CN108052760A - A kind of gear pair nonlinear kinetics computational methods - Google Patents
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Abstract
The invention discloses a kind of gear pair nonlinear kinetics computational methods.Consider that vibration displacement to the feedback effect of flank of tooth dynamic Contact situation during gear pair real-world operation, calculates dynamic mesh stiffness and dynamic comprehensive meshing error non-linear excitation.Gear pair dynamic contact analysis with system dynamics solution procedure is combined, realizes the calculation process of system " excitation-response-feedback " phase closed loop.This method can obtain the gear pair nonlinear dynamic phenomenon as caused by the parameters such as tooth surface error and correction of the flank shape, can more realistically dynamic behavior of the simulation system during dynamic Contact, improve dynamics calculation accuracy.
Description
Technical field
The invention belongs to dynamics technology fields, are related to a kind of gear pair Nonlinear dynamic models method.
Background technology
Gear system dynamic problem is always industrial quarters and the hot issue of academia's research.Gear pair is in real-world operation
In the process, influenced, shown multiple be subject to factors such as time-variant mesh stiffness, gear error, correction of the flank shape, backlash and mesh impacts
Miscellaneous dynamic response.The basic law of the vibration of different situations lower tooth wheel set, impact and noise is studied, to designing low-vibration noise
There is important directive significance with the gear train assembly of high reliability.
Gear time-variant mesh stiffness and tooth surface error are the important driving sources of two classes for causing system vibration.Previous research is usually
Mesh stiffness and gear error excitation are separately considered, but are actually interactional between the excitation of this two class.On the one hand, by
The influence of tooth surface error, installation error and support system deformation, the flank of tooth are likely to occur part contact phenomena, so as to influence comprehensive change
The size of shape and mesh stiffness.And on the other hand, due to registration and the joint effect of engagement wheel between cog deformation, the reality of error
Actuating quantity should be less than original tooth surface error amplitude.Using error free Gear Meshing Stiffness and assume that comprehensive meshing error amplitude substitutes into
Dynamics calculation will certainly influence computational accuracy.
The gear pair contact analysis method established at present is all based on static meshing state, and method there is no to consider gear vibration
The feedback effect of displacement.In gear real-world operation, the vibration displacement of variation can make the actual contact condition of the flank of tooth with being produced during static state
Raw difference, influences the vibrational excitations such as actual dynamic mesh stiffness and dynamic comprehensive meshing error, further changes gear pair dynamic
Response.When vibration displacement is larger, the flank of tooth may be changed by being partially disengaged contact in vibration completely disengages contact, generates by force
Strong nonlinear response.It focuses mostly on both at home and abroad on the dynamic (dynamical) research of mission nonlinear in as caused by backlash at present,
And have experimental study and show that anti backlash gear pair flank of tooth drop out of gear, response amplitude jump, chaos and fork etc. may also occur by force
Nonlinear dynamic phenomenon.The complete drop out of gear phenomenon of the gear teeth occurred in big load condition test, conventional with people recognize very
Conflict to generating.These nonlinear kineticses as caused by non-backlash factor respond, it is necessary to which foundation is more really examined
Considering the kinetic model of the actual dynamic Contact state of gear can accurately be obtained.
The content of the invention
It is an object of the invention to consider feedback influence of the gear pair dynamic response to vibrational excitation, by gear pair contact point
Analysis is combined with system dynamics solution procedure, a kind of gear pair nonlinear dynamic analysis method is established, with truer mould
Intend gear pair dynamic Contact process so as to study mission nonlinear vibration characteristics.
In order to achieve the above objectives, the specific technical solution of the present invention is：
A kind of gear pair nonlinear kinetics computational methods, comprise the following steps：
Step 1：The gear pair dynamic contact analysis model for considering Basic parameters of gear and flank of tooth foozle is established, is calculated
Dynamic mesh stiffness and dynamic comprehensive meshing error；
Step 2：Gear minor bend-torsion-axial direction Coupling Dynamic Model is established using concentrated quality method；
Step 3：Gear pair dynamic contact analysis with kinetics equation solution procedure is combined, is integrated using Newmark
Nested Secant Method solves gear pair dynamic Contact problem, realizes the analysis process of " excitation-response-feedback " phase closed loop.
The realization process of step 1 is：
(1.1) the mesh stiffness average of error free gear is calculated, while calculates each position of engagement theoretical contact line length, is obtained
Mesh stiffness onto contact unit line length；
(1.2) flank of tooth foozle is converted to flank of tooth normal direction and is measured and be overlapped, it is initial to obtain each contact point
Normal gap；
(1.3) judge the contact condition of each point reality, actual contact line length is obtained, by the contact unit of (1.1) step
Line length rigidity value calculates actual dynamic mesh stiffness and dynamic comprehensive meshing error.
The realization process of step 2 is：
Using the concentrated quality method in Theory of Vibration, driven wheel engagement is considered as comprising bending-torsion-axial direction mostly certainly
By the mass-spring system spent, the factor of Gear Meshing Stiffness, comprehensive meshing error and bearing rigidity is included in, using newton the
Two laws establish the system motion differential equation.
The realization process of step 3 is：
(3.1) given system displacement, speed and acceleration initial value are progressively accumulated using the implicit Newmark of unconditional stability
Point-score solving system differential equation of motion；
(3.2) for Newmark integrations each time step, it is necessary to solve nonlinear displacement balance equation, using changing
Into Newton methods-Chord iterative method solve；
(3.3) in each iteration of Secant Method, transmission of the conversion to path of contact normal direction is calculated according to each immediate movement value
Error, and substitute into the contact equation calculation in step 1 new dynamic mesh stiffness and stiffness matrix, carry out next iteration until
Meet iteration precision, then carry out Newmark integration future time steps and calculate；
(3.4) by two neighboring intrinsic displacement mesh cycle response difference, judge whether Newmark integrations reach stable state：
If reaching stable state, the results such as the response of output system final mean annual increment movement, dynamic load distribution；Otherwise, future time step meter is continued
It calculates.
Compared with prior art, beneficial effects of the present invention are：
The present invention considers feedback effect of the vibration displacement to flank of tooth dynamic Contact situation during gear pair real-world operation, meter
Dynamic mesh stiffness and dynamic comprehensive meshing error non-linear excitation are calculated.By gear pair dynamic contact analysis and system dynamics
Solution procedure is combined, and realizes the calculation process of system " excitation-response-feedback " phase closed loop.The computational methods are more realistically
The dynamic engagement process of gear pair is simulated, real-time vibrational excitation is obtained by gear pair dynamic contact analysis, and is included in
In dynamic analysis process analysis procedure analysis, avoid in traditional power credit analysis and consider that static state connects without considering gear pair contact condition or only
The calculation error that the situation of touching is influenced and brought more accurately obtains mission nonlinear dynamic response to study nonlinear vibration motivation
Reason.This method can obtain the gear pair nonlinear dynamic phenomenon as caused by the parameters such as tooth surface error and correction of the flank shape, can be more realistically
Dynamic behavior of the simulation system during dynamic Contact improves dynamics calculation accuracy.
Description of the drawings
Fig. 1 is nonlinear dynamic analysis calculation flow chart；
Contact point distribution map when Fig. 2 is a certain position of engagement on plane of action；
Fig. 3 is gear pair kinetic model schematic diagram；
Fig. 4 is different rotating speeds lower tooth wheel set dynamic response result；
Fig. 5 is different rotating speeds lower tooth wheel set dynamic mesh stiffness；
Fig. 6 is distributed for different rotating speeds dedendum flank dynamic load.
Specific embodiment
As shown in Figure 1, a kind of gear pair nonlinear kinetics computational methods of the present invention, establish gear pair dynamic Contact with being
System Vibration-coupling model, specifically includes following steps：
Step 1：The gear pair dynamic contact analysis model for considering Basic parameters of gear and flank of tooth foozle is established, is calculated
The vibrational excitations such as dynamic mesh stiffness and dynamic comprehensive meshing error；Specially：
(1) the mesh stiffness average of error free gear is calculated using GB3480-1997, while it is theoretical to calculate each position of engagement
Contact line length obtains the non-linear excitation in contact unit line length；
(2) by flank of tooth foozle (including total profile deviation, circular pitch deviation and spiral deviation etc.) conversion to flank of tooth normal direction
It is measured and is overlapped, obtain the initial normal gap of each contact point；
(3) judge the contact condition of each point reality, obtain actual contact line length, by the contact unit line length of (1) step
Rigidity value is spent, calculates actual dynamic mesh stiffness and dynamic comprehensive meshing error.
Step 2：Gear minor bend-torsion-axial direction Coupling Dynamic Model is established using concentrated quality method；Specially：
Using concentrated quality method classical in Theory of Vibration, driven wheel engagement is considered as comprising bending-torsion-axial direction
Multivariant mass-spring system is included in Gear Meshing Stiffness, the comprehensive factors such as meshing error and bearing rigidity, using ox
The second law that pauses establishes the system motion differential equation.
Step 3：Gear pair dynamic contact analysis with kinetics equation solution procedure is combined, is integrated using Newmark
Nested Secant Method solves gear pair dynamic Contact problem, realizes the analysis process of " excitation-response-feedback " phase closed loop.Specifically
For：
(1) given system displacement, speed and acceleration initial value, using the implicit Newmark step_by_step integrations of unconditional stability
Method solving system differential equation of motion；
(2) for Newmark integrations each time step, it is necessary to solve nonlinear displacement balance equation, can be used and change
Into Newton methods-Chord iterative method solve；
(3) in each iteration of Secant Method, the transmission that conversion to path of contact normal direction are calculated according to each immediate movement value misses
Difference, and the contact equation calculation in step 1 new dynamic mesh stiffness and stiffness matrix are substituted into, next iteration is carried out until full
Sufficient iteration precision then carries out Newmark integration future time steps and calculates；
(4) by two neighboring intrinsic displacement mesh cycle response difference, judge whether Newmark integrations reach stable state.If
Reach stable state, the results such as the response of output system final mean annual increment movement, dynamic load distribution；Otherwise, continue future time step to calculate.
Technical scheme is described in more detail with reference to specific embodiment.
As shown in Figure 1, a kind of nonlinear dynamic analysis method that the present invention is established, is as follows：
(1) the average value L of gear pair contact length is calculated firstm, such as formula (1)：
Lm=εαb/cosβb (1)
Wherein, εαFor transverse contact ratio, b is the facewidth, βbFor Base spiral angle.
Fig. 2 is contact line and contact point distribution map on a certain position of engagement plane of action of helical gear, p in figurebtFor end face base
Knuckle-tooth is away from liTo be segmented the length of contact line shared by contact point i.
The total profile deviation, circular pitch deviation and spiral deviation of contact point are superimposed, can obtain initial normal gap εi.Judge
The initial normal gap ε of contact pointiWith the size of transient behavior transmission error DTE, if εi<DTE illustrates that contact point i is contacted；Otherwise,
Then contact point i not in contact with.According to the number of actual contact point, actual contact line total length L can be acquireda.By Basic parameters of gear
The mesh stiffness average k of error free gear pair is calculated according to GB3480-1997m0.Due to mesh stiffness and actual contact line length
Trend be consistent, gear pair dynamic mesh stiffness k can be obtainedmdFor：
The instantaneous dynamic comprehensive meshing error e of gear pairmdFor：
(2) gear pair kinetic model is modeled using concentrated quality method, and kinetic model is as shown in Figure 3.Under in figure
It marks p and represents driving wheel, subscript g represents driven wheel.By each gear vibration displacement to path of contact direction projection, it is opposite path of contact can be obtained
It is to dynamic total deformation：
δd=DTE-emd=VX-emd (4)
Wherein, X={ xp,yp,zp,θzp,xg,yg,zg,θzg}T, the displacement column vector of two gears of expression；DTE=VX is tooth
Wheel set transient behavior transmission error；V is the projection vector that all directions displacement is converted to path of contact direction, be can be represented by the formula
In formula, rpAnd rgThe respectively base radius of driven wheel；βbFor Base spiral angle；Angleα is engagement
Angle, φ install phase for driven wheel.
According to Newton's second law, can establish system motion differential equation group is：
In formula, mi(i=p, g) is respectively driving and driven gear quality；Izi(i=p, g) is respectively driven wheel around z-axis
Rotary inertia；cmdDynamically to engage damping；kij,cij(i=p, g；J=x, y, z) be respectively gear i along j to support stiffness
And damping；T1And T2The respectively torque of gear 1 and gear 2.
Formula (4) is substituted into equation group (6), being organized into matrix form is：
In formula：M, C and K are respectively mass of system matrix, damping matrix and stiffness matrix；P is external applied load column vector；E is
Comprehensive meshing error vector；F is total Vibrating Load vector.
(3) it is Newmark integrations is nested with Secant Method progress, consider that instant contact situation changes when solving dynamic response
The analysis process of " excitation-response-feedback " phase closed loop is realized in excitation variation caused by becoming.Specific embodiment is：
1) to the dynamic differential equation group of such as formula (7), initial engagement rigidity and comprehensive meshing error is given, is formed and always swashed
Shake load vectors initial value F0, while given displacement initial value X0, velocity original valueAnd calculate acceleration initial value
2) integration step Δ t, Newmark integral parameter α, β are selected, and calculates integral constant (β >=0.5 and α >=0.25
(0.5+β)2)
3) stiffness matrix K, mass matrix M and the damping matrix C of t+ time Δts are formed, calculates effective load of t+ time Δts
Lotus：
4) effective stiffness matrix is formedFormer differential equation of motion (7) is converted to the form of formula (11).
Equation group format surface shown in formula (11) is system of linear equations, but from Such analysis, system dynamic respond
Can influence dynamic Contact situation makes dynamic mesh stiffness kmdIt changes, and then changes effective rigidity matrixSo this side
Journey is essentially Nonlinear System of Equations, can be solved by Secant Method, and basic ideas are：
A. formula (11) is rewritten asFor simplicity, t+ Δs t is omitted herein；
B. iterations k=1 is made, gives iteration initial value X0And X1, calculate f (X0) and f (X1)；
C. according to Chord iterative method equations Xk+1：
D. the transmission error DTE of k+1 iteration is calculatedk+1=VXk+1, and it is dynamic to substitute into the solution of the contact model in step (1)
State mesh stiffness kmdWith dynamic comprehensive meshing error emd, according to formula (6), (9)~(10) calculate Keqv,And calculate f (Xk+1)；
E. judge | f (Xk+1) | ＜ ε (εFor the convergence precision of setting) it is whether trueIf so, Xt+Δt=Xk, iteration terminates；
Otherwise, k=k+1, return to step c.
5) acceleration and speed of t+ time Δts are calculated：
6) judge the dynamic respond in former and later two mesh cycles, if the two is close enough, exit calculating；Otherwise, return
3) step, carries out the calculating of subsequent time.
The gear pair nonlinear kinetics computational methods proposed according to the present invention are carried out using the spur gear pair shown in table 1
Example calculation.Gear wheel has 5 μm of barrel shape line deviation during calculating, and pinion gear is error free.
Table 1
Fig. 4 is the corresponding transmission error root-mean-square value of different rotating speeds under the conditions of raising speed and reduction of speed.It can be seen from the figure that it rises
There is apparent response amplitude chattering when speed and reduction of speed, illustrate through this paper algorithms in computing system nonlinear response
It is effective.
Dynamic mesh stiffness curve of the gear pair in 100rpm, 2100rpm and 2150rpm when Fig. 5 is raising speed.From Fig. 5
In as can be seen that the rectangular ripple of dynamic mesh stiffness of the gear pair in 100rpm, it is bent with the mesh stiffness under dead weight
Line is identical.Dynamic mesh stiffness during 2100rpm declines in the middle part of bidentate area, and dynamic mesh stiffness during 2150rpm
Numerical value in part bidentate area is 0, illustrates that complete drop out of gear state occurs in this region.
2100rpm (Fig. 6 a) and 2150rpm (Fig. 6 b) in comparison diagram 6 although when contact force state can be seen that the two
Rotating speed difference is little, but real load distribution gap is apparent.
Those listed above is a series of to be described in detail only for feasibility embodiment of the invention specifically
Bright, they are not to limit the scope of the invention, all equivalent implementations made without departing from skill spirit of the present invention
Or change should all be included in the protection scope of the present invention.
Claims (4)
1. a kind of gear pair nonlinear kinetics computational methods, which is characterized in that comprise the following steps：
Step 1：The gear pair dynamic contact analysis model for considering Basic parameters of gear and flank of tooth foozle is established, calculates dynamic
Mesh stiffness and dynamic comprehensive meshing error；
Step 2：Gear minor bend-torsion-axial direction Coupling Dynamic Model is established using concentrated quality method；
Step 3：Gear pair dynamic contact analysis with kinetics equation solution procedure is combined, is integrated using Newmark nested
Secant Method solves gear pair dynamic Contact problem, realizes the analysis process of " excitation-response-feedback " phase closed loop.
A kind of 2. gear pair nonlinear kinetics computational methods according to claim 1, which is characterized in that the reality of step 1
Now process is：
(1.1) the mesh stiffness average of error free gear is calculated, while calculates each position of engagement theoretical contact line length, obtains list
Mesh stiffness on the contact line length of position；
(1.2) flank of tooth foozle is converted to flank of tooth normal direction and is measured and be overlapped, obtain the initial normal direction of each contact point
Gap；
(1.3) judge the contact condition of each point reality, obtain actual contact line length, by the contact unit line length of (1.1) step
Rigidity value is spent, calculates actual dynamic mesh stiffness and dynamic comprehensive meshing error.
A kind of 3. gear pair nonlinear kinetics computational methods according to claim 1, which is characterized in that the reality of step 2
Now process is：
Using the concentrated quality method in Theory of Vibration, driven wheel engagement is considered as comprising bending-torsion-axial direction multiple degrees of freedom
Mass-spring system, be included in the factor of Gear Meshing Stiffness, comprehensive meshing error and bearing rigidity, it is fixed using newton second
Rule establishes the system motion differential equation.
A kind of 4. gear pair nonlinear kinetics computational methods according to claim 1, which is characterized in that the step 3
Realization process be：
(3.1) given system displacement, speed and acceleration initial value, using the implicit Newmark step by step integrations of unconditional stability
Solving system differential equation of motion；
(3.2) for Newmark integration each time step, it is necessary to solve nonlinear displacement balance equation, use is improved
Newton methods-Chord iterative method solves；
(3.3) in each iteration of Secant Method, calculated and converted to the transmission error of path of contact normal direction according to each immediate movement value,
And the contact equation calculation in step 1 new dynamic mesh stiffness and stiffness matrix are substituted into, next iteration is carried out until meeting
Iteration precision then carries out Newmark integration future time steps and calculates；
(3.4) by two neighboring intrinsic displacement mesh cycle response difference, judge whether Newmark integrations reach stable state：If it reaches
To stable state, the results such as the response of output system final mean annual increment movement, dynamic load distribution；Otherwise, continue future time step to calculate.
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