ENGI 1040 - Case Study PDF

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Slider Crank Shaking Forces ENGI 1040: Thinking Like an Engineer August 7, 2020 Darlene Spracklin-Reid Donald Ubah |201924644 Ganna Ahmed | 201914785

Introduction As a vital part of the “Thinking Like an Engineer” course, this case study will follow the basis of analyzing the motion and shaking forces of a slider-crank. Thus the prompt includes analyzing and concluding the velocity, acceleration, and the shaking forces of a slider-crank relative to a piston in a situation similar to that of a car engine. To fulfil the latter, the workings of this case study will be met through employing a Sketch, Observations/Objectives, List, Variables, Equations, Manipulate (SOLVEM) approach in addition to using Microsoft Excel as an engineering tool. Analyzing the motion and the shaking forces of a slider-crank is a typical situation in which the study of mechanics and dynamics can be fulfilled. Being a common phenomenon, it is necessary to analyze the shaking forces of a slider-crank before implementing design ideas that utilize the tool. As for background information, the shaking force developed through a slider-crank is a result of a linked mass (most times a piston) moving up and down as it undergoes acceleration and deceleration. As proven through Newton’s second law of motion, the slider-crank motion and velocity changes due to the pulling and pushing effect the piston has on it. Thereby, as the piston pushes on the slider-crank, the slider-crank inverts a similar force on the piston. The acceleration of the piston will thus be used to determine the shaking forces created by it, this is given that the connecting rod’s mass is negligible and the shaking forces are proportional to the piston’s acceleration. Furthermore, as for analyzing the slider crank as to derive an equation for the position of the slider, multiple variables must be established and the geometry of the slider crank must be evaluated. With the angular speed depicting the rod’s rotary speed, its units are given in radians per second. On the other hand, with the angle (theta) between the rod’s length and the rod’s horizontal component being in radians, it can then be found as the product of angular speed and time. The angle Theta is then applied to trigonometric equations to find the “x” and “y” components of the rod length “r”. As of this, the parameters of the slider crank mechanisms can be further evaluated to derive an equation for the position of the slider, which will help steer the basis of this report and yield results for the shaking forces present.

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SOLVEM As to move further with the process of analyzing the slider’s shaking forces, a “SOLVEM” approach is implemented to start the problem-solving stage. The “SOLVEM” method is a systematic approach to problem-solving through the steps of providing a sketch, stating observations and/or objectives, listing variables/constants and equations, and finally manipulating the result. For this particular problem in question, which can be identified as a “Parametric” problem, an exact answer for the position of the piston will be evaluated to determine velocity, acceleration, and the shaking forces. Thus, with a methodical approach being applied, one can guarantee appropriate and accurate results. Firstly, the crank alongside the rods, links, and slider is sketched to mimic the scenario exactly, as shown in figure 2. The sketch shows the crank being supported by fixed support at D, a connecting rod attached to the crank at B, and the slider that moves in the horizontal direction at A. Thereby, the sketch provided exhibits with a simple single slider crank. Secondly, as to proceed with “SOLVEM”, the following objective is established: an equation for the slider’s position relative to the crank must be established to further determine the velocity, acceleration, and the shaking forces. Accordingly, one can observe that length of both the crank (labelled “r”) and the connecting rod (labelled “L”) create a right triangle with their vertical and horizontal components. In addition to both lengths sharing their vertical component. Furthermore, the distance between the crank and the slider geometrically represents the slider’s position. Thirdly, a list of variables is established, as shown in figure 2. It can then be deduced that the slider’s position can be found using the workings shown in figure 3, where the entire horizontal distance, “s”, equates to the sum of “x” and “x’”. This marks the fourth step of SOLVEM. Lastly, the general equation for the position of the slider is finalized to be utilized with different values for “r” and “L”.

