THE THEORY OF MECCANO GEARS

Part 2: Helical Gears

1) Introduction

Presumably helical gears were introduced into the Meccano system to provide an alternative crossed axis drive system to the worm gear, and in response to requests for a skew gear in the system. Also, helical gears can be ‘back driven’ whereas single start worms gears generally cannot be. The original Meccano helical gears were in the form of the 14 and 35 tooth gears, parts 211a & 211b, giving a reduction ratio of 2.5:1, a much smaller numerical ratio than usually obtained with a worm and wheel. (See figure 1). Licensed and reproduction Meccano specialists have added to the available range of compatible helical gears.

2) General Helical Gear Principles

Helical gears have teeth cut at an angle to their cylindrical axis, as opposed to spur gears having so called ‘straight cut’ teeth (parallel to their axis). The teeth can be envisaged as a spiral running round a cylinder, and can be left or right hand helices (direction of wind). This is rather similar to a ‘quick’ screw thread, as opposed to the normal screw thread or worm.

The helical angle can vary according to the application requirements. The Meccano helical gears (211a/211b) have a helix angle of 45 degrees, and are both right hand helices. This means that they will mesh on axes at 90 degrees to each other (crossed axes). Gears having the same helix angle, but of opposite hand, can be meshed together on parallel axes.

The commercial advantages of helical gears are that the teeth are stronger than the equivalent spur gear due to the longer and thicker (in the plane of rotation) tooth form, and they run quieter owing to the additional contact ratio achieved through the helical overlap. The helix angle of parallel axis gearing is generally less than 20 degrees, and is determined by the facewidth and the degree of overlap required. Otherwise the helix angle can be set to create gears that mesh on axes at any angle. The disadvantage of helical gearing is that axial thrust is created, the magnitude of which depends upon on the helix angle, and this thrust load has to be borne by the bearings and mounting arrangements. Also, in non-parallel axis helical gearing, sliding friction is significant.

The direction of thrust created by the helical action depends upon the direction of rotation and the hand of the helix. There are too many variations and arrangements of helical gears to describe the thrust direction, but it can easily be assessed by imagining the helix of the driver as a screw thread, and the direction of the resultant axial thrust this would produce on the drive shaft. Similarly, the reactive thrust direction on the driven shaft can be determined. In reversible drives, the thrust will act in both directions.

3) Helical Gear Proportions, Definitions & Terminology

Compared to straight spur gears, most of the proportions of helical gears are modified by the helix angle. For example: for a given pitch, all the radial proportions and center distances of helical gears are larger by a function of the helix angle. Tooth proportions and functions can be described in terms of the normal pitch (at right angles to the tooth) or transverse (in the plane of rotation). Figure 2 shows definitions for the helical gear¹.

The influence of the helix angle on the radial proportions of a helical gear, render it more difficult to obtain exact or integer values of pitch diameter and center distance. For example, if a specific pitch diameter and/or center distance is required, involute profile shifting or a compromise helix angle can achieve them. The latter affords the designer some flexibility in achieving specific pitch diameters or center distance by making the helix angle the variable in the calculations. Thus, the helix angle need not be an integer figure, but the required fractional/decimal value dictated by other parameters. Because the helix angle increases the pitch diameter (relative to straight spur gears), for a given center distance, the equivalent helical gear is courser in pitch. In the case of Meccano spur gears, 38DP equates to a one-inch center distance. For the 45° helicals (211a&b) the pitch is 35DP.

Where common, the notations are the same as those used in part 1(qv).

Helix Angle (σ)

This is the angle the involute tooth form makes with the transverse plane (plane of rotation) at the pitch radius.

Diametral Pitch (DP) (P_n)

This is the pitch normal (at right angles) to the tooth. It can be expressed in Module or Circular terms, also in the Transverse plane.

Direction of Helix (RH) or (LH)

The direction of wind of the helix. Can also be expressed as Lead, the axial advance of the tooth in one revolution (as in thread pitch).

Pitch Circle (PCD) (D)

The standard pitch circle diameter is given by: D = N/P_n /cos σ......(1)

Operating Pitch Circle (D_e)

The effective rolling diameter D_e = 2NC_e /(N+n)......(2)

Pressure Angle (PA) (ψ_n)

The pressure angle normal to the tooth. This is usually the pressure angle of the cutter (of generation)

Transverse pressure angle (ψ_t)

The pressure angle in the plane of rotation at the standard PCD. (Note: Helical gear teeth appear thicker in this plane than in the ‘normal’ plane).

Ψ_t = arc tan(tan ψ_n/cos σ) ......(3)

Centre Distance – Standard (C)

C = (N+n)/cos σ/2/P_n ......(4)

Extended Center Distance (C_e)

The intended operating center distance - extended or contracted standard center distance.

Addendum (A)

A = 1/P_n ......(5)

Outside Diameter (OD)(D_o)

D_o = D + 2A ......(6)

Dedendum (B)

Assuming the same root clearance of 0.4/P as for straight Meccano spur gears. B = 1.4/P_n

Whole Depth of Tooth (h)

h = A + B ......(7)

Root Diameter (D_r)

D_r = D – 2B ......(8)

Contact Ratio (C_r)

This is the sum of the involute tooth overlap and the helical overlap, hence a higher value than can be achieved with straight spur gears.

Profile Shift (k)

The amount the involute profile is displaced radially from standard. It can be applied to straight spur gears to meet specific operating conditions. It is invariably applied to helical gears for the same reason and to compensate for the radial shift created by the helix angle.

4) Helical Gear Design

The design of helical gears involves complex calculation to determine the optimum geometry, operating conditions and inspection measurements. Attempting to ‘reverse’ engineer a pair of gears without any knowledge of the original design intent or manufacturing methods is fraught with difficulties.

Gears may be initially designed in accordance with established standards and commercial practice, when a technical specification is drawn-up in the form of a working drawing. Such gears would be designed around available machinery and cutters, or maybe special cutters manufactured for the purpose – usually very expensive. Alternatively, they may be ‘created’ in a prototype workshop using available cutting tools and equipment, with the overall proportions established by trial and error. When the desired proportions are established, they, the machine settings and the tooling geometry would be recorded for future reference and manufacture of further batches.

It is probable that the design and conception of Meccano helical gears falls somewhere in between the two extremes mentioned above. Thus, not being aware of the design or manufacturing philosophy, some assumptions have to be made. I have assumed that the Meccano helical gears conform to standard gear design practice and manufacture for machine cut gears.

A copy of the spreadsheet output from my ‘helicalgearcalc’ programme² for the 14/35t Meccano helical gears (211a&b) is presented below (Fig 3.) This version of the programme adopts Imperial proportions for pitch in the form of diametral pitch and standard proportions according to the following definition: “20 degree pressure angle, full depth, machine cut involute tooth form in accordance with British Standard: BS 436 (and others)”.

An initial study of my pair of Meccano helical gears indicates that they are 20-degree pressure angle, of involute tooth form, having a pitch of 35 DP. This data is entered into the programme, together with the tooth numbers, which then calculates the remaining data. The dimensions predicted by the programme are confirmed by the actual sample measurements shown in italics, the small differences being within normal manufacturing tolerances. This agreement is despite the fact that my sample 35-tooth gear has an oversized root diameter, so much so that in close mesh, the crests of the14t pinion teeth interfere with the root diameter of the 35t gear!

The programme output is more comprehensive than is required for Meccano gears because it was developed for commercial engineering applications, and the presentation is automatic for the given input data. Note: there are some differences in nomenclature between the computer programme and the above notes.

Postscript: It may be (and highly probable) that less theory was applied in the design of Meccano gears that I would like to believe. In which case, maybe a better title for these articles would be: "The (assumed) Theory of Meccano Gears!"

Feedback: Please direct any queries or comments to the "Other / General Resources / Prototype Information & Research" section of the Meccanoscene public forum.

References:

1) GEAR HANDBOOK – Darle W Dudley

2) Computer programme: ‘helicalgearcalc’ by A Wenbourne

NB: Numbers have been associated with formulae to aid subsequent reference.

© A Wenbourne 2008