Introduction and Background

One application in which mechanisms can be useful is with timing the opening and closing of valves. For example, in car engines, the fuel valves must open and close at the proper moments during the ignition cycle in order for the engine to run smoothly. Since this opening and closing is powered by the continuous motion of the engine itself, the trick then becomes to design a mechanism where, given the continuous rotational motion of the input, the output will remain stationary (i.e. closed) for a portion of the range of motion of the input. One solution is to use a camshaft; another more complex solution using a 5-bar mechanism is presented here.

Design Process

Initial design of the mechanism was performed using the PMKS application which allowed me to quickly insert links and joints of different types and view the resulting kinematics. I eventually settled on a 5-bar linkage as shown below, after I observed that when the left end of Link 4 is close to the origin, Link 3’s center of rotation will nearly coincide with that of Link 2, resulting in little motion in the rest of the system. Assuming that Link 3 is slightly longer than Link 2, as long as Link 2 is pointed to the right, there should be little to no movement in links 4 and 5.

The mechanical design of the mechanism was driven primarily by the availability of materials. Birch plywood was selected as the primary building material since arbitrary 2D links can be easily cut using a laser cutter. Screws seemed to be an ideal choice for the joints since they can fasten linkages together while also allowing for relative rotation.

Kinematic Analysis and Synthesis

Below are the equations used to find the position, velocity, and acceleration of all links. There are five links of lengths g, a, b, d, and f respectively:

Link 1 is grounded and connects to joints O and D (33.0 cm).
Link 2 connects to ground through the rotational joint at O (5.0 cm).
Link 3 connects to Link 2 through rotational joint A (6.0 cm).
Link 4 connects to Link 3 through rotational joint B and can also slide and rotate about O (25.3 cm).
Link 5 connects to Link 4 through rotational joint C and to ground through rotational joint D (21.5 cm).

Link 2 is the input link and is assumed to rotate about O at a constant 2 rad/s for purposes of analysis. The positions of links 3, 4, and 5, as well as the distance c between B and O are unknown. The equations below show how the unknown positions as well as the velocities and accelerations can be calculated. Eqs. (1) and (2) are used to numerically find c and theta_4 for a given theta_2. Then Eqs. (3) and (4) can be used to find theta_3 and theta_5. The system of linear equations (5) is then solved to find the unknown velocities, and (6) is solved to find the unknown accelerations.

The angular position of output Link 5 is plotted in the figure. As can be seen, the angle remains within a 5 degree range for 168 degrees of the rotation of the input compared to the 24 degree range moved through during the rest of the input rotation.

Manufacturing and Assembly

As described in the Design Process, the main materials used were laser cut plywood and M3 screws. The ground link is a larger rectangular base plate to which everything else is attached. A screw is used at point D to attach to Link 5 and another at point O acts as both a rotational and sliding joint for Link 4. Two more screws attach Link 3 to Link 4 and Link 2. The links are stacked so that Link 4 is nearest to the base, Links 3 and 5 are in the next plane up and then Link 2 is in the plane above that. In this way, all the links can move without collision. However, since Link 3 is between Links 4 and 2 and must move freely through point O, a second grounded support is nailed to the main base with a hole aligned with point O. A wooden dowel fits through this hole that connects rigidly to Link 2 and acts as a crank for manual operation of this mechanism.

Software

The PMKS application was used for early design of the mechanism. An early draft of the mechanism is shown below. The later kinematic analysis and visualizations were completed using Matlab. The files used are attached below.

Mechanism Visualization

Final Prototype

The final prototype can be seen in motion below. Notice that for approximately half of the rotation of the input, the rightmost link does not significantly move, while for the rest of the rotation, it rocks back and forth.

Final Video

Conclusions and Future Work

The finished mechanism successfully accomplishes the motion that it was designed to do. When given a constant speed input, the output rocks back and forth for part of the rotation in order to open an attached valve, but then remains relatively stationary to keep the valve closed. There are some obvious improvements that could be made to the design. The fasteners were kept somewhat loose to reduce the friction in the joints, but a better solution would be to include bearings in the joints so that the joints remain low friction without sacrificing precision in alignment. A further improvement would include attaching a motor to the input shaft that would be speed controlled for more consistent behavior.

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