DECEMBER 27th, 2018
Consider the following simplified diagram of a connecting rod-crank mechanism:
I have two constraints, the crankpin (the crankshaft) and the prismatic coupling (the cylinder). That system is subjected to two external forces, the gas pressure on the piston FG and the torque M. The piston makes a translational motion, the crank makes a rotational motion around the crankpin, and the connecting rod makes a rototranslational motion. Since the motion of the connecting rod is complicated, it is convenient to use "substitution masses". I replace the mass of the connecting rod with more masses arranged appropriately, I can do this provided that the new masses have:
Then each rigid body can be represented by 4 equivalent masses. The connecting rod needs only 3, one at the barycenter, one rotating mass (integral to the crank) and one translating mass (integral to the piston). The barycentric mass still makes a complicated trajectory, so we represent only two masses at the ends and a substitution inertia, that is an additional inertia that serves to make the inertia of the two masses coincide with the initial inertia of the connecting rod. Wanting to further simplify the problem, I also eliminate this replacement inertia keeping only the two masses, the error that is committed is small. The crank is now composed of its mass plus the mass of the connecting rod. By placing a suitable mass at a suitable distance, opposite to the crank + connecting rod masses, we balance the system. Consider now the external forces: The force that pushes down the piston, pushes up the engine block, so it is balanced. If I consider the engine to be rotating at a constant speed, the torque is constant. The translating mass of the piston plus the translating mass of the connecting rod are the only source of vibration. I represent the system this way:
This is a simple oscillator excited by harmonic forcing. I calculate the distance between the bench pivot and the piston:
yR = r*cos(θ) + (l2-r2*sin(θ))0.5
Which I can rewrite as:
yR = r*cos(ωt) + (l2-r2*sin(ωt))0.5
This function can be developed in Fourier series. The first approximation represents the first harmonic, i.e., first-order alternating forces:
yR = r*cos(ωt) + k
yR'' = -ω2r*cos(ωt)
A boxer engine, or an inline 4-cylinder, has these forces perfectly balanced. In the case of a single-cylinder engine, it can be balanced by adding an equal and opposite force. This is an alternating force, it can be created by two unbalanced counter-rotating masses. One is created by adding mass to the crank, then adding a counter-rotating shaft having an unbalanced mass equal to that added to the crank. The second Fourier series approximation represents second-order alternating forces. Since this is an unbalance at twice the frequency of crankshaft rotation, it can be balanced with shafts rotating at twice the speed. In single-cylinder engines it is never done, but it can be done in 4-cylinder engines.
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