Spinning Gear

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Spinning GearSOLVED WITH COMSOL MULTIPHYSICS 3.5a

spinning_gear.book Page 1 Wednesday, December 3, 2008 2:24 PM

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S p i n n i n g Gea r

Introduction

One way to fasten a gear to a shaft is by thermal interference. In preparation of the assembly, the shaft diameter is oversized and the gear thermally expanded in a heat treating oven. At an appropriate expansion state, the gear is removed from the oven, slanpcoco

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2

spinning_gear.book Page 1 Wednesday, December 3, 2008 2:24 PMS P I N N I N G G E

id onto the shaft, and allowed to cool. As the gear temperature drops, the gear shrinks d comes into contact with the shaft before it reaches its original shape. From this

oint on, additional gear shrinkage results in hoop stresses in the gear as well as normal mpression of the shaft. At thermal equilibrium, an intimate bond between the two mponents is reached.

uch an assembly can operate safely in many situations. However, there are operating nditions under which the fastening stresses become insufficientfor instance, when inning the assembly at high rpm.

he goal of this analysis is to determine the critical spinning frequency at which gear d shaft separate.

ote: This model requires the Structural Mechanics Module.

odel Definition

he model computations consist of two steps:

Thermal interference fit

- Import the gear geometry from a given CAD file and draw the shaft using COMSOL Multiphysics solid modeling tools.

- Fasten the gear to the shaft by thermal interference: Initially, both shaft and gear reside at room temperature (23 C). Then, the gear is heated to 700 C, positioned on the shaft, and allowed to cool.

Spinning the shaft-gear assembly

- Spin the shaft-gear assembly and determine the separation frequency.

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For performing Step 2, the model is available in two versions that demonstrate different ways of determining the separation frequency:

Doing a parametric sweep over a frequency range containing the separation frequency.

Solving an inverse problem using the Optimization application mode. Note that this version requires the Optimization Lab.

In the modeling instructions that follow, the two approaches are described in turn.

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ssume plane stress conditions for all computations and neglect contact phenomena uring separation.

E O M E T R Y

he geometry in Figure 1 consists of a shaft and a gear.

haft specifications:

Material: Steel AISI 4340

Radius: 0.015 m

Length: 0.1 m

ear specifications:

CAD file: gear.dxf. This file is included in the model folder and was taken from Ref. 1.

Material: Steel AISI 4340

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Thickness: 0.01 m

F

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igure 1: Gear geometry.

O M A I N E Q U A T I O N S

he given problems are solved by computing the stress and deflection fields of the eady thermal interference and critical separation states. Starting with the stress-strain lation

d the thermal strain relation

(1)

here the subscript el stands for elastic and t is the coefficient of thermal pansion, you can state that

xyxy

D

xyxy el

=

xyxy el

xyxy

tt0

T Tref( )=

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where

Fm

Info

w

B

T(Dsysy

M

Bththreth

xyxy

D

xyxy

tt0

T Tref( ) x

yxy res

+=

1 0


or more information about the underlying equations for the Plane Stress application ode, see the Structural Mechanics Module Users Guide.

the second part of the analysis, the forcing term, F, represents the centripetal body rce:

here is the angle from the positive x-axis and f denotes the rotation frequency.

O U N D A R Y C O N D I T I O N S

o prevent rigid body translation and rotation, you must impose some constraints irichlet conditions): For computational efficiency, the analysis only includes a

mmetric quarter of the geometry. By setting the normal displacements on the mmetry boundaries to zero, it is easy to constrain the model.

odeling in COMSOL Multiphysics

ecause the analysis neglects contact phenomena, the gear geometry is modeled at the ermal expansion of 700 C, at which it fits precisely on the shaft. The model assumes at the gear expands freely in the heat-treating oven and that the heating profile moves all internal stresses. When the assembly is spun, the gear expands more quickly an the shaft and reaches a critical separation point.

D E

1 2---------------

1 00 0 1

2------------

=

F ( ) F ( )cosF ( )sin ,= F

2r 42f2 x2 y2+= =

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Results

I N T E R F E R E N C E

In the first analysis step, you obtain the stress distribution of the thermal interference. Figure 2 illustrates the hoop stresses in the gear, which increase gradually toward the interface between shaft and gear. As a result, the shaft is exposed to normal compression.

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igure 2: von Mises stresses superimposed on shaft and gear. Note the hoop stresses.

E P A R A T I O N

he parametric analysis spins the prestressed assembly at various frequencies, and you n plot the displacement between the shaft and the gear. Figure 3 and Figure 4

lustrate an advanced displacement state at 1600 Hz and a displacement versus equency plot, respectively. The separation frequency occurs at the minimum of about

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1550 Hz (Figure 4). This result agrees is confirmed by the solution of the inverse problem, which gives the value 1550.1886 Hz.

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igure 3: von Mises stresses and displacement at 1600 Hz.

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igure 4: Displacement vs. frequency; separation occurs at 1550 Hz.

eference

. http://claymore.engineer.gvsu.edu/~schmitte/assign5.html.

Spinning GearIntroductionModel DefinitionModeling in COMSOL MultiphysicsResultsReference

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Spinning Gear