Method for carrying out a non-linear analysis of gears in the frequency domain

Abstract

A method for carrying out a non-linear analysis of at least two gears in the frequency domain provides for generating a gear mathematical model, define a motion equation of such model and solve such equation with a method of the coupled harmonic balancing, the implementation of method comprising the solution of such equation simultaneously considering a static condition and a dynamic condition of the interaction between the gears.

Description

[0001] METHOD FOR CARRYING OUT A NON-LINEAR ANALYSIS OF GEARS IN

[0002] THE FREQUENCY DOMAIN

[0003] The present invention relates to a method for carrying out an analysis of the non-linear forced response of gears .

[0004] The gears are widespread components and act as critical components in various mechanical systems . However, in their service life they are subject to vibrations which can lead to such components yielding under fatigue and high number of cycles ; such vibrations are mainly generated by the variation in the equivalent contact stiffness which couples the wheels during their rotation . As a matter of fact , it has long been known that the possibility of having, in various instants , a number of pairs of teeth in variable contact causes a stiffness fluctuation, which results in a variable dynamic behaviour which induces loads that can significantly deteriorate the fatigue resistance of the gears and also vibrations which increase the noise level .

[0005] The dynamics of the gears is also affected by the type of contact which occurs between the pairs of teeth of meshing gears , intermittent-type contact of the teeth which therefore operate by compression when they mesh with each other or which can also be lost in some operating conditions . This defines a highly non-linear behaviour which characterises these mechanical components (the gears ) .

[0006] Therefore, there it is essential to be in a position to simulate the operating condition of the gears so as to be able to estimate the amplitude of the oscillations or of the vibrations and therefore predict the service life of such components .

[0007] In order to be able to understand the behaviour of the gears it is essential to be able to predict and control the vibration levels in various operating conditions and meshing configurations which comprise such mechanical components .

[0008] There are known many methods for carrying out non-linear analysis on the gears .

[0009] Generally, these methods can be classified into approaches in the time domain and approaches in the frequency domain . The methods in the time domain, which involve the direct integration over time, are predominant and widely implemented on algorithms or commercial software . However, approaches in the frequency domain offer some advantages compared to methods in the time domain . For example, the harmonic balancing method (HBM) may obtain the response in the standard operating condition of the system without solving the transient behaviour, resulting in acceleration calculations .

[0010] Despite these advantages , the methods based on HBM reveal drawbacks . In order to precisely capture the intermittent contact behaviour there may arise the need to solve the equations associated with many multiple harmonics of the fundamental one . Furthermore, the use of HBM involves the limitation of having to work on the assumption that the response to be calculated is periodic, but this is not always true .

[0011] In addition, these methods generally depend on quasistatic preliminary analysis to obtain the static transmission error ( STE ) . Such error is defined as the difference between the current actual position of the gears during the rotation and the ideal position they would take should the wheels be perfect stiff bodies . Such difference may derive from production errors or from the elastic deformation of the teeth that inevitably occurs during the contact .

[0012] Upon obtaining the STE values for a motion period, in the HBM-based methods such values are used as dynamic exciters to overcome the dynamic problem obtaining the dynamic transmission error (DTE ) . Therefore, the two results are combined to obtain the overall transmission error .

[0013] Therefore, it is clear that each prior art HBM-based method requires time to be implemented and it always depends on the preliminary statistical assessment or analysis .

[0014] Furthermore, the STE as source of excitation to solve the dynamic problem represents an approximation and requires a sequence of preliminary analysis which can be computationally difficult to solve (especially on large gear models , industrial standards ) .

[0015] The solution methods based on the direct integration of motion equations of the models over time provide for solving the transient starting from the initial arbitrary conditions up to obtaining the standard operating condition . Given that the models in question are scarcely damped, the duration of the transient may even be very high . The need to solve the transient result s in significant computational costs even for methods based on integration over time .

[0016] Therefore, the present invention relates to providing a method which allows to predict the dynamic behaviour of gears in an accurate and computationally efficient manner, so as to be able to predict the vibration levels thereof .

[0017] In particular, an object of the invention is to provide a method of the type mentioned above which allows to carry out non-linear dynamic analysis of gears during the design thereof , analysis which can be carried out within a short period of time with respect to those required to carry out the analysis through the methods disclosed in the prior art and mentioned above .

[0018] Another object is to provide a method of the type mentioned above which allows to carry out the analysis mentioned above accurately so as to be a valid support to the design of the gears and the preliminary study thereof within mechanical structures .

[0019] A further object is to provide a method of the type mentioned above which, thanks to its rapidity of performance, allows within a short time to perform a number of simulations on the gears that is greater than the number of simulations which can be carried out , over the same period of time, with the methods disclosed by the prior art and described above ; this translates in a clear saving in terms time required in the analysis with clear cost-effectiveness .

[0020] These and other objects which shall be apparent to the person skilled in the art are attained by a method according to the attached claims .

[0021] For a better understanding of the invention, the following drawings are attached hereto, purely by way of non-limiting example, wherein : figure 1A shows a flow chart showing various implementation steps of the method according to the invention; figure IB shows a numerical embodiment of the invention according to Figure 1A; figure 2 shows a gear with some parts omitted for the sake of greater clarity and the forces which act thereon during use thereof ; figure 3 shows a model with concentrated parameters of two gears adapted to cooperate with each other, with some parts omitted for the sake of greater clarity; figure 4 shows in a chart a comparison between the Root Mean Square (RMS ) of the oscillating component of the transmission error as a function of the excitation frequency obtained through the method according to the invention ( solid line curve P ) and through the direct integration over time (dotted line curve Q) , using a known model to solve non- linear contact problems of the "penalty-based" type and in the presence of variable play over time ; figure 5 shows a comparison of the solutions obtained with the method according to the invention ( solid line curve P ) with the experimental results represented by the dotted line curve K .

[0022] The present invention relates to a method for carrying out the non-linear dynamic analysis of gears by using a microprocessor system or device . In particular, such method operates according to the harmonic balancing method (HBM) where, however, the static transmission error STE is evaluated simultaneously with the dynamic transmission error, that is STE and DTE are simultaneously determined during the step of solving of the motion equation of a model of the gears in question . It is no longer necessary to distinguish the transmission error between static and dynamic, given that the transmission error calculated with the method according to the invention automatically takes both contributions into account .

[0023] Therefore, such method will be defined as the coupled harmonic balancing method (HBM) .

[0024] Firstly, as shown in figure 1A and with reference to figure 2 , using prior art methods there is created a mathematical model of a gear 10 (block 100 in figure 1 ) which is shown in figure 2 ; in particular, figure 2 also shows the external generic stresses acting on the wheel (Fext) and the non-linear loads (FCon) resulting from the contact with other gears 21 . In figure 2 , the wheel 10 is shown having only one tooth for the sake of simplicity .

[0025] Therefore, there are defined, in a per se known manner, the mass , stiffness and damping structural matrices (block 110 of figure 1A) of a pair of gears 10 and there is defined (block 120 ) the motion equation of the wheels in the time domain .

[0026] Such equation can be shows as :

[0027] MX + CX + KX = Fext+ Fcon

[0028] 1 wherein the generalised displacements of the mathematical model and the derivatives thereof are respectively collected in the vectors X, X and X .

[0029] It should be observed that the expression generalised displacements is used to indicate for example linear displacements , the rotations , the modal displacements , etc . The matrices M, C and K represent the mass , damping and stiffness matrices .

[0030] The vector Fconcollects the generalised contact forces applied to the teeth 20 of the wheel 10 due to the contact that the latter have with other bodies (that is the teeth of the other wheels ) . The vector Fextcontains all the other generalised external forces applied to the degrees of freedom of the wheel, ( for example from the hub-shaft interface which drives it in rotation) . It should be observed that the expression "generalised forces" is used to indicate, besides the forces strictly also referred to as torques , the modal forces and other physical quantities which are capable of modifying the energy of the system through a displacement of the degrees of freedom thereof .

[0031] Usually, the vector Fextis known a priori and it represents the load that the external elements exert on the wheel ; it may also depend or not depend on time, but it is in any case a period function .

[0032] The vector Fconis not known a priori and it depends on the displacement X of the degrees of freedom of the wheel 10 and the derivatives thereof , besides time . Considering the speed of rotation of the wheel as constant and equal to O), after reaching the standard operating condition each point of the wheel will occupy a determined position at regular time intervals T = 2nlc>) . This will require that also the values of the contact forces will be periodic with the same period T . Having established that all quantities present in the equation 1 are periodic, they can be expressed as Fourier series :

[0033] The vectors Xh rFext,h and Pcon.h (determined in the block 130 of the method of figure 1A) are the complex Fourier coefficients in the / i-ith order of the displacements and of the forces . The Fourier series are truncated in the H-ith order which is arbitrarily large . The choice for H of a very high value (>15 ) ensures a high accuracy of the solution . On the contrary, very small values of H (<3 ) can lead to an excessively approximate solution . The choice of the total number of harmonics used for the solution of the equation must also take into account the fact that together with H there also increases the computational cost : it usually 3 < H < 6 represents a good compromise between the accuracy of the solution and calculation time required to solve the problem . The h = 0 order represents the static problem, and it is addressed in this formulation simultaneously with all the other orders .

