Formulation of kinematic efficiency of inline crank mechanism ijaerdv03i1131689

Scientific Journal of Impact Factor (SJIF): 4.14

e-ISSN (O): 2348-4470 p-ISSN (P): 2348-6406

International Journal of Advance Engineering and Research Development Volume 3, Issue 11, November -2016

Formulation of Kinematic Efficiency of Inline Crank Mechanism Nitin Metange1, Dr. S.P.Trikal2 1 2

Mechanical Engineering Department, SSGMCE Shegaon Mechanical Engineering Department, SSGMCE Shegaon

Abstract —This Crank Mechanism is a four bar mechanism being used universally in engines, compressors, expanders and other applications for conversion of linear motion to rotary and vice versa. It is widely accepted and is unavoidably used due to its unique excellence like simplicity, robustness and ability to provide fixed compression / expansion ratio for wide range of speeds and forces. Efforts have been made to increase transmission efficiency by various methods, some of which include providing offsets, incorporation of articulated connecting rod, etc. These efforts have been resulted in advantages like dwells at dead centers, variable compression ratio, better recovery, increased speed ratio coefficient / time ratio, increased power transmission etc. The relative gain in efficiency due to above methods has to be expressed in terms of efficiency of a basic planer inline mechanism (without any modifications). Hence it is inevitable to derive efficiency of a basic mechanism. The mechanism has inherent kinematic limitation which does not allow 100% power transmission from input link to output link. This upper limit of power transmission can be considered as kinematic efficiency of the mechanism and can be derived by comparing output power with input power to the mechanism.

Further, it is seen that the equation of efficiency converges to a smallest factor which contains only geometric parameters like length of the links and offset if provided. In this paper, it is envisaged to,

a) Formulate kinematic efficiency of crank mechanism for transmission of power from input link to output link for various strokes of a crank mechanism, b) Formulate smallest equation containing geometrical parameters that defines efficiency of a stroke, c) Formulate relation between efficiencies of various strokes, d) Formulate geometrical limitation of mechanism to which maximum kinematic efficiency is restricted.

Keywords- force transmission, mechanical advantage, power transmission efficiency, slider crank mechanism.

I. INTRODUCTION Transmission efficiency of crank mechanism has been estimated by various methods [1], [2]. Efforts have also been made to increase the same. Some of which include providing offset [3], [4] incorporation of articulated connecting rod [5] etc. It has been resulted into increased transmission efficiency with some other advantages like dwells at top dead centers that results into injection of fuel at constant volume process in case of IC engines. However, while discussing about increase in transmission of power from input link to output link of a modified mechanism, the maximum limit of power transmission from input link to output link in case of inline planer crank mechanism should be a known with which comparison of increased efficiency of modified mechanism can be made. Hence it is inevitable to derive maximum power transmission efficiency the inline crank mechanism.

Predominantly, slider crank mechanism is used in following two ways.

1. When linear motion of slider is converted to rotational motion of crank. i.e. when slider works as driving and crank as driven link.

2. When rotary motion of crank is converted to linear motion of slider. i.e. when crank works as driving and slider as driven link.

In any exiting application of crank mechanism for operating in cycle, it has to use one of the ways mentioned above. For example in steam engine or air expander, in both the strokes (primary as well as secondary) piston works as driving link. Whereas in a four stroke cycle engine, during power stroke, piston would be driving link whereas for rest of strokes, crank would act as driving.

Efficiency of power transmission is closely related to mechanical advantage. For a given mechanism, when input and output links are changed, mechanical advantage changes. Similarly, power transmission efficiency would be changed if input and output links are changed. On these grounds, for given mechanism, different types of kinematic efficiencies can be derived for different strokes. Once efficiencies for all individual strokes comprising a cycle are known, and if power at output link is known, it would be easier to formulate overall kinematic efficiency of a cycle.

