5.2 Time Switching Constraints

A TimeSwitch constraint is a compound constraint that contains several normal Mech constraints and determines which one is active by the value of time T. This function can be used to model any mechanism that switches from one mode of operation to another with the progression of time.
A TimeSwitch constraint is basically a special case of a StageSwitch constraint that has switching capability explicitly dependent on time. All of the functionality of TimeSwitch can be duplicated with StageSwitch, except that the TimeSwitch constraint can be used with another special function, FindSwitchTime, that is used to seek the point in time at which two adjacent constraints should switch from one to the next.

5.2.1 TimeSwitch

The time-dependent, stage-switching function.

Each of the constraints supplied to the TimeSwitch must be a Mech constraint object, or a list of constraint objects. Each constraint or list of constraints must constrain the same number of degrees of freedom. Note that the constraint numbers of the constraints that are inside of the TimeSwitch are irrelevant. The constraint number of the entire TimeSwitch is the only one used to reference the entire constraint.

Examples

This loads the Modeler2D package.

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The following example shows how a TimeSwitch constraint (constraint 1) might be used to model a pin on a slider that slides along a horizontal track on the ground until the value of time reaches 2.0, at which time the pin begins to slide up an inclined track.
This constraint could replace the StageSwitch constraint used in the slider-ramp model in Section 5.1.

Here is a valid TimeSwitch constraint.

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The following example shows how a TimeSwitch constraint that constrains two degrees of freedom might be implemented. The first constraint in the TimeSwitch constraint forces the origin of the slider to be coincident with the origin of the ground. If the value of time rises above 4.0, the second constraint pair becomes active, which forces the origin of the slider to lie upon an inclined line, while the Y coordinate of the origin of the slider is a function of time.

Here is a valid TimeSwitch constraint.

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5.2.2 Detent Mechanism Example

To demonstrate the use of the Modeler2D TimeSwitch function a 2D model of a detent mechanism is developed. The model consists of two moving bodies, a slider and a detent plunger.
The detent mechanism goes through seven distinct stages of operation as the detenter passes across the slider, down into the detent notch, and back out. These stages are progressively activated via the TimeSwitch function as follows.

In stage 1 the detenter travels across the right top surface of the slider.
In stage 2 the detenter travels around the corner that leads into the detent notch.
In stage 3 the detenter travels down the inclined right side of the detent notch.
In stage 4 the detenter travels around the round bottom of the detent notch.
In stage 5 the detenter travels up the inclined left side of the detent notch.
In stage 6 the detenter travels around the corner that leads out of the detent notch.
In stage 7 the detenter travels across the left top surface of the slider.

Here is the 2D detent model graphic.

Bodies

Three body objects are defined for the detent mechanism model.

Names for each body number in the model are defined.

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Here are all three bodies in the detent mechanism model.

The body objects are incorporated into the model.

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Constraints

Three regular constraints and one TimeSwitch constraint are used to model the detent mechanism.

The TimeSwitch constrains the one remaining degree of freedom:
Four of the seven stages of operation (1, 3, 5, and 7) are modeled by PointOnLine1 constraints that place the center of the detenter tip a specified distance (detenter tip radius) away from various lines on the slider.
The second and sixth stages, in which the detenter tip moves around sharp corners on the slider, are modeled with RelativeDistance1 constraints that place the center of the detenter tip a specified distance (detenter tip radius) away from the sharp corners on the slider.
The fourth stage, in which the detenter tip is traveling around the arc at the bottom of the detent notch, is also modeled with a RelativeDistance1 constraint that places the center of the detenter tip a specified distance (detent notch radius minus detenter tip radius) away from the center point of the arc.

The exact values of the switch times given as arguments to the TimeSwitch constraint are not known. The symbols given in place of switch times (tm2, tm3, etc.) are set to approximate values before running the model. The exact switch times will be determined later with FindSwitchTime.

Here are the three regular constraints.

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Here is the seven-stage TimeSwitch constraint.

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The constraints are incorporated into the current model.

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Runtime

Before the detent model can be run, the switch times used in the TimeSwitch constraint (tm2, tm3, etc.) must be given numeric values.

Here are guesses for switch times.

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Now the model is run at T = - 1.3.

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Here is the detent mechanism at T = - 1.3.

Now the model is run twice more at T = - 0.5 and T = - 0.1.

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Here is the detent mechanism at the two new solution points.

Note again that the current values of the switch times (tm2, tm3, etc.) are merely guesses. If the model is run at values very close to those times, incorrect results will be generated whenever the model uses the wrong constraint. The exact values of the switch times are found in Section 5.2.3 with the FindSwitchTime function.

