Kinematic Comparison of Scotch Yoke with Single Slider Crank Mechanism
Introduction
Scotch Yoke and Single Slider Crank Mechanism are the most fundamental of all mechanism. Numerous machines and applications are based on their very foundation. Both the mechanisms perform the same function, i.e., conversion of reciprocating motion into rotary motion and vice versa. Hence, from functional point of view, these different mechanisms do not vary much. But underneath their indifferent functional appearance, lies a stark set of difference or dissimilarities. These dissimilarities encompass a number of aspects such as constructional features, total number of components, working principles, kinematic and dynamic nature, etc. Due to these vast differences, these mechanisms cannot be used alternatively in place of each other. They have their own particular area of applications, based off of their inherent differences mentioned previously. In this blogpost, we explore to highlight the differences between the scotch yoke and single slider crank mechanism from kinematic point of view. The obvious benefit of using a scotch yoke over a slider crank is the simplicity of its design. The scotch yoke has less moving parts and may be easier to maintain. The scotch yoke also provides pure sinusoidal motion of the slider. Although the scotch yoke provides several benefits, they come with several disadvantages. Among these include a fast wear rate at the pin-in-slot joint and increases in friction due to the sliding of the pin in the slot.
Kinematic Analysis of Scotch Yoke Mechanism
The simplicity of the scotch yoke mechanism allows for a very simple kinematic analysis to be done using geometric relationships. Consider an arbitrary point S on the slider a distance Δx from the pin-in-slot joint as shown below in Figure x. The crank length is given by the distance from revolute A to pin B, and is denoted as r. Displacement equation is the analytical expression describing the x-position of point S at any crank angle, θ. It should be noted that the slider is fixed in the y-direction and thus every point on the slider will have a constant y-position and zero velocity and acceleration in the y-direction. By differentiating the displacement equation, expressions for the velocity and acceleration of the slider are obtained.
Kinematic Analysis of Single Slider Crank Mechanism
We are only considering the kinematics of the output end of the mechanisms under study. Hence, for kinematic analysis of the single slider crank mechanism. We will only consider kinematics of piston end. Here, l = length of connecting rod, r = crank radius, n = obliquity ratio = l/r.
For comparison purpose, consider equivalent systems having the following specifications.
Displacement Plots
The nature of displacement plot of scotch yoke is a pure sinusoidal wave, while the displacement curve of the slider crank deviates a little from pure sine wave. It is because the piston displacement for slider crank is a function of crank angle (θ) and connecting rod angle (Φ). Due to this extra Φ term, the ultimate displacement deviates from pure sinusoidal nature.
The slider crank reaches the bottom of its path (the closest distance to the crank) first and also spends less time dwelling at the bottom. The scotch yoke reaches the top of its path (the farthest distance from the crank) before the slider crank and dwells at the top longer. As noted earlier, this shorter time spent at the bottom of the path makes the scotch yoke mechanism undesirable for two-stroke engines since it reduces blowdown time. However, the longer dwell at the top of the cycle can increase efficiency in engines.
Velocity Plots
Velocity is a derivative of displacement. Hence, for scotch yoke mechanism, naturally, it is a pure cosine curve as its displacement is a sinusoidal curve. While, for slider crank mechanism, the small deviations of displacement from previous displacement plot are magnified here, due to which, the nature of velocity plot is largely different as compared to scotch yoke’s.
Acceleration Plots
Acceleration plot of scotch yoke mechanism is a pure sinusoidal curve as it is a derivative of velocity, which in turn is a cosine curve. The deviations in the velocity plots are further magnified here. Comparing the both plots simultaneously, it is observed at negative acceleration (retardation) in case of slider crank mechanism is more than compared to scotch yoke. While on the other hand, the positive acceleration of scotch yoke is more than slider crank. Also, the slider crank curve shows a period of dwell as it reaches the inner dead center.
Jerk & Snap Plots
It is evident from the jerk and snap plots that the motion of slider crank mechanism is nowhere near as smooth as that of the scotch yoke mechanism. This nature also highlights the undesirable kinematic effects that could arise in the machine from even the minutest of manufacturing errors in case of slider crank mechanism.
Conclusion
Scotch yoke mechanism and slider crank mechanism are discussed from their similarities and dissimilarities point of view, as both the mechanisms are used for conversion of rotary motion to reciprocating motion and vice versa.
The dissimilarities are highlighted from their kinematics point of view.
It is clear from the kinematic plots that even though the distance travelled by both the mechanisms is the same, the path taken by each one is different.
The deviations in the path taken by each mechanism in displacement plots, get magnified in the further derivatives such as acceleration, jerk and snap.
Since the displacement of scotch yoke mechanism is a sinusoidal function, all of its derivatives are smooth sinusoidal or cosine curves. This is very desirable from kinematics point of consideration.
On the other hand, the derivatives of slider crank mechanism get disruptive and noisy as the derivative moves higher up the order.
