, the lock is applied to the relative rotation of the destination attachment about its own axis. There are two ODE variables used as locking indicator. They are updated using
The moment required to enforce the lock on the axial rotation is:
To apply the lock, a virtual work contribution is added for the cosine of the axial rotation of the destination boundaries: