Non-Circular Gearing Analytical Design for Driven Slider- Crank Mechanisms Balancing

Non-Circular Gearing Analytical Design for Driven Slider-Crank Mechanisms Balancing

Cesare Maglioni a *, Giovanni Molari a

aDEIAgra – Department of Agricultural Economics and Engineering – University of Bologna, Via G. Fanin, 50 I-40127 Bologna, Italy

__________________________________________________________________________________________ Abstract

In this paper, a solution for balancing operated slider – crank mechanisms is proposed and analytically developed. The solution design provides the application of an unusual configuration of a pair of toothed non-circular gears, together with the use of a standard flywheel. The pinion gear is mounted on the mechanism crank shaft, while the driven one is keyed on a secondary parallel shaft, which does not participate directly in the transmission of motion and carries the flywheel. The gearing model is developed and constraints satisfaction analyzed. The system dynamics and equation of motion are therefore developed adopting a specific case study, which regards an agricultural reciprocating cutter mechanism previously analyzed by the authors. This kind of non-circular gearing implementation, which can be thought as a purely mechanical controller of inertia momentums, is generally applicable to operated slider – crank mechanisms and avoids deep changes in the original system architecture. The paper represents the analytical preamble to the upcoming numerical analysis. PACS: 05.45.a, 07.10.Lw, 45.40.f Keywords: slider – crank mechanism balancing; non-circular gears; agricultural reciprocating cutter; __________________________________________________________________________________________ 1. Introduction

In driving slider – crank mechanism, like internal combustion engines, the generated force on the slider and the mechanism architecture make the driving torque represent the disequilibrium source. Hence, since the operated machine has to run smoothly, the purpose of the balancing is to regularize the driving torque, to have the engine output rotational speed as more uniform as possible and reduce chassis vibrations. A large flywheel is set on the engine output rotational axis, the basic principia being the introduction of an additive (inertia) torque between the driving and the resistance ones [1, 10]. _______ Corresponding author. Tel.: +41 22 767 3673; fax: +41 22 767 5760; email: [email protected] C. Maglioni, 34 rue du Quart F-01710 Thoiry

In case the slider – crank mechanism is part of the operated machine, driven by any other kind of engine, the disequilibrium is still caused by the force acting on the slider and therefore it comes from the resistance torque. The supporting structure hence vibrates and the rotational speed assumes a time-variable shape [2, 10]. In spite of that it is nowadays possible to find applications where, for balancing purposes, a flywheel is set on the engine rotational axis even if the mechanism is driven. As it has been previously shown with the numerical analysis of an agricultural application, this configuration is totally useless, since the engine itself is not the source of but suffers the disequilibrium, and another kind of balancing system has to be worked out [11].

In this paper, on the same basic principia on which the flywheel works, a different solution is proposed to introduce the additive balancing torque. The idea is to make use of a flywheel together with an unusual configuration of a pair of toothed non-circular gears, but where the purpose was usually to regularize the driving torque here it is to regularize the resistance one.

Although variable radius gears are not very diffuse, they are essential for applications where there is not only a need for speed variation control, but also if a purely mechanical system is desired [6, 7]. Until recently, except for elliptical gears [4, 18, 19, 20], variable radius gears have been limited in their diffusion because of production costs and lack of generalized knowledge about production methodologies. Anyway, different and costless methodologies has been recently implemented for cylindrical gears cutting, like wire cutting, which has changed the prospective upon the non-circular gears utilization even for cheap mechanisms [3, 16].

Together with the wish for contributing in the development of this sort of gearing, the above considerations led the authors in making their choice for the balancing solution. In the follow the gearing mathematical model is therefore implemented in the most general case. Secondly, system dynamics and new equation of motion are analytically developed adopting a specific case study which regards the agricultural reciprocating cutter previously analyzed by the authors. Finally, gearing constraints satisfaction is analyzed and discussed.

2. Analytical model 2.1. Functional condition

Figure 1 shows the standard configuration of an eccentric slider – crank mechanism,

powered by engine E with driving torque C(t). Naming with ϑ the single degree of freedom of the system, assuming a not-constant rotational speed dtdϑϑ = and defining the motor rotational axis as the primary shaft (P), the rotational equilibrium gives

( ) ( ) ϑ*c JtbFtC 1=⋅− (1)

Fig. 1 – Simplified standard configuration of a driven eccentric slider – crank mechanism.

where Fc is the axial force acting along the conrod, b(t) the normal distance between the conrod and the engine rotational axis (x3), *

1J the inertia momentum of the primary axis and 22 dtd ϑϑ = the crank rotational acceleration

Since the mechanism is driven, the dynamic analysis of the system allows individuating the component

( ) ( )tbFJtC c ⋅=− ϑ*1 (2)

as the main and geometrically unavoidable unbalancing source (see equation (20) and (30) in [11]), which then makes the chassis structure vibrate. Therefore the resistance torque acting on the mechanism ( )tbFc ⋅ is the component which has to be balanced.

To attain this purpose, a non-circular pinion gear is set on the primary axis [7, 8, 9 14 and 17], the rotational equilibrium becoming:

( ) ( ) ( ) ϑ1JtbFtCtC cADD =⋅−+ (3)

where J1 is the inertia momentum of the primary shaft in the new configuration (adding the pinion inertia), while CADD(t) is an additive time-variable torque led through the pinion. This additive torque is supplied by a secondary parallel shaft (S), which carries the meshing driven gear and a flywheel (Fig. 2), and it has to be designed such as the resistance torque becomes time-constant

( ) ( ) ( ) constant1 =−+=⋅ ϑJtCtCtbF ADDc (4)

P

Fig. 2 – Simplified design configuration of a driven slider – crank mechanism. Gear teeth are

not showed for clearness reason.

Equation (4) represents the functional condition needed for balancing purposes. The fundamental idea laying under this application is to supply the cyclically time-variable additive torque CADD(t) by means of the constant inertia torque of the flywheel modified by a cyclically time-variable gear ratio [7, 8, 9, 12, 14 and 17]. 2.2. Gearing modelling

The kinematic energy theorem applied to the secondary shaft gives

dtdEWW OUTIN

2=+ (5)

where WIN is the input power, adsorbed by the secondary and coming from the primary shaft, WOUT is the output power, supplied by the secondary to the primary shaft, while E2 is the kinematic energy of the secondary shaft.

With the commonly adopted sign convention, the adsorbed power is positive while the supplied one is negative, so that the energy contents of the secondary shaft decreases (increases) when the power flows from it (from the primary shaft) to the primary shaft (to it). Anyway, the power exchange between the two shafts occurs only through the non-circular

S

gearing and hence the power cannot be adsorbed and supplied simultaneously. When the power is adsorbed it cannot be supplied, and vice versa. Hence, equation (5) can be written as:

22222222 2

1 ϑϑϑϑ JJdtdC =⎟

⎠⎞

⎜⎝⎛=⋅ (6)

where C2, 2ϑ and J2 are respectively the torque, instantaneous rotational speed and total inertia momentum of the secondary shaft. It follows that

222 ϑ⋅= JC (7) Moreover, because of the power conservation between the meshed gears it is

( ) ϑϑ ⋅=⋅ tCC ADD22 (8)

and hence, the additive torque supplied from the secondary to the primary shaft results as

( ) ( )tJCtCADD τϑϑϑ

222

2 =⋅= (9)

where τ(t) is the time-variable gear ratio.

Equation (4) is satisfied writing1:

( ) 0=⋅bFdtd

c (10)

and so:

( ) ( )( ) 0221 =+− tJJtCdtd τϑϑ (11)

In the follow each term of equation (11) will be analyzed separately and written in

function of the single degree of freedom ϑ, its derivatives and in function of the gear ratio. For the first term some assumptions to describe the driving torque have to be made.

Considering a periodic running condition it can be reasonably assumed the independence of the driving torque from the rotational position of the axis, linearizing it as a function of the rotational speed ϑ [10]. Hence, for small values of rotational speed fluctuation, the driving torque trend can be developed as a Taylor series about the rotational speed average value ω , neglecting the terms over the first:

( ) ( ) ( ) ( )ωkCωddCωCtC

ω

−+=−⎥⎦⎤

⎢⎣⎡+≈ ϑϑ

ϑ (12)

1 The value of the constant in equation (4) is not important since the necessity is to annihilate the variation of the function. A constant non-zero value translate the equilibrium point of the mechanism but does not perturb it.

where C is the driving torque average value, that is the torque evaluated for the average rotational speed, while k is the (constant) engine stiffness, representing the slope of the considered curve ( )ϑCC = within the engine functional diagram, calculated in the abscissa point where ω=ϑ 2.

With these assumptions, anytime the driving torque can be expressed as in (12), the first term of equation (11) becomes:

( ) ϑ⋅= ktCdtd (13)

while the second term is

( ) ϑϑ 11 JJdtd

−=− (14)

where ϑ is the rotational jerk of the primary shaft. Moreover, the rotational speed of the secondary shaft is:

( ) ϑτϑ ⋅= t2 (15)

and hence its rotational acceleration and jerk are respectively:

( )( ) ϑτϑτϑτϑ ⋅+⋅=⋅=dtdt

dtd

2 (16)

and

ϑτϑτϑτϑτϑτϑϑ ⋅+⋅⋅+⋅=⎟⎠⎞

⎜⎝⎛ ⋅+⋅==

dtd

dtd

dtd

dtd

dtd 22

22

2 (17)

Considering then the last term of equation (11):

( )( ) ⎟⎠⎞

⎜⎝⎛ ⋅+⋅=

dtdJtJ

dtd τϑτϑτϑ 22222 (18)

and substituting it becomes:

( )( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛⋅+⋅+⋅+⋅=

22

2

2

222 3dtd

dtd

dtdJtJ

dtd τϑτϑττϑττϑτϑ (19)

Finally, introducing equations (13, 14 and 19) into equation (11) and collecting in respect

of ϑ derivatives, it becomes

2 Particularly, k identifies the engine tendency to maintain a given rotational speed with variations in the resistance torque.

03 3

3

2

122

2

2

2

2

2

=⋅⎥⎦

⎤⎢⎣

⎡−+⋅⎥

⎦

⎤⎢⎣

⎡++⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

dtd

JJ

dtd

Jk

dtd

dtd

dtd

dtd ϑτϑττϑτττ (20)

that could be further simplified with a variable change. Naming

ττγdtd

dtd

= (21)

it is consequently:

2

2

2

2

2

⎟⎠⎞

⎜⎝⎛+=

dtd

dtd

dtd τττγ (22)

and the dimensionless substituting variable is

2

21τττγγ ∫ ∫ === dt

dtddt

dtd (23)

Substituting these quantities into equation (20) it finally becomes:

023 3

3

2

12

2

22

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡++⎟

⎠⎞

⎜⎝⎛

dtd

JJ

dtd

Jk

dtd

dtd

dtd ϑγϑγϑγ (24)

which is the functional equation of the system, describing the time-variable gear ratio in function of the unique degree of freedom ϑ and its derivatives. The trends of the latter quantities, which are described by the motion equation of the slider – crank mechanism, depends upon the resistant torque and hence upon the mechanism application.

A specific case study is chosen to introduce the motion equation, and describe the coupling between independent variable and gear ratio. 2.3. New motion equation

The motion equation can be developed from equation (3). It is anyway necessary to know the time-shape of the resistant torque, and so the shape of the force acting on the slider. The chosen case study refers to an agricultural application, where the slider – crank mechanism is the blade of a reciprocating cutting machine, driven by a hydraulic motor (Fig. 3). This particular application well evidenced the unbalancing problems, since its low working average rotational speed and weakness of engine stiffness. The whole analysis of the original system can be found in reference [11].

Fig. 3 – Application of driven slider – crank mechanism to a reciprocating cutter.

The introduction of the additive torque leads to some changes in the original motion

equation of the mechanism. The kinematic energy theorem is reapplied to the primary shaft, giving

dtdEWWWW ADDdrm

1=+++ (25)

where Wm, Wr and Wd are respectively the driving, the resistance and the dissipative power, while the term on the right side represents the variation in the kinematic energy of every movable part of the mechanism [11]. The differences between equation (25) and the original equation are the new inertia momentum of the primary shaft (with the pinion gear added and without the crank disc) and the term WADD, which is the additive power introduced by the secondary shaft

ADDW ( ) ϑ⋅= tCADD (26) Considering equation (9) it is

( ) ( ) ϑτϑϑ ⋅=⋅ tJtCADD 22 (27) and then, from (16)

ϑτϑτϑτ⎟⎠⎞

⎜⎝⎛ ⋅+⋅=

dtdJWADD 2 (28)

which becomes, with the change of variable (21) introduced in paragraph 2.2:

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛=

dtd

dtdJ

dtd

dtdJWADD

ϑϑγϑγ2

2

22 2 (29)

Finally, taking into consideration the original and already developed equation of motion3, equation (25) becomes:

( ) ( )[ ] ( ) ( ) ( )[ ]

( ) ( ) ( )( ) 0

2

22

23

2

2

22

1

=+⎟⎠⎞

⎜⎝⎛⋅⎥⎦

⎤⎢⎣⎡ ++

+⎟⎠⎞

⎜⎝⎛⋅−⎟

⎠⎞

⎜⎝⎛⋅′+⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅++

rmADdtdB

dtdJ

dtdk

dtdffA

dtd

dtdJfAJ

Tϑϑϑϑγ

ϑϑϑϑϑϑϑγϑϑ (30)

where f(ϑ), f’(ϑ), g(ϑ), D(ϑ), A(ϑ) and B(ϑ) are known functions of the single degree of freedom.

2.4. Gear ratio periodicity condition

Functional condition (24) and new motion equation (30) form an ordinary strongly non-

linear and non-homogeneous differential equation system, with coupled equations and ϑ -dependent coefficients. This system, which is of second order in respect of γ(t) and of third order in respect of ϑ(t), does not admit a closed form solution [1, 5, 15] but can be numerically integrated. The solution is represented by the new motion law of the primary shaft ϑ = ϑ(t) together with the gear ratio trend in time ( ) ( )tt γττ 2== . It can be noticed, however, that ϑ does not appear directly in equation (24), meaning that the coupling is between ( )tτ and ( )tϑ . Moreover, since time t does not appear directly in the system while every coefficient is constant or ϑ -periodic with period T = 2π/ω , these two coupled variables will be ϑ -periodic as well, with a proportional period. Hence both gear ratio and rotational speed can be written as functions of the degree of freedom

( )ϑττ = and ( )ϑϑϑ = (31) From a functional point of view, have a periodic variation of the gear ratio τ(ϑ) means

that the ratio between the lengths of the two pitch lines must be a rational number. Hence, the average gear ratio τ , evaluated on the basis of a single cycle of period T, is a rational number too. As a consequence, each tooth will mesh always with another tooth, or with a fixed number of teeth if τ is greater than 1. That is, at any angle ϑ of the driving gear will correspond a specific radius or a fixed number of radii of the driven gear, this number depending upon the chosen value for τ [7, 13]. Therefore, the condition of the gear ratio average value to be a rational number has to be verified by the solution

( ) q21 2

== ∫+ πϑ

ϑϑϑτ

πτ d (32)

3 See [11], equation (28) on page 760.

where q is a rational number. 2.5. Generation of variable radius pitch lines

Both the two non-circular gear pith lines can be obtained from solution of system (24, 30)

together with respecting a series of geometrical and kinematic constraints [7, 17]. The first condition regards the wheelbase satisfaction, i.e. the distance between the two gear axes must remain constant and equal to a certain given value at any instant:

( ) ( ) Δ=+ 221 ϑϑ RR (33) where Δ is the (constant) wheelbase value while R1(ϑ) and R2(ϑ2) are corresponding values of respectively the primary and secondary gear radii. But since

( )ϑϑ

ϑϑ

ϑϑϑτ

dd

ddt

dtd 222 === (34)

it is also ϑ2 = ϑ2(ϑ) and equation (33) becomes

( ) ( ) Δ=+ ϑϑ 21 RR (35)

The second condition regards the gearing mesh continuity, which means at any rotation of the primary shaft pitch line has to correspond a well defined rotation of the secondary one. This condition, which means also the absence of sliding between the two gears, has to be satisfied independently from the two pitch lines overall lengths. Reasoning for infinitesimal quantities it is:

21 dsds = (36)

where dsi represent the arc length of the pitch line movement, on the i – shaft, that is: ( ) ( ) 221 ϑϑϑϑ dRdR = (37)

So, from condition (37) it is

( )( ) ( )ϑτ

ϑϑ

ϑϑ

==dd

RR 2

2

1 (38)

and substituting into condition (35) it is

( ) ( )ϑτϑ

+Δ

=12R (39)

Consequently again from condition (37) it is:

( ) ( )( )ϑτϑτϑ

+Δ

=11R (40)

The two obtained radii (39, 40) describe the pitch lines of a non-circular gear pair that is

able to generate the prescribed additive torque, which annihilate the resistance torque variance as needed.

3. Discussion 3.1. Other constraints

Some observation has still to be marked about constraints that have to be satisfied for a good design of the gearing [7]. First of all, it is necessary the two pitch lines to be both closed geometries, always to guarantee the continuity of motion. But while the closeness constraint upon the driving gear is obviously guaranteed by the periodicity of the degree of freedom itself, the pitch closeness for the driven gear can be expressed by the following boundary condition:

( )( )( )( )⎩

⎨⎧

+=+=

πϑπϑϑϑϑϑ

q2q2 2002

2002

tt

(41)

where t0 is an arbitrary point in time. But since ϑ2 = ϑ2(ϑ), the driven gear angle of rotation is periodic as well. Therefore, the periodic feature of the differential system coefficients implies equation (41) to be satisfied too.

In the same way, the periodicity guarantees also the continuity (and differentiability up to the second order) for the driven gear angle of rotation ϑ2, essential conditions for the smoothness of running. In fact, while this condition upon ϑ is guaranteed by the integrability of the motion law, the same condition upon the driven gear could be written as

( ) ( )

( ) ( )⎪⎪

⎩

⎪⎪

⎨

⎧

=

=

+

+

πϑϑ

πϑϑ

ϑϑ

ϑϑ

ϑϑ

ϑϑ

q222

2

22

2

q2

22

00

00

tt

tt

dd

dd

dd

dd

(42)

and remembering equation (38):

( )( ) ( )( )( )

( )

( )( )

⎪⎩

⎪⎨

⎧

=

+=

+ πϑϑ ϑϑτ

ϑϑτ

πϑτϑτ

q2

00

00

q2

tt dd

dd

tt (43)

These two equations represent respectively the periodicity and first derivability of the gear ratio, in respect to ϑ. But while the first one is already satisfied, the second equation can be rewritten as:

( )( )

( )( ) πϑϑ ϑ

ϑτϑ

ϑτ

q200 +

⎟⎠⎞

⎜⎝⎛ ⋅=⎟

⎠⎞

⎜⎝⎛ ⋅

tt ddt

dtd

ddt

dtd (44)

that is:

( ) ( ) πϑϑ ϑτ

ϑτ

q200 +

=tt

(45)

This equation is satisfied always by the periodicity of τ, which implies the periodicity of

its first derivative τ , and by the periodicity of ϑ too, which as seen implies the periodicity of its first derivative ϑ .

Last but not least, the pitch lines must not contain reverse portions, to avoid retrograde motion for the driven gear. This condition is satisfied by imposing the gear ratio to be positive everywhere upon its domain:

( ) ϑϑτ ∀> 0 (46)

and hence by imposing the integer number q = τ to be positive. Moreover, to have a sort of uniformity in the gear dimensions, and to limit the system overall dimensions, it is helpful to have τ not too far from the unity.

3.1. Influencing parameters

As noticed, ϑ does not appear directly in equation (24), which can be rewritten as

[ ]dtd

Jk

dtd

JJ

dtd

dtd

dtd

dtd ϑϑγϑγϑγϑ ⋅−⋅=⋅⎥

⎦

⎤⎢⎣

⎡+⋅⎥

⎦

⎤⎢⎣

⎡+⋅

22

2

2

12

2

2

2

23 (47)

This is, precisely, an ordinary linear non – homogeneous second order differential

equation in γ, with coefficients dependent upon the instantaneous rotational speed dtdϑϑ = and its first and second derivatives. If the two solution variables could be thought as uncoupled, equation (47) would represent an hypothetical system which linearly vibrates in γ, with forcing function

( )kJJGdtd

Jk

dtd

JJG ,,,, 21

22

2

2

1 ϑϑϑϑ=⋅−⋅= (48)

In this simplified line of reasoning the gear ratio, and so both the pitch line profiles (39)

and (40), has a periodic trend which depends on the rotational speed ( )ϑϑϑ = , on the

rotational acceleration and jerk, on the primary and secondary inertia momentums J1 and J2 and on the motor stiffness k:

( )21,,,,,2 JJkϑϑϑτγτ == (49) Looking at these parameters and how they appear in the forcing function, it is however:

22

1

Jk

JJ

<< (50)

meaning that the jerk component of the forcing function 33

21 dtdJJ ϑ⋅ could be neglected in respect to the acceleration component 22

2 dtdJk ϑ⋅− . Even if the system (24, 30) is coupled, equation (24) is the governing equation for the gear

ratio, therefore it can be thought as the most influencing one for this variable. On the other hand condition (50) shows that the strongest influence on the gear ratio is given by the rotational acceleration and by the ratio between the motor stiffness k and the secondary inertia momentum J2, the forcing function G diminishing with k decreasing and J2 increasing, while the inertia momentum of the primary shaft doesn’t have any considerable effect.

So, having a stiffer engine or a bigger secondary inertia momentum leads to a less variable gear ratio, while the balancing remains in any case satisfied. This behaviour leaves to the designer one more degree of freedom on these three design parameters. Besides, having a certain engine (i.e. reasoning for a fixed value of k) means that the gear ratio trend remains mainly defined by the value of the secondary shaft inertia momentum J2.

4. Conclusions

An unconventional solution has been proposed for equilibrating driven slider – crank

mechanisms. The design provides the engine shaft with a pinion non-circular gear, which is used also as crank, while a standard flywheel and the driven gear are mounted on a secondary parallel shaft. The solution supplies the time-variable balancing torque by means of the flywheel constant inertia through the time-variable gear ratio of the non-circular gearing.

The application has been analytically developed, on the basis of the unique functional condition and with motion equation of a specific case study, while the gear meshing profiles have been implemented using some constitutive conditions. Finally, different constraint satisfactions have been verified and a qualitative analysis has been discussed.

The gear profiles depend upon the specific application, the gear ratio being coupled to the modified mechanism motion law. This solution, anyway, allows the complete balancing of the system by means of a purely mechanical implement, which leads to a minimum amount of changes in the original architecture of the mechanism, so that the overall cost should not presumably grow up too much.

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