The base of dynamic mechanism operation of engine is slider crank mechanism, which consists of crankshaft, connecting rod and piston. Slider crank mechanism, arrangement of mechanical parts designed to convert straight-line motion to rotary motion, as in a reciprocating piston engine, or to convert rotary motion to straight-line motion, as in a reciprocating piston. The reciprocating engine mechanism is often analyzed, since it serves all the demands required for the convenient utilization of natural sources of energy such as gaseous, liquid fuels and steam, for generation of power. To convert rotary motion into reciprocating motion, the slider crank is part of a wide range of machines, typically pumps and compressors. Another use of slider crank is in toggle mechanisms, also called knuckle joints. The driving force is applied at the crankpin so that, at TDC, a much larger force is developed at the slider.

A slider mechanism is shown in below figure. Derive an expression for the motion of the piston P in terms of crank length r, connecting rod length l, and constant angular velocity of the crank ω. Find the value of l/r for which the amplitude of every higher harmonic is smaller than that of the first harmonic by a factor of at least 25.

The procedure to solve this problem is divided into two parts and they are, 1. Analytical Method The procedure adopted in solving this problem is, i. To derive an expression for the motion of the piston P in terms of crank length r, connecting rod length l, and constant angular velocity of the crank ω. ii. To find the value of l/r for which the amplitude of every higher harmonic is smaller than that of the first harmonic by a factor of at least 25. 2. Animation Method

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Derivation of the slider crank mechanism: Consider the slider crank mechanism shown in above figure. Let θ be the angle turned through by the crank OA = r when the slider B has moved by an amount x to the right, and ∅ the angle which the connecting rod AB=l makes with the line of stroke.

The above equation is used for finding the l/r ratio by writing simple MATLAB program.

The MATLAB program for above equation is as below, M-file program: n=.6667; r=12.5; a=1-cosd(t); b=1-cosd(2*t); c=4*n; d=r-(r*(a+(b/c))); plot (t,d) xlabel (‗crank angle‘) ylabel (‗n‘) title(‗synthesis of slider crank mechanism‘) When this program is executed the following graph appears. In the above graph the maximum displacement from the mean position represents the amplitude of the wave form (i.e A=13.5mm).

From the above plot graph the value of n=x1/xh as follows, Let, x1=amplitude of first harmonic. xh=amplitude of higher harmonic to be calculated. y=amplitude of higher harmonic at BDC. Now from the graph, x1= 13.5mm xh= (x1-y)/2 = (13.5-12.5)/2

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Therefore we‘ve n=x1/xh=13.5/0.5=27 Also we‘ve, n=l/r For n=0.6667 and r=12.5mm we get the value of ‗l‘ as, l= n × r =0.6667×12.5=8.33375mm Hence for the ratio, n=x1/xh=13.5/0.5=27 We achieved the condition that the amplitude of every higher harmonic is smaller than that of the first harmonic by a factor of at least 25. Therefore, the value of ‗l‘ is equal to 8.33375mm and the value of ‗r‘ is equal to 12.5mm.

The project problem is validated by this below m-script written for the animation of simple slider crank mechanism. From the animation it can be clearly understood that how exactly the piston reciprocates for different crank angles and for different n=l/r values.

The following graph appears when the animation m-script file is executed. From the graph it is observed that the wave form is not sinusoidal and at particular angle amplitude varies to small interval of time within the completion of one complete cycle.

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For n=0.007 and l=.0875mm, when r=12.5mm

For n=0.01 and l=0.125mm, when r=12.5mm.

For n=0.04 and l=0.5mm, when r=12.5mm.

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For n=0.09 and l=1.125mm, when r=12.5mm.

For n=0.5 and l=6.25mm, when r=12.5mm.

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For n=0.6667 and l=8.33375mm, when r=12.5mm.

For n=1 and l=12.5mm, when r=12.5mm.

For n=2 and l=25mm, when r=12.5mm.

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For n=2.5 and l=31.25mm, when r=12.5mm.

For n=3 and l=37.5mm, when r=12.5mm.

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It is concluded from the graphs obtained for this simple slider crank mechanism analysis made for the conditions defined in the problem; the following results are found and validated from the MATLAB animation. It is not possible to get the sinusoidal wave for the ratio of n=l/r values ranging from 0.007 to 3 where ‗l‘ is connecting rod length and ‗r‘ is crank radius. It is found that when connecting rod length(l)= crank radius(r), the piston is stationary for 180° when the crank is rotating from the angle 90° to 270° in one complete cycle. It is concluded from analysis that we get a sinusoidal wave when n=l/r ratio is less than 0.007 and more than 3.

Mechanical Vibrations by Singiresu. S. Rao. 5th edition, Printice Hall Publications, Page no.120. Getting started with MATLAB by Rudra Pratap, OXFORD university press. H.D.Desai, ―Computer aided kinematic and dynamic analysis of a horizontal slider crank mechanism for single cylinder four stroke IC engine‖ proceedings of the world congress on Engineering 2009,VOL II, WCE 2009,July 1-3, 2009, London, U.K. Mohammad Ranjbarkohan, Mansour Rasekh, Abdol Hamid Hoseini, Kamran Kheiralipour, and Mohammad Reza Asadi ―Kinematics and kinetic analysis of the slider crank mechanism in otto linear four cylinder Z24 engine‖ Journal *H. Jiménez¹, I. Hilerio2, M. Gómez3, B. Vázquez4, G. D. Álvarez5, G. Rosas6 ―SLIDER-CRANK MECHANISM‖, 8th World Congress on Computational Mechanics (WCCM8). 5th. European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008) June 30 – July 5, 2008 Venice, Italy. Chih-Cheng Kaoa,c, Chin-Wen Chuangb, Rong-Fong Funga*, ―The self-tuning PID control in a slider–crank mechanism systemby applying particle swarm optimization approach‖, Elsevier Mechatronics 16 (2006) 513–522, Received 14 March 2006.

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