1、90 FTM $1AI- Kinematic Analysis of Transmissions -Based on the Finite Element Methodby: A. L. Sytstra, Delft University of TechnologyAmerican Gear Manufacturers AssociationI IIITECH N ICAL PAPERKinematic Analysis of Transmissions - Based on the Finite Element Method AA. L. SytstraDelft University of
2、 TechnologyThe Statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.ABSTRACT:In order to evaluate the kinematic properties of a design of a transmission in its early stages, a compu
3、terprogram has been developed. By means of geometric reasoning a finite element model is deduced fromthe conceptual design which has been built using an experimental 3D object editor. Since theinterrelations between the objects are not specified by the designer they are found by the computer bysuppl
4、ying a set of rules. The following kinematic analysis uses a mixed Euler/Lagrange description anddetects mechanisms (possible infinitesimal displacements of the nodal points without causing strain in theelements) in the transmission. The mechanisms found are visualized by the object editor by means
5、ofalternating images of the design on the screen which gives a real idea of motion. A well-founded decisioncan be made whether the conceptual model has to be changed, rejected, or can be accepted. Themethod of analysis used gives a good start for a static and dynamic analysis.Copyright 1990American
6、Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 1990ISBN: 1-55589-568-9i. INTRODUCTION.- This chapter describes the reasons underlying the kinematicanalysis of transmissions (why), the model of the transmissionsv used (what), and the method of analysis (h
7、ow).WHY: Overdimensioning transmissions as a result of designing forextreme load situations can be avoided at the cost of a completedynamic analysis. If the response due to the stochastically varyingpower can be determined, more efficient and lighter constructionswith a longer life expectancy can be
8、 designed. The Finite ElementMethod (F.E.M) and availability of computers can be successfullyapplied here.However, it is possible that dynamic calculations based on thefinite element method show unexpected results, or that the programsused fail during the calculation. This can be caused by:- Model d
9、efinition errors.- Seen from a kinematic viewpoint the model is not properlydefined. Unforeseen mechanisms (fields of possible infinitesimaldisplacements without strain in the elements) are present.These situations result in ili conditioned or even singularmatrices, and attempts to invert them resul
10、t in program failures.Tracing the causes of these errors is a very difficult and timeconsuming job. This is why the following ideas were proposed:- use the computer in generating the model of a transmission.- After definition of the model, its kinematics should first beanalyzed before we continue wi
11、th a dynamic analysis. We will firsthave to know what the possible mechanisms are, what is going tomove, and in what rate these movements may occur.WHAT: I DRIVE I ITRANSMI88ION I IMACHINE 1A transmission can be described as the part between the drive andthe driven machine needed to couple two non-f
12、itting characteristics.Since this covers too wide an area a restriction has to be made. Afirst set up was to build up transmissions with shafts, cones,toothed wheels (gears), fixed bearings in space and beam elements(connecting rods between objects or a model of the gear case).HOW: The analysis is b
13、ased on the finite element method because:- complete automatization of the kinematic analysis is possible;- a part of the basis for the static and dynamic calculations isobtained, to which the kinematic analysis can be extended.- there are no restrictions on orientations in space.- the analysis can
14、be generalized to more element types.The results of the analysis should contain at least:- Possible infinitesimal displacements without deformation of theelements (“mechanisms“).- The distribution of the velocities in such a possible motion.We are interested in which parts are moving and at what rat
15、e.22. MODEL DEFINITION AND GENERATION. (THE USE OF COMPUTER PROGRAMS).This chapter describes how the computer can aid the non F.E.M.specialist designer in generating a finite element model for theanalysis of transmissions.Initially the computer has to be given information about theconstruction. A ha
16、nd made input file containing the finite elementmodel can only be used by specialists, since designers do not thinkin terms of finite elements (see fig. la). In order to meet prevai-ling standards about user friendliness we made use of an experimen-tal 3 dimensional menu driven object editor (writte
17、n by K. van derWerff D:X-E ; _ E ; _ _ X ; (3)This relation is called THE 0_ ORDER CONTINUITY EQUATION.Since D is non-linear, system behavior is approximated by lineariza-tion for simplicity._=DD(_)_; DD:X-E; DD is linear ; (4)D is the differentiation operator and by a “ “ (dot) we meandifferentiati
18、on with respect to time.This relation is called THE Ith ORDER CONTINUITY EQUATION whichexpresses the rate of change of deformations in terms of the rate ofchange of position parameters in a current position.The problem we want to solve is : what rate of change of positionparameters x do not result i
19、n a rate of change of deformations _ ofthe elements. This amounts to determining the kernel of the firstorder continuity equation and can be achieved by solving thefollowing equation:= DD (x)_ (5)Before solving this system, different X spaces and the conceptskinematically indeterminates and first or
20、der transfer functionare introduced (seel,4,6).X can be subdivided into three different subspaces: (6)X :the space of constant position parameters; These are given andconstant in time.Xm :the space of prescribed position parameters; These can be arbi-trarily prescribed by the designer. (e.g. the inp
21、ut velocity ofa transmission)Xc :the space of remaining position parameters;The three subspaces do not overlap; X is the Cartesian product ofX, Xa and Xc.The kinematically indeterminates are: the set of position parametersof which the velocities have to be prescribed in order to eliminate4all unwant
22、ed degrees of freedom (they then become the Xa space). Therate of change of all position parameters can be calculated in termsof the rate of changes of the kinematically indeterminates (theinputs): This is called the first order transfer function (DF)._=DF5 m ; DF:_-X ; DF is linear. (7)The first or
23、der transfer function is deduced from the first ordercontinuity equation by two successive steps:- subdivide the DD(_) matrix into three submatrices (DD(_), .DD(_h,D_D(_) corresponding to the subdivision of the X space into x ,Xc and Xm.= DD(_)5 = DCD(_)_ c + DmD(_)5 m + DD(_)5 (8a)- substitute 5=_
24、in (8a) (constant position parameters)_c -1 “this leads to : x = -DCD(_) DmD(_)_ m = DF5 m (9)For the sake of clearness we rewrite (Sa) and (9) into (8b) and (i0)using a new notation in which the vectors _ ,_m,_o are placed abovetheir corresponding matrices._c _m 5oX X XU = DD(_) = DCD(_) DmD(_) DD(
25、_) (8b)Inverting the DCD(_) submatrix in the left part of (i0) results inthe right part which is similar to equation (9).5 5 5 5 5“ 5DCD(_) DmD(_) DD(_) =_ I -DF * =_ (I0)The * means : “is not of interest here“.The remaining problem is that the Am space has to be detected by thecomputer itself since
26、 we are in search for it. This can be achievedby decomposing the original X space of the system in a X space andthe remaining part called xr“_ By a sweeping matrix-procedurethe Xr_al“ space can be decomposed into a Xc space and the requiredX_ space. By sweeping a matrix we mean adding rows and multi
27、plyingthem by a scalar until unit columns appear, as many as possible. Toobtain the identity matrix I in (ii) we may have to interchangethe D=_ai_Dcolumns of (5) and the corresponding variables with them.Solving the system O = DD(_)_ can now be represented with the aidof the consecutive matrices:_r.
28、ain 50 5 A 5 sweeping _ I B = I DranD(X) DD(x) I :_ 0 : F I (11)L J EL D C J_c _m _oIf the matrix obtained is x xA compared with (i0) it appears _ that we have foundthat: 0 = I I -DF * I (i0)- - The variables of the Xc space marked by A are a set of kinemati-cally indeterminates.- The submatrix B co
29、ntains the first order transfer function.- The submatrices C and D contain only zeros.- The submatrix F is not of interest here.- The variables of the E space marked by E are found when moreequations are present than needed to solve the problem. The set ofstresses related with these deformations are
30、 called the staticallyindeterminates.Conclusion:By calculating the kernel of the first order transfer function it ispossible to determine:- The kinematically indeterminates.(they can be considered as aprescribed input of the possible motion). A number of differentsets are possible but the computer f
31、inds only one of them.- The first order transfer function.(how is the motion of thetransmission as a result of that input motion)- The statically indeterminates. (pre stresses in the construction)The results of the kinematic analysis can be used as part of theinput for static/dynamic calculations.4
32、THE EULERAND LAGRANGE DESCRIPTIONS.We did not state in chapter 3 how the X space can be described. Inthis chapter three possibilities are given:- Description of the X space in a fixed global frame of reference(Lagrange).- Description of the X space in a local coordinate system attachedto the element
33、s (Euler).- A mixed Euler-Lagrange description.In the Lagrange description the X space is completely described ina global fixed coordinate system. Since the generalized deformationsof the elements are set up in a local coordinate system attached tothe body (where the x axis is mostly in the longitud
34、inal directionof the object) transformation matrices are needed to obtain thedesired result. These matrices express displacements(rotations) inthe local frame of reference in terms of displacements(rotations) inthe global system.In the Euler description the X space is completely described inlocal fr
35、ames of reference attached to the elements. This means thatan element and related points are referred to in the same coordinatesystem. Use of coordinate transformations is reduced to a minimum.An example of the kinematic analysis of a shaft(P,Q) in free space6is given according to the theory of chap
36、ter 3. The 6 degrees offreedom found are visualized in the figs 2 and 3.Though the results of the kinematic analysis by means of an Eulerdescription have to be translated from local to the global frame of vreference, the method has some major advantages.- the use of transformation matrices is reduce
37、d to a minimum.- when calculating gear ratios, the longitudinal rotations neededare found easily if the local x-axis is chosen in this direction.- the displacement parameters of the nodal points of the elementscan be subdivided into the X and Xr“ spaces which enables aneasy definition of kinematic b
38、oundary conditions and implemen-tation of different types of bearings.In the Euler description the coupling of two elements with differentlocal coordinate systems cannot be achieved by common nodal points(a nodal point may refer to one frame of reference only). Aconnecting element is needed but the
39、use of this element results inexplosive growth (12 displacement parameters and 6 deformationsextra per element) of the first order continuity equation. By mixingthe Euler and Lagrange description it is possible to avoid thisdisadvantage. In our case this amounts to the following:The Euler descriptio
40、n can be retained without any problem forshafts,cones,bearings and wheels. Local coordinate systems of theseelements always coincide with those of their nodal points.A beam element can be attached to objects with different directions.If we leave out the connecting element, the first order continuity
41、equation of this object has to be formed in terms of the local fra-mes of reference of the points between which the beam is situated.These coordinate systems can be different and may even be the globalframe of reference. This is why it is called mixed Euler/Laaranqe.For our purpose the mixed Euler/L
42、agrange method is most favorableabove the other two.5 THE FINITE ELEMENTS.The finite element model is constructed with five types ofelements: the shaft, cone, bearing, gear and truss elements. Thegear pair element is described in more detail.The shaft and cone elements. (see fig 4)Seen from a kinema
43、tic point of view these elements are the same. Sixdeformations can be constructed for this element, in its local frameof reference (1,3).eI = uq - up (elongation) ik = length ofe2 = ik(_q - _p) (torsion ) the elemente3 = wp - wq - l_Sp (bending )e4 =-Wp + Wq + ik8q (bending ) i0 = reference_5 =-vp +
44、 vq - IkOp (bending ) length.e6 = vp - Vq + l_q (bending )7The bearing element. (see fig 5)A In order to set up a system of deformations for this element, thedisplacement parameters of the outer ring are transformed to thev local frame of reference (which must coincide with the local frameof the sup
45、ported shaft) attached to the inner ring. Since the localX axis of both rings coincide, the relative angle u is of importanceonly.Three suitable deformations are suitable, viz.:- local Xp direction i = -up + uq- local yp direction 2 = -vp + vq*cos(u) - wq*sin(u)- local z_ direction e3 -wp + vq*sin(_
46、) + Wq*COS(_)This model is appropriate only for bearings which permit a certaininclination of the inner ring relative to the outer one. Formodeling bearings which do not permit this torsion, two moredeformations are needed concerning relative angular displacements:- local yp direction 4 = 10(-Sp+Sq*
47、cos(u) - 0q.sin(u)- local Zp direction es 10(-0p+8_*sin(_) + _q*cos(u)Fixation of the bearings in space is attained by marking theposition parameters of the outer-ring as “O“ instead of “C“ ; theyare added to the X space. Various types of bearings can be imple-mented by assigning different “C“ and “
48、O“ variables to the nodalpoint of the outer ring (see figs 6,7,8).The gear pair element.This element is constructed from two wheel elements and establishesthe kinematic relation between the two. Setting up deformations fordifferent types of gear pairs is a very complicated 3D-matter 5.Therefore we u
49、se an approach in which the first order continuityequation is derived from the equilibrium equation of the gear pair.This is possible because of the duality between displacements andforces, and between deformations and stresses (3,6). If wesucceed in determining the direction of the normal force between theteeth (the direction of the line of action), we are able to set upthe external forces and moments needed for equilibrium of eachwheel. We will show in the following part that an easy
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