However, to formulate efficiency of cycle is not in the scope of this text. Kinematic efficiency is purely considered as ability of mechanism to transfer power kinematically from input link to output link and while formulation of efficiency, @IJAERD-2016, All rights Reserved

62

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406 inertia and friction forces are not taken into consideration. For formulation of efficiency, attempt is made to calculate output power available on output link for a given input power. For calculation of input or output power at input or output links, it is necessary to have,  Definite relation between velocity at point of application of input force / torque on input link and velocity at point on output link where output force / torque is measured.  Definite relation between input and output and force / torque. i.e. Mechanical Advantage.

Accordingly, relation between angular velocity of crank and linear velocity of piston is frmulated in section II. Section III is the introduction to section IV and V. Section IV introduces derivation of efficiency for conversion of linear motion to rotary. Section V is meant for derivation of efficiency for conversion of rotary motion to linear. In section VI, conclusions and relation between above two efficiencies are derived.

II. FORMULATING DEFINITE RELATION (RELATION WITHOUT ANY APPROXIMATIONS) BETWEEN VELOCITIES OF INPUT AND OUTPUT LINK

The configuration of inline slider crank is illustrated in Fig 1. The geometrical parameters would be length of connecting rod (l.cr) and crank radius r . The angular movement of crank would be represented by θo and the angle made by connecting rod with line of action of slider motion would be represented by ϕo.

Figure 1. Schematic diagram of inline slider crank mechanism Let „x‟ be the distance travelled by piston for θo of crank rotation. It can be written in terms of geometrical parameters as mentioned below

(1) Let ′n′ be the ratio between length of connecting rod (l.cr) and that of crank (r). n = l. cr/r Following relations can be derived from the geometry for ϕ as function of θ.

and Hence, ϕ can be written in terms of θ as a function of θ as below

Where asin = sin-1

(2)

Let „Vp‟ be the slider velocity and „ω‟ be the angular velocity of crank at given θ. By using value of ϕ from (2) and differentiating (1), with respect to θ, expression can be derived for instantaneous slider velocity in terms of ω @IJAERD-2016, All rights Reserved

63

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406 (angular velocity of crank), r (the crank radius), l.cr (length of connecting rod) and θ (the instantaneous position of crank w.r.t. positive x axis).



vp (  )  r  sin(  ) 

   

sin(  ) cos(  )



1

2  ( sin(  ) ) 2   n 1    2 n   

(3)

Where the term in bracket can be considered as velocity factor, „V f‟ and is a function of θ i.e. Vf(θ). If slider velocity (Vp) is known, ω, the angular velocity of crank can be written in terms of Vp as function of θ as mentioned below.

Vp ( )  r  Vf ( ) III.

CALCULATION OF KINEMATIC EFFICIENCY OF INLINE CRANK MECHANISM

For transmission of linear motion to rotary and vice versa, crank mechanism can be used in four distinct ways depending upon the type of force / torque applied over the input link (either slider or crank) and the degrees that the crank traces.

This can be more clarified as below.

When slider is input link, during forward stroke, slider pushes the connecting rod till first 180o of crank rotation, and then it pulls the C.R. from 180o to 360o of crank rotation for reverse stroke. E.g. in steam engine, one can find both the cases.

In case of offset mechanism the angle traced during both the strokes will be different.

Similarly, when crank is input link, it would either push the connecting rod or pull it.E.g.in case of compressor cycle, both cases can be observed.

Any mechanism operating in a cycle would definitely use at least two ways / strokes amongst the four mentioned above. E.g. in Otto cycle, for power stroke, slider/piston would be input link whereas rest of strokes, crank would be input link. Hence for deriving the efficiency of overall cycle, it is envisaged to find the efficiency for all the four strokes individually, however, to find the efficiency of complete cycle is not in the scope of this text.

It is observed that in case of crank mechanism, keeping the length of links unaltered; efficiency of a stroke depends upon degrees of crank rotation during the stroke irrespective of push or pull forces on connecting rod. For inline mechanism, when slider is driving, whereever it is pushing or pulling the connecting rod, efficiency of transmission would be same for both the strokes as equal angle is traced by crank during both strokes. Similarly when crank is driving, for inline mechanism, for both the strokes i.e. crank is either pushing or pulling the connecting rod; efficiency of transmission would be same.

However, for a given mechanism, it is found that,  Efficiency when slider is driving would differ from efficiency when crank is driving.  Instantaneous efficiency is found to be a pure function of geometric parameters like length of links forming a mechanism and position of crank (θ) irrespective of form of input power.  Overall efficiency of a stroke can be found out by integrating instantaneous efficiency over the stroke.  A definite correlation is found between overall efficiency when crank is driving to the efficiency when slider is driving.

As mentioned earlier, for any link as input, efficiency would be same where ever connecting rod is experiencing compressive or tensile forces. One would naturally question, when both efficiencies are same, what is the necessity to mention these cases as different. The answer is that, the efficiency when connecting rod is in compression would vary from efficiency when it is in tension in case of offset mechanism i.e. when angle traced by crank for forward stroke would be different from angle traced in reverse stroke. However, study of offset crank mechanism is not in the scope of this text hence only two types of efficiencies are to be formulated for inline crank mechanism.

In following sections, it is envisaged to formulate the efficiency for inline crank mechanism for both the cases, i.e. when slider is driving and when crank is driving.

IV.

FORMULATION OF POWER TRANSMISSION EFFICIENCY OF INLINE CRANK MECHANISM WHEN DRIVING ELEMENT IS SLIDER

When driving element is slider, for calculating kinematic efficiency for both forward as well as reverse stroke, power output available at crank is compared with input power at slider. Relation (3) is already set between instantaneous angular velocity of crank (ω) and instantaneous velocity of slider (Vp) as a function of θ. Hence relation between input @IJAERD-2016, All rights Reserved

64

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406 force on slider and output torque on crank has to be set for deriving relation between output to input power. By comparing power at both the links, instantaneous kinematic efficiency can be derived in terms of θ. Kinematic efficiency of a complete stroke can be found by integrating instantaneous efficiency over the degree of rotation that crank traces.

Moreover, efficiency can also be found by integrating instantaneous output power over degrees of crank rotation during the stroke and then dividing it by input power.

For simplicity, input force on piston / slider is assumed to be constant throughout the stroke and slider is assumed to be moved with constant velocity throughout the stroke. This assumption of force whether it is constant / variable / fluctuating in nature, and whether slider is moving with constant / variable velocity will not affect kinematic efficiency as it is a relative term (ratio between output to input power) and is sole function of geometrical parameters and is proved in following text.

4.1. Formulation of efficiency of forward stroke (from 0o to 180o of crank rotation) when slider is driving.

(Connecting rod is experiencing compressive forces) Let F1 be the constant force applied at slider throughout the stroke and F2 be the force along the connecting rod and F3 be the tangential force on crank. Figure 2 shows a schematic of forces and their components.

Figure 2. Two different positions showing forces and their components when slider is driving element Resolving input force F1 in the direction of connecting rod, force on connecting rod F2 can be obtained. F2 can further be resolved into two components. One component would be in the direction perpendicular to crank and another in the direction of crank. The force F3 in the direction perpendicular to crank is the useful force which drives the crank to get useful power at crank. The force Fr in the direction of crank would result into either compression or tensile force in crank and is transferred on bearings.

F2 ( )  F1 cos( ( ))

(5)

Where F2 is the force in the direction of connecting rod and is function of θ.

The useful force perpendicular to the crank (F3) as a function of θ can be written as

F3 ( )  F2 ( ) sin(   ( ))



2

F3 ( )  F1  sin( ) cos( ( ))  sin( ( )) cos( ) cos( ( ))

(6)



Hence F3 can be written in terms of F1 as follows Using equation (2), F3, the force tangential to connecting rod can be written in terms of θ as mentioned below. .5  2  sin(  ) 3  sin(  ) cos (  )  ( sin(  ) )   F3 (  )  F1  sin(  )  1     2   2 n n     n 

@IJAERD-2016, All rights Reserved

(7) 65

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406 From above equation, variation of tangential force (F3) or torque over the 180o of rotation of crank can be obtained and can be shown graphically as below where F3 is plotted for unit value of F1.

Figure 3. Variation of tangential force over crank with respect to θ o (θ in radians) Now, Instantaneous efficiency is the ratio of instantaneous output power available at crank to input power at slider. Instantaneous output power at crank is product of instantaneous output torque at crank and angular velocity of crank at that position. Let Pout be instantaneous output power available at crank which is function of θ and Pin be power input at slider. and Instantaneous kinematic efficiency in terms of θ can be written as mentioned below.

(9)

Putting the values from 7, 9 & 3, one would obtain function of instantaneous efficiency at given θ.

1 2

2  sin(  ) 3  sin(  ) cos(  )  ( sin(  ) )  sin(  )  1    2  2 n n    n   (  )  sin(  ) cos(  ) sin(  )  1

 ( sin(  ) ) 2  n 1   2 n  

2

(10)

From equation 10, it is clear that the efficiency is purely the function of geometrical parameters. Hence earlier assumption that, uniform / variable input force and uniform / variable input velocity of slider won‟t affect the efficiency of mechanism stands true. Above equation of instantaneous efficiency can converge to a smallest factor in terms of θ. For resolving above equation, let us put

Hence,

After resolving, one would get, @IJAERD-2016, All rights Reserved

66

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406

Let us put value of „Z‟ as

Hence

Above equation results into Putting value of „Y‟,

(11)

(12) Equations 11 and 12 are equations of instantaneous kinematic efficiency of mechanism in terms of θ & ф respectively. For a typical value of „n‟ as 3.75, instantaneous efficiency can be graphically represented as below.

Figure 4. Graph between instantaneous Kinematic efficiency and θ when slider is driving. (Where θ is in radians and n=3.75) Integrating above function of instantaneous efficiency over 180o of crank rotation, average kinematic efficiency (η.I) for a complete stroke can be obtained as follows

(12) As mentioned initially, efficiency can also be derived by integrating instantaneous output power over 180o of crank rotation and dividing it with input power as mentioned below.

Let Pout be the instantaneous power output (product of instantaneous output force (F.3) and instantaneous output velocity) at crank and let P.average be average power available at crank. This can be written in equation form as mentioned below

@IJAERD-2016, All rights Reserved

67

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406

Average kinematic efficiency for a complete stroke can be written as ratio of average power available on output link (Paverage) to input power (Pin).

(13) Above equation would yield the same result as equation 12.

Table 1 is plotted for average kinematic efficiency of a complete stroke for the range of „n‟ from 1 to 4.9. It is observed that, for n=1, i.e. for a imaginary case when crank length is equal to connecting rod, the efficiency approaches almost 50%. i.e only 50% power can be transmitted to crank from slider when n=1. Whereas for another case when connecting rod length is infinitely long (an imaginary case) i.e. when n = ∞, efficiency approaches 100%. Variation of average kinematic efficiency of a complete stroke when slider is driving, with respect to „n‟ (from 1 to 4.9) is mentioned in following table Table 1: Variation of overall average efficiency of a stroke with respect to ‘n’ when slider is driving n ηI = n ηI = n ηI = n ηI = 1

0.50

2

0.875

3

0.944

4

0.969

1.1

0.587

2.1

0.887

3.1

0.948

4.1

0.970

1.2

0.653

2.2

0.897

3.2

0.951

4.2

0.972

1.3

0.704

2.3

0.905

3.3

0.954

4.3

0.973

1.4

0.745

2.4

0.913

3.4

0.957

4.4

0.974

1.5

0.778

2.5

0.920

3.5

0.959

4.5

0.975

1.6

0.805

2.6

0.926

3.6

0.961

4.6

0.976

1.7

0.827

2.7

0.931

3.7

0.963

4.7

0.977

1.8

0.846

2.8

0.936

3.8

0.965

4.8

0.978

1.9

0.861

2.9

0.941

3.9

0.967

4.9

0.979

It is observed that the average efficiency when slider is driving increases exponentially w.r.t. „n‟ as shown in following plot.

