A slider-crank linkage (also commonly referred to as a crank-slider linkage) is a four-link mechanism with three revolute joints and one prismatic (sliding) joint. The naming convention of slider-crank and crank-slider is generally used to refer to the functional [input]-[output] of the linkage. In a crank-slider, the rotation of the crank drives the linear movement of the slider, and in a slider-crank, the expansion of gases against a sliding piston in a cylinder can drive the rotation of the crank. There are two types of slider-cranks: in-line and offset.
In-line: An in-line slider-crank has its slider positioned so the line of travel of the hinged joint of the slider passes through the base joint of the crank. This creates a symmetric slider movement back and forth as the crank rotates. Offset: If the line of travel of the hinged joint of the slider does not pass through the base pivot of the crank, the slider movement is not symmetric. It moves faster in one direction than the other. This is called a quick-return mechanism. There are also two methods to design each type: graphical and analytical.
In-line kinematics
The displacement of the end of the connecting rod is approximately proportional to the cosine of the angle of rotation of the crank, when it is measured from top dead center (TDC). So the reciprocating motion created by a steadily rotating crank and connecting rod is approximately simple harmonic motion:
x = r cos α + l {\displaystyle x=r\cos \alpha +l}
where x is the distance of the end of the connecting rod from the crank axle, l is the length of the connecting rod, r is the length of the crank, and α is the angle of the crank measured from top dead center (TDC). Technically, the reciprocating motion of the connecting rod departs from sinusoidal motion due to the changing angle of the connecting rod during the cycle, the correct motion, given by the Piston motion equations is:
x = r cos α + l 2 − r 2 sin 2 α {\displaystyle x=r\cos \alpha +{\sqrt {l^{2}-r^{2}\sin ^{2}\alpha }}}
As long as the connecting rod is much longer than the crank l >> r {\displaystyle l>>r} the difference is negligible. This difference becomes significant in high-speed engines, which may need balance shafts to reduce the vibration due to this "secondary imbalance". The mechanical advantage of a crank, the ratio between the force on the connecting rod and the torque on the shaft, varies throughout the crank's cycle. The relationship between the two is approximately:
τ = F r sin ( α + β ) {\displaystyle \tau =Fr\sin(\alpha +\beta )\,}
where τ {\displaystyle \tau \,} is the torque and F is the force on the connecting rod. But in reality, the torque is maximum at crank angle of less than α = 90° from TDC for a given force on the piston. One way to calculate this angle is to find out when the Connecting rod smallend (piston) speed becomes the fastest in downward direction given a steady crank rotational velocity. Piston speed x' is expressed as:
x ′ = ( − r sin α − r 2 sin α cos α l 2 − r 2 sin 2 α ) d α d t {\displaystyle x'=\left(-r\sin \alpha -{\frac {r^{2}\sin \alpha \cos \alpha }{\sqrt {l^{2}-r^{2}\sin ^{2}\alpha }}}\right){\frac {d\alpha }{dt}}}
For example, for rod length 6" and crank radius 2", numerically solving the above equation finds the velocity minima (maximum downward speed) to be at crank angle of 73.17530° after TDC. Then, using the triangle sine law, it is found that the crank to connecting rod angle is 88.21832° and the connecting rod angle is 18.60639° from vertical (see Piston motion equations#Example). When the crank is driven by the connecting rod, a problem arises when the crank is at top dead centre (0°) or bottom dead centre (180°). At these points in the crank's cycle, a force on the connecting rod causes no torque on the crank. Therefore, if the crank is stationary and happens to be at one of these two points, it cannot be started moving by the connecting rod. For this reason, in steam locomotives, whose wheels are driven by cranks, the connecting rods are attached to the wheels at points separated by some angle, so that regardless of the position of the wheels when the engine starts, at least one connecting rod will be able to exert torque to start the train.
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