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A 1/4 -in drill rod was heat-treated and ground. The measured hardness was found to be 490 Brinell. Estimate the endurance strength if the rod is used in rotating bending.
Estimate S′e for the following materials:
(a) AISI 1020 CD steel.
(b) AISI 1080 HR steel.
(c) 2024 T3 aluminum.
(d) AISI 4340 steel heat-treated to a tensile strength of 250 kpsi.
Estimate the fatigue strength of a rotating-beam specimen made of AISI 1020 hot-rolled steel corresponding to a life of 12.5 kilocycles of stress reversal. Also, estimate the life of the specimen corresponding to a stress amplitude of 36 kpsi. The known properties are Sut = 66.2 kpsi, σ0 = 115 kpsi, m = 0.22, and εf = 0.90.
Derive Eq. (6–17). For the specimen of Prob. 6–3, estimate the strength corresponding to 500 cycles.
For the interval 103 ≤ N ≤ 106 cycles, develop an expression for the axial fatigue strength (S′f ) ax for the polished specimens of 4130 used to obtain Fig. 6–10. The ultimate strength is Sut = 125 kpsi and the endurance limit is (S′e) ax = 50 kpsi.
Estimate the endurance strength of a 32-mm-diameter rod of AISI 1035 steel having a machined finish and heat-treated to a tensile strength of 710 MPa.
Two steels are being considered for manufacture of as-forged connecting rods. One is AISI 4340 Cr-Mo-Ni steel capable of being heat-treated to a tensile strength of 260 kpsi. The other is a plain carbon steel AISI 1040 with an attainable Sut of 113 kpsi. If each rod is to have a size giving an equivalent diameter de of 0.75 in, is there any advantage to using the alloy steel for this fatigue application?
A solid round bar, 25 mm in diameter, has a groove 2.5-mm deep with a 2.5-mm radius machined into it. The bar is made of AISI 1018 CD steel and is subjected to a purely reversing torque of 200 N • m. For the S-N curve of this material, let f = 0.9.
(a) Estimate the number of cycles to failure.
(b) If the bar is also placed in an environment with a temperature of 450°C, estimate the number of cycles to failure.
A solid square rod is cantilevered at one end. The rod is 0.8 m long and supports a completely reversing transverse load at the other end of ±1 kN. The material is AISI 1045 hot-rolled steel. If the rod must support this load for 104 cycles with a factor of safety of 1.5, what dimension should the square cross section have? Neglect any stress concentrations at the support end and assume that f = 0.9.
A rectangular bar is cut from an AISI 1018 cold-drawn steel flat. The bar is 60 mm wide by
10 mm thick and has a 12-mm hole drilled through the center as depicted in Table A–15–1. The bar is concentrically loaded in push-pull fatigue by axial forces Fa , uniformly distributed across the width. Using a design factor of nd = 1.8, estimate the largest force Fa that can be applied ignoring column action.
Bearing reactions R1 and R2 are exerted on the shaft shown in the figure, which rotates at 1150 rev/min and supports a 10-kip bending force. Use a 1095 HR steel. Specify a diameter d using a design factor of nd = 1.6 for a life of 3 min. The surfaces are machined.
A bar of steel has the minimum properties Se = 276 MPa, Sy = 413 MPa, and Sut = 551 MPa. The bar is subjected to a steady torsional stress of 103 MPa and an alternating bending stress of 172 MPa. Find the factor of safety guarding against a static failure, and either the factor of safety guarding against a fatigue failure or the expected life of the part. For the fatigue analysis use:
(a) Modified Goodman criterion.
(b) Gerber criterion.
(c) ASME-elliptic criterion.
Repeat Prob. 6–12 but with a steady torsional stress of 138 MPa and an alternating bending stress of 69 MPa.
Repeat Prob. 6–12 but with a steady torsional stress of 103 MPa, an alternating torsional stress of 69 MPa, and an alternating bending stress of 83 MPa.
Repeat Prob. 6–12 but with an alternating torsional stress of 207 MPa.
Repeat Prob. 6–12 but with an alternating torsional stress of 103 MPa and a steady bending stress of 103 MPa.
The cold-drawn AISI 1018 steel bar shown in the figure is subjected to an axial load fluctuating between 800 and 3000 lbf. Estimate the factors of safety ny and nf using (a) a Gerber fatigue failure criterion as part of the designer’s fatigue diagram, and (b) an ASME-elliptic fatigue failure criterion as part of the designer’s fatigue diagram.
Repeat Prob. 6–17, with the load fluctuating between .800 and 3000 lbf. Assume no buckling.
Repeat Prob. 6–17, with the load fluctuating between 800 and -3000 lbf. Assume no buckling.
The figure shows a formed round-wire cantilever spring subjected to a varying force. The hardness tests made on 25 springs gave a minimum hardness of 380 Brinell. It is apparent from the mounting details that there is no stress concentration. A visual inspection of the springs indicates that the surface finish corresponds closely to a hot-rolled finish. What number of applications is likely to cause failure? Solve using:
(a) Modified Goodman criterion.
(b) Gerber criterion.
The figure is a drawing of a 3- by 18-mm latching spring. A preload is obtained during assembly by shimming under the bolts to obtain an estimated initial deflection of 2 mm. The latching operation itself requires an additional deflection of exactly 4 mm. The material is ground high-carbon steel, bent then hardened and tempered to a minimum hardness of 490 Bhn. The radius of the bend is 3 mm. Estimate the yield strength to be 90 percent of the ultimate strength.
(a) Find the maximum and minimum latching forces.
(b) Is it likely the spring will fail in fatigue? Use the Gerber criterion.
Repeat Prob. 6–21, part b, using the modified Goodman criterion.
The figure shows the free-body diagram of a connecting-link portion having stress concentration at three sections. The dimensions are r = 0.25 in, d = 0.75 in, h = 0.50 in, w1 = 3.75 in, and w2 = 2.5 in. The forces F fluctuate between a tension of 4 kip and a compression of 16 kip. Neglect column action and find the least factor of safety if the material is cold-drawn AISI 1018 steel.
The torsional coupling in the figure is composed of a curved beam of square cross section that is welded to an input shaft and output plate. A torque is applied to the shaft and cycles from zero to T. The cross section of the beam has dimensions of 5 by 5 mm, and the centroidal axis of the beam describes a curve of the form r = 20 + 10 θ/π, where r and θ are in mm and radians, respectively (0 ≤ θ ≤ 4π). The curved beam has a machined surface with yield and ultimate strength values of 420 and 770 MPa, respectively.
(a) Determine the maximum allowable value of T such that the coupling will have an infinite life with a factor of safety, n = 3, using the modified Goodman criterion. (b) Repeat part (a) using the Gerber criterion.
(c) Using T found in part (b), determine the factor of safety guarding against yield.
Repeat Prob. 6–24 ignoring curvature effects on the bending stress.
In the figure shown, shaft A, made of AISI 1010 hot-rolled steel, is welded to a fixed support and is subjected to loading by equal and opposite forces F via shaft B.A theoretical stress concentration Kt s of 1.6 is induced by the 3-mm fillet. The length of shaft A from the fixed support to the connection at shaft B is 1 m. The load F cycles from 0.5 to 2 kN.
(a) For shaft A, find the factor of safety for infinite life using the modified Goodman fatigue failure criterion.
(b) Repeat part (a) using the Gerber fatigue failure criterion.
A schematic of a clutch-testing machine is shown. The steel shaft rotates at a constant speed ω. An axial load is applied to the shaft and is cycled from zero to P. The torque T induced by the clutch face onto the shaft is given by
T = f P (D + d)
4 Where D and d are defined in the figure and f is the coefficient of friction of the clutch face. The shaft is machined with Sy = 800 MPa and Sut = 1000 MPa. The theoretical stress concentration factors for the fillet are 3.0 and 1.8 for the axial and torsional loading, respectively.
(a) Assume the load variation P is synchronous with shaft rotation. With f = 0.3, find the maximum allowable load P such that the shaft will survive a minimum of 106 cycles with a factor of safety of 3. Use the modified Goodman criterion. Determine the corresponding factor of safety guarding against yielding.
(b) Suppose the shaft is not rotating, but the load P is cycled as shown. With f = 0.3, find the maximum allowable load P so that the shaft will survive a minimum of 106 cycles with a factor of safety of 3. Use the modified Goodman criterion. Determine the corresponding factor of safety guarding against yielding.
For the clutch of Prob. 6–27, the external load P is cycled between 20 kN and 80 kN. Assuming that the shaft is rotating synchronous with the external load cycle, estimate the number of cycles to failure. Use the modified Goodman fatigue failure criteria.
A flat leaf spring has fluctuating stress of σmax = 420 MPa and σmin = 140 MPa applied for 5 (104) cycles. If the load changes to σmax = 350 MPa and σmin = .200 MPa, how many cycles should the spring survive? The material is AISI 1040 CD and has a fully corrected endurance strength of Se = 200 MPa. Assume that f = 0.9.
(a) Use Miner’s method.
(b) Use Manson’s method.
A machine part will be cycled at ±48 kpsi for 4 (103) cycles. Then the loading will be changed to ±38 kpsi for 6 (104) cycles. Finally, the load will be changed to ±32 kpsi. How many cycles of operation can be expected at this stress level? For the part, Sut = 76 kpsi, f = 0.9, and has a fully corrected endurance strength of Se = 30 kpsi.
(a) Use Miner’s method.
(b) Use Manson’s method.
A rotating-beam specimen with an endurance limit of 50 kpsi and an ultimate strength of 100 kpsi is cycled 20 percent of the time at 70 kpsi, 50 percent at 55 kpsi, and 30 percent at 40 kpsi. Let f = 0.9 and estimate the number of cycles to failure.
Solve Prob. 6–1 if the ultimate strength of production pieces is found to be Sut = 245LN (1, 0.0508) kpsi.
The situation is similar to that of Prob. 6–10 wherein the imposed completely reversed axial load Fa = 15 LN (1, 0.20) kN is to be carried by the link with a thickness to be specified by you, the designer. Use the 1018 cold-drawn steel of Prob. 6–10 with Sut = 440 LN (1, 0.30) MPa and Syt =370 LN (1, 0.061). The reliability goal must exceed 0.999. Using the correlation method, specify the thickness t.
A solid round steel bar is machined to a diameter of 1.25 in. A groove 1/8 in deep with a radius of 1/8 in is cut into the bar. The material has a mean tensile strength of 110 kpsi. A completely reversed bending moment M = 1400 lbf • in is applied. Estimate the reliability. The size factor should be based on the gross diameter. The bar rotates.
Repeat Prob. 6–34, with a completely reversed torsional moment of T = 1400 lbf • in applied.
A 5/4 -in-diameter hot-rolled steel bar has a 1/8 -in diameter hole drilled transversely through it. The bar is nonrotating and is subject to a completely reversed bending moment of M = 1600 lbf • in in the same plane as the axis of the transverse hole. The material has a mean tensile strength of 58 kpsi. Estimate the reliability. The size factor should be based on the gross size. Use Table A–16 for Kt.
Repeat Prob. 6–36, with the bar subject to a completely reversed torsional moment of 2400 lbf • in.
The plan view of a link is the same as in Prob. 6–23; however, the forces F are completely reversed, the reliability goal is 0.998, and the material properties are Sut = 64LN (1, 0.045) kpsi and Sy = 54LN (1, 0.077) kpsi. Treat Fa as deterministic, and specify the thickness h.
A 1/4 by 3/2 -in steel bar has a 3/4 -in drilled hole located in the center, much as is shown in Table A–15–1. The bar is subjected to a completely reversed axial load with a deterministic load of 1200 lbf. The material has a mean ultimate tensile strength of Sut = 80 kpsi.
(a) Estimate the reliability.
(b) Conduct a computer simulation to confirm your answer to part a.
From your experience with Prob. 6–39 and Ex. 6–19, you observed that for completely reversed axial and bending fatigue, it is possible to • Observe the COVs associated with a priori design considerations. • Note the reliability goal. • Find the mean design factor .nd which will permit making a geometric design decision that will attain the goal using deterministic methods in conjunction with .nd. Formulate an interactive computer program that will enable the user to find .nd .While the material properties Sut , Sy , and the load COV must be input by the user, all of the COVs associated with Φ0.30 , ka , kc , kd , and Kf can be internal, and answers to questions will allow Cσ and CS , as well as Cn and .nd , to be calculated. Later you can add improvements. Test your program with problems you have already solved.
A shaft is loaded in bending and torsion such that Ma = 600 lbf • in, Ta = 400 lbf • in, Mm =
500 lbf • in, and Tm = 300 lbf • in. For the shaft, Su = 100 kpsi and Sy = 80 kpsi, and a fully corrected endurance limit of Se = 30 kpsi is assumed. Let Kf = 2.2 and Kf s = 1.8. With a design factor of 2.0 determine the minimum acceptable diameter of the shaft using the
(a) DE-Gerber criterion.
(b) DE-elliptic criterion.
(c) DE-Soderberg criterion.
(d) DE-Goodman criterion.
Discuss and compare the results
The section of shaft shown in the figure is to be designed to approximate relative sizes of d = 0.75D and r = D/20 with diameter d conforming to that of standard metric rolling-bearing bore sizes. The shaft is to be made of SAE 2340 steel, heat-treated to obtain minimum strengths in the shoulder area of 1226-MPa ultimate tensile strength and 1130-MPa yield strength with a Brinell hardness not less than 368. At the shoulder the shaft is subjected to a completely reversed bending moment of 70 N • m, accompanied by a steady torsion of 45 N • m. Use a design factor of 2.5 and size the shaft for an infinite life.
The rotating solid steel shaft is simply supported by bearings at points B and C and is driven by a gear (not shown) which meshes with the spur gear at D, which has a 6-in pitch diameter. The force F from the drive gear acts at a pressure angle of 20o. The shaft transmits a torque to point A of TA = 3000 lbf • in. The shaft is machined from steel with Sy = 60 kpsi and Sut = 80 kpsi. Using a factor of safety of 2.5, determine the minimum allowable diameter of the 10 in section of the shaft based on
(a) a static yield analysis using the distortion energy theory and
(b) a fatigue-failure analysis. Assume sharp fillet radii at the bearing shoulders for estimating stress concentration factors
A geared industrial roll shown in the figure is driven at 300 rev/min by a force F acting on a 3-in-diameter pitch circle as shown. The roll exerts a normal force of 30 lbf/in of roll length on the material being pulled through. The material passes under the roll. The coefficient of friction is 0.40. Develop the moment and shear diagrams for the shaft modeling the roll force as
(a) a concentrated force at the center of the roll, and
(b) a uniformly distributed force along the roll. These diagrams will appear on two orthogonal planes
The figure shows a proposed design for the industrial roll shaft of Prob. 7–4. Hydrodynamic film bearings are to be used. All surfaces are machined except the journals, which are ground and polished. The material is 1035 HR steel. Perform a design assessment. Is the design satisfactory?
In the double-reduction gear train shown, shaft a is driven by a motor attached by a flexible coupling attached to the overhang. The motor provides a torque of 2500 lbf • in at a speed of 1200 rpm. The gears have 20o pressure angles, with diameters shown on the figure. Use an AISI 1020 cold-drawn steel. Design one of the shafts (as specified by the instructor) with a design factor of 1.5 by performing the following tasks.
(a) Sketch a general shaft layout, including means to locate the gears and bearings, and to transmit the torque.
(b) Perform a force analysis to find the bearing reaction forces, and generate shear and bending moment diagrams.
(c) Determine potential critical locations for stress design.
(d) Determine critical diameters of the shaft based on fatigue and static stresses at the critical locations.
(e) Make any other dimensional decisions necessary to specify all diameters and axial dimensions. Sketch the shaft to scale, showing all proposed dimensions.
(f) Check the deflection at the gear, and the slopes at the gear and the bearings for satisfaction of the recommended limits in Table 7–2.
(g) If any of the deflections exceed the recommended limits, make appropriate changes to bring them all within the limits.
In the figure is a proposed shaft design to be used for the input shaft a in Prob. 7–7. A ball bearing is planned for the left bearing, and a cylindrical roller bearing for the right.
(a) Determine the minimum fatigue factor of safety by evaluating at any critical locations. Use a fatigue failure criteria that is considered to be typical of the failure data, rather than one that is considered conservative. Also ensure that the shaft does not yield in the first load cycle.