1 0.9



0to  1 n

0.8 0.7 0.6 0.5

1

2

3

4

5

6

n

Figure 5. Variation of average efficiency for a complete stroke w.r.t. ‘n’ when slider is driving element From above graph the conclusion can be drawn that, when slider is driving element, efficiency of crank mechanism would always be more than 50%.

4.2. Formulation of efficiency for a reverse stroke (from 180o to 0o of crank rotation) when slider is driving.

(Connecting rod is experiencing tensile forces)

@IJAERD-2016, All rights Reserved

68

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406

Figure 6. Two different positions showing forces and their components when slider is a driving element By comparing resolved forces with those in Fig 2, and from numerical analysis, it is clear that F 3, the tangential force on crank would vary in the same fashion and yield same results when connecting rod experiences compressive forces. Hence equation 11 & 12 (variation of instantaneous efficiency) will be applicable for case of reverse stroke too as in case of forward stroje. Hence overall efficiency would also be same and would be given by equation 12 or 13 and its variation w. r. t. „n‟ would be same as in Fig 5.

V.

FORMULATION OF POWER TRANSMISSION EFFICIENCY OF INLINE CRANK MECHANISM WHEN DRIVING ELEMENT IS CRANK

For calculating kinematic efficiency when driving element is crank, (crank either pushes or pulls the connecting rod), power output at slider is compared with input power at crank. For simplicity, input torque on crank is assumed to be constant throughout its rotation and crank is assumed to be rotating with constant angular velocity throughout the stroke.

As mentioned in earlier case, whether driving torque is constant / variable (increasing / reducing / fluctuating) in nature and whether the crank is rotating with constant / variable angular velocity will not affect the efficiency as efficiency is purely a function of geometrical parameters. In case when crank is driving, similar to previous case when slider as driving, mechanism would experience two strokes. i.e forward and reverse stoke.

5.1 Formulation of efficiency of forward stroke (from 180 o to 360o of crank rotation) when crank is driving.

(Connecting rod is experiencing compressive forces) Fig 7 shows schematic of forces when crank is driving element. Let F3 be the tangential force due to input torque „τin‟. F2 be the component of F3 force in the direction of connecting rod. F1 be the useful force which is component of F2 in the direction of slider velocity.

@IJAERD-2016, All rights Reserved

69

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406

Figure 7. Two different positions showing forces and their components when crank is driving element

From geometry, it is clear that, F2, the force in the direction parallel to crank is the component of F3 as mentioned below.

F1 be the force in the direction parallel to motion of slider (14) From equation 3, velocity of piston / slider (V.p) can be written in terms of ω as Vp( )  r   sin ( ) 

  

sin ( )  cos ( )

  2   ( sin ( ) )  n 1    2  n   



.5 

  

(15)

Let P.out be the output power which is the product of useful output force available on slider and slider velocity Hence Pout ()  F1() Vp()

(16) Let P.in be the input power input to the crank Pin   in 

(17)

Where τ.in is the torque input to crank rotating at speed of ω rad/s. Instantaneous kinematic efficiency can be written as a ratio of output to input power in terms of θ.  ( ) 

Pout ( ) Pin

Putting values for 16 and 17 in above equation,  ( ) 

@IJAERD-2016, All rights Reserved

F1( )  vp( ) F3 r  

70

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406

 F3 sin (  ( )  )  cos (  ( ) )  vp( )   F3 r    

Hence

 ( )  

Using value of v.p, in terms of θ,  ( )  sin (    ( ) )  cos (  ( ) )  sin ( ) 

  

 .5    2   ( sin ( ) )    n 1  2   n     sin ( )  cos ( )

This is the equation of instantaneous efficiency of crank w.r.t.θ. From this equation, it is clear that the efficiency when crank is driving is also a pure function of geometrical parameters. and can converge to a smallest factor in terms of θ as follows.

