Kinematic Analysis of Mechanism

The following kinematic analysis includes velocity and acceleration analysis of all moving links, as well as points A, B, and P1. It should be noted that all velocities and accelerations are shown as functions of the input crank angle. Please refer to "Overview of Mechanism" section to get more information about link and point names.

Angular Velocities:

Figure 3-Angular velocity of link 3 (Left)

Figure 4-Angular velocity of link 4 (Right)

As shown in Figure 3, the angular velocity of link 3 fluctuates between negative 3.2 (clockwise) and positive 2.7 (counter-clockwise). It hits its maximum around 135 degrees, and its minimum around 360 degrees. Similarly, the angular velocity of link 4 oscillates between negative 4.5 and positive 4. It’s minimum occurs at 250 degrees, and its minimum occurs around 30 degrees (Figure 2).

Angular Accelerations:

Figure 5-Angular acceleration link 3 (Left)

Figure 6-Angular acceleration link 4 (Right)

As shown in Figure 5, the angular acceleration of link 3 varies between -22 radians/second^2 at a crank angle of 320, and +44 radians/second^2, at a crank angle of 48 degrees. Figure 6 displays the angular acceleration of link 4, as a function if the crank angle. This ranges from between -62 radians/second^2 at a crank angle of 330 degrees, and 32 radians/second^2 at a crank angle of 160 degrees.

Point A Kinematics:

Figure 7-Velocity magnitude of Point A (Left)

Figure 8-Velocity direction of Point A (Right)

Link 2 is rotating at a constant angular velocity (ω2=10 rad/s). Since the magnitude of velocity at point A must be length of link 2*ω2, this results in a constant velocity at point A, as shown in Figure 7. However, the direction of velocity is continuously changing, as the link rotates about point O2, as shown in Figure 8.

Figure 9-Acceleration magnitude of Point A (Left)

Figure 10-Acceleration direction of Point A (Right)

Similar to the velocity, the acceleration at point A must be constant because there is no angular acceleration. In this case, the acceleration does not affect the magnitude of velocity at point A, but it accounts for the change in velocity direction (shown in Figures 9 and 10).

Point B Kinematics:

Figure 11-Velocity magnitude of Point B (Left)

Figure 12-Velocity direction of Point B (Right)

As shown in Figure 11, the velocity of point B reaches zero at two points: once when it reaches its highest point, once when it reaches its lowest point. The motion of point B is not entirely linear, as shown in Figure 12.

Figure 13- Acceleration magnitude of Point B (Left)

Figure 14- Acceleration direction of Point B (Right)

P1 Kinematics:

Figure 15-Velocity magnitude of Point P1 (Left)

Figure 16-Velocity direction of Point P1 (Right)

The velocity magnitude and direction of P1 are shown in Figures 15 and 16, respectively. As shown in Figure 15, the velocity of P1 dips down to zero at two points: the lowest and highest point of the pump-jack. The velocity is zero at the points because P1 is changing direction from up to down, or vice-versa. On the way up, P1 is moving at an angle of 100 degrees from the+X-axis, whereas on the way down, it is moving at an angle of 280 degrees. The mechanism is intended to be a 1-degree-of-freedom mechanism which moves linearly, so these results align with the expected performance of this mechanism. 

Figure 17- Acceleration magnitude of Point P1 (Left)

Figure 18- Acceleration direction of Point P1 (Right)