on gear modelling in multistage rotary vane engines - b. librovich.pdf

7/27/2019 On Gear Modelling in Multistage Rotary Vane Engines - B. LIBROVICH.pdf

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Meccanica 39: 4761, 2004.

c 2004 Kluwer Academic Publishers. Printed in the Netherlands.

On Gear Modelling in Multistage Rotary Vane Engines

B. LIBROVICH, R.W. TUCKER and C. WANGDepartment of Physics, Lancaster University, Lancaster LA1 4YB, UK

(Received: 28 November 2002; accepted in revised form: 21 January 2003)

Abstract. A discussion of the dynamics of a multistage rotary vane engine is given in terms of a simplified model

for combustion driving torques, power dissipation, and torque transmission. Torque transmission is effected by

conjugate gear pairs attached to each unit of the engine. An argument for the design of such pairs is presented so

that unwanted torque fluctuations in a flywheel attached to a member of the pair can be significantly attenuated. It

is suggested that a variant of simple Cosserat dynamics offers a useful modelling tool for discussing the complex

interaction between interacting gear teeth. A quasi-stationary analysis is used to place bounds of a particular choice

of conjugate gear coupling in the presence of such interactions. It is concluded that a multistage rotary vane engine

with at least two units can be usefully coupled to a single flywheel via a well-defined conjugate gear system that

attenuates unwanted torque fluctuations over a broad range of rotary speeds.

Key words: Internal combustion engines, Gear modelling, Non-circular gears, Rotary engines.

1. Introduction

The conventional reciprocating internal combustion engine is often regarded as one of the

most efficient mechanical devices for producing rotary motion ever invented. Despite con-

tinuous development few rotary engines (RE) have challenged the market supremacy of the

reciprocating engine [15]. However with increasing concern over the environment and the

need to conserve energy resources a number of RE designs are again being actively recon-

sidered here and overseas. There are a number of reasons for the resurgence of interest inthis engine. Early designs were often given only cursory investigation and the limited success

of the Wankel-type engine may have had an adverse effect on the development of viable

alternatives. It also appears from the published literature that many alternative proposals were

not exposed to a unified dynamical analysis based on firm physical principles that correlate

the design of the combustion chamber with effects of inertial forces and gas dynamics [6].

Furthermore in recent years a number of developments have made feasible the possibility

of more critical scientific assessments of the basic concepts behind the RE and the means

to enhance its relevance in the new millennium. These developments include the fabrica-

tion of new engineering materials, improvements in sealing techniques, the sophistication

of computer controlled ignition, fuel injection and torque feedback, and the degree to which

mathematical simulations based on computationally inexpensive methods can replace costlybench testing of performance characteristics. This paper is concerned mainly with the latter.

A mathematical model of a novel RE is explored, based on the authors experience with

vibrational analysis and the control of torsional vibrations in rotary feedback systems [7].

Preliminary mathematical investigations presented here suggest that a number of innovations

Author for correspondence: Tel.: +44-1524-593281, Fax: +44-1254-844037,

e-mail: [email protected]


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48 B. Librovich et al.

offer dramatic improvements in both the stability characteristics of the system and the control

of torque vibrations.

The analysis is based on a system of non-linear differential equations that model the

rigid body dynamics of the engine. The deviation from rigidity due to gear deformation is

estimated by modelling gear teeth as Cosserat rods. A quasi-stationary analysis used to place

bounds on such a deformation shows that the rigidity assumption is well maintained for a

wide range of practical flywheel rotary speeds. Each rotary vane unit consists of a cylinder

(with appropriate ignition, intake and exhaust ports), partitioned by two pairs of vanes (rotary

pistons) into four compact combustion chambers as shown in Figure 1. Each vane-pair rotates

through a finite angle on independent concentric shafts aligned along the axis of the cylinder.

Each shaft is coupled to a common flywheel via a pair of conjugate gears (see Figure 2).

