Determine functional
5.6 Engineering analysis of preliminary design
Once the preliminary design has been determined, it is necessary to analyze the design to ensure that it will meet the load capacity and contact ratio requirements. Refer to clause 6 for the equations for calculating bending stress, contact stress and contact ratio. If the analysis of the preliminary design reveals that it would not be adequate for the application, the designer should revise the original design. The cycle of design followed by analysis should be repeated until the preliminary analysis verifies that the design is adequate.
This clause contains a brief synopsis of some of the types of engineering analysis that should be done at the preliminary design stage. The primary objective of the preliminary design analysis is to ensure that the gears have adequate strength for the applica- tion. More detailed analysis will be done later during the detailed design stage (refer to clause 6). At the detailed design stage the specific tooth geometry and meshing parameters will be determined to optimize the design for its particular application. The reader is also referred to the publications in annex A for additional information.
5.6.1 Bending stress
Bending stress can be determined using a variety of techniques. Each method has some advantages and disadvantages.
-- Lewis form factor. This method treats a
gear tooth as a stubby cantilever beam. Figure 33 maps the geometry of the gear tooth to that of a cantilever beam. This method usually predicts conservative (higher) than actual bending stress for unmodified gears. This method evaluates a normalized measure called the Lewis form equation based on the dimensions of the beam. Machine design text books give a detailed ex- planation of this method. This method does not account for load sharing and evaluates stresses at the root by applying the load at the tip of the gear tooth.
W l F W t (a) t a x l (b) rf
Figure 33 -- Gear tooth as a simple beam
-- AGMA geometry factor method. This method is also based on construction. However, the load is applied at the highest point of single tooth contact for spur gears and inscribes a pa- rabola inside the gear tooth. Figure 34 illustrates the construction. The point where the parabola is tangent to the tooth profile is called the critical point and is where the maximum stress occurs. This method evaluates a normalized measure called the geometry factor which accounts for the stress concentration at the root of the gear tooth and the load sharing between the mating gears. For helical gears, the load is applied at the tip of the gear tooth. Clause 6 uses this technique to evaluate bending stresses at the root of the gear tooth. C Gear toothL Vertex Load Inscribed parabola Critical point
Figure 34 -- Tooth load acting at inscribed parabola
-- Finite element model. This method uses
the finite element method to model the actual
gear tooth geometry. This method will provide more accurate results than the Lewis form factor, especially for modified gear tooth geometry. The effects of thin rim gears can also be analyzed us- ing this method. The tooth model is usually loaded at the highest point of single tooth contact (HPSTC).
-- Boundary element model. This method,
like the finite element method, will provide more accurate results than the Lewis form factor, especially for modified gear tooth geometry.
5.6.2 Contact stress
Contact stress occurs when two bodies come in contact under a force. The involutes of external gear pairs can be approximated as a cylinder on a cylinder. The involutes of internal--external gear pairs can be approximated as a cylinder inside a cylinder. The Hertzian (contact) stress can be calculated at any point of contact by knowing the radius of curvature of each involute at the point of contact.
The two most common places to calculate contact stresses are:
-- Pitch diameter. At the pitch diameter the
contact stress is less than the maximum, but it is the point where pitting usually occurs. Pitting is a surface fatigue failure due to repetitions of high contact stress. Pitting occurs near the pitch di- ameter because the relative sliding between the pinion and gear changes direction as the contact passes through the pitch point. This change in
sliding creates frictional subsurface shear stresses which can eventually remove material and form the surface cavities known as pitting; -- Lowest point of single tooth contact.
This is the point where the contact stress is at its maximum. It occurs here because the radii of curvature of the involutes are at their most opposite extremes, when the tooth is under its maximum load (when there is only one pair of teeth carrying the load). For external gear pairs, the contact on the driver is near its root, (relative- ly small radius of curvature) and the contact on the driven gear is near its tip (relatively large ra- dius of curvature). If the contact stresses exceed the surface endurance strength of the material, surface failure will result.