(b) Check the design for adequacy with respect to deformation, according to the recommendations in Table 7–2
An AISI 1020 cold-drawn steel shaft with the geometry shown in the figure carries a transverse load of 7 kN and a torque of 107 N • m. Examine the shaft for strength and deflection. If the largest allowable slope at the bearings is 0.001 rad and at the gear mesh is 0.0005 rad, what is the factor of safety guarding against damaging distortion? What is the factor of safety guarding against a fatigue failure? If the shaft turns out to be unsatisfactory, what would you recommend to correct the problem?
A 1-in-diameter uniform steel shaft is 24 in long between bearings.
(a) Find the lowest critical speed of the shaft.
(b) If the goal is to double the critical speed, find the new diameter.
(c) A half-size model of the original shaft has what critical speed?
Demonstrate how rapidly Rayleigh’s method converges for the uniform-diameter solid shaft of
Prob. 7–14, by partitioning the shaft into first one, then two, and finally three elements
Compare Eq. (7–27) for the angular frequency of a two-disk shaft with Eq. (7–28), and note that the constants in the two equations are equal.
(a) Develop an expression for the second critical speed.
(b) Estimate the second critical speed of the shaft addressed in Ex. 7–5, parts a and
For a uniform-diameter shaft, does hollowing the shaft increase or decrease the critical speed?
The shaft shown in the figure carries a 20-lbf gear on the left and a 35-lbf gear on the right.
Estimate the first critical speed due to the loads, the shaft’s critical speed without the loads, and the critical speed of the combination.
A transverse drilled and reamed hole can be used in a solid shaft to hold a pin that locates and holds a mechanical element, such as the hub of a gear, in axial position, and allows for the transmission of torque. Since a small-diameter hole introduces high stress concentration, and a larger diameter hole erodes the area resisting bending and torsion, investigate the existence of a pin diameter with minimum adverse affect on the shaft. Then formulate a design rule. (Hint: Use Table A–16.)
A guide pin is required to align the assembly of a two-part fixture. The nominal size of the pin is 15 mm. Make the dimensional decisions for a 15-mm basic size locational clearance fit.
An interference fit of a cast-iron hub of a gear on a steel shaft is required. Make the dimensional decisions for a 45-mm basic size medium drive fit.
A pin is required for forming a linkage pivot. Find the dimensions required for a 50-mm basic size pin and clevis with a sliding fit.
A journal bearing and bushing need to be described. The nominal size is 1 in. What dimensions are needed for a 1-in basic size with a close running fit if this is a lightly loaded journal and bushing assembly?
A gear and shaft with nominal diameter of 1.5 in are to be assembled with a medium drive fit, as specified in Table 7–9. The gear has a hub, with an outside diameter of 2.5 in, and an overall length of 2 in. The shaft is made from AISI 1020 CD steel, and the gear is made from steel that has been through hardened to provide Su =100 kpsi and Sy = 85 kpsi.
(a) Specify dimensions with tolerances for the shaft and gear bore to achieve the desired fit.
(b) Determine the minimum and maximum pressures that could be experienced at the interface with the specified tolerances.
(c) Determine the worst-case static factors of safety guarding against yielding at assembly for the shaft and the gear based on the distortion energy failure theory.
(d) Determine the maximum torque that the joint should be expected to transmit without slipping, i.e., when the interference pressure is at a minimum for the specified tolerances.
An uncrowned straight-bevel pinion has 20 teeth, a diametral pitch of 6 teeth/in, and a transmission accuracy number of 6. Both the pinion and gear are made of through-hardened steel with a Brinell hardness of 300. The driven gear has 60 teeth. The gearset has a life goal of 109 revolutions of the pinion with a reliability of 0.999. The shaft angle is 90◦; the pinion speed is 900 rev / min. The face width is 1.25 in, and the normal pressure angle is 20◦. The pinion is mounted outboard of its bearings, and the gear is straddle-mounted. Based on the AGMA bending strength, what is the power rating of the gearset? Use K0 = 1, SF = 1, and SH = 1
For the gearset and conditions of Prob. 15–1, find the power rating based on the AGMA surface durability.
An uncrowned straight-bevel pinion has 30 teeth, a diametral pitch of 6, and a transmission accuracy number of 6. The driven gear has 60 teeth. Both are made of No. 30 cast iron. The shaft angle is 90◦. The face width is 1.25 in, the pinion speed is 900 rev/min, and the normal pressure angle is 20◦. The pinion is mounted outboard of its bearings; the bearings of the gear straddle it. What is the power rating based on AGMA bending strength? (For cast iron gearsets reliability information has not yet been developed. We say the life is greater than 107 revolutions; set KL = 1, CL = 1, CR = 1, KR = 1; and apply a factor of safety. Use SF = 2 and SH = √2.)