The geometry of the gear mechanism is largely responsible for the nature of the vibrational

characteristics induced by the driving torques in the thermodynamic cycle. A major concern

with most designs to date has been the presence of vibrations arising from the variable angular

speeds of the rotary pistons leading to amplified stresses among the coupled gears and torque

fluctuations in the flywheel as rotary speeds increase. The purpose of this investigation is

to ameliorate these problems by exploring the following two directions: A multistage rotary

vane engine (MRVE) is envisaged by connecting thermally isolated rotary vane units via aspecially designed gear mechanism in such a way that torque vibrations interfere to minimise

torque fluctuations. Secondly a mechanism is proposed for maintaining a regular oscillatory

motion superimposed on a steady rotation of the rotary pistons based on recent technological

developments in the manufacture of robust non-circular gears [810].

Figure 1. A schematic representation of a rotary vane unit used in a MRVE. It consists of a cylinder with intake and

exhaust ports, partitioned by two pairs of vanes into four combustion chambers: I for intake, II for compression,

III for combustion and IV for exhaust. 1 denotes the angular position of vane-pair 1 consisting of vanes 1 and 3

while 2 denotes the angular position of vane-pair 2 consisting of vanes 2 and 4 as indicated with circled numbers.


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On Gear Modelling in Multistage Rotary Vane Engines 49

Figure 2. A MRVE composed of two combustion units. This represents a possible layout of the power transmission

connecting rotary vanes with the flywheel (shown on the right side). The outer walls of the units are omitted in

order to show the vane-pairs. Dark and light vane-pairs are connected to the dark and light non-circular gears,

respectively.

2. Equation of Motion for a Multistage Rotary Vane Engine

A simple mathematical model based on rigid body dynamics is constructed here to simulate

the dynamics of a MRVE consisting of m rotary vane units. The kinematics is formulated in

terms of the angular positions of the flywheel, (t), and of the rotary vane k in unit , k;(t),

at time t. Here 1 k 4 and 1m. Within each unit the constraints:

3;(t) = 1;(t) + , 4;(t) = 2;(t) + (1)

hold and hence the pairs of vanes 1, 3 and 2, 4 are referred to as vane-pairs 1 and 2, respec-

tively. In this manner the vane angles k;(t) for k = 1, 2 are treated as vane-pair anglesand they are measured counterclockwise as opposed to the flywheel angle (t) being measured

clockwise. This simplifies the formulation of the coupling of these angles through conjugate

gears:

k;(t) = k;((t)) (2)

for some coupling functions k;( ) to be specified.

Suppose the flywheel has an effective rotary inertia I and is coupled to the rotary vane

units via a gear whose axis is parallel to that of the rotary vane units and with a separation L.

When a combustion chamber is undergoing a combustion phase, the resulting increase in gas

pressure applies a driving torque on the corresponding rotary vanes. This torque will depend

on the actual ignition processes as well as specific geometries of the vanes. In modelling thedynamic torque acting on any rotary vane assumed to have identical shapes, it is useful to treat

the torque due to atmospheric pressure (in the absence of ignitions) as a reference value and

denote it by Q. We introduce scales of length, time and mass:

L = L, T =

I

Q, M =

I

L2. (3)


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Henceforth quantities are non-dimensionalised using these scales. Thus the equation of the

rotary motion for the flywheel subject to a reaction torque K(t) due to engine output and

torques Ck;(t) transmitted from vane-pairs k in units is

(t) =

m

=1

2

k=1

Ck;(t) + K(t), (4)

where the dot in superscript denotes the time derivative. The equation of motion for vane-pair

k in unit with common effective rotary inertia I subject to the gas dynamic driving torques

Gk;(t) and reaction torque Ck;(t) from coupling with the flywheel is similarly expressed as

Ik;(t) = Ck;(t) + Gk;(t). (5)

Each Gk;(t) contains contributions from torques Gij,(t) (>0) due to gas dynamics in the

individual chamber between vanes i and j according to

G1;(t) = G2;(t) = G12;(t) + G23;(t) G34;(t) + G41;(t). (6)

The torque transmission between the flywheel and the vane-pairs is modelled by the set of(power balance) equations

Ck;(t)(t) + Ck;(t)k;(t) = Fk;(t)(t), (7)

where the (non-positive) torque Fk;(t) represents the loss of power due to gearing friction.

The time derivative of (2) yields

k;(t) = k;((t))(t) (8)

where denotes differentiation and k; ((t)) represents the corresponding rotary speed ratio.

It is assumed that (t) > 0 and k;((t)) > 0 hold so that the vane-pairs never reverse the

directions of shaft rotation. It follows from (7) and (8) that

Ck;(t) = k;((t))Ck;(t) + Fk;(t). (9)

Furthermore by differentiating (8),

k;(t) = k;((t)) +

k;((t))(t)

2 (10)

and using (5) we have

Ck;(t) = Ik;((t)) +I

k;((t))

2 Gk;(t). (11)

Substituting (9) and (11) into (4) gives the flywheel rotary equation of motion in the form

1 +

m=1

2k=1

Ik;((t))2

(t)

=

m=1

2k=1

Ik;((t))

k;((t))(t)

2 + k;((t))Gk;(t) + Fk;(t)

+

+ K(t), (12)


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which incorporates the effects of gas dynamics, gearing friction and external loading on the

flywheel in term of functions Gk;(t), Fk;(t) and K(t), which in general depend on and .

Since () is bounded in if the size of these terms do not increase with faster than 2,

the asymptotic rotary motion of the flywheel for (t) 1 is governed by (12) by neglectingthem. Therefore the angular speed of the flywheel for 1 tends to a constant if the coupling

functions k;( ) satisfy the relation

m=1

2k=1

k;()k;( ) = 0, (13)

that is,

m=1

2k=1

k;( )2 = constant (14)

for all t. Solutions to these having the properties k;() > 0, 2;( ) =

1;( + /2) and

2;( ) = 1;( ) require at least m = 2. For a MRVE with two units (14) can be satisfied

with the choice

1;1( ) = (), (15)

2;1( ) =

+

2

, (16)

1;2( ) =

+

4

, (17)

2;2( ) =

+

3

4

, (18)

where

() = + sin(2 ) (19)

for some constant amplitude parameter with < /4. The form (19) will be referred to as

a harmonic coupling function. A possible layout of the power transmission connecting the

rotary vanes with the flywheel using the above coupling functions is illustrated in Figure 2.

In order to compare the performance offered by (19) with the dynamical behaviour associated

with a non-harmonic coupling function we also consider

() = + {(1 + a) sin(2 ) + a sin(6 )} (20)

with a (small) parameter a. If a = 0 then (20) simply reduces to (19). Hence the above repre-sents a one-parameter family of coupling functions that deviates from the harmonic coupling

in (19). This choice allows the effect of non-harmonic coupling functions to be assessed by

varying the parameter a.

By taking advantage of the two expansions followed by two contractions of chamber

volumes per revolution of rotary vanes based on either (19) or (20), a four-stroke type internal

combustion cycle may be initiated in each unit.


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3. Construction of Gear Pitch Curves

The coupling functions introduced in (2) are used to generate the corresponding pitch curves

of conjugate gears. To illustrate this procedure consider two families of closed curves in a

Euclidean plane with Cartesian coordinates (x,y):

P : (x = 1 R() cos( ),y = R() sin( )), (21)

P : (x = R() cos( ),y = R() sin( )) (22)

for , R and some functions R(), R(), where 0, < 2 (see Figure 3). In thexy plane the family P represents the counterclockwise rotation of a rigid shape (associated

with R()) about the origin by increasing whereas the family P represents the clockwise

rotation of another rigid shape (associated with R()) about the point (1, 0) by increasing .