5.6.3 Contact ratio
Contact ratio can be visualized as the average number of tooth pairs in contact during the mesh cycle. For example, if the contact ratio for a gear mesh is 1.75, then two tooth pairs will be in contact 75% of the time and only one tooth pair will be in contact 25% of the time. In general, gear meshes with more tooth pairs in contact exhibit smoother operation. For most applications, it is recom- mended that the contact ratio be at least 1.4. Contact ratio is a function of center distance. Contact ratio decreases as the center distance increases from the nominal center distance. It is important to determine the allowable deviation from the nominal center distance that will maintain an acceptable contact ratio for the application. Finer diametral pitch gears have a smaller allowable center distance deviation. It is important to consider the expected center distance tolerance stack up, of the entire system, early in the design process. The designer should then design the system so the contact ratio is acceptable throughout the complete center distance tolerance zone.
5.6.4 Fatigue strength
Many machine design texts, such as “Mechanical Engineering Design” by J.E. Shigley [6], contain information on how to determine fatigue strengths, or endurance limits, for ferrous materials. It is important to realize that the analytical approaches do not yield absolutely accurate results. The results should only be used as a guide, as something that
indicates what is important and what is not impor- tant, in designing to avoid fatigue failure. Many materials, such as plastics, aluminum and bronze do not have true endurance limits so they cannot be designed to have infinite life. Fatigue curve information must be obtained from the material supplier and verified by testing in order to design the gears to meet life requirements. It is extremely important to confirm the design by conducting a testing program on the materials that will be used. Heat treatment of ferrous materials will increase fatigue strength. Refer to clause 15 for more information on load rating and testing procedures.
5.6.5 Surface durability
Surface durability, also known as pitting resistance, is the capacity to resist the kind of failure which results from repeated surface or subsurface stresses. See ANSI/AGMA 2001--C95 for more information on surface durability ratings for spur and helical gears. These rating methods assume the design provides adequate lubrication. Inadequate lubrication can lead to other modes of surface failure (wear), which are not covered by these rating methods. The load rating procedure in ANSI/AGMA 2001--C95 is not suitable for every fine--pitch application. The rating procedure is based primarily on experience with coarse pitch gears. As with fatigue strength, available data on material properties are limited to the more traditional gear materials. Heat treatment of ferrous materials will increase surface durability. Material property information should be obtained from the material supplier and testing should be done to confirm the design.
5.6.6 Gear system assembly
Before the detailed design can be performed, the designer must consider the method of mounting to be used. Proper installation of the gear system is essential for achieving good performance. Some items of consideration should be:
-- Mounting of gears on the shafts. There
are several methods used to mount gears to shafts. Many designs offer various degrees of precision, cost, reliability and ease of assembly. The designs can be classified into two main types: removable fastenings and permanent fastenings.
Removable fastenings
pinning key way
clamping spline
set screws taper and screw
Permanent fastenings:
press fit staking shrink fit pegging molded assembly riveting cementing compounds spinning
For more information on mounting gears refer to “Precision Gearing, Theory and Practice” by Michalec [7];
-- Shaft alignment. A deviation in the alignment of shafts is composed of two compo- nents: in--plane deviation and out--of--plane deviation. The in--plane deviation is measured in the common plane of axes and out--of--plane deviation is measured in the plane (skew plane) perpendicular to the common plane of axes. Fig- ure 35 shows the in--plane and out--of--plane deviations for two shafts A and B;
In--plane deviation Out--of--plane deviation Plane of axes Plane of axes Shaft B Shaft A Shaft B Shaft A
Figure 35 -- Shaft alignment deviations
-- Couplings. The coupling must have some
degree of flexibility to accommodate the types of misalignment mentioned previously. It must be able to transmit torque, yet limit the forces on ma- chine components such as shafts and bearings that result from misalignment. However, it is im- portant to understand the effects the coupling has on smooth motion when misalignment is present;
-- Gear--box housing. The housing must be
designed to remain sufficiently rigid during oper- ating conditions. It is possible to encounter a resonant condition, where the natural frequency of the complete system assembly coincides with an operating frequency.
It is beyond the scope of this Information Sheet to provide detailed information on vibration control and analysis. It is suggested that the designer consult someone experienced in this field or a text on the subject.