For the gearset and conditions of Prob. 15–3, find the power rating based on AGMA surface durability. For the solutions to Probs. 15–3 and 15–4, what is the power rating of the gearset?
An uncrowned straight-bevel pinion has 22 teeth, a module of 4 mm, and a transmission accuracy number of 5. The pinion and the gear are made of through-hardened steel, both having core and case hardnesses of 180 Brinell. The pinion drives the 24-tooth bevel gear. The shaft angle is 90◦, the pinion speed is 1800 rev/min, the face width is 25 mm, and the normal pressure angle is 20◦. Both gears have an outboard mounting. Find the power rating based on AGMA pitting resistance if the life goal is 109 revolutions of the pinion at 0.999 reliability.
For the gearset and conditions of Prob. 15–5, find the power rating for AGMA bending strength.
In straight-bevel gearing, there are some analogs to Eqs. (14–44) and (14–45). If we have a pinion core with a hardness of (HB) 11 and we try equal power ratings, the transmitted load Wt can be made equal in all four cases. It is possible to find these relations:
Refer to your solution to Probs. 15–1 and 15–2, which is to have a pinion core hardness of 300 Brinell. Use the relations from Prob. 15–7 to establish the hardness of the gear core and the case hardnesses of both gears.
Repeat Probs. 15–1 and 15–2 with the hardness protocol
A catalog of stock bevel gears lists a power rating of 5.2 hp at 1200 rev/min pinion speed for a straight-bevel gearset consisting of a 20-tooth pinion driving a 40-tooth gear. This gear pair has a 20◦ normal pressure angle, a face width of 0.71 in, and a diametral pitch of 10 teeth/in and is through-hardened to 300 BHN. Assume the gears are for general industrial use, are generated to a transmission accuracy number of 5, and are uncrowned. Given these data, what do you think about the stated catalog power rating?
Apply the relations of Prob. 15–7 to Ex. 15–1 and find the Brinell case hardness of the gears for equal allowable load Wt in bending and wear. Check your work by reworking Ex. 15–1 to see if you are correct. How would you go about the heat treatment of the gears?
Use your experience with Prob. 15–11 and Ex. 15–2 to design an interactive computer-aided design program for straight-steel bevel gears, implementing the ANSI/AGMA 2003-B97 standard. It will be helpful to follow the decision set in Sec. 15–5, allowing the return to earlier decisions for revision as the consequences of earlier decisions develop.
A single-threaded steel worm rotates at 1725 rev/min, meshing with a 56-tooth worm gear transmitting 1 hp to the output shaft. The pitch diameter of the worm is 1.50. The tangential diametral pitch of the gear is 8 teeth per inch and the normal pressure angle is 20◦. The ambient temperature is 70◦F, the application factor is 1.25, the design factor is 1, the gear face is 0.5 in, the lateral case area is 850 in2, and the gear is sand-cast bronze.
(a) Determine and evaluate the geometric properties of the gears.
(b) Determine the transmitted gear forces and the mesh efficiency.
(c) Is the mesh sufficient to handle the loading?
(d) Estimate the lubricant sump temperature.
As in Ex. 15–4, design a cylindrical worm-gear mesh to connect a squirrel-cage induction motor to a liquid agitator. The motor speed is 1125 rev/min, and the velocity ratio is to be 10:1. The output power requirement is 25 hp. The shaft axes are 90◦ to each other. An overload factor Ko (see Table 15–2) makes allowance for external dynamic excursions of load from the nominal or average load Wt. For this service Ko = 1.25 is appropriate. Additionally, a design factor nd of 1.1 is to be included to address other unquantifiable risks. For Probs. 15–15 to 15–17 use the AGMA method for (Wt G) all whereas for Probs. 15–18 to 15–22, use the Buckingham method. See Table 15–12.
The figure shows an internal rim-type brake having an inside rim diameter of 12 in and a dimension R = 5 in. The shoes have a face width of 1 ½ in and are both actuated by a force of 500 lbf. The mean coefficient of friction is 0.28.