If the rotations of P and P are to describe rolling of the two shapes with a contact point

along the x-axis, where = and = , then

R( ) + R() = 1. (23)

If the angles and are correlated by

= () (24)

for some function () (with (0) = 0, (2 ) = 2 and ( ) = ( + 2 )) it followsfrom (23) and the rolling condition that

( ) =d

d=

R()

R(()). (25)

Figure 3. The closed curve P represents the counterclockwise rotation of a rigid shape about the origin by

increasing whereas closed curve P represents the clockwise rotation of another rigid shape about the point

(1, 0) by increasing . They give rise to pitch curves if the rotations are related by rolling as described in

Section 3.


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Figure 4. Possible conjugate gear profiles associated with the pitch curves illustrated in Figure 3. The gears

represented by the profiles on the left and right may be connected to a rotary vane-pair and the flywheel of a

MRVE, respectively.

Hence (23) and (25) yield

R() =()

1 + (), (26)

R() =1

1 + (1()). (27)

Given a coupling function (), (26) and (27) define a pair of coupled pitch curve families,

P and P . In the MRVE modelling these are used to define the pitch curves associated with

the conjugate gears connected to the flywheel and a vane-pair, respectively. This is illustrated

in Figure 4.

4. Numerical Simulations Based on a Simple Gas Dynamic Model

To explore the behaviour of the MRVE model in the presence of driving and loading terms we

choose a viscous type external torque

K(t) = (t) (28)

with a (non-negative) damping parameter .

Furthermore a simple model is adopted for gas dynamics within the combustion chambers

between adjacent vanes with angles 1 and 2 belonging to a vane-pair satisfying (2). This

model expresses the driving torques:

G12;(t) = G(1;(t),2;(t)), (29)

G23;(t) = G(2;(t),3;(t)), (30)

G34;(t) = G(3;(t),4;(t)), (31)

G41;(t) = G(4;(t),1;(t) + 2 ) (32)


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in terms of some function G(1, 2). Its form depends on the angular span of any rotary

vane, (0 < < /2), as well as the ratio of the maximum to minimum angular span

of any chamber denoted by ( 1). The chamber between 1 and 2 has an angular span2 1 bounded by / and , where

=( 2)

+ 1

. (33)

Since the volume enclosed by the chamber is proportional to its variable angular span, the

parameter may be identified as the compression ratio for internal combustion processes. It

follows that

= 1

4. (34)

In terms of these parameters we take

G(1, 2) =

I: 1 if 0 < 2

II:

21

if

2 <

III: h 21

if < 32

IV: 1 if 32 < 2

(35)

where = ((1 + 2)/2 /2) mod2 , is the ratio of the principal specific heat capacitiesof the gas and the ignition factor h a positive parameter with typical value between 1 and

7. It models the increase of pressure due to ignition. As indicated the function G(1, 2) is

expressed in terms of four phases (IIV) for each chamber per revolution and accommodates

the conventional 4-stroke type thermodynamic cycle (cf. Figure 1). Transitions between these

phases take place at the extrema of the chamber volumes (proportional to 2 1 ). Asillustrated in Figure 5 each phase simulates the driving torque applied to the corresponding

Figure 5. Behaviour of the gas dynamic torque G(1, 2) given in (35) with parameters = 3/2, h = 5,

= 2/5, and = 9 versus /2 where = ((1 + 2)/2 /2). This simulates the torque produced

by a chamber between adjacent vanes with angles 1 and 2 which are in turn determined by the flywheel angle

via a coupling function. There are 4 phases: I (intake), II (compression), III (combustion) and IV (exhaust).


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vane due to

Phase I: an intake stroke where the chamber is open through an intake port. The resulting

gas torque is approximately that produced by atmospheric pressure. This gives 1

in the units adopted here.

Phase II: a compression stroke modelled by an adiabatic process where the gas torque in-

creases from 1.Phase III: a combustion stroke also modelled by an adiabatic process. The energy input is rep-

resented by an initial gas torque which is h times that at the end of the compression

stroke.

Phase IV: an exhaust stroke where the chamber is re-open through an exhaust port with a

released gas torque 1, associated with atmospheric pressure.

Numeric integration of (12) is performed based on (28), (35) and neglecting gearing

dissipation:

Fk;(t) = 0. (36)

In these simulations the ignition factor h is chosen to be 5, viscous parameter chosen to be

1/5 and the vane angular span and compression ratio are set to be 2/5 and 9, respec-

tively, corresponding to amplitude parameter = /25. For the adiabatic processes = 3/2is adopted. It is interesting to compare the simulation results using the harmonic coupling

function (19) with those using the non-harmonic coupling function (20) with a = 1/25 and

different initial flywheel rotary speeds.

With the above parameter values and initial conditions (0) = 0 and (0) = 1, the equationof motion (12) using either (19) or (20) yields numeric solutions that give rise to similar

behaviours in the flywheel rotary speed (t) and acceleration (t) (see Figure 6).

Figure 6. The simulated flywheel rotary accelerations (t) versus time in T based on harmonic (full) and

non-harmonic (dotted) gear coupling with the initial rotary speed (0) = 1 produce fluctuations with comparable

amplitudes. It indicates that there is no significant effect of gearing geometry on the MRVE dynamics at low rotary

speeds.


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For the same parameter values but with initial conditions (0) = 0 and (0) = 57.5,the equation of motion (12) using (20) produces numeric data for the flywheel rotary speed

and acceleration that exhibit fluctuations noticeably greater than those using (19). These are

Figure 7. Comparison between the simulated flywheel rotary speeds (t) versus time in T based on a harmonic

(full) and non-harmonic (dotted) gear coupling with the initial rotary speed (0) = 57.5. Both curves have steady

mean angular speeds due to the external dissipation (28). However the rotary speed based on the non-harmonic

coupling displays a much larger fluctuation. This highlights the advantage of the harmonic over non-harmonic

gear coupling at high rotary speeds.

Figure 8. Comparison between the simulated flywheel rotary accelerations (t) versus time in T based on a

harmonic (full) and non-harmonic (dotted) for gear coupling with the initial rotary speed (0) = 57.5. Both

curves have approximately zero means due to the external dissipation (28). However the rotary acceleration based

on the non-harmonic coupling displays a much large fluctuation. This highlights the advantage of the harmonic

over non-harmonic gear coupling at high rotary speeds.


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demonstrated in Figures 7 and 8. Such a dramatic contrast highlights the effect of gearing

geometry on the vibrational characteristics of MRVEs.

5. Estimate of Gear Deflection

The above analysis has been based on the imposition of algebraic constraints between the

flywheel and vane-pair angles through (2). Given a prescribed coupling function such as (19),

the corresponding pair of pitch curves are constructed as outlined in (21)(27). The actual

conjugate gears are obtained by fitting tooth profiles around the pitch curves such that the

coupled angles of rotation are correlated by the prescribed coupling function assuming rigidity

of these gears (cf. Figure 4).

In reality the meshing of gears with teeth is imperfect. Even as rigid bodies they experience

periods of slipping that give rise to friction losses. As deformable structures the teeth undergo

a degree of bending and shear leading to further elastic hysteresis losses in general. In such

circumstances the idealised motion of the contact point along the line joining the centres of

rotation of the pitch curves can change. A complete description of this motion is a complicated

problem since it depends on a piecewise formulation of the gear system dynamics including

details of the free boundary elastodynamics of each gear under time varying transmissiontorques and non-linear friction [11]. Such effects cause deviations from the pitch curve angular

correlation = ().One can attempt to model the salient deformation features of the teeth by regarding them

as rigidly mounted elastic beams with free ends. A quasi-static approximation may be based

on the assumption that each gear-tooth tip responds to a slowly varying y-component of a

contact force derived from the torque being transmitted by the gear to which it is attached.

The deflections of each gear-tooth tip, attached to the gears with pitch curves P and P, are

then expected to be dominantly in the y-direction. Denoting these deflections to be y (t) and

y(t), respectively, to a first approximation one expects that the angular correlation is modified

to

() = () + y (t)R(())

+ y(t)R()

. (37)

It remains to estimate y (t) and y(t) in terms of the acting torques.