(18) As ϕ is the function of θ and can be expressed in terms of θ, instantaneous kinematic efficiency can be supposed to be a sole function of θ as mentioned below.

sin ( )       sin    asin       n  

2

(19) Equation 18 or 19 is the equation of instantaneous efficiency in terms of θ and / or ϕ. For a typical value of „n‟ as 3.75, the variation of instantaneous efficiency can be graphically represented as below.

Figure 8. Graph between instantaneous kinematic efficiency and θ when crank is driving (Where θ is in radians and n=3.75) Integrating momentary kinematic efficiency over 180o of crank rotation, π to 2 π, average efficiency of a complete stroke ηII can be obtained.

(20)

@IJAERD-2016, All rights Reserved

71

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406 Table 2 is plotted for average kinematic efficiency a complete stroke for the range of „n‟ from 1 to 4.9. It is observed that, for n=1, i.e. for a imaginary case when crank length is equal to connecting rod, the efficiency approaches almost 25%. i.e only 25% power can be transmitted from crank to slider when n=1. . And when n = ∞, another ideal case, efficiency approaches 50%. Variation of average kinematic efficiency of a complete stroke with respect to „n‟ (from 1 to 4.9) when crank is driving, is mentioned in following table Table 2. Variation of overall efficiency with respect to ‘n’ when crank is driving n

ηII =

n

ηII =

n

ηII =

n

ηII =

1

0.25

2

0.438

3

0.472

4

0.484

1.1

0.293

2.1

0.443

3.1

0.474

4.1

0.485

1.2

0.326

2.2

0.448

3.2

0.476

4.2

0.486

1.3

0.352

2.3

0.453

3.3

0.477

4.3

0.486

1.4

0.372

2.4

0.457

3.4

0.478

4.4

0.487

1.5

0.389

2.5

0.46

3.5

0.48

4.5

0.488

1.6

0.402

2.6

0.463

3.6

0.481

4.6

0.488

1.7

0.413

2.7

0.466

3.7

0.482

4.7

0.489

1.8

0.423

2.8

0.468

3.8

0.483

4.8

0.489

1.9

0.431

2.9

0.47

3.9

0.484

4.9

0.49

It is observed that the efficiency increases exponentially as ratio of length of connecting rod to crank radius increases. However, its minimum value is 25% for „n‟ equal to unity whereas, for infinite length of connecting rod, the maximum value efficiency approaches is 50 %.

0.5

 

0.4

to2  2 n

0.3 0.2

1

2

3

4

5

6

n Figure 9. Variation of average efficiency w.r.t. ‘n’ when crank is driving Above graph represents variation of average efficiency of a complete stroke with respect to „n‟ when crank is driving element. The conclusion can be drawn that the efficiency when crank is driving the slider is always between 25 to 50%.

5.2. Formulation of efficiency of a reverse stroke (from 0 o to 180o of crank rotation) when crank is driving.

(Connecting Rod is experiencing tensile forces)

@IJAERD-2016, All rights Reserved

72

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406

Figure 10. Two different positions showing forces and their components when crank is driving element

By comparing resolved forces with those in Fig 7, and from numerical analysis, it is clear that F3, the tangential force on crank would vary in the same fashion and yield same results as earlier case of forward stroke. Hence equation 18 will be applicable for instantaneous efficiency and variation of instantaneous efficiency over the degrees of crank rotation and variation of average efficiency for a complete stroke w.r.t „n‟ would also be same.