(a) Find the maximum pressure and indicate the shoe on which it occurs.
(b) Estimate the braking torque effected by each shoe, and find the total braking torque.
(c) Estimate the resulting hinge-pin reactions.
For the brake in Prob. 16–1, consider the pin and actuator locations to be the same. However, instead of 120°, let the friction surface of the brake shoes be 90° and centrally located. Find the maximum pressure and the total braking torque.
In the figure for Prob. 16–1, the inside rim diameter is 280 mm and the dimension R is 90 mm. The shoes have a face width of 30 mm. Find the braking torque and the maximum pressure for each shoe if the actuating force is 1000 N, the drum rotation is counterclockwise, and f = 0.30.
The figure shows a 400-mm-diameter brake drum with four internally expanding shoes. Each of the hinge pins A and B supports a pair of shoes. The actuating mechanism is to be arranged to produce the same force F on each shoe. The face width of the shoes is 75 mm. The material used permits a coefficient of friction of 0.24 and a maximum pressure of 1000 kPa.
(a) Determine the actuating force.
The block-type hand brake shown in the figure has a face width of 30 mm and a mean coefficient of friction of 0.25. For an estimated actuating force of 400 N, find the maximum pressure on the shoe and find the braking torque.
Suppose the standard deviation of the coefficient of friction in Prob. 16–5 is = 0.025, where the deviation from the mean is due entirely to environmental conditions. Find the brake torques corresponding
The brake shown in the figure has a coefficient of friction of 0.30, a face width of 2 in, and a limiting shoe lining pressure of 150 psi. Find the limiting actuating force F and the torque capacity.
Refer to the symmetrical pivoted external brake shoe of Fig. 16–12 and Eq. (16–15). Suppose the pressure distribution was uniform, that is, the pressure p is independent of θ. What would the pivot distance a′ be? If θ1 = θ2 = 60◦ , compare a with a′.
The shoes on the brake depicted in the figure subtend a 90◦ arc on the drum of this external pivoted-shoe brake. The actuation force P is applied to the lever. The rotation direction of the drum is counterclockwise, and the coefficient of friction is 0.30.
(a) What should the dimension e be?
(b) Draw the free-body diagrams of the handle lever and both shoe levers, with forces expressed in terms of the actuation force P.
(c) Does the direction of rotation of the drum affect the braking torque?
Problem 16–9 is preliminary to analyzing the brake. A molded lining is used dry in the brake of Prob. 16–9 on a cast iron drum. The shoes are 7.5 in wide and subtend a 90◦arc. Find the actuation force and the braking torque.
The maximum band interface pressure on the brake shown in the figure is 90 psi. Use a 14-indiameter drum, a band width of 4 in, a coefficient of friction of 0.25, and an angle-of-wrap of 270◦. Find the band tensions and the torque capacity.
The drum for the band brake in Prob. 16–11 is 300 mm in diameter. The band selected has a mean coefficient of friction of 0.28 and a width of 80 mm. It can safely support a tension of 7.6 kN. If the angle of wrap is 270◦, find the lining pressure and the torque capacity
The brake shown in the figure has a coefficient of friction of 0.30 and is to operate using a maximum force F of 400 N. If the band width is 50 mm, find the band tensions and the braking torque
The figure depicts a band brake whose drum rotates counterclockwise at 200 rev/min. The drum diameter is 16 in and the band lining 3 in wide. The coefficient of friction is 0.20. The maximum lining interface pressure is 70 psi.
(a) Find the brake torque, necessary force P, and steady-state power.
(b) Complete the free-body diagram of the drum. Find the bearing radial load that a pair of straddle-mounted bearings would have to carry.
(c) What is the lining pressure p at both ends of the contact arc?
The figure shows a band brake designed to prevent “backward” rotation of the shaft. The angle of wrap is 270◦, the band width is 2 1/8 in, and the coefficient of friction is 0.20. The torque to be resisted by the brake is 150 lbf . ft. The diameter of the pulley is 8 ¼ in.
(a) What dimension c1 will just prevent backward motion?
(b) If the rocker was designed with c1 = 1 in, what is the maximum pressure between the band and drum at 150 lbf ?ft back torque?
(c) If the back-torque demand is 100 lbf . in, what is the largest pressure between the band and drum?
A plate clutch has a single pair of mating friction surfaces 300 mm OD by 225 mm ID. The mean value of the coefficient of friction is 0.25, and the actuating force is 5 kN.
(a) Find the maximum pressure and the torque capacity using the uniform-wear model.
(b) Find the maximum pressure and the torque capacity using the uniform-pressure model.
A hydraulically operated multidisk plate clutch has an effective disk outer diameter of 6.5 in and an inner diameter of 4 in. The coefficient of friction is 0.24, and the limiting pressure is 120 psi. There are six planes of sliding present.
(a) Using the uniform wear model, estimate the axial force F and the torque T.
(b) Let the inner diameter of the friction pairs d be a variable. Complete the following table:
Look again at Prob. 16–17.
(a) Show how the optimal diameter d∗ is related to the outside diameter D.
(b) What is the optimal inner diameter?
(c) What does the tabulation show about maxima?
(d) Common proportions for such plate clutches lie in the range 0.45 ≤ d / D ≤ 0.80. Is the result in part a useful?
A cone clutch has D = 330 mm, d = 306 mm, a cone length of 60 mm, and a coefficient of friction of 0.26. A torque of 200 N ?m is to be transmitted. For this requirement, estimate the actuating force and pressure by both models
Show that for the caliper brake the T / (f F D) versus d / D plots are the same as Eqs.
(b) and (c) of Sec. 16–5.
A two-jaw clutch has the dimensions shown in the figure and is made of ductile steel. The clutch has been designed to transmit 2 kW at 500 rev/min. Find the bearing and shear stresses in the key and the jaws.
A brake has a normal braking torque of 320 N m and heat-dissipating surfaces whose mass is 18 kg. Suppose a load is brought to rest in 8.3 s from an initial angular speed of 1800 rev/min using the normal braking torque; estimate the temperature rise of the heat-dissipating surfaces.
A cast-iron flywheel has a rim whose OD is 60 in and whose ID is 56 in. The flywheel weight is to be such that an energy fluctuation of 5000 ft ?lbf will cause the angular speed to vary no more than 240 to 260 rev/min. Estimate the coefficient of speed fluctuation. If the weight of the spokes is neglected, what should be the width of the rim?
A single-geared blanking press has a stroke of 8 in and a rated capacity of 35 tons. A cam-driven ram is assumed to be capable of delivering the full press load at constant force during the last 15 percent of a constant-velocity stroke. The camshaft has an average speed of 90 rev/min and is geared to the flywheel shaft at a 6:1 ratio. The total work done is to include an allowance of 16 percent for friction.
(a) Estimate the maximum energy fluctuation.
(b) Find the rim weight for an effective diameter of 48 in and a coefficient of speed fluctuation of 0.10.
Using the data of Table 16–6, find the mean output torque and flywheel inertia required for a three-cylinder in-line engine corresponding to a nominal speed of 2400 rev/min. Use Cs = 0.30
When a motor armature inertia, a pinion inertia, and a motor torque reside on a motor shaft, and a gear inertia, a load inertia, and a load torque exist on a second shaft, it is useful to reflect all the torques and inertias to one shaft, say, the armature shaft. We need some rules to make such reflection easy. Consider the pinion and gear as disks of pitch radius.
• A torque on a second shaft is reflected to the motor shaft as the load torque divided by the negative of the stepdown ratio.
• An inertia on a second shaft is reflected to the motor shaft as its inertia divided by the stepdown ratio squared.
• The inertia of a disk gear on a second shaft in mesh with a disk pinion on the motor shaft is reflected to the pinion shaft as the pinion inertia multiplied by the stepdown ratio squared.
(a) Verify the three rules.
(b) Using the rules, reduce the two-shaft system in the figure to a motor-shaft shish-kebob equivalent. Correctly done, the dynamic response of the shish kebab and the real system are identical.
(c) For a stepdown ratio of n = 10 compare the shish-kebab inertias.
Apply the rules of Prob. 16–26 to the three-shaft system shown in the figure to create a motor shaft shish kebab.
(a) Show that the equivalent inertia Ie is given by
(b) If the overall gear reduction R is a constant nm, show that the equivalent inertia becomes
(c) If the problem is to minimize the gear-train inertia, find the ratios n and m for the values of
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