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Lc 2 2 CE > k Sy
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The radius of gyration k, cross-sectional area A, and second moment of area I are related by I = Ak2, simplifying the above expression to Lc 1 2 2CE > dr 4 Sy
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(13.15)
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For a steel screw whose yield strength is 60 000 psi and whose end-condition constant is 1.0, the critical slenderness ratio is about 100, and Lc /dr is about 25. For steels whose slenderness ratio is less than critical, the Johnson parabolic relation can be used: Fc 1 Sy Lc = Sy A CE 2 k
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(13.16)
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TABLE 13.2 Buckling End-Condition Constants End condition Fixed-free Rounded-rounded Fixed-rounded Fixed-fixed C
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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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which can be solved for a circular cross section of minor diameter dr as dr = Sy L2 Fc c + 2 Sy CE (13.17)
The load should be externally guided for long travels to prevent eccentric loading. Axial compression or extension can be approximated by = FLc 4FLc = AE d 2 E r (13.18)
And similarly, angle of twist , in radians, can be approximated by = TLc 32TLc = JG d 4G r (13.19)
13.5 STRESSES
Using St. Venants principle, the nominal shear and normal stresses for cross sections of the screw rod away from the immediate vicinity of the load application may be approximated by = x = Tr 16T = d 4 J r 4F F = A d 2 r (13.20) (13.21)
Failure due to yielding can be estimated by the ratio of Sy to an equivalent, von Mises stress obtained from = 4F d 2 r
16T d 3 r
F d2 r
+ 48
T d3 r
(13.22)
The nominal bearing stress b on a nut or screw depends on the number of engaged threads Ne = h/p of pitch p and engaged thickness h and is obtained from b = F 4F = Aprojected (d 2 d 2 ) r p h (13.23)
Threads may also shear or strip off the screw or nut because of the load force, which is approximately parabolically distributed over the cylindrical surface area Acyl. The area depends on the width w of the thread at the root and the number of engaged threads Ne according to Acyl = dwNe. The maximum shear stress is estimated by = 3 F 2 Acyl (13.24)
For square threads such that w = p/2, the maximum shear stress for the nut thread is = 3F dh (13.25)
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
POWER SCREWS 13.10
GEARING
To obtain the shear stress for the screw thread, substitute dr for d. Since dr is slightly less than d, the stripping shear stress for the screw is somewhat larger. Note that the load flows from the point of load application through the thread geometry to the screw rod. Because of the nonlinear strains induced in the threads at the point of load application, each thread carries a disproportionate share of the load. A detailed analytical approach such as finite-element methods, backed up by experiments, is recommended for more accurate estimates of the above stresses and of other stresses, such as a thread bending stress and hoop stress induced in the nut.
13.6 BALL SCREWS
The design of ball screw assemblies is similar to that of machine screw systems. Kinematic considerations such as screw or nut travel, velocity, and acceleration can be estimated following Sec. 13.2. Similarly input torque, power, and efficiency can be approximated using formulas from Sec. 13.3. Critical buckling loads can be estimated using Eq. (13.12) or (13.16). Also, nominal shear and normal stresses of the ball screw shaft (or rod) can be estimated using Eqs. (13.20) and (13.21). Design for strength, however, is typically completed using a catalog selection procedure rather than analytical stress-versus-strength analysis. Ball screw manufacturers usually list static and dynamic load capacities for a variety of screw shaft (rod) diameters, ball diameters, and screw leads; an example is shown in Table 13.3. The static capacity for basic static thrust capacity Ti , lbf, is the load which will produce a ball track deformation of 0.0001 times the ball diameter. The dynamic capacity