CN108052760B - Nonlinear dynamics calculation method for gear pair - Google Patents
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Abstract
The invention discloses a gear pair nonlinear dynamics calculation method. And the feedback effect of the vibration displacement on the dynamic contact condition of the tooth surface in the actual running process of the gear pair is considered, and the dynamic meshing rigidity and the dynamic comprehensive meshing error nonlinear excitation are calculated. The dynamic contact analysis of the gear pair is combined with the system dynamics solving process, and the calculation process of the closed loop of the system 'excitation-response-feedback' is realized. The method can obtain the nonlinear dynamics phenomenon of the gear pair caused by parameters such as tooth surface errors, modification and the like, can simulate the dynamics behavior of the system in the dynamic contact process more truly, and improves the dynamics calculation accuracy.
Description
Technical Field
The invention belongs to the technical field of dynamics, and relates to a gear pair nonlinear dynamics modeling method.
Background
The problem of gear system dynamics has been a hot issue of research in the industry and academia. In the actual operation process of the gear pair, the gear pair is influenced by factors such as time-varying meshing rigidity, gear errors, modification, backlash, meshing impact and the like, and shows complex dynamic response. The basic rules of the vibration, the impact and the noise of the gear pair under different conditions are researched, and the method has important guiding significance for designing a gear transmission system with low vibration noise and high reliability.
Gear time-varying meshing stiffness and tooth surface error are two important types of excitation sources that cause system vibration. Previous studies often considered meshing stiffness and gear error excitations separately, but in practice these two types of excitations are interacting. On the one hand, under the influence of tooth flank errors, mounting errors and support system deformations, partial contact phenomena of the tooth flanks may occur, which affect the magnitude of the combined deformation and the meshing stiffness. On the other hand, the actual contribution of the error should be smaller than the original tooth flank error magnitude due to the combined influence of the contact ratio and the tooth flank deformation of the meshing wheel. The calculation accuracy is influenced by adopting the meshing rigidity of the error-free gear and assuming that the comprehensive meshing error amplitude is substituted into the dynamic calculation.
The existing established gear pair contact analysis method is based on a static meshing state, and the feedback effect of gear vibration displacement cannot be considered. When the gear actually runs, the changed vibration displacement can cause the difference between the actual contact state of the tooth surface and the static state, so that the vibration excitation such as the actual dynamic meshing rigidity, the dynamic comprehensive meshing error and the like is influenced, and the dynamic response of the gear pair is further changed. When the vibration displacement is large, the tooth surface may transition from partially out of contact to fully out of contact during vibration, producing a strong nonlinear response. At present, the research on the nonlinear dynamics of the system at home and abroad is mostly focused on the research caused by the backlash, and the existing experimental research shows that the gear pair without the backlash can have strong nonlinear dynamics phenomena such as tooth surface disengagement, response amplitude jump, chaos, bifurcation and the like. The complete disengagement phenomenon of the gear teeth during the test under the heavy load condition conflicts with the conventional knowledge of people. The nonlinear dynamic response caused by the non-backlash factors can be accurately obtained only by establishing a more real dynamic model considering the actual dynamic contact state of the gear.
Disclosure of Invention
The invention aims to consider the feedback influence of the dynamic response of the gear pair on vibration excitation, combine the gear pair contact analysis with the system dynamics solving process, and establish a gear pair nonlinear dynamics analysis method so as to simulate the gear pair dynamic contact process more truly and study the nonlinear vibration characteristics of the system.
In order to achieve the purpose, the specific technical scheme of the invention is as follows:
a gear pair nonlinear dynamics calculation method comprises the following steps:
step 1: establishing a gear pair dynamic contact analysis model considering gear basic parameters and tooth surface manufacturing errors, and calculating dynamic meshing rigidity and dynamic comprehensive meshing errors;
step 2: establishing a gear pair bending-torsion-axial coupling dynamic model by using a concentrated mass method;
and step 3: the dynamic contact analysis of the gear pair is combined with the dynamic equation solving process, and the dynamic contact problem of the gear pair is solved by adopting a Newmark integral nested frustum method, so that the analysis process of an excitation-response-feedback phase closed loop is realized.
The implementation process of the step 1 is as follows:
(1.1) calculating the mean value of the meshing stiffness of the error-free gear, and simultaneously calculating the length of a theoretical contact line at each meshing position to obtain the meshing stiffness on the length of a unit contact line;
(1.2) converting the tooth surface manufacturing error to a tooth surface normal direction for measurement and superposing to obtain an initial normal gap of each contact point;
and (1.3) judging the actual contact state of each point to obtain the actual contact line length, and calculating the actual dynamic meshing rigidity and dynamic comprehensive meshing error according to the unit contact line length rigidity value in the step (1.1).
The implementation process of the step 2 is as follows:
a centralized mass method in a vibration theory is adopted, the meshing of a driving wheel and a driven wheel is regarded as a mass-spring system containing bending-torsion-axial multi-degree of freedom, the factors of gear meshing rigidity, comprehensive meshing error and bearing rigidity are included, and a system motion differential equation is established by adopting a Newton's second law.
The implementation process of the step 3 is as follows:
(3.1) giving initial values of system displacement, speed and acceleration, and solving a system motion differential equation by adopting an unconditionally stable implicit Newmark step-by-step integral method;
(3.2) for each time step of the Newmark integration, solving a nonlinear displacement balance equation, and adopting an improved Newton method-chord method to carry out iterative solution;
(3.3) when the truncation method is iterated every time, calculating a transfer error converted to the normal direction of the meshing line according to an instantaneous displacement value every time, substituting the transfer error into the contact equation in the step 1 to calculate a new dynamic meshing rigidity and rigidity matrix, carrying out next iteration until the iteration precision is met, and then carrying out Newmark integration and next time step calculation;
(3.4) judging whether the Newmark integration reaches a steady state or not through the displacement response difference in two adjacent meshing periods: if the steady state is reached, outputting the final displacement response, dynamic load distribution and other results of the system; otherwise, continuing to calculate the next time step.
Compared with the prior art, the invention has the beneficial effects that:
the invention considers the feedback effect of vibration displacement on the dynamic contact condition of the tooth surface in the actual running process of the gear pair, and calculates the dynamic meshing rigidity and the nonlinear excitation of dynamic comprehensive meshing error. The dynamic contact analysis of the gear pair is combined with the system dynamics solving process, and the calculation process of the closed loop of the system 'excitation-response-feedback' is realized. The calculation method simulates the dynamic meshing process of the gear pair more truly, obtains real-time vibration excitation through the dynamic contact analysis of the gear pair, and takes the vibration excitation into the dynamic analysis process, avoids the calculation error caused by the fact that the contact condition of the gear pair is not considered or only the influence of the static contact condition is considered in the traditional dynamic analysis, and obtains the nonlinear dynamic response of the system more accurately to study the nonlinear vibration mechanism. The method can obtain the nonlinear dynamics phenomenon of the gear pair caused by parameters such as tooth surface errors, modification and the like, can simulate the dynamics behavior of the system in the dynamic contact process more truly, and improves the dynamics calculation accuracy.
Drawings
FIG. 1 is a flow chart of a nonlinear dynamics analysis calculation;
FIG. 2 is a graph showing a distribution of contact points at a certain engagement position on the engagement plane;
FIG. 3 is a schematic view of a gear pair dynamics model;
FIG. 4 shows the dynamic response results of the gear pair at different rotational speeds;
FIG. 5 is the dynamic meshing stiffness of the gear pair at different rotational speeds;
FIG. 6 is a dynamic load distribution of the tooth surface at different rotational speeds.
Detailed Description
As shown in FIG. 1, the method for calculating the nonlinear dynamics of the gear pair of the present invention establishes a gear pair dynamic contact and system vibration coupling analysis model, and specifically comprises the following steps:
step 1: establishing a gear pair dynamic contact analysis model considering gear basic parameters and tooth surface manufacturing errors, and calculating vibration excitations such as dynamic meshing rigidity, dynamic comprehensive meshing errors and the like; the method specifically comprises the following steps:
(1) calculating the mean value of the meshing stiffness of the error-free gear by using GB3480-1997, and simultaneously calculating the length of a theoretical contact line at each meshing position to obtain nonlinear excitation on the length of a unit contact line;
(2) converting the tooth surface manufacturing errors (including tooth profile deviation, tooth pitch deviation, spiral line deviation and the like) into the tooth surface normal direction for measurement and superposing to obtain initial normal gaps of all contact points;
(3) and (3) judging the actual contact state of each point to obtain the length of the actual contact line, and calculating the actual dynamic meshing rigidity and the dynamic comprehensive meshing error according to the rigidity value of the unit contact line length in the step (1).
Step 2: establishing a gear pair bending-torsion-axial coupling dynamic model by using a concentrated mass method; the method specifically comprises the following steps:
a classical mass concentration method in a vibration theory is adopted, meshing of a driving wheel and a driven wheel is regarded as a mass-spring system containing bending-torsion-axial multi-degree of freedom, factors such as gear meshing stiffness, comprehensive meshing error and bearing stiffness are included, and a system motion differential equation is established by adopting a Newton's second law.
And step 3: the dynamic contact analysis of the gear pair is combined with the dynamic equation solving process, and the dynamic contact problem of the gear pair is solved by adopting a Newmark integral nested frustum method, so that the analysis process of an excitation-response-feedback phase closed loop is realized. The method specifically comprises the following steps:
(1) initial values of system displacement, speed and acceleration are given, and a system motion differential equation is solved by adopting an unconditionally stable implicit Newmark step-by-step integration method;
(2) for each time step of the Newmark integral, a nonlinear displacement balance equation needs to be solved, and an improved Newton method-chord cut method can be adopted for iterative solution;
(3) when the intercept method is iterated every time, calculating a transfer error converted to the normal direction of the meshing line according to an instantaneous displacement value every time, substituting the transfer error into the contact equation in the step 1 to calculate a new dynamic meshing rigidity and rigidity matrix, carrying out next iteration until the iteration precision is met, and then carrying out Newmark integration and next time step calculation;
(4) and judging whether the Newmark integral reaches a steady state or not through the displacement response difference in two adjacent meshing periods. If the steady state is reached, outputting the final displacement response, dynamic load distribution and other results of the system; otherwise, continuing to calculate the next time step.
The technical solution of the present invention will be described in further detail with reference to specific examples.
As shown in fig. 1, the nonlinear dynamics analysis method established by the present invention specifically comprises the following steps:
(1) first, the average value L of the contact length of the gear pair is calculatedmAs shown in formula (1):
Lm＝εαb/cosβb (1)
wherein epsilonαIs the end face contact ratio, b is the tooth width, betabIs a base circle helix angle.
FIG. 2 is a diagram showing the distribution of contact lines and contact points on the meshing plane at a certain meshing position of helical gears, where p isbtIs the pitch of the end face base circleiThe length of the segmented contact line occupied by the contact point i.
The tooth profile deviation, the tooth pitch deviation and the spiral line deviation of the contact point are superposed to obtain an initial normal clearance epsiloni. Judging the initial normal clearance epsilon of the contact pointiWith the magnitude of the instantaneous dynamic transfer error DTE, if ei<DTE, which indicates contact point i touches; otherwise, the contact point i is not contacted. According to the number of actual contact points, the total length L of the actual contact line can be obtaineda. Calculating the mean value k of the meshing rigidity of the error-free gear pair according to GB3480-1997 by the basic parameters of the gearm0. The meshing rigidity k is obtained due to the fact that the meshing rigidity is consistent with the trend of the actual contact line lengthmdComprises the following steps:
instantaneous dynamic comprehensive meshing error e of gear pairmdComprises the following steps:
(2) the gear pair dynamic model is modeled by adopting a concentrated mass method, and the dynamic model is shown in figure 3. In the figure, the subscript p denotes the primary pulley and the subscript g denotes the secondary pulley. The vibration displacement of each gear is moved to the direction of the meshing line for projection, and the obtained dynamic total deformation of the meshing line relative to the meshing line is as follows:
δd＝DTE-emd＝VX-emd (4)
wherein X ═ { X ═ Xp,yp,zp,θzp,xg,yg,zg,θzg}TRepresenting the displacement column vector of the two gears; DTE is VX and is the instantaneous dynamic transmission error of the gear pair; v is a projection vector for converting each direction to the direction of the meshing line, and can be represented by the following formula
In the formula, rpAnd rgThe base circle radius of the driving wheel and the driven wheel respectively; beta is abIs a base circle helical angle; angle of rotation
Alpha is the engagement angle and phi is the driven wheel installation phase.
According to Newton's second law, a system motion differential equation can be established as follows:
in the formula, mi(i ═ p, g) each is mainlyThe mass of the driven gear; i iszi(i ═ p, g) are the moments of inertia about the z axis of the primary and secondary wheels, respectively; c. CmdDynamic meshing damping is adopted; k is a radical ofij,cij(i ═ p, g; j ═ x, y, z) respectively represent the support stiffness and damping of gear i in the j direction; t is1And T2The torque of gear 1 and gear 2 respectively.
Substituting the formula (4) into an equation system (6) and arranging the equation into a matrix form:
in the formula: m, C and K are respectively a system mass matrix, a damping matrix and a rigidity matrix; p is an external load column vector; e is a comprehensive meshing error vector; and F is the total excitation load vector.
(3) The Newmark integration and the truncation method are nested, and excitation change caused by change of instantaneous contact condition is considered when dynamic response is solved, so that an analysis process of 'excitation-response-feedback' phase closed loop is realized. The specific implementation mode is as follows:
1) for a dynamic differential equation set as the formula (7), initial meshing stiffness and comprehensive meshing error are given to form a total excitation load vector initial value F0While simultaneously giving an initial value X of the displacement0Initial value of speed
And calculating an initial value of acceleration
2) Selecting integration step length delta t and Newmark integration parameters alpha and beta, and calculating integration constants (beta is more than or equal to 0.5 and alpha is more than or equal to 0.25(0.5+ beta)2)
3) Forming a rigidity matrix K, a mass matrix M and a damping matrix C at the time of t + delta t, and calculating the effective load at the time of t + delta t:
4) forming an effective stiffness matrix
The original differential equation of motion (7) is converted into the form of equation (11).
The equation set surface form shown in equation (11) is a linear equation set, but from the foregoing analysis, the system displacement response influences the dynamic contact condition to make the dynamic meshing stiffness kmdChanges occur, thereby changing the effective stiffness matrix
Therefore, the equation is essentially a nonlinear equation system, and can be solved by means of a truncation method, and the basic idea is as follows:
a. rewriting formula (11) to
For simplicity, t + Δ t is omitted here;
b. let the iteration number k equal to 1, give an iteration initial value X0And X1Calculating f (X)0) And f (X)1)；
c. Solving for X according to a truncated method iterative formulak+1：
d. Calculating the transfer error DTE of k +1 iterationsk+1＝VXk+1And substituting the contact model in the step (1) to solve the dynamic meshing stiffness kmdAnd dynamic combined meshing error emdK is calculated from the expressions (6), (9) to (10)eqv，
And calculating f (X)k+1)；
e. Judgment of | f (X)k+1)|＜ε(εConvergence accuracy of settings)? If true, Xt+Δt＝XkEnding the iteration; otherwise, k is k +1, and the step c is returned.
5) Calculating the acceleration and the speed at the moment t + delta t:
6) judging the displacement response in the front and back meshing periods, and if the two are close enough, exiting the calculation; otherwise, returning to the step 3) to calculate the next moment.
According to the nonlinear dynamics calculation method of the gear pair, the straight gear pair shown in the table 1 is adopted for example calculation. The big gear wheel has drum shape spiral line deviation of 5 μm, and the small gear wheel has no error.
TABLE 1
FIG. 4 shows the root mean square values of the transfer errors corresponding to different rotation speeds under the conditions of increasing and decreasing speed. As can be seen from the figure, a distinct response amplitude jump occurs at both ramp-up and ramp-down, demonstrating that the algorithm herein is effective in computing the system non-linear response.
FIG. 5 is a graph of dynamic mesh stiffness at 100rpm, 2100rpm and 2150rpm for a gear pair at ramp-up. As can be seen from fig. 5, the dynamic mesh stiffness of the gear pair at 100rpm is a rectangular wave, which is the same as the mesh stiffness curve under static load. The dynamic meshing stiffness at 2100rpm was decreased in the middle of the double tooth zone, while the dynamic meshing stiffness at 2150rpm was 0 in the partial double tooth zone, indicating that the fully disengaged state occurred in this region.
Comparing the tooth surface load states at 2100rpm (fig. 6a) and 2150rpm (fig. 6b) in fig. 6 shows that although the two rotation speeds are not greatly different, the actual load distribution difference is obvious.
The above-listed detailed description is only a specific description of a possible embodiment of the present invention, and they are not intended to limit the scope of the present invention, and equivalent embodiments or modifications made without departing from the technical spirit of the present invention should be included in the scope of the present invention.
Claims (3)
1. A gear pair nonlinear dynamics calculation method is characterized by comprising the following steps:
step 1: establishing a gear pair dynamic contact analysis model considering gear basic parameters and tooth surface manufacturing errors, and calculating dynamic meshing rigidity and dynamic comprehensive meshing errors;
step 2: establishing a gear pair bending-torsion-axial coupling dynamic model by using a concentrated mass method;
and step 3: combining the dynamic contact analysis of the gear pair with the dynamic equation solving process, solving the dynamic contact problem of the gear pair by adopting a Newmark integral nested frustum method, and realizing an analysis process of an excitation-response-feedback phase closed loop;
the implementation process of the step 3 is as follows:
(3.1) giving initial values of system displacement, speed and acceleration, and solving a system motion differential equation by adopting an unconditionally stable implicit Newmark step-by-step integral method;
(3.2) for each time step of the Newmark integration, solving a nonlinear displacement balance equation, and adopting an improved Newton method-chord method to carry out iterative solution;
(3.3) when the truncation method is iterated every time, calculating a transfer error converted to the normal direction of the meshing line according to an instantaneous displacement value every time, substituting the transfer error into the contact equation in the step 1 to calculate a new dynamic meshing rigidity and rigidity matrix, carrying out next iteration until the iteration precision is met, and then carrying out Newmark integration and next time step calculation;
(3.4) judging whether the Newmark integration reaches a steady state or not through the displacement response difference in two adjacent meshing periods: if the steady state is reached, outputting the final displacement response and dynamic load distribution result of the system; otherwise, continuing to calculate the next time step.
2. The method for calculating the nonlinear dynamics of the gear pair according to claim 1, wherein the step 1 is realized by the following steps:
(1.1) calculating the mean value of the meshing stiffness of the error-free gear, and simultaneously calculating the length of a theoretical contact line at each meshing position to obtain the meshing stiffness on the length of a unit contact line;
(1.2) converting the tooth surface manufacturing error to a tooth surface normal direction for measurement and superposing to obtain an initial normal gap of each contact point;
and (1.3) judging the actual contact state of each point to obtain the actual contact line length, and calculating the actual dynamic meshing rigidity and dynamic comprehensive meshing error according to the unit contact line length rigidity value in the step (1.1).
3. The method for calculating the nonlinear dynamics of the gear pair according to claim 1, wherein the step 2 is realized by the following steps:
a centralized mass method in a vibration theory is adopted, the meshing of a driving wheel and a driven wheel is regarded as a mass-spring system containing bending-torsion-axial multi-degree of freedom, the factors of gear meshing rigidity, comprehensive meshing error and bearing rigidity are included, and a system motion differential equation is established by adopting a Newton's second law.
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