A variant of the simple theory of Cosserat rods [12, 13] offers one modelling strategy. Its

advantage over simpler beam models is that it can effectively accommodate the constitutive

properties of the gear tooth (including visco-elastic damping) and in principle predict the

effects of elastic wave excitations of different type in the gear tooth. In this article we do not

exploit this degree of generality, being satisfied with a rough estimate of bounds based on a

quasi-static analysis. The tip deformation of each contacting tooth is estimated from solutions

to the Cosserat equations of motion with a time dependent tip force where the acceleration and

velocity of the tooth (relative to the rigid gear to which it is attached) is neglected. It is part

of a continuing research programme to validate such an approximation scheme by exploringbeam deflections in a fully dynamic environment in terms of Cosserat rod modelling.

A Cosserat rod of unstrained length is described by its axis rrr = rrr(s,t) parametrisedby arc length parameter s (0 s ) at time t. The ends where s = 0 and s = definethe root and tip of the gear tooth, respectively. The orientation of each cross-section with

area A = A(s) at rrr(s,t) is specified by two orthogonal unit vectors, ddd1 = ddd1(s,t) and

ddd2 = ddd2(s,t), such that the unit vector ddd3 = ddd1 ddd2 is normal to the cross-section. Within


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such a cross-section each material point may be labelled by the coordinates (1, 2, s) such

that this point has the position rrr(s,t) + 1 ddd1(s,t) + 2 ddd2(s,t) in space.Thus the axis rrr and triad of directors, {ddd1, ddd2, ddd3}, constitute six continuous degrees of

freedom associated with a gear tooth that can sustain deformations such as stretch, dilation,

shear, flexure and torsion. Given a constant gear mass density it is convenient to orient the

directors {ddd1, ddd2, ddd3} with the principal axes associated with the cross-section such that in thedirector basis the moment of inertia per unit length has non-zero components:

I11 = 1, I22 = 2, I33 = 3 (38)

in terms of the area moments:

1 =

A

22 d1 d2, 2 =

A

21 d1 d2, 3 = 1 + 2. (39)

In terms of the partial differential operator strain variables are chosen as vvv = vvv(s,t) =srrr and uuu = uuu(s,t) such that sdddi = uuu dddi for i = 1, 2, 3. The angular velocity www = www(s,t)of each triad is similarly defined such that tdddi = www dddi . Constitutive relations correlatethese strains to a contact force nnn = nnn(s,t) and contact torque mmm = mmm(s,t) and their director

components in turn give rise to tension, shear force, bending moment and twisting couple. Thedynamics of a Cosserat tooth in the presence of a body force fff = fff (s,t) and body torquelll = lll(s,t) follows from Newtons laws:

At trrr = snnn + fff , (40)

t(III www) = smmm + vvv nnn + lll. (41)

The partial differential system is closed by employing the classical Kirchhoff constitutive laws

for elastic steel:

m1 = E1u1, m2 = E2u2, m3 = G3u3 (42)

n1 = GAv1, n2 = GAv2, n3 = EA(v3 1) (43)

in terms of the Youngs modulus E, shear modulus G and vector components mi = mmm dddi,ni = nnn dddi , ui = uuu dddi, vi = vvv dddi in the director basis.

The above general formalism can be used to estimate the static deflection of a gear tooth

due to a contact force with magnitude F in the gear plane transverse to the tooth axis. For

simplicity this tooth is idealised to have constant cross-section area A and area moments

1, 2, 3. As shown in Figure 9 we parameterise the configuration of this tooth in a local

Cartesian basis by

rrr = {s + x(s)}iii + y(s)jjj (44)

and

ddd1 = sin (s)iii + cos (s)jjj , (45)

ddd2 = kkk, (46)

ddd3 = cos (s)iii + sin (s)jjj , (47)


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Figure 9. Deformation of an idealised gear tooth in the xy plane withbasis {iii, jjj} rigidly attached to the gear body.

Its tip (on the right) is subject to a force in the y-direction with magnitude F. This force induces a displacement

xiii + yjjj as well as rotation of directors ddd1, ddd3 with an angle .

where the planar translational and rotational displacements of the tooth cross-sections are

represented by x(s), y(s) and (s), respectively. The position and orientation at s = 0 are

constrained:

rrr(0) = 0, ddd1(0) = jjj , ddd2(0) = kkk, ddd3(0) = iii, (48)

whereas the other end at s = is subject to the contact force and moment:

nnn() = F jjj , mmm() = 0. (49)

It follows from (48) and (49) that

0 =

(0) = x(0) = y(0), (50)0 = s () = s x() = GA{s y() ()} F. (51)

Substituting (44)(47) into (40), (41) for fff = lll = 0 and neglecting second and higher orderterms in x(s), y(s) and (s) gives

ss x(s) = 0, (52)

ss y(s) = s (s), (53)

ss (s) =GA

E{sy(s) + (s)}, (54)

where = 2. Solving (52)(54) with boundary conditions (50) and (51) yields

x(s) = 0, (55)

y(s) =F

GAs +

F

2Es2

F

6Es3, (56)

(s) =F

Es

F

2Es2, (57)


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giving rise to the maximum transverse displacement

y() =F

GA+

F 3

3E. (58)

Although based on a static analysis, (58) may provide a first approximation to the tooth de-

flection with coupled gear wheels at a rotary speed that does not excite significant elastic

deformation. In order not to excite elastic waves the maximum rotary speed of the gears

should be less than the angular frequency of the lowest eigenmode of a gear tooth. For an

axial excitation this demands

2

E

. (59)

In this case if (58) is used to estimate the tooth deflection on a gear connected to, say, the 1st

vane-pair in unit 1, then the force magnitude F may be approximated by F C()/R( ) atthe vane angle () in terms of the associated reaction torque C( ), radius R() and

coupling function (). For 1 with harmonic coupling it follows from (11) and (15)(19)that

C(t) I((t))(t)2, (60)

and therefore (27) yields

F C()

R() I()(1 + ())2. (61)

The tooth deflection leads to a contribution to the deformation in the vane angle given by

=y()

R()=

F

R( )

GA+

3

3E

(62)

in coupling with the flywheel. For || 1 it follows from (27), (58) and (61) that thecondition on the flywheel rotary speed is given by

4I

GA+

3

3E

1/2(63)

provided < 1. With I 1, 0.01 0.1 and E, G 105 in units adopted here thisindicates (for deflections, based on the approximations above), that the harmonic coupling

scheme is effective in smoothing torque oscillations of a typical MRVE with high angular

speed.

6. Conclusions

A discussion of the dynamics of a multistage rotary vane engine has been given in terms of a

simplified model for the combustion driving torques, power dissipation, and torque transmis-

sion. The torque transmission is effected by conjugate gear pairs in each unit. An argument

for the design of such pairs has been given so that unwanted torque fluctuations in a flywheel

attached to a member of the pair can be significantly attenuated. Effects due to the depar-

ture of a gear pair as idealised rolling rigid pitch curves have been discussed in terms of


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elastic deformations of gear teeth modelled as attached rods. It is suggested that a variant of

simple Cosserat dynamics offers a useful modelling tool for discussing the complex interac-

tion between interacting gear teeth. A quasi-stationary analysis has been used to place bounds

on the usefulness of a particular choice of conjugate gear coupling in the presence of such

interactions. It is concluded that a MRVE with at least two units can be usefully coupled to

a single flywheel via a well-defined conjugate gear system that attenuates unwanted torque

fluctuations over a broad range of rotary speeds.

Acknowledgements

We are grateful to R. Carter and M. Widden for stimulating interactions and to EPSRC for

financial support in this investigation.

References

1. Wankel, F. and Ansdale, R.F., Rotary Piston Machines: Classification of Design Principles for Engines,

Pumps and Compressors, Iliffe Books, London, 1965.

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