Concept explainers
The demonstration of how rapidly Rayleigh’s method converges for the uniform-diameter solid shaft.
Answer to Problem 29P
The Rayleigh method for uniform diameter shaft is converging rapidly by using a static deflection beam equation.
Explanation of Solution
Write the expression for moment of inertia.
Here, the diameter of the shaft is
Write the expression for area of the shaft.
Write the expression for weight of the shaft.
Here, the specific weight is
Write the expression for influence coefficient.
Here, the length of the shaft is
Write the expression for deflection at point 1.
Write the expression for Rayleigh method.
Write the expression for Rayleigh method.
Write the expression for first critical speed.
Draw the diagram for the two elements system.
Figure-(1)
The figure-(1) shows the required dimension.
Write the expression for the deflection at point 1 for two element.
Write the expression for the deflection at point 2 for two element.
Write the expression for Rayleigh method for two element.
Write the expression for Rayleigh method for two element.
Write the expression for first critical speed for two element.
Draw the diagram for the three element system.
Figure-(2)
The Figure-(2) shows all the dimensions for the three elements.
Write the expression for the deflection at point 1 for three elements
Write the expression for the deflection at point 2 for three element.
Write the expression for the deflection at point 2 for three element.
Write the expression for Rayleigh method for three elements.
Write the expression for Rayleigh method for three elements.
Write the expression for first critical speed for three elements.
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
Calculate the square of the deflection at point 1 of element 1.
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Calculate the square of the deflection at point 1 of element 2.
Calculate the square of the deflection at point 2 of element 2.
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Calculate the square of the deflection at point 1 of element 3.
Calculate the square of the deflection at point 2 of element 3.
Calculate the square of the deflection at point 3 of element 3.
Substitute
Substitute
Substitute
Since the static bending equation is available, and satisfied the moment-free and deflection-free ends, so the convergence is rapid using a static deflection beam equation.
Thus, the Rayleigh method for uniform diameter shaft is converging rapidly by using a static deflection beam equation.
Want to see more full solutions like this?
Chapter 7 Solutions
Shigley's Mechanical Engineering Design (in Si Units)
- A hollow shaft has to be designed for a marine engine delivering 1200 kW when running at 120 r/min. The maximum allowable shear stress is 50 MPa and the maximum torque to be transmitted by the shaft is 30 % greater than the mean torque. The internal diameter of the shaft is 50 % that of the external diameter. Calculate the shaft dimensions and the angle of twist of the shaft over 3 m when transmitting the above torque (G = 80 GPa).arrow_forwardA 5 ft long solid SAE4340 OQT1000 steel shaft is required to transmit a power of 180 HP. The shaft speed is 3,000 rpm. The allowable shear stress is 50 ksi and the allowable angle of twist per foot of length is 0.04 radians. Determine the minimum diameter that meets the design requirements.arrow_forwardA Hollow power transmission shaft of length ‘L’ 1103 mm, outer diameter ‘D’ 73mm and inner diameter 'd' 50. is required to transmit a power of P 60kW at a speed of 2950rpm. The modulus of rigidity of hardened carbon steel, which the shaft is made of, is 79.5 GPa Calculate the angle of twist of the above stating your answer in degrees?arrow_forward
- A hollow shaft with a length of 1103mm and outer diameter 73mm and inner diameter 50mm is required to tranmit a power of 60kW at a speed of 2950 rpm if the modulus of rigidity is 79.5GPa calculate the maximum shear stress in the shaftarrow_forwardA solid circular shaft has a diameter of d mm and is made from steel, which fail when tested in simple tension test at a stress of 150 MPa. The shaft was subjected by bending moment and torque which are 22.2 kNm and 44.4 kNm respectively. Calculate the minimum allowable shaft diameter, d according to:- (a) Tresca failure criterion (b) Von Mises theory of elastic failurearrow_forwardPlease do it properly otherwise leave to other expert. A hollow shaft of diameter ratio 3/5 is required to transmit 400 kW at 140 r.p.m with a uniform twisting moment. The shear stress in the shaft must not exceed 60 MPa and the twist in a length 2.5 m must not exceed 1°. Calculate the minimum external diameter of the shaft. Take C = 8 * 10 ^ 4arrow_forward
- A hollow shaft has an inner diameter of 0.035 m and an outerdiameter of 0.06 m. Compute for the torque if the shear stress isnot to exceed 120 MPa in Nm.arrow_forwardA hollow shaft with a length of 1103mm and outer diameter 73mm and inner diameter 50mm is required to tranmit a power of 60kW at a speed of 2950 rpm if the modulus of rigidity is 79.5GPa calculate the polar second moment of area of the shaftarrow_forwardA steel shaft 100mm in diameter is loaded and support in shaft bearing 0.4m apart. The shaft carries three loads: first mass 12kg at the centre, second mass 10kg at a distance 0.12m from the left bearing and third mass of 7kg at a distance 0.09m from the right bearing. Find the value of the critical speed by using Dunker ley’s method if E = 2X1011N/m2.arrow_forward
- A solid and a hollow shaft, connected by a coupling, transmit power. The outside diameters of the shafts are the same and the inside diameter of the hollow shaft is 75 % of its outside diameter. The solid shaft is three times as long as the hollow shaft. Take the modulus of rigidity for both shafts as 80 GPa. (a) If the angle of twist for the solid shaft is 1°, find the angle of twist for the hollow shaft. (b) If the maximum shear stress in the solid shaft is 30 MPa, find the maximum stress in the hollow shaft.arrow_forwardA 16 ft steel line shaft has no bending action except its own weight. What power in hp can shaft deliver at a speed of 200 rpm. Consider that the torsional deflection wm not exceed 0.08o/ft of length.arrow_forwardDetermine the minimum diameter that can be used for a solid circular steel shaft which is limited to an allowable shear stress of 70 MPa and a maximum angle of twist of 1.5o (degrees) per meter length. Applied maximum torque is 25 KN-m and the overall shaft length is 1.25 meters. Shear modulus = G= 80 GPa Note: When using torsion stress formulas, you must convert the angle of twist from degrees to radians.arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY