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Shaft Design Considerations Design Considerations: to minimize both shaft deflections and stresses, shafts should be as short as possible cantilevers are not recommended unless required for serviceability (i.e. access to v-belts, etc.) avoid stress concentrations at regions of high bending moments; minimize their effect through generous reliefs S. Waldman MECH 323 Shaft Design Constraints Design Constraints: deflections at gears should not exceed: 0.1 mm relative slopes between mating gear axes should be held to less than: 0.0005 rad shaft slope at bearings should be kept less than: • cylindrical roller bearings: 0.0001 rad • tapered roller bearings: 0.0005 rad • deep-groove roller bearings: 0.004 rad • spherical ball bearings: 0.0087 rad S. Waldman MECH 323 Multiaxial Loading What happens if there is loading in more than one plane (i.e. vertical and horizontal planes)? the simplest method is to create separate bending moment diagrams for each plane and then determine the resultant moment at sites of interest using Pythagorem’s theorem: M resultant M vertical M horizontal 2 2 the same procedure should be used for determining shaft slopes and deflections: tan resultant tan 2 vertical tan 2 horizontal resultant vertical2 horizontal 2 S. Waldman MECH 323 Design for Fluctuating Bending and Torsion The von Mises stress amplitude component sa´ and mean component sm´ are given by: 4K f M a 3K fsTa 3 16 16 A s a s x,a 3 xy,a 2 2 2 2 d 3 d 3 4K f M m 3K fsTm 3 16 16B s m s x ,m 3 xy,m 2 2 2 2 d d where A and B are the radicals in the above equations. The Gerber fatigue failure criterion is defined by: 2 2 2 Sa Sm ns a ns m 16nA 16nB 3 3 1 S e Sut S e Sut d S e d Sut S. Waldman MECH 323 Design for Fluctuating Bending and Torsion solving for the shaft diameter d: 1 1 3 8nA 2 BSe 2 2 d AS 1 1 Se ut or, solving for the factor of safety, n: 1 2 2 BSe 2 1 8A AS 1 1 n d Se 3 ut where: A 4K f M a 3K fsTa 2 2 B 4K f M m 3K fsTm 2 2 S. Waldman MECH 323 Determining Shaft Deflections Various method exist to determine the deflections of beams due to bending. The complicating factor for the design of shafts is typically the presence of step changes in shaft diameter along its length (shoulders, etc.). Thus, one commonly used method is the Integration Method with aid of Singularity Functions. d4y distributed load function: q EI 4 dx d3y shear force function: V EI 3 dx d2y moment function: M EI 2 dx dy slope function: dx S. Waldman MECH 323 Determining Shaft Deflections Integrating: V qdx C1 M Vdx C1 x C2 M dx C1 x 2 C2 x C3 EI y dx C1 x3 C2 x 2 C3 x C4 The integration constants C1 and C2 are the boundary conditions on the shear and moment function, which are simply the reaction forces imposed on the beam. Thus, if the reaction forces are used in the analysis (which is a very good idea): C1 C2 0 S. Waldman MECH 323 Singularity Functions Singularity functions are use to represent discrete entities (loads, moments, etc.) applied in a discontinuous fashion over the beam length. Denoted by the binominal function in angled brackets: xa n where: x is the variable of interest a is a user defined parameter to denote where in x the singularity acts n is the power of the function S. Waldman MECH 323 Singularity Functions Commonly used functions are: f x a : f 0 for x a ; f x a for x a 2 2 unit parabolic function: f x a : f 0 for x a ; f x a for x a 1 unit ramp function: f x a : f 0 for x a ; f 1 for x a 0 unit step function: 1 unit impulse function: f xa : f 0 for x a ; f 0 for x a Integration of Singularity Functions: 1 n 1 x a dx xa , n0 n n 1 n 1 xa , n0 S. Waldman MECH 323 Torsional Deflection Angular deflection of a shaft from torsional loads is: Tl GJ where: T is the torque l is shaft length G is the shear modulus J is the polar moment If the shaft is stepped or has multiple torques applied to it, the angular deflection can be determined from the sum of the deflections of each shaft segment: Ti li where, i is the shaft segment Gi J i S. Waldman MECH 323 Example S. Waldman MECH 323

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failure theories, Fatigue Failure, Machine Design, ductile materials, brittle materials, machine elements, Mechanical Design, factor of safety, principal stresses, Stress Theory

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posted: | 4/9/2010 |

language: | English |

pages: | 11 |

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