SHAFT DESIGN WITH HELICAL GEAR, BEBEL GEAR, AND TWO SUPPORTING BEARINGS

Virna V. RodríguezAisha M. NievesEmmanuel Rosa

Wilfredo MercadoMechanical Engineering Department, University of Puerto Rico

Dr. Pablo Caceres

Abstract: A detailed analysis was performed in order to complete a shaft design with a helical gear, bevel gear, two supporting bearings and a section change. The unknown forces were calculated by sums of forces and moments, which were further used to determine the shear stresses, bending moments and principal stresses. Taking into consideration the actual uses of the shaft, an appropriate material was assumed to calculate a diameter using material indices. The characteristics in consideration were a stiff, strong, and light material. Calculations for acting plane and principal stresses, fracture toughness, along with material selection properties agreed with the selection of a low carbon steel (AISI 1025) as the best material for the shaft.

Introduction A shaft design will be completed in which the team will determine the

combined loads that are acting on the component, the maximum stresses and the stresses

with stress concentrators. A material selection and a fatigue analysis due to fluctuating

loads will be done in order to find the radius of the shaft. The team will research among

existing devices which work with such components, incorporating the helical gear, bevel

gear and the supporting bearings. Transmission shafts that transmit torque from one

location to another, such as the Caterpillar tractor transmission is an application for the

shaft design. Therefore, the analysis will help in the design of an efficient component

that might be used in real life.

Schematic:

The analysis begins by noticing the forces acting on the shaft, and performing a series of calculations to determine the unknowns.

Analysis of forces acting on the shaftAssumptions:

1) Thrust load at bearing A.2) Both gears will have forces acting on the three directions.3) Bearing C will not have any forces acting on the ‘x’ direction4) Uniform diameter throughout the shaft5) Torque input magnitude at point D equal to torque output at point B

Given: First gear located on Point D:

a. Fx=2.57 KNb. Fy=2.57 KNc. Fz=10.0 kN

Torque at point D (Input)

Therefore;

Tinput=Toutput

With this value we can proceed to calculate the force in the ‘z’ and ‘x’ direction at point B.

Forces and Momentum on the different sections of the shaft:

We calculated the Momentum in point A, in the ‘y’ direction, to find the Force acting on point C in the ‘z’ direction. By obtaining this we will be able to calculate the Force acting on point A in the ‘z’ direction.

Therefore;

We calculated the Momentum in point A, in the ‘z’ direction, to find the Force acting on point C in the ‘y’ direction. By obtaining this we will be able to calculate the Force acting on point A in the ‘y’ direction.

Therefore;

As part of the analysis for the shaft chosen, it includes the search for the different forces exerted in the bearings. Also, how these forces act on the shaft. By finding these different forces and developing the Shear and Moment Diagrams we are able to find out which are the critical points.

y

x

Fay Fby Fcy Fy

V 3.168 0 2.57

-4.332

M 0 .8988 -2.166

Z

X Faz Fbz Fcz Fz

10

V 0 -2.3576 -5.113

M 0

-2.5569

-3.5

Critical Section:

At Point B:

=-120.575MPa

With Stress concentration:σs.c= Kfbσnom= -233.314MPa

At Point C:

= -165.049 MPa

Therefore the critical Section will be at point B

Principal Stresses

Stresses

For the ‘zx’ plane the critical zone occurs at point c: Moment in ‘y’

Vmax=10kNMmax=-3.5kN*m

There are two shear stresses caused by Torsion. They are the following:

The total stresses are the following:

Normal stresses for ‘zx’ and ‘yx’ are: -43831.2713275 kPa

The shear stresses for ‘yx’ plane and due to the torsion acting on critical sections B and C:τtotal= 436.296750663-3272.22562997-9549.29658551= -12385.2254648 kPa

Invariants

We substitute in the following equation to find the principal stresses acting in the shaft.

Corrected Stresses with Stress Concentrators

Fatigue Analysis

Material Selection:

In order to select our material, we determined that the shaft should be stiff. It should be able to hold a large amount of force to prevent it from bending and it should have a small angle of twist. If it has a big deflection, the gears could separate from each other, causing them to not function properly or to wear off. This would also cause failure of the bearings. In order to achieve these criteria, the following calculations were performed.

For the bending forces:

For angle of twist:

After obtaining these results, we went to the tables (“Young’s modulus of elasticity vs Density” and “Strength vs Density”) to maximize the followings

relationships: for bending forces, and for the angle. Because is a

function of: , we maximize .

The following diagram was used to determine the materials applicable to the shaft design:

From the table, we obtained certain materials that could be use for the shaft.

Strenght Angle of twist Al2O3 CFRP SiC Ti alloys Composites Mg alloys Glass Al alloys Al alloys Al2o3 CFRP SiC Steels Steels

As we can observe, some of these material can be discarded such as glass and ceramics because of their tendency to fail due to the appearance of cracks. After analyzing these facts, we determine that Al and steels are good candidates for the shaft, but aluminum was discarded because it can be seen in the fatigue diagram that aluminum will always fail after certain number of cycles, while steel supports higher stresses fluctuations before failing. Research was performed and it was founded that low carbon steel has the specifications desired for a shaft design.

An analysis between several low carbon steels was done.

Material

AISI 1010 Steel 205GPa 7.87g/cc 305 GPa 57.53 5757.38AISI 1015 Steel 205GPa 7.87g/cc 300 GPa 57.53 5694.28AISI 1022 Steel 205GPa 7.85g/cc 300 GPa 57.61 5702.98AISI 1025 Steel 200GPa 7.85g/cc 310 GPa 56.91 5829.02

After comparing various types of carbon steels we have selected AISI 1025 which is a low-carbon steel. This material will be machined to achieve the desired shape.

AISI 1025 Steel

Physical Properties Metric English Comments

Density 7.858   g/ cc

0.284 lb/in³

 Chemical composition of 0.23% C, 0.635% Mn, 0.11% Si, annealed at 925°C

Mechanical Properties

Hardness, Brinell 111 111  Hardness, Knoop 129 129  Converted from Brinell hardness.Hardness, Rockwell B 64 64  Converted from Brinell hardness.Hardness, Vickers 115 115  Converted from Brinell hardness.Tensile Strength, Ultimate

380   MPa 55100 psi  

Tensile Strength, Yield 310   MPa 45000 psi  Elongation at Break 15   % 15 %  Reduction of Area 35   % 35 %  Modulus of Elasticity 200   GPa 29000 ksi  Typical for steelBulk Modulus 140   GPa 20300 ksi  Typical for steel.Poisson's Ratio 0.29 0.29  Typical For SteelShear Modulus 80   GPa 11600 ksi  Typical for steel.

Stress concentrations: Section y-x: Critical Point B

Using the following diagrams:

r/d = 1.8/60= 0.03D/d = 90/60 = 1.5

Kt,bending = 2.35Kt,shear = 1.9Kt, axial = 2.7

Sut steel = 900 MPa

Kf (bending) = 1.94Kf (shear) = 1.62Kf (axial) = 2.18

From Materials selection Sut= 900MPa:Se’ = 0.5Sut = 0.5 (900) = 450MPa

Se corrected:Ksurf=0.93Ksize:A95= 0.0766d^2 = 0.0766 (0.06)^2 = 2.758e-4 m2

d equivalent =

Ksize = 1.189(d)^*-0.097 = 0.8

Se= (0.93)(0.897)(0.8)(1.5621)= 300.3156 Mpa

Corrected stresses :

σs.c= (0.696 MPa) (2.18) + (0.909) (2.18) + (11.265) (1.94) + (90.895) (1.94) = -201.6893MPaτs.c = (1.62)(-44.20) + 1.3 + 5.827 – 4.715 = 234.55 MPa

nf= Se/σVM= 300.3156/234.55 = 1.3

ConclusionA complete analysis of the forces acting on the shaft, the calculation of the maximum stresses acting with stress concentrators and a fatigue analysis was done. Low carbon steel AISI 1025 has been selected as the proper material that agrees with the material selection analysis drawn. The material selected is stiff, holds a considerably high static force, has a small twist angle, and supports considerably high load fluctuation. A diameter of 60mm has been selected for the shaft taking into consideration the material selected along with the loads acting on it. A safety factor was computed using Goodman theory in a fatigue analysis. The fatigue analysis was done for infinite life. These calculations corroborated the safe operation of the shaft we designed.

References:

http://www.matweb.com/index.asp?ckck=1

http://academic.uprm.edu/pcaceres/

http://www.google.com