AGMA 90FTMS1-1990 Kinematic Analysis of Transmissions - Based on the Finite Element Method《传输的运动学分析.基于有限元方法》.pdf
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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(
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