Figure 2: simplified sketch of slider crank scenario with list of variables

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Figure 3: workings for the final derived slider position equation for input values for variables “r” and “L”, alongside sketch of single slider crank

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Spreadsheets In regards to the development of the spreadsheet, Microsoft Excel was used as an engineering tool to gather and present data in terms of the output parameters in question. This was done by inserting the final slider position equation which was derived through SOLVEM, as shown in figure 3, in the cell F5. Whilst the slider position equation had to be translated to excel language, the equation stored in cell f5 referenced cell B5, which held the value for “r”, cell E5, which held the values for the angles theta, and cell F3, which held the value for “L”. This allowed one to then apply the equation in cell F5 for all the values listed for time and theta by selecting the cell and dragging it down the column. By doing so, the initial referenced cells were adjusted in regards to all the values for theta in column E, keeping the values for “r” and “L” constant. This same approach was done for the four other values for “L” in terms of displacement. Furthermore, for velocity and acceleration, numerical differentiation was used to evaluate each value. This was done through directing the cells listed under velocity to take the difference between the next greater position value and the corresponding position value, as to divide it by the change in time respectively. Similar was done for each acceleration value, as it equated to the change in velocity divided by the corresponding change in time. Lastly, to evaluate the corresponding shaking forces, a general equation of force equates to the acceleration multiplied by a piston mass of 0.5 kilograms was applied to all cells for different values of “L”.

Figure 4: Excel spreadsheet of position, velocity, acceleration, and force values evaluated for different values for “r” using the general equation for the position of slider-crank as derived through SOLVEM

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Analysis of Position and Velocity Through employing the spreadsheet in figure 4, position vs. time was graphed for each rod length value, “L”, as shown in figure 5. As for the position of the slider when the rod length is 0.8 meters, the maximum position reached is 1 meter whilst the 0.22 meter rod reaches a maximum position of 0.41 meters. This is coherent with the fact that the slider reaches a further horizontal distance in one rotation with a longer rod than when it does with a shorter rod. Thus, the graph implies a linear relationship between the rod length and position, a change in the length of the rod will cause an equivalent change in the position reached by the slider. Furthermore, one can see that each plot oscillates as a result of the changes in position, depicting a parabolic graph demonstrating simple harmonic motion. This can be further justified through visualizing the circular movement of the rod implemented to push the piston, as shown in figure 6. Where the rod, “B”, has a rotational motion of radius “AB”. When “B” is at position “h”, the piston reaches maximum displacement at “H”, and minimum displacement when it is at “G” due to “B” being at “g”. Thus, the graph presented agrees with the circular motion of the slider crank, with a period of approximately 0.17 seconds for each rotation.

Figure 5: Position of Slider Crank for 4 Rotations. Position of the slider changes with a periodic effect, as rod length decreases, the maximum position (amplitude) of the slider decreases and the periods for each rotation stays unchanged. Local minimas plateau more drastically for shorter rod lengths.

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Figure 6: Slider Crank Mechanism. Figure shows displacement of piston at G and H in relation with B’s position along path of rotation.

In graphing the velocity vs. time graph, the velocity of the slider at different rod lengths is graphed against time. As velocity is the derivative of position, the graph shown in figure 7 is shown to be a graph of the different slopes portrayed throughout the position vs. time graph. This is further supported by the mathematical translation of the local maximas and minimas of the position vs. time graph to the velocity vs. time graph. Whereas a local maxima/minima for the graph in figure 5 translates to a saddle point for the graph in figure 7, and vice versa. This feature can be seen more clearly for a rod length of 0.22 meters, where the plot transcends the y-axis at zero for a longer period. This matches the position plot for the 0.22-meter rod where the local minimas show no change in position for a longer period in comparison to the other rod lengths, exhibiting a slope of zero. Furthermore, as a result of the translations between the position and velocity graph, the velocity graph is shown to be a negative sine curve. This is proven through the mathematical relationship between position and velocity for a particle moving in a circular motion, as shown in figure 8 [1]. In regards to maximum velocity reached by the different rods, it can be deduced that a longer rod length results in increased velocity. Whereas when comparing the 0.3-meter rod and the 0.6-meter rod, velocity is increased exponentially for the longer rod length.

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Figure 7: Velocity of Slider for 4 Crank Rotations. Velocity of the slider is shown to start off at the negative y-axis, and oscillates similar to the position vs. time graph, thus it portrays the slopes for the graph in figure 5. As rod length increases, velocity increases with time. The plot for each rod length intersects with the x-axis to show a slope of zero when the slider crank is in between rotations.

Figure 8: Mathematical Relationship Between Position and Velocity of A Particle/Object Moving In Circular Motion.

