II. Medios de contribuir a la difusión del
1. Medios de comunicación
The polar moment of inertia for a solid 0.75-in.-diameter shaft is J 32 (0.75 in.)4 0 0310631. in.4
Therefore, the maximum shear stress produced in the shaft is Tc
J
( lb-ft)(0.75 in./2)(12 in./ft) 0.0310
11 4890.
6
631 in. 1,664 psi
4
Ans.
A 2-m-long hollow steel [G 5 75 GPa] shaft has an outside diameter of 75 mm and an inside diameter of 65 mm. If the maximum shear stress in the shaft must be limited to 50 MPa and the angle of twist must be limited to 18, determine the maximum power that can be transmitted by this shaft when it is rotating at 600 rpm.
MecMovies Example M6.16
A motor shaft is being designed to transmit 40 kW of power at 900 rpm. If the shear-ing stress in the shaft must be limited to 75 MPa, determine
(a) the minimum diameter required for a solid shaft.
(b) the minimum outside diameter required for a hollow shaft if the shaft inside diameter is assumed to be 80 percent of its outside diameter.
MecMovies Example M6.17
The motor shown supplies 15 hp at 1,800 rpm at A. Shaft (1) is a solid 0.75-in.-diameter shaft, and shaft (2) is a solid 1.50-in.-diameter shaft.
Both shafts are made of steel [G 5 12,000 ksi]. The bearings shown permit free rotation of the shafts. Determine
(a) the maximum shear stress produced in each shaft.
(b) the rotation angle of gear D with respect to fl ange A.
MecMovies Example M6.18
⬘ c06Torsion.indd Page 177 1/27/12 10:27 AM user-F393
c06Torsion.indd Page 177 1/27/12 10:27 AM user-F393 /Users/user-F393/Desktop/Users/user-F393/Desktop
178
Two solid 25-mm-diameter steel shafts are connected by the gears shown. A motor supplies 20 kW at 15 Hz to the system at A. The bearings shown permit free rotation of the shafts. Determine (a) the torque available at gear D.
(b) the maximum shear stress magnitudes in each shaft.
Plan the Solution
The torque in shaft (1) can be calculated from the power trans-mission equation. The torque in shaft (2) can then be deter-mined from the gear ratio. Once the torques are known, the maximum shear stress magnitudes will be determined from the elastic torsion formula.
SOLUTION
The torque in shaft (1) can be calculated from the power transmission equation. The power supplied by the motor is 20 kW, or
P (20 kW) 1,000 W
kW 20,000 W 20,000 N-m s
1
The motor rotates at 15 Hz. This rotation speed must be converted to units of rad/s:
The torque in shaft (1) is therefore
T P
1 94 2477820,000 N-m/s 212 2066
rad/s N-m
. .
The torque in shaft (2) will be increased because gear C is larger than gear B. Use the number of teeth on each gear to establish the gear ratio, and compute the torque magni-tude in shaft (2) as
Note: Only the torque magnitude is needed in this instance; consequently, the absolute value of T2 is computed here.
The torque available at gear D in this system is therefore TD 5 340 N-m. Ans.
Shear Stresses
The polar moment of inertia for the solid 25-mm-diameter shafts is J 32 (25 mm)4 38,349.5 mm4
The maximum shear stress magnitudes in each segment can be calculated by the elastic torsion formula:
c06Torsion.indd Page 178 1/27/12 10:27 AM user-F393
c06Torsion.indd Page 178 1/27/12 10:27 AM user-F393 /Users/user-F393/Desktop/Users/user-F393/Desktop
FIGURE M6.15
M6.15 Six basic calculations involving power transmission in three shafts connected by gears.
MecMovies Exercises
M6.14 Six basic calculations involving power transmission in two shafts connected by gears.
FIGURE M6.14
179
PROBLEMS PROBLEMS
P6.36 The driveshaft of an automobile is being designed to transmit 180 hp at 3,500 rpm. Determine the minimum diameter required for a solid steel shaft if the allowable shear stress in the shaft is not to exceed 6,000 psi.
P6.37 A solid 20-mm-diameter bronze shaft transmits 11 kW at 25 Hz to the propeller of a small sailboat. Determine the maximum shear stress produced in the shaft.
P6.38 A tubular steel shaft is being designed to transmit 225 kW at 1,700 rpm. The maximum shear stress in the shaft must not exceed 30 MPa. If the outside diameter of the shaft is D 5 75 mm, determine the minimum wall thickness for the shaft.
P6.39 A solid 3-in.-diameter bronze [G 5 6,000 ksi] shaft is 7 ft long. The allowable shear stress in the shaft is 8 ksi, and the angle of twist must not exceed 0.03 rad. Determine the maximum horse-power that this shaft can deliver
(a) when rotating at 150 rpm.
(b) when rotating at 540 rpm.
P6.40 A tubular steel [G 5 80 GPa] shaft with an outside diameter of D 5 100 mm and a wall thickness of t 5 6 mm must not twist more than 0.05 rad in a 7-m length. Determine the maximum power that the shaft can transmit at 375 rpm.
P6.41 A hollow titanium [G 5 43 GPa] shaft has an outside diameter of D 5 50 mm and a wall thickness of t 5 1.25 mm. The maximum shear stress in the shaft must be limited to 150 MPa.
Determine
(a) the maximum power that can be transmitted by the shaft if the rotation speed must be limited to 20 Hz.
(b) the magnitude of the angle of twist in a 700-mm length of the shaft when 30 kW is being transmitted at 8 Hz.
P6.42 A tubular steel [G 5 80 GPa] shaft is being designed to transmit 150 kW at 30 Hz. The maximum shear stress in the shaft must not exceed 80 MPa, and the angle of twist is not to exceed 68
in a 4-m length. Determine the minimum permissible outside diam-eter if the ratio of the inside diamdiam-eter to the outside diamdiam-eter is 0.80.
P6.43 A tubular aluminum alloy [G 5 4,000 ksi] shaft is being designed to transmit 400 hp at 1,500 rpm. The maximum shear stress in the shaft must not exceed 6 ksi, and the angle of twist is not to exceed 58 in an 8-ft length. Determine the minimum permissible outside diameter if the inside diameter is to be three-fourths of the outside diameter.
P6.44 The impeller shaft of a fl uid agitator transmits 28 kW at 440 rpm. If the allowable shear stress in the impeller shaft must be limited to 80 MPa, determine
(a) the minimum diameter required for a solid impeller shaft.
(b) the maximum inside diameter permitted for a hollow impeller shaft if the outside diameter is 40 mm.
(c) the percent savings in weight realized if the hollow shaft is used instead of the solid shaft. (Hint: The weight of a shaft is proportional to its cross-sectional area.)
P6.45 A pulley with a diameter of D 5 8 in. is mounted on a shaft with a diameter of d 5 1.25 in. as shown in Figure P6.45. Around the pulley is a belt having tensions of F1 5 120 lb and F2 5 480 lb. If the shaft turns at 180 rpm, calculate
(a) the horsepower being transmitted by the shaft.
(b) the maximum shear stress in the shaft.
F2 c06Torsion.indd Page 179 1/27/12 10:28 AM user-F393
c06Torsion.indd Page 179 1/27/12 10:28 AM user-F393 /Users/user-F393/Desktop/Users/user-F393/Desktop
P6.46 A conveyor belt is driven by an 8-hp motor turning at 1,500 rpm. Through a series of gears that reduce the speed, the motor drives the belt drum shaft at a speed of 10 rpm. If the allow-able shear stress is 8,000 psi and both shafts are solid, calculate (a) the required diameter of the motor shaft.
(b) the required diameter of the belt drum shaft.
P6.47 A solid steel [G 5 80 GPa] shaft with a diameter of 40 mm and a length of 1.8 m transmits 30 kW of power from an electric motor to a compressor. If the allowable shear stress is 60 MPa and the allowable angle of twist is 1.58, what is the slowest allowable speed of rotation?
