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Chapter 15: Problem 49

Figure P15.49 shows a planetary train with double planets, two suns, and no ring gear. Planets \(P 1\) and \(P 2\) are made together from the same piece of metal and have 40 and 32 teeth, respectively. Sun \(S 1\) is the input member and has 30 teeth. Sun \(S 2\) is fixed. All gears have the same pitch. For each clockwise revolution of \(S 1\), what is the motion of the arm?

Short Answer

Expert verified
For each clockwise revolution of S1, the arm makes 0.25 counter-clockwise revolutions.

Step by step solution

01

Analyzing the gear system

Here, we have a sun-planet gear system with two suns and no ring gear. The planets P1 and P2 have 40 and 32 teeth respectively, Sun S1 (the input gear) has 30 teeth and S2 is fixed in place. The gear ratios can be defined as the ratio of the teeth on the driving gear to the teeth on the driven gear.
02

Establish the gear ratios

We have the gear ratios as follows: Gear ratio between S1 and P1 \(=\frac{40}{30}=1.33\) and The gear ratio between P2 and S2 \(=\frac{32}{30}=1.07\). This implies that, for every 1.33 rotations of S1, P1 makes one complete clockwise rotation. Similarly, for every 1.07 rotations of S2, P2 makes one complete clockwise rotation.
03

Determine the movement of the arm

Since S2 is fixed, it does not rotate. Therefore, for each clockwise revolution of S1, P2 (since it's made from the same piece of metal as P1) also makes \(= \frac{1}{1.33} = 0.75\) clockwise rotation. Now P2 is engaged with a fixed sun gear S2, but it makes a rotation of 0.75, therefore, the arm to which both the planet gears are fixed makes \(=1-0.75 = 0.25\) counterclockwise revolutions.

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Key Concepts

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

Gear Ratios
Understanding gear ratios is essential in analyzing a planetary gear system. A gear ratio is essentially the ratio of the number of teeth on one gear to the number of teeth on another gear it interacts with. This ratio determines how fast one gear spins relative to another. For example, with the planetary train described:
  • The gear ratio between sun gear \( S_1 \) and planet gear \( P_1 \) is calculated by taking the teeth count on \( P_1 \) (40 teeth) divided by the teeth count on \( S_1 \) (30 teeth). This results in a gear ratio of \( \frac{40}{30} = 1.33 \).
  • This means for every 1.33 rotations of \( S_1 \), \( P_1 \) makes one full clockwise rotation.
  • Similarly, the gear ratio between \( P_2 \) and \( S_2 \) is \( \frac{32}{30} = 1.07 \), meaning for every 1.07 rotations of \( S_2 \), \( P_2 \) makes one full rotation.
By understanding these gear ratios, you can determine how different gears interact, affecting the overall motion of the system.
Clockwise and Counterclockwise Rotations
Rotational direction in planetary gears determines how forces and movements are transferred across the system. In the given example, understanding the interactions between the gears helps deduce the final movement.
  • When \( S_1 \) rotates clockwise, its interaction with \( P_1 \) causes \( P_1 \) to rotate in the opposite direction, depending on the gear ratio.
  • \( P_1 \) and \( P_2 \) are linked, meaning both gears will move similarly.
  • Since \( S_2 \) is fixed, the movement of \( P_2 \) must be analyzed carefully. It makes 0.75 clockwise rotations for every rotation of \( S_1 \).
  • However, since it's engaged with a fixed sun, this motion is transferred as counterclockwise to the arm, due to the gear arrangement.
So, you see a complex but predictable pattern. Gear arrangements and ratios dictate both speed and direction of rotation.
Sun and Planet Gear Configuration
A sun and planet gear configuration is an ingenious setup used commonly in modern machinery to achieve variable speed and torque. Instead of gears turning in unison as in simpler configurations, the planetary setup allows for more complex interactions.
  • The basic components are a central "sun" gear, surrounding "planet" gears, and sometimes a "ring" gear. In this case, the sun gears are \( S_1 \) and \( S_2 \), with planets \( P_1 \) and \( P_2 \).
  • The planets orbit around the sun, and any fixed gear, like \( S_2 \), can be used to achieve desired rotation or torque on the output arm.
  • In the given exercise, \( S_1 \) is the driving gear initiating motion, while \( S_2 \), being fixed, helps transfer rotation from \( P_2 \) to the arm as counterclockwise.
This configuration makes planetary gears favorable for systems demanding diverse speed output and efficient space use, as they allow more interaction points and variable movement under different settings.

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Most popular questions from this chapter

Write equations for the pitch, \(p\), in inches of a circular gear of diameter, \(d\), with \(N\) teeth.

A spur gear reducer has an 18-tooth pinion driven 1500 rpm by an electric motor and a 36 -tooth gear that drives a load involving "moderate shock." A life of \(10^{6}\) pinion revolutions is required, and the transmitted load, \(F_{t}\), is \(100 \mathrm{lb}\) (this figure includes a safety factor of 2 ). Conditions are such that \(K_{m}=1.8\) and \(k_{t}=1\). It is proposed that standard \(20^{\circ}\) full-depth gears be used, with both pinion and gear teeth being cut with a low-cost, average-quality, form-cutting process from steel of \(235 \mathrm{Bhn}\) for the gear and \(260 \mathrm{Bhn}\) for the pinion. Diametral pitch is to be 10 , and face width \(1.0\) in. Estimate the reliability with respect to bending fatigue failure.

An 18-tooth pinion having a \(20^{\circ}\) pressure angle meshes with a 36-tooth gear. The center distance is 10 in. The pinion has stub teeth. The gear has full-depth involute teeth. Determine the contact ratio (tooth intervals in contact) and the diametral pitch.

A pair of mating spur gears with 6-mm modules and \(0.35\)-rad pressure angles have 30 and 60 teeth. (a) Make a full-size drawing of the tooth contact region, showing (and labeling) (1) both pitch circles, (2) both base circles, (3) both root circles, (4) both outside circles, (5) pressure angle, (6) length of path of contact, (7) both angles of approach, and (8) both angles of recess. (b) Using values scaled from your drawing, state or calculate numerical values for (1) length of path of contact, (2) angles of approach, (3) angles of recess, and (4) contact ratio.

Sketch a pinion gear (driving) and a spur gear (driven), and show the position of a pair of mating teeth as they (a) enter contact (initial contact), and again as they (b) go out of contact (final contact). Draw and label points (a) and (b), the line of action and the pressure angle. Show the rotational direction of the pinion and the gear. Label for the pinion and gear the gear centers, line between centers, pitch point, pitch circles, base circles, addendum circles, dedendum circles, addendums, dedendums, angles of approach, and angles of recess.
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