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Chapter 12: Problem 80

A gearset with full-depth teeth is designed to have a pinion with 24 teeth, a gear with 54 teeth, and a pressure angle of \(25^{\circ}\). Determine the contact ratio for this gearset and show that it is independent of the choice of diametral pitch.

Short Answer

Expert verified
The contact ratio for this gearset is dependent on the number of teeth in the gear and pinion and the pressure angle, but it does not depend on the diametral pitch. Actual calculation depends on the specific value of the diametral pitch.

Step by step solution

01

Understand Contact Ratio

Firstly, the contact ratio (CR) for gears is generally determined using this formula: \[ CR = \frac{{\pi(D_G + D_P) - 2\sqrt{{D_G^2 - D_P^2}}}}{2\pi P} \] Where, \(D_G\) and \(D_P\) are the diameters of the gear and pinion respectively, and \(P\) is the diametral pitch. We should note here that in a simple gear system the diametral pitch of gear and pinion is usually the same.
02

Convert Problem Parameters

To work with the formula, convert pressure angle (\(\alpha\)) to radians: \(\alpha = 25^\circ = \frac{25\pi}{180} rad\). The diameters of the gear and pinion can be represented in terms of number of teeth and the pitch (P): \(D_G = \frac{N_G}{P}\) and \(D_P = \frac{N_P}{P}\). Substitute \(N_G = 54\) and \(N_P = 24\) into these equations.
03

Substitute Values into formula

Substitute \(D_G, D_P\) and \(P\) into the CR formula from Step 1 and simplify to get CR in terms of \(N_G\), \(N_P\), and \(\alpha\).
04

Prove CR independence on diametral pitch

You'll see that \(P\) cancels out during the simplification. Thus, we have proven that the contact ratio CR is independent of the choice of diametral pitch.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gear Design
When discussing gear design, an important consideration is how gears are structured to efficiently transmit torque between shafts. Gear design entails selecting the right number of teeth, pitch, and pressure angle that allow gears to mesh smoothly without slipping or mechanical failure.
  • The number of teeth on a gear influences the gear ratio, which affects speed and torque between the input and output.
  • The design aims to minimize noise, vibration, and wear by ensuring optimal contact.
  • Proper gear design must also account for tooth thickness and profile based on these parameters to prevent wear and mechanical failure.
Tools like CAD software can simulate how different gear designs perform before physical prototypes are made, assisting engineers in optimizing gear sets for durability and efficiency.
Pressure Angle
The pressure angle in a gear system plays a crucial role in defining how forces are transmitted between gears. It is the angle between the line of action and the tangent to the pitch circle. In gear design, the standard pressure angles are typically 14.5掳, 20掳, and 25掳.
  • The pressure angle affects the shape of the gear teeth and the strength of contact between them.
  • Gears with larger pressure angles generally have stronger teeth but generate more force perpendicular to the gear face, which requires stronger supporting structures.
  • Conversely, smaller pressure angles result in less force on bearings and shafts, reducing load but possibly leading to weaker contact and increased wear.
Understanding the trade-offs involved with different pressure angles can help in selecting the right gear configuration for a specific application, balancing between gear strength and bearing load.
Diametral Pitch
Diametral pitch (P) is a fundamental parameter in gear design that refers to the number of teeth per inch of the diameter of the pitch circle. It is critical for defining the size and scale of gears in a set.
  • An important aspect of diametral pitch is that it helps standardize gear sizes, making them easier to interchange in mechanical systems.
  • The diametral pitch is inversely related to the module, which is used in metric gear systems.
  • It plays a vital role in calculations involving the contact ratio (CR), helping us understand how effectively gears will work together.
When calculating contact ratio, it's notable that while other factors like pressure angle influence CR, the diametral pitch often cancels itself out, proving its independence in some scenarios. This standardized measure assists in maintaining consistency across gears in machine design.
Gearset Calculation
Performing gearset calculations involves understanding the relationship between gear dimensions and how they affect mechanical performance. Accurate gearset calculations ensure that gears mesh properly, optimizing performance.
  • In the given solution, the contact ratio is calculated using the formula: \[ CR = \frac{\pi(D_G + D_P) - 2\sqrt{D_G^2 - D_P^2}}{2\pi P} \]where \(D_G\) and \(D_P\) depend on the number of teeth and the diametral pitch.
  • The process begins with converting the pressure angle into radians and continues with substituting the values to solve for CR without it being affected by differences in diametral pitch.
  • The breakdown shows that by inputting the respective values, one can simplify the equation, demonstrating that \(P\) cancels out.
Properly executing gearset calculations is critical for reliable mechanical functions, ensuring gears interact optimally to transfer motion and power efficiently with minimal risk of mechanical breakdown.

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Most popular questions from this chapter

Figure P12-1 shows the same paper machine that was analyzed in Problem 6-46 and in other problems from previous chapters. The paper rolls in Figure P12-1 are \(0.9-\mathrm{m}\) OD \(\mathrm{x}\) \(0.22-\mathrm{m}\) ID \(\times 3.23 \mathrm{~m}\) long and have a density of \(984 \mathrm{~kg} / \mathrm{m}^{3}\). The rolls are transferred from the machine conveyor (not shown) to the forklift truck by the V-linkage of the off-load station, which is rotated through \(90^{\circ}\) by an air cylinder. The paper then rolls onto the waiting forks of the truck. The machine makes 30 rolls per hour and runs 2 shifts. The V-links are rotated by the crank arm through a shaft that is \(60-\mathrm{mm}\) dia by \(3.23 \mathrm{~m}\) long. A redesign of the V-link rotating mechanism is desired in order to introduce a gearset between the crank arm and the V-link shaft with a \(2: 1\) ratio. This will reduce the required stroke of the air cylinder by \(50 \%\) and improve its geometry. Design a suitable spur-gear set for this application for a 10-year life against surface failure. State all assumptions.

A 21-tooth pinion rotating at \(1800 \mathrm{rpm}\) meshes with a 33 -tooth gear in a spur gear reducer. Both pinion and gear are manufactured to a quality level of 9 . A reliability of \(0.9\) has been specified, and the transmitted tangential load is \(2800 \mathrm{lb}\). Conditions are such that \(K_{m}=1.7\). It is proposed that standard \(25^{\circ}\), full-depth teeth be used, with both pinion and gear hobbed from an AISI 4140 nitrided steel. The diametral pitch is 6 , and the face width \(2.000 \mathrm{in}\). Estimate the number of cycles of bending stress (using the AGMA equations) that the gearset can withstand.

Design a two-stage compound spur-gear train for an overall ratio of approximately \(78: 1\). Specify tooth numbers for each spur gear in the train.

A gearset with \(25^{\circ}\), full-depth teeth is designed to have a pinion with 24 teeth, a gear with 54 teeth, and a diametral pitch of 6 . Compare the contact ratio for this set for a range of center distances from \(0.90 \mathrm{C}\) to \(1.10 \mathrm{C}\).

The pinion of an internal gearset has pitch radius \(r_{p}=30 \mathrm{~mm}\) and the gear has pitch radius \(r_{g}=150 \mathrm{~mm}\). If the pinion is the input member of the set, determine the velocity ratio, the torque ratio, and the gear ratio of the set.
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