__Note:__

For the calculation below the Maximum Axial Position Changing RPM Method (see Maximum Axial Position Changing RPM Method) is used.

**Axial Position Changing Force Calc.**

A sample calculation for a “Lever Indexing Mechanism 2” that is used to move a “cone with one torque transmitting member” axially is described below. Let’s say for the cone with one torque transmitting member (also referred to as “cone” or “the cone” in this section), the smallest transmission diameter, Dsmall, is 3 inches and the largest transmission diameter, Dlarge, is 6 inches. In addition, the length of the cone from Dsmall to Dlarge is 6 inches, and the pitch length of a tooth of the torque transmitting member is 0.25 inches. For this setup, the number of teeth at Dsmall are approximately 75, and the number of teeth at Dlarge are approximately 150. As such the difference in the amount of teeth from Dsmall to Dlarge is 75 teeth. Since a difference of 75 teeth is achieved over a length of 6 inches. The length (axial distance), S, that the cone has to be moved for a transmission diameter increase or decrease of 1 tooth is: S = 6 inches / 75 teeth = 0.08 inches.

And let’s say the maximum rotating speed at which axial position changing of the cone is allowed, rpm_max, is 3000 rpm, and axial position changing of the cone has to be performed during a duration of 3/4 of a revolution of the cone. Then the maximum time, t, at which the axial position of the cone can be changed is: t = (3/4) / (3000 rpm) = 0.00025 minutes = 0.015 seconds.

In order to calculate the acceleration, a, needed to move the cone we use: S = 0.5 * a * t^2, from which we get: a = 2 * S / t^2. Using our earlier results we get: a = 2 * 0.08 / 0.015^2 = 711 inches/seconds^2 = 18 meters/seconds^2.

Assuming that the mass of the cone and its mechanisms to be moved axially, m, is 5 kg; then the Force, F, needed to move the cone is: F = m * a = 5 * 18 = 90 Newtons = 20 lbs.

For a “Lever Indexing Mechanism 2”, the force to move a cone is provided by a pre-tensioned tension spring, and the force of the pre-tensioned tension spring decreases as its actuator lever rotates towards its neutral position. The 20 lbs force required, as calculated above, assumes that a constant 20 lbs force is applied on the cone. In order to compensate for this, the pre-tensioned tension spring needs to provide an initial force, F_initial, of: 2 * F = 2 * 20 lbs =

**40 lbs**. Here if the final force provided by the pre-tensioned tension spring is 0, then the average acceleration provided by the pre-tensioned tension spring is equal to the average acceleration provided by a constant 20 lbs force. The calculation described here is only a ballpark estimation; the actual initial force needed can be easily obtained and refined through experimentation.

The impact velocity of the cone as it hits its stopping/final position is: v_impact = a * t = 18 * 0.015 = 0.27 meters/second. The dropping height, h_dropping, for the impact velocity can be determined from the following: t = v_impact / gravity = 0.27 / 9.81 = 0.028 seconds, and h_dropping = 0.5 * gravity * t^2 = 0.5 * 9.81 * 0.028^2 = 0.0038 meters =

**0.38 cm**.

Using the same equations above except for different rpm values, we get the following values for rpm_max = 4000 rpm: F_initial = 72 lbs, h_dropping = 0.67 cm; and the following values for rpm_max = 6000 rpm: F_initial = 162 lbs, h_dropping = 1.5 cm.

The values calculated in this section are only ballpark estimates from which more accurate values can be obtained through experimentation. For reliability purposes, it is also recommended that a factor of safety, such as factor of safety of 1.5 or 2 for example, is used for the values for F_initial.

In order to damp the impact that occurs when a moved cone hits its stopping position, a Recovery System described earlier can be used to damp/brake the cone before it hits its stopping position. For a Recovery System, the braking force is provided by the pressurized gas or liquid in Recovery Accumulator 46. In order to determine the right amount the pressure in Recovery Accumulator 46, trial and error experimentation can be used.

For example, during trial and error experimentation, the pressure in Recovery Accumulator 46 can be increased/decreased in increments (such as 1 lbs, 2 lbs, 5 lbs, etc., increments for example), while adjusting the force for F_initial and “the duration when damping is active” until an acceptable “axial position changing duration” and “stopping impact force” are obtained.

“The duration when damping is active” depends on when the pressure in Recovery Accumulator 46 is started to be used to damp/brake the cone moved during axial position changing of the cone. If the tension in the transmission belt of the cone is reduced so that the resisting force for moving a cone is negligible (refer to U.S. Patent Applications #14/182,306 and #14/186,853); then it is preferred that a timer is used to trigger when the pressure in Recovery Accumulator 46 is started to be used to damp/brake the cone moved during axial position changing of the cone. The timer can be a time delay that activates damping/braking after a pre-determined duration from when axial position changing of the cone was started.