I.

Conclusion

It is observed that the transmission efficiency is independent of the nature of input power. i.e. the efficiency is not affected whether the forces and velocity at input link are constant or varying.

For inline crank mechanism, 1. When piston is driving crank, it is observed that the efficiency of transmission of power would always be more than 50%. It is minimum for ideal case for „n‟ equal to 1 i.e., when length of connecting rod equals crank length and its maximum value is 100% for ideally infinite length of C.R. 2. When crank is driving, it is observed that efficiency would vary from 25 to 50%. It is minimum 25% when „n‟ is 1 and increases with „n‟ and its maximum value is limited to 50% for ideally infinite length of C.R. 3. For the inline slider crank mechanism, efficiency of a stroke when slider is driving would be exactly double of the efficiency when crank is driving. This can be illustrated with the help of an example.

Let us consider a crank radius „R‟ as 1m, Length of connecting rod, l.cr = 3.75m, Let slider be driving the crank and θ1, angle made by crank with positive x axis varies from 0 to π.

The instantaneous efficiency for six positions of crank from 0 to 180o (30o apart) would be given by

1

1 

 2

sin 1 n

0

2



1

deg

30

0.982

60

0.947

90

0.929

120

0.947

150

0.982

180

1

Fig11. Graph for various values of instantaneous efficiency w.r.t. θ when slider is driving The average efficiency of a complete stroke ηI would be given by @IJAERD-2016, All rights Reserved

73

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406      I 



1

sin 12 n

2

0



d1  0.964

(21)

For the same mechanism when crank would be driving the slider, for six positions of crank from 180 to 360 o (30o apart) would be given by



2  180

deg

 2 

sin 2   2 0

210

0.144

240

0.529

270

0.929

300

0.918

330

0.373

360

0

Fig 12. Graph for various values of instantaneous efficiency w.r.t. θ when crank is driving The average efficiency of a stroke ηII would be given by

(22) Since equation 21 and 22 belongs to a same mechanism, the relation can be formulated between the two as shown in 23.

(23)

This relation is the ratio of average efficiency of a stroke when slider is driving to the efficiency of a stroke when crank is driving. Equation 23 would be interestingly producing a result that efficiency of a stroke when slider is driving is twice the efficiency of stroke when crank is driving.

@IJAERD-2016, All rights Reserved

74

International Journal of Advance Engineering and Research Development (IJAERD) Volume 3, Issue 11, November -2016, e-ISSN: 2348 - 4470, print-ISSN: 2348-6406

REFERENCES [1]

[2] [3]

[4]

[5]

Relly Victoria Petrescu, Florian Ion Petrescu, “Determining the Mechanical Efficiency of Otto Engine‟s Mechanism,” The ninth iftomm international symposium on theory of machines and mechanisms bucharest, romania, september 1 - 4, 2005 syrom 2005 Relly Victoria Petrescu, Florian Ion Petrescu, “Forces of Internal Combustion Heat Engines,” International Review on Modelling and Simulations (I.RE.MO.S.), Vol. 7, n. 1 February 2014.

Dr. Taj Elssir Hassan, “Theoretical Performance Comparison between Inline, Offset and Twin Crankshaft Internal Combustion Engines,” Proceedings of the World Congress on Engineering 2008 Vol II, WCE 2008, July 2 - 4, 2008, London, U.K. Bharat K. Gupta, A. Rehman and N.D. Mittal, “Validating experimentally the gain in torque due to crankshaft offset of an internal combustion engine,” International Journal of Engineering, Science and Technology, Vol. 6, No. 2, 2014, pp. 76-88 Vaibhav Ahlawat, Sandeep Tripathi and Naveen Kumar, “Compressed air engine:A bridge for sustainable development,” Asian Journal of Scientific Research 6(2): 254-262, 2013

@IJAERD-2016, All rights Reserved

75