5.2.3 Finding Switch Times

The Mech FindSwitchTime function is used to seek the time at which two adjacent constraints in a TimeSwitch are simultaneously valid, that is, to numerically find the correct switch time.

A function for seeking stage transition times.

FindSwitchTime works by adding one degree of freedom to the model (T becomes a dependent variable) and adding one extra constraint by using the pair of adjacent constraints simultaneously. FindSwitchTime uses the same solution block as SolveMech so it uses the most recent solution returned by SolveMech (LastSolve[]) for its initial guess.
FindSwitchTime accepts two options to deal with special situations that may arise in a pair of adjacent constraints, TangentConstraint and RequiredEquations.

Options for FindSwitchTime.

To understand the need for the RequiredEquations option, imagine a two degree of freedom TimeSwitch. If both sides of the adjacent two degree of freedom constraint pair are used simultaneously, two extra constraints are added to the model, but only one degree of freedom (time) is added. This results in an overconstrained model to which no solution can be found.
The default setting RequiredEquations -> {All, Sum} deals with this problem in a general but unsophisticated manner. All of the constraint equations from the first side of the TimeSwitch pair are used, as well as a single equation formed from the sum of all the equations from the second side, resulting in the required one extra constraint equation. This combination of constraints is valid if all of the equations in the second side of the constraint pair except one are redundant with respect to the equations in the first side. Thus the redundant equations in the sum are satisfied because of the first side of the constraint pair, and the one nonredundant equation is satisfied by the satisfying the sum. However, if there is no redundancy in the pair this will not work, and the solution will fail. If the default setting fails, the user must explicitly specify which equations to use to seek the switch time.
For example, take the following TimeSwitch example that was given in the beginning of this section.

Here is a TimeSwitch example.

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To seek the correct value of the switch time, which is currently set to 3.5, the following FindSwitchTime example could be used with all of the default option settings. Note that the TimeSwitch constraint is constraint number 3.

Here is a command to find the correct value of the switch time.

The default setting for RequiredEquations works fine in this case because the first constraint equation in the second side of the TimeSwitch constraint (PointOnLine1) is redundant with the first side of the TimeSwitch constraint (Revolute2), because PointOnLine1 must be satisfied for Revolute2 to be satisfied.
The following example shows a setting for RequiredEquations that explicitly specifies which equations to use. This setting would cause only the RelativeY1 constraint to be used from the second side of the TimeSwitch constraint.

Here is a setting of the RequiredEquations option.

When the RequiredEquations option is used with a TimeSwitch that contains multiple degree of freedom constraint functions, such as Revolute2 (2D) or Cylindrical4 (3D), it is necessary to know exactly what each of the 2, 3, 4, or 5 equations in the constraint are doing, so that you can know which ones need to be specified with the RequiredEquations option. To find out "which equation does what", see the notes on the particular constraint function in the appendix.

Detent Mechanism Example

In the detent mechanism example in Section 5.2.2, the values given for the switch times that were assigned variable names (tm2 = - 0.8, tm3 = - 0.4, etc.) were merely guesses. FindSwitchTime is used to find the correct values of the switch times.
The first and sixth switch times were set to values of - 1 and 1, respectively, which are exact values evident from the geometry of the detenter and notch.
The value of the second switch time must be found with FindSwitchTime. This is the time at which the detenter stops traveling around the sharp corner on the upper-right side of the detent notch and passes onto the inclined right side of the notch. These two adjacent constraints come into conjunction tangentially because there is no velocity discontinuity as the detenter passes onto the inclined flat side of the detent notch. Thus, FindSwitchTime must actually search for the point of tangency. This requires that the option TangentConstraint -> True be specified. Before FindSwitchTime is invoked, SolveMech is used to move the mechanism close to the correct position to provide better initial guesses.

This finds switch time 2 in constraint 4.

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This sets the value of tm2.

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Here is the detent mechanism at the tangency point associated with tm2.

The third switch point occurs when the detenter passes from the inclined right side of the detent notch onto the round bottom of the notch. This is also a tangent constraint pair, so TangentConstraint -> True is used again.

This finds and sets switch time 3 in constraint 4.

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Here is the detent mechanism at the tangency point associated with tm3.

Because of symmetry, the values of tm4 and tm5 are known to be the negatives of tm3 and tm2, respectively. With all of the switch times set to valid values, the detent mechanism model can be run at any value of time.

Set the last two switch times.

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Undocumented Graphics Generation

The following inputs generate the graphics in this section. These Modeler2D commands are defined in Chapter 3.

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