Harmonic Motion Drives

Applications requiring the linear transfer of a load under controlled acceleration and deceleration are quite common. Within limits, this type of motion can be achieved thru a harmonic motion drive. An actuator driven, scotch yoke arrangement as shown in Figure 1 imparts this type motion. The scotch yoke converts the constant speed rotating motion to a sinusoidal motion producing maximum linear force for acceleration, maximum linear speed thru the middle of the actuator stroke, and maximum decelerating forces to slow and stop the load.



The following equations assume a constant actuator rotational velocity. This is sometimes difficult to achieve, particularly for short cycle times that result in a large load velocity. The inertia of the load will tend to drive the actuator during the deceleration phase. These forces may cause cavitation or physical damage to the actuator. Therefore, under certain conditions the actuator may require external assistance in decelerating the load.

A flow control in the discharge side of the actuator provides this assistance, assuring a positive-pressure throughout the cycle. The added resisting torque resulting from the discharge metering must be added to the driving torque requirement.

Equations of Motion
The equation of motion for a Scotch Yoke mechanism can be developed as follows:



The velocity of link 2, and thus load W, may be found by differentiating the movement with respect to time.



The acceleration of load W is found by differentiating its velocity with respect to time:





required to transfer the load a distance of 2r. Therefore,



This relation applies for any load W.

FIGURE 1. TYPICAL HARMONIC MOTION DRIVE ARRANGEMENT



Required Torque
Consider an actuator powered Scotch Yoke mechanism moving a load as shown in Figure 1. Assume for simplicity that the system is frictionless. The forces acting on the actuator crank (link 1) are also shown in Figure 1.



Therefore, the required actuator torque at any time during the cycle is:



The maximum torque requirement may be found by differentiating equation (10) with respect to time and setting the result equal to 0 as follows:



Therefore, the maximum actuator torque requirement is:



This expression may be used to determine the maximum actuator torque requirement for a frictionless system by knowing the load weight, crank arm length and the time required for 180º crank rotation.
In systems where friction must be considered, the required actuator torque will obviously be greater than that given by equation 13. The derivation of torque equations which consider the effects of friction becomes somewhat mathematically involved and will therefore not be repeated here.

However, by considering only friction of the moving load and neglecting the crank friction forces along the vertical axis (vertical friction forces have little effect on torque) it can be shown that the maximum actuator torque is approximately: