GEARX Logo Study on Noncircular Gear's Dynamics Characters


Noncircular gear transmission is an important branch of gear transmission,is characterized by its compact structure,good dynamic equilibration and other advantages,and can be used in automobileengineering machine,ship,machine tool,aviation and spaceflight field etc.Currently,the studying work of noncircular gear concentrates on geometry modeling,kinematics,machining etc,while the studying work on dynamics is much less.Studying on the dynamics feature of noncircular gear transmission can improve the ability to carry loads of, reduce the vibration and noise of, increase the life of noncircular gear transmission machine,provides guidance for the design of noncircular gear,and has significant theories and practical meanings.


The pitch curve of noncircular gear is noncircular,which makes the design of noncircular gear difficult.The keys Of noncircular gear design are to determine the position on the pitch curve of each tooth,tooth top curve,tooth root curve and the tooth profile curve of noncircular gear.The tooth profile curve of noncircular gear can be realized by two methods:programming,which is difficult to common designer;equivalent method,which use the involute of the equivalent cylindrical gear to substitute the tooth profile curve of noncircular gear,and make the model imprecise.All these make the analysis of noncircular gear difficult.This paper aims at this problem,takes oval gear as an example,uses the tooth profile curve formula,combines mathematic software MathCAD and 3D software UG,and realizes
the precise model of oval gear.

The pitch curve formula of oval gear Call be written as
r=a(1-K2)/(1-kcos(2(ф)) (1)
r—radius of the pitch
a—radius of the long axis,mm
ф—polar angle,deg
The pitch curve of oval gear is symmetrical with the X-axis and y-axis of Cartesian Coordinate.For the design convenience.the
tooth number Z=4C+2(C is positive integer),and the sections at long and short axes should be tooth and alveolus respectively.

The steps of oval gear modeling are shown as following.
1 Give design parameters: tooth number Z,modulus m, eccentricity k,and solve a.
2 Determine the polar angles of the intersection points of the pitch carve and the tooth profile curves.
3 Use MathCAD and the tooth profile curve formula to product coordinate data of key points on the tooth profile curves.
4 Import the point data into UG.obtain the tooth profile curve by fitting with cube spline.obtain the plane sketch of oval gear by arc length,corner,trim,and mirror function,get the 3D model of oval gear by extrude function.
The parameters of the oval gear in this paper:
tooth number z=22,modulus m=lmm,eccentricity e=0.1,tooth addendum ha=lmm,tooth
height h=2.25mm,a=10.9457mm,tooth width B=5mm,and inner radius rin=5mm.The two oval gears are same.


In the 3D software UG finish the assembly model of oval gears,save the assembly file as parasolid(x-t)file,and import the assembly model into ADAMS to establish the virtual prototype.
In ADAMS,apply restriction,excitation,drive,load and contact to oval gears.Two revolute joints are applied to active gear and driven gear,and contact force is defined between active gear and driven gear.

Definition of contact force
In ADAMS.the definition of contact force has two sorts:contact force based on Impact Function,which uses stiffness and damping to calculate the contact force;contact force base on Restitution Function,which uses restitution coefficient to calculate the contact force.The paper uses Impact Function to calculate contact force.

2.1 Stiffness
The exciting force aroused by impact between teeth can be considered as impact problem of two cylinders,the curvature radius of which are variable.To solve the problem can be received from Hertz static elasticity contact theory.According to Hertz theory. considering the contact square is circle. The value of tooth height is much small than that of pitch curve radius,the variable scope of which is not big,so it can be substituted with the value of pitch curve. The tolerance of this simplification is small.

Analysis the relation curve between stiffness and polar angle of certain oval gear,It shows that stiffness of oval gear changes periodically with change of polar angle.the change period is ∏.When the pitch curve radius of the active gear and the driven gear equals,stiffness gets the maximum;when pitch curve radius of the active gear and the driven gear get the biggest and smallest values respectively, tiffness gets the minimum.

