5. Propuesta educomunicativa: Andalucía Diversa
5.1. Bases
FIGURE 6.14 Basic gear assembly.
FIGURE 6.15 Free-body diagrams of gears A and B.
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Gears are a fundamental component found in many types of mechanisms and devices—
particularly those devices that are driven by motors or engines. Gears are used for many purposes, such as
•
transmitting torque from one shaft to another,•
increasing or decreasing torque in a shaft,•
increasing or decreasing the rate of rotation of a shaft,•
changing the rotation direction of two shafts, and•
changing rotational motion from one orientation to another; for instance, changing rotation about a horizontal axis to rotation about a vertical axis.Furthermore, since gears have teeth, shafts connected by gears are always synchronized exactly with one another.
A basic gear assembly is shown in Figure 6.14. In this assembly, torque is transmitted from shaft (1) to shaft (2) by means of gears A and B, which have radii of RA and RB, re-spectively. The number of teeth on each gear is denoted by NA and NB. Positive internal torques T1 and T2 are assumed in shafts (1) and (2). For clarity, bearings necessary to sup-port the two shafts have been omitted. This confi guration will be used to illustrate basic relationships involving torque, rotation angle, and rotation speed in torsion assemblies with gears.
Torque
To illustrate the relationship between the internal torques in shafts (1) and (2), free-body diagrams of each gear are shown in Figure 6.15. If the system is to be in equilibrium, then each gear must satisfy equilibrium. Consider the free-body diagram of gear A. The internal torque T1 acting in shaft (1) is transmitted directly to gear A. This torque causes gear A to rotate counterclockwise. As gears A and B rotate, the teeth of gear B exert a force on gear A that acts tangential to both gears. This force, which opposes the rotation of gear A, is
6.7 Gears in Torsion Assemblies
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denoted by F. A moment equilibrium equation about the x axis gives the relationship
Next, consider the free-body diagram of gear B. If the teeth of gear B exert a force F on gear A, then the teeth of gear A must exert on gear B a force that is equal in magnitude, but acts in the opposite direction. This force causes gear B to rotate clockwise. A moment equilib-rium equation about the x9 axis gives
ΣMx F RBT2 0 (b) If the expression for F determined in Equation (a) is substituted into Equation (b), then the torque T2 required to satisfy equilibrium can be expressed in terms of torque T1:
T ⬖
The magnitude of T2 is related to T1 by the ratio of the gear radii. Since the two gears rotate in opposite directions, however, the sign of T2 is opposite from the sign of T1.
Gear Ratio. The ratio RByRA in Equation (c) is called the gear ratio, and this ratio is the key parameter that dictates relationships between shafts connected by gears. The gear ratio in Equation (c) is expressed in terms of the gear radii; however, this parameter can also be expressed in terms of gear diameters or gear teeth.
The diameter D of a gear is simply two times its radius R. Accordingly, the gear ratio in Equation (c) could also be expressed as DByDA, where DA and DB are the diameters of gears A and B, respectively.
For two gears to interlock properly, the teeth on both gears must be the same size. In other words, the arclength of a single tooth, which is termed the pitch p, must be the same for both gears. The circumference C of gears A and B can be expressed either in terms of gear radius,
CA 2RA CB 2RB or in terms of the pitch p and the number of teeth N on the gear,
CA pNA CB pNB
The circumference expressions for each gear can be equated and solved for the pitch p on each gear:
Moreover, since the tooth pitch p must be the same for both gears, R
In summary, the gear ratio between any two gears A and B can be expressed equivalently by either gear radii, gear diameters, or numbers of gear teeth:
Gear ratio R
GEARS IN TORSION ASSEMBLIES
MecMovies 6.9 presents an animation that illustrates basic gear relationships for torque, rotation angle, rotation speed, and power transmission.
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TORSION Rotation Angle. When gear A turns through an angle of A as shown in Figure 6.16, the arclength sA along the perimeter of gear A is sA 5 RAA. Similarly, the arclength sB along the perimeter of gear B is sB 5y RBB. Since the teeth on each gear must be the same size, the arclengths that are turned by the two gears must be equal in magnitude. The two gears, however, turn in opposite directions. If sA and sB are equated and rotation in the op-posite direction is accounted for, then the rotation angle A can be expressed as
R R R
A A B B A RB
A B
⬖ (e)
Note: The term RByRA in Equation (e) is simply the gear ratio; therefore,
A (Gear ratio)B (f )
Rotation Speed. Rotation speed is the rotation angle turned by the gear in a unit of time; therefore, the rotation speeds of two interlocked gears are related in the same man-ner as described for rotation angles.
A (Gear ratio)B (g)
FIGURE 6.16 Rotation angles for gears A and B.
Two solid steel [G 5 80 GPa] shafts are connected by the gears shown. Shaft (1) has a diameter of 35 mm, and shaft (2) has a diameter of 30 mm. Assume that the bearings shown allow free rotation of the shafts. If a 315 N-m torque is applied at gear D, determine
(a) the maximum shear stress magnitudes in each shaft.
(b) the angles of twist 1 and 2.
(c) the rotation angles B and C of gears B and C, respectively.
(d) the rotation angle of gear D.
Plan the Solution
The internal torque in shaft (2) can easily be determined from a free-body diagram of gear D; however, the internal torque in shaft (1) will be dictated by the ratio of gear sizes. Once you have determined the internal torques in both shafts, calculate the angles of twist in each shaft, paying particular attention to the signs of the twist angles.
The twist angle in shaft (1) will dictate how much gear B rotates, which in turn will dictate the rotation angle of gear C. The rotation angle of gear D will depend upon the rotation angle of gear C and the angle of twist in shaft (2).
SOLUTION Equilibrium
Consider a free-body diagram that cuts through shaft (2) and includes gear D. A positive internal torque will be assumed in shaft (2). From this free-body diagram, a moment equilibrium equation about the x 9 axis can be written to determine the internal torque T2
in shaft (2).
ΣMx 315N-mT2 0 ⬖T2 315N-m (a) Next, consider a free-body diagram that cuts through shaft (2) and includes gear C. Once again, a positive internal torque will be assumed in shaft (2). The teeth of
850 mm
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gear B exert a force F on the teeth of gear C. If the radius of gear C is denoted by RC, a moment equilibrium equation about the x9 axis can be written as
ΣM T F R F T
x C R
C
2 0 ⬖ 2 (b)
A free-body diagram of gear B that cuts through shaft (1) is shown. A positive internal torque T1 is assumed to act in shaft (1). If the teeth of gear B exert a force F on the teeth of gear C, then equilibrium requires that the teeth of gear C exert an equal mag-nitude force in the opposite direction on the teeth of gear B. With the radius of gear B denoted by RB, a moment equilibrium equation about the x axis can be written as
ΣMx T1 F RB 0 ⬖ T1 F RB (c) The internal torque in shaft (2) is given by Equation (a). The internal torque in shaft (1) can be determined by substituting Equation (b) into Equation (c):
T F R T between gears B and C. Since the teeth on both gears must be the same size in order for the gears to mesh properly, the ratio of teeth on each gear is equivalent to the ratio of gear radii. Consequently, the torque in shaft (1) can be expressed in terms of NB and NC, the number of teeth on gears B and C, respectively:
T T R
The maximum shear stress magnitude in each shaft will be calculated from the elastic tor-sion formula. The polar moments of inertia for each shaft will be required for this calcula-tion. Shaft (1) is a solid 35-mm-diameter shaft, which has a polar moment of inertia of
J1 4 4
32 147 324
(35 mm) , mm
Shaft (2) is a solid 30-mm-diameter shaft, which has a polar moment of inertia of
J2 4 4
32 79 552
(30 mm) , mm
To calculate the maximum shear stress magnitudes, the absolute values of T1 and T2 will be used. The maximum shear stress magnitude in the 35-mm-diameter shaft (1) is
1 1 1
and the maximum shear stress magnitude in the 30-mm-diameter shaft (2) is
2 2 2 long, and its shear modulus is G 5 80 GPa 5 80,000 MPa. The angle of twist in this shaft is
1 1 1 1 1
T L 405 J G
( N-m)(600 mm)(1,000 mm/m)
147,324 mmm 80,000 N/mm rad rad
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