Problem Set 9
650:342 Design of Mechanical Components, Fall 2012 Due: Monday, 12/10/12 Problems copied, selected, and/or modified from Juvinall and Marshek’s Fundamentals Marshek’s Fundamentals of Machine Component Design, Design, 5th edition, 2012. 1. Determine the thickness of a spur gear tooth with with a diametral pitch of 8, measured along the pitch circle.
2. Two gears in a 3:1 ratio gearset and with a diametral pitch of 4 are mounted at a center distance of 6 in. find the number of teeth teeth in each gear.
3. A 17-tooth pinion meshes with an 84-tooth gear. The full-depth involute involute gear teeth have 20o pressure angle and a diametral pitch of 32. Determine the base pitch and the contact ratio. Also, calculate the addendum, dedendum, dedendu m, circular pitch, tooth thickness, and the base diameter for the
pinion and gear. If the center distance is increased by 0.125 in., what are the new values for the contact ratio and the pressure angle?
4. Juvinall Figure P15.23 shows a two-stage gear reducer. Identical pairs of gears are used. (This enables input shaft a and output shact c to be colinear, which facilitates machining of the housing.) Shaft b, called the countershaft, turns freely in bearings A and B, except for the geartooth forces. (a) Determine the rpmm of shafts b and c, the pitch diameters of the pinion and gear, and the circular pitch. (b) Determine the torque carried by each of the shafts a, b, and c: (i) assuming 100% gear efficiency, and (ii) assuming 95% efficiency of each gear pair. (c) For the 100% gear efficiency case, determine the radial loads applied to bearings A and B, and sketch the countershaft as a free body in equilibrium. (Note: This problem illustrates a machine designed using SI units except for the gear teeth, which are dimensioned in inches.)
5. Juvinall Figure P15.29 shows an electric motor driving a machine by means of three straighttooth spur gears having 16, 32, and 24 teeth. The gears have P=8, and a pressure angle of 20o. The idler shaft is supported by bearings A and B. (a) For the direction of motor rotation shown, determine the radial load carried by each bearing. (b) Determine the bearing loads for the opposite direction of motor rotation. (c) Explain briefly why the answers to parts a and b are different.
6. A simply supported steel shaft in Juvinall Figure P17.3 is connected to an electric motor with a flexible coupling. Find the value of the critical speed of rotation for the shaft.
7. Determine the critical speed of rotation for the steel shaft shown in Figure P17.15 using a 3in.-diameter shaft instead of the 2-in.-diameter shaft shown.
8. A mass of 120 lb at location 1 and an 80-lb mass at location 2 are attached to the 2-n.diameter shaft as shown in Juvinall Figure P17.15. By means of a deflection analysis, the influence of coefficients for the shaft were determined as a11=0.000308 in./lb, a12=0.000321 in./lb, a21=0.000321 in./lb, and a22=0.000510 in./lb. Note that a11 is the deflection at location 1 caused by a 1-lb force at that location, etc. Determine the first critical speed, ignoring the mass of the shaft, using both the Rayleigh equation and the Dunkerley equation.
9. The six shafts represented in Juvinall Figure P17.24 carry various combinations of static and alternating bending, axial, and torsional loads. State the loading involved for each of the shafts, and give a short one-sentence explanation for the cause of the loading.
Extra Problems (Do not turn in) 1. 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) both pitch circles, both base circles, both dedendums, both addendums, pressure angle, length of path of contact, both angles of approach, and 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.
2 A pair of standard 20o spur gears with 10-in. center distances has a velocity ratio of 4.0. The pinion has 20 teeth. (a) Determine P, p, and P b. (b) Begin a full-size layout showing partial pitch circles, partial base circles, pressure angle, addendum, and dedendum. Label each of these on your drawing. (c) Show on your drawing the interference-limiting maximum addendum radii, r ag,max, and r ap,max. Scale their numerical values from the drawing. Will interference be encountered with teeth of standard proportions? (d) Measure on the drawing the length of the path of contact for standard tooth proportions and, from this, compute the contact ratio.
3. Juvinall Figure P17.25 shows the load components acting on a helical gear mounted on a simply supported shaft. Bearing B takes thrust. A flexible coupling for transmitting torque attaches to the right end of the shaft. The left end is free. (a) Draw load, shear force, and bending moment diagrams for the shaft, in both the horizontal and vertical planes. Also draw diagrams showing the torsional loading and the axial force loading. (The desired diagrams include the seven shown in Juvinall Figure 17.7c. Add a similar diagram showing axial load, with tension plotted positive and compression negative.) (b) What radial and thrust loads are applied to the bearings? (c) Identify the most critically loaded shaft cross section, and for this location determine the diameter theoretically required for infinite life. Assume that the shaft will be machined from steel having Su=150 ksi and Sy=120 ksi, and that K f =2.0, 1.5, and 2.0 will apply to bending, torsional, and axial loading, respectively, at the critical location.