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Analysis of Acceleration In graphing the slider acceleration vs. time graph, the acceleration of each rod length is graphed against time as shown in figure 9. Similar to the relationship between position and velocity, the derivative of velocity equates to the acceleration of the slider. Thus, figure 9 shows periodic motion for the acceleration of the slider at different rod lengths, as a result of the changes in velocity during its rotations. Hence, the graph obtained is a negative cosine curve due to the rod’s circular motion. As rod length increases, a higher acceleration is reached by the slider, in addition to it changing drastically during one rotation for shorter rod lengths. This is due to how the velocity changes in between rotations, reaching zero when the slider crank does not move for a couple of seconds, and then begins to move again. This allows one to visualise how acceleration changes throughout the rotation period of the rod, where at maximum speed, acceleration is negative, and at minimum speed, acceleration is positive.

Figure 9: Acceleration of Slider for 4 Crank Rotations. Acceleration oscillates with change in velocity during the 4 rotations. Whilst longer rod lengths have higher acceleration, as rod length decreases the change in acceleration during each rotation is more drastic. The graph is also the cosine curve for the velocity graph in figure 8.

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As to determine the shaking forces of the crank slider, acceleration was used to graph the shaking forces vs. time as shown in figure 8. As of Newton’s second law, where force is proportional to the acceleration, the graph representing the shaking forces is very similar to the graph representing the acceleration of the slider. Whilst the mass of the slider is 0.5 kilograms, the shaking forces due to acceleration were cut by half, however, the same trend shown for the acceleration graph applies for the force graph. This means that increasing the rod length has a direct effect on the shaking forces experienced in the slider-crank, as they increase when rod length is increased as well. From this, one can deduce that due to the proportional relationship between force and acceleration, it is difficult to reach a scenario of insignificant shaking forces and maintain high acceleration. In general, it is best to implement an engine design that has insignificant shaking forces to ensure stability and convenience. Thus, a recommended design that maintains a healthy r/L ratio but still manages to maintain insignificant shaking forces would be one that minimizes the masses of the slider-crank mechanisms. This design can thus be implemented by increasing the deck height of the engine to allocate space for the long rods [2]. Where the rods will be attached to the piston higher up in the piston body. By using long rods, a shorter and more lightweight piston can be used, therefore minimizing the piston mass and allocating for the rods’ increased mass. With this, one can guarantee that the slider crank will produce an insignificant magnitude of shaking force.

Figure 8: Shaking Forces for 4 Crank Rotations. Shaking force is proportional to the acceleration of the slider. As rod length increases, the shaking force increases. For shorter rod lengths there is a fluctuation in

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the forces present during each rotation as the slider crank halts for a period of time, thus eliminating shaking forces when it does so.

Conclusion For the purpose of analyzing the shaking forces of a slider-crank, a “SOLVEM” approach alongside the tool Excel was utilized to meet appropriate results. Therefore, the report contains thorough workings of the derivation of an equation for the position of the slider, the development of excel spreadsheets with data for 4 output parameters, and the creation and analysis of the graph for each parameter. Thereby, this report presents conclusions for the shaking force present due to a slider-crank at five different rod lengths. Whilst a range of rod lengths was evaluated, the report concludes that a longer rod length results in higher velocity, acceleration, and thus stronger shaking forces. In meeting the latter conclusion, the report makes clear the mathematical relationships between position, velocity, and acceleration, without failing to apply them to the physical mechanics of the slider-crank in question. As such, with proving the connecting rod’s rotational circular motion, the graphs for the position, velocity, and acceleration were analyzed carefully as to recommend a design with minimum shaking forces and a maximum r/L ratio. With the conclusion that an increase in rod length increases acceleration, and therefore in shaking forces as well, the design recommended aims at minimizing the masses of the slider-crank mechanism. With doing so, a high acceleration value can be achieved, which is then minimized by the low piston mass, maintaining the use of longer rods and the presence of insignificant shaking forces.

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References 1. Libretexts. “4.5: Uniform Circular Motion.” Physics LibreTexts, Libretexts, 13 July 2020, phys.libretexts.org/ 2. Builder, ByEngine, et al. “How Rod Lengths and Ratios Affect Performance.” Engine Builder Magazine, 13 Feb. 2020, www.enginebuildermag.com/2016/08/understanding-rod-ratios/

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