P6.48 A 1.50-in.-diameter solid bronze [G 5 6,500 ksi] shaft is used to transmit 15 hp. The length of the shaft is 42 in. If the allow-able shear stress is 6,000 psi and the allowallow-able angle of twist is 2.58, calculate the slowest permissible speed of rotation in Hz.
P6.49 A motor supplies 200 kW at 6 Hz to fl ange A of the shaft shown in Figure P6.49/50. Gear B transfers 125 kW of power to operating machinery in the factory, and the remaining power in the shaft is transferred by gear D. Shafts (1) and (2) are solid aluminum [G 5 28 GPa] shafts that have the same diameter and an allowable shear stress of 5 40 MPa. Shaft (3) is a solid steel [G 5 80 GPa]
shaft with an allowable shear stress of 5 55 MPa. Determine (a) the minimum permissible diameter for aluminum shafts
(1) and (2).
(b) the minimum permissible diameter for steel shaft (3).
(c) the rotation angle of gear D with respect to fl ange A if the shafts have the minimum permissible diameters as determined in (a) and (b). shown in Figure P6.49/50. Gear B transfers 40 kW of power to op-erating machinery in the factory, and the remaining power in the shaft is transferred by gear D. Shafts (1) and (2) are solid 65-mm-diameter aluminum [G 5 28 GPa] shafts, and shaft (3) is a solid 40-mm-diameter steel [G 5 80 GPa] shaft. Calculate
(a) the maximum shear stress in the aluminum shafts.
(b) the maximum shear stress in the steel shaft.
(c) the rotation angle of gear D with respect to fl ange A.
P6.51 A motor supplies suffi cient power to the system shown in Figure P6.51/52 so that gears C and D provide torques of TC 5 800 N-m and TD 5 550 N-m, respectively, to machinery in a factory.
Power shaft segments (1) and (2) are hollow steel tubes with an outside diameter of D 5 60 mm and an inside diameter of d 5 50 mm. If the power shaft [i.e., segments (1) and (2)] rotates at 40 rpm, determine (a) the maximum shear stress in power shaft segments (1) and (2).
(b) the power (in kW) that must be provided by the motor as well as the rotation speed (in rpm).
(c) the torque applied to gear A by the motor.
TD
P6.52 A motor supplies 9 kW to the system shown in Figure P6.51/52. Sixty-fi ve percent of the power supplied by the motor is taken off by gear C, and the remaining 35 percent of the power is taken off by gear D. Power shaft segments (1) and (2) are hollow steel tubes with an outside diameter of D 5 60 mm and an inside diameter of d 5 50 mm. If the allowable shear stress for the steel tubes is 55 MPa, calculate the slowest permissible rotation speed for the motor.
P6.53 A motor supplies 25 hp at 6 Hz to gear A of the drive system shown in Figure P6.53/54. Shaft (1) is a solid 2.25-in.-diameter alumi-num [G 5 4,000 ksi] shaft with a length of L1 5 16 in. Shaft (2) is a solid 1.5-in.-diameter steel [G 5 12,000 ksi] shaft with a length of L2 5 12 in. Shafts (1) and (2) are connected at fl ange C, and the bear-ings shown permit free rotation of the shaft. Determine
(a) the maximum shear stress in shafts (1) and (2).
(b) the rotation angle of gear D with respect to gear B.
TD
P6.54 A motor supplies 15 hp to gear A of the drive system shown in Figure P6.53/54. Shaft (1) is a solid 2.25-in.-diameter aluminum [G 5 4,000 ksi] shaft with a length of L1 5 16 in. and an allowable shear stress of 6,000 psi. Shaft (2) is a solid 1.5-in.- diameter steel [G 5 12,000 ksi] shaft with a length of L2 5 12 in.
180
c06Torsion.indd Page 180 1/27/12 10:28 AM user-F393
c06Torsion.indd Page 180 1/27/12 10:28 AM user-F393 /Users/user-F393/Desktop/Users/user-F393/Desktop