2.2 Damping
The damping of cylindrical gear can be concluded by analysis the math model and express as following:
damping of oval gear changes periodically with change of polar angle.the change period is ∏.When the pitch curve radius of the active gear and the driven gear equals,damping gets the maximum;when the pitch curve radius of the active gear and the driven gear get the biggest and smallest values respectively,damping gets the minimum.

2.3 Force exponent
From equation(4),the force exponent call be obtained as follows:


3.1 Simulation condition
This paper sets input rotate speed ω=60r/min.resistant torque T=150000Nmm,Young's modulus E1=E2=2.07x105N/mm,Poisson’s ratio ч1=ч2=0.29.and damping ratio =0.15,To avoid saltation when aping load,Step Function is used to make the load increase evenly in 0.05s, namely Step(time,0,,150000).The simulation time is 1S,and the number of simulation steps is 10000.

3.2 Simulation result and analysis
In ADAMS,simulation experiment and analysisof the dynamic virtual prototype for oval gear cluster are carried,
conclusions can be educed as follows:
The dynamic parameters(force and torque)of oval gear cluster change periodically with change of polar angle,the change period is ∏.Compared with those of cylindrical gear, the dynamic parameters of oval gear cluster fluctuate around trigonometric functions between the maximum and minimum,while those of cylindrical gear fluctuate around the average
The maximums of the dynamic parameters for the active gear appear about the long axis,and the minimums appear about the short axis,while the maximums of the dynamic parameters for the driven axis gear appear about the short axis,and the minimums appear about the long axis.So the danger section of oval should be the teeth about the short and long axis.
The active gear rotates uniformly, so the forces and torques acting on the active gear satisfy static equation;while the driven gear rotates unevenly, so the force acting on the driven gear satisfy static equation,but the torques don’t satisfy static equation.
The torque acting on the active gear changes with polar angle,so the power that electromotor provides to the oval gear cluster changes with polar angle too.
The components of the contact force are obtained,and can be used as load conditions for FEM analysis.

3.3 Deformation
According to equation,the relation curve between integrated deformation δ and time can be obtained, the integrated deformation δ of oval gear cluster changes periodically with the change of time,the change period is ∏ ,the maximum appears about the long axis of active gear while the minimum appears about the short axis.

3.4 Influent factors of contact force
Study oil the influent factors of contact force has important meaning to reduce the impact vibration and noise of, to increase the life of oval gear transmission machine,and to provide guidance for the design of oval gear.

3.4.1 Damping ratio
Using virtual prototype experiment,under the condition of equal input rotate speed,resistant torque and stiffness,the relations between maximum,average,RMS of contact force and damping ratio by changing damping ration.we could get the conclude data for the maximum,average and RMS of contact force increase along with reduction of damping ratio.

3.4.2 Stiffness
Using virtual prototype experiment,under the condition of equal input rotate speed.resistant torque and damping ratio,the relations between maximum,average.RMS of contact force and stiffness.we could get the conclude data for the maximum,average and RMS of contact force increase along with increase of damping ratio.

Analysis shows the influence on contact force of oval gear is consistent with that on contact force of cylindrical gear,Under the precondition that the tooth has adequate ability to carry loads,increase of damping ratio and decrease of stiffness are propitious to lower fatigue breaking of teeth and impact of system,increase ife of oval gear. ‘

In this paper, a method for precise modeling of oval gear is provided by combining MathCAD and UG,gear is established under ADAMS,the parameters for the virtual prototype are determined according to Hertz Impact theory;the dynamic simulation experiment is carried on。the simulation results are analyzed,the comparison with that of cylindrical gear is made,the influent factors and change law of dvnamics of oval gear are discussed,and the dynamic features of oval gear are achieved.This Paper provides an analytical method for dynamic design of oval gear.(Ranxiaohu Lichao Liugang)

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