[0034] In the formulae, a) represents the angular velocity of the wheels and ] = V— 1 represents the imaginary unit .

[0035] Therefore, the equation 1 can be rewritten by replacing the quantities in terms of Fourier series . As disclosed by the harmonic balancing principle , the single terms of the sums for each order h are equivalent (block 140 figure 1A) . Therefore, for a generic order it is observed that :

[0036] (—h2a>2M +jha>C + K)Xh- Fcon h- Fext h= 0

[0037] Given that the vector FCon,h isafunction of Fourier coefficients of the displacement of all the orders , the system of equations thus expressed is potentially fully coupled .

[0038] The motion equation in the frequency domain may be resolved with an iterative method, for example with the Newton-Raphson method . The Fourier coefficients of the contact non-linear forces can be obtained, for example, through Fourier integers or with an Alternating Frequency Time (AFT) scheme .

[0039] By way of example, herein described in greater detail is a possible iterative algorithm with AFT for the numerical solution of the equation 7. The description below is provided purely by way of non-limiting example of a possible algorithm adapted to solve the equation 7 . However, the equation may also be solved with different algorithms or methods , without this affecting the implementation and validity of the method according to the invention .

[0040] At each iteration of the iterative method :

[0041] 1 . once all the harmonic components of the tentative vector solution Xhhave been known, one can calculate the vector of the displacements in the time domain X(t) through the Inverse Fast Fourier Transform ( IFFT) .

[0042] 2 . Once X(t) has been known, one can calculate, through the implemented contact model (per se known) , the contact force vector in the time domain

[0043] 3 . Using the equation 6 (using the Fast Fourier Transform, FFT, alone ) one can return to the motion equation in the frequency domain and proceed with the subsequent iteration .

[0044] Therefore, according to the invention, the method provides for solving the problem of the gears also using the 0 order, associated with the static . Therefore, such problem is not solved a priori, in a manner separated from the rest of the equations , but simultaneously with the rest of the equations .

[0045] The prior art methods that solve the dynamics of the gears in the frequency domain, first solve the 0 harmonic to obtain the STE, which is in turn used to obtain the DTE which depends on all harmonics h > 0.

[0046] Therefore, the invention sets out to overcome the distinction between the static and the dynamic case simultaneously solving the whole problem keeping it coupled .

[0047] The method reveals various advantages :

[0048] • the solution obtained is more accurate from a theoretical standpoint , given that it allows that the static solution can be affected by the dynamic solution,

[0049] • it allows to avoid any preliminary analysis ,

[0050] • being very general, the formulation can be applied to various types of gears (that is cylindrical gears , conical gears , epicyclic gears etcetera) .

[0051] However, there arises the need for an expression of the contact forces Fconas a function of the displacements , a matter solved in literature with various contact models . In order to better describe the present invention with reference to figure IB, now let us consider the following embodiment (provided solely by way of non-limiting example ) of an analysis carried out on a pair of gears as shown in figure 3 .

[0052] Therefore, considering the model shown in figure 3 , the motion equations of the two wheels can be written as : l

[0053] The inertias of the two wheels are respectively / xand / 2, c is a generic proportional damping coefficient (considered equal for the two wheels ) , Fcis the contact force exchanged by the two wheels and T is the torque applied to the torsional degrees of freedom . Multiplying the motion equations by the radius of the single wheels and subtracting member by member, shows :

[0054] By way of non-limiting example, the problem was solved by way of example for a specific case in which the wheels are identical, therefore resulting in the simplified equation :

[0055] Introducing an auxiliary coordinate x = 02r—$irthe equation 11 can be rewritten as :

[0056] There has been expressed the motion equation in a structure referring to that of the equation 1, therefore, from here one can proceed by applying the steps outlined according to the blocks 130 and 140 mentioned further above .

[0057] Lastly, in order to be able to solve the motion equation, there has been introduced a non-linear contact model of the penalty-based type in the presence of a play variable over time . This allows to obtain the contact force exchanged between the two wheels Fc ras a function of the ancillary coordinate x .

[0058] Figure IB shows a flow chart 150 relating to a numerical solution example of the differential equations associated with the block 140 .

[0059] The flow chart 150 shows that the equations can be solved iteratively . However, it should be observed that the content of the description in the flow chart 150 is a per se known operative method and it is only an example of how to solve the equations according to block 140 .

[0060] According to the described iterative method, there are used different values of the variable (angular velocity of the gears at contact ) and the motion equation 7 is solved using them so that the result of such motion equation is close to 0. There is defined a tolerance that is the maximum value of result that deviates from 0 and the calculation of equation through different values of the variable is stopped when the error becomes smaller than the tolerance .

[0061] Therefore, more particularly, figure 1A shows how the method subject of the invention is solved : there is identified the angular velocity of interest (block 150A) , corresponding to a standard condition of the gear, and it is introduced into the equation 7; in block 150B the tentative solution is updated (a zero vector is usually used at the first iteration) . In the subsequent block 150C the non-linear contact forces indicated above are calculated after expressing the result of the block 150B in the time domain with an Inverse Fast Fourier Transform . The result of the block 150C is once again expressed in the frequency domain through a Fast Fourier Transform so as to calculate the residue of the motion equation (block 150D) . The residue of the motion equation represents the error between the tested tentative solution and the exact solution . As the iterations of the numerical method proceeds , the residue will tend to progressively decrease, indicating that the tested approximate solutions are increasingly closer to the actual solution .

[0062] Therefore, block 150E is evaluated : if the residue is greater than the maximum acceptable error, there is generated a new tentative solution ( returning to block 150B) . Otherwise, the obtained solution is saved and it is evaluated whether other angular velocities of interest (block 15 or F) ; in case of positive response, one can proceed with the subsequent angular velocity of interest (block 150A) . Otherwise the calculation ends (block 150G) .

[0063] In order to verify the effectiveness of the method according to the invention there were carried out non-linear dynamic analysis on gear models through direct integration of the motion equation over time, using the Runge-Kutta method, used as reference, and through the present coupled harmonic balancing method . The results of such verification are shown in figure 4 where there is observed the perfect superimposition of the simulations carried out for the two cases .

[0064] Furthermore, given that for the studied wheels in literature there are available experimental test results , figure 5 shows a comparison of the simulations obtained with the method according to the invention with the experimental results mentioned above . As observable, the method offers a method that replicates , in terms of quality, the obtained experimental data .

[0065] Although a preferred embodiment of the invention has been described, it shall be solely considered by way of nonlimiting example, given that the present invention is limited to the scope of protection of the claims that follow .

Claims

CLAIMS1 . Method for carrying out , through a system or device provided with a microprocessor, a non-linear analysis of the forced response to the stresses of a pair of gears ( 10 ) in a design step thereof , said method comprising :- a definition of such gears through a mathematical model (100 ) ;- the subsequent definition ( 120 ) of equations of the motion of such model of gears ;- the solution of such equations in the frequency domain by means of the method for the coupled harmonic balancing,- characterised in that the solution of the equations occurs simultaneously considering, for each predefined angular speed at which the gears rotate reached at standard operating condition, a static transmission error and a dynamic transmission error .2 . Method according to claim 1 , characterised in that the equation of the motion isMan X + CX + KX = Fext+ FconX, X and X = Vector of the generalised displacements of the mathematical model and derivatives thereof ;M = Mass matrix of the wheel model ;C = Damping matrix of the wheel model ;K = Stiffness matrix of the wheel model ;Fcon=Vector of the generalised contact forces acting on the wheels ;Fext=Vector of the generalised external forces applied to the wheels , said quantities of said equation therefore being expressed as Fourier series , each series comprising an initial order (h = 0) whose solution represents a static transmission error .3 . Method according to claim 2 , characterised in that the coefficients of the equation of the motion expressed as Fourier series are expressed as follows :Wherein h = generic order of each Fourier series , h = 0 order associated with the static equation, whose solution if the static transmission error, a) = angular speed of the gears] = V—1 imaginary unit4 . Method according to one or more of the preceding claims , characterised in that there is used, for the nonlinear analysis of the forced response of the gears , the motion equation expressed in terms of Fourier series having, for a generic order, the following form(—h2a>2M +jha>C + K)Xh- Fcon h- Fext h= 05 . Method according to any one of the preceding claims , characterised in that there is carried out ( 150C) thecalculation of the non-linear contact forces , and there is calculated ( 150D) the residue of the motion equation, said residue being compared with a tolerance value, that is maximum acceptable error of the aforementioned residue, as a function of the comparison therefore the angular velocity value of the wheel used for carrying out said calculation of such contact forces being varied or the calculation of such forces after updating the angular velocity used for the calculation of the non-linear contact forces is reiterated ( 150B) .6 . Method according to claim 4 , characterised in that the motion equation is solved with an iterative method with an Alternating Frequency Time scheme .7 . Method according to claim 6, characterised in that for each iteration of the iterative method, starting from the motion equation :- there calculated a displacement vector in the time domain through the Inverse Fast Fourier Transform,- once the displacement vector is known, the contact force vector in the time domain is calculated,- by using a Fast Fourier Transform, the contact force vector is transformed in the frequency domain and the motion equation in the frequency domain is calculated,- subsequent iteration is carried out .8 . Method according to claim 1 , characterised in that the mathematical model is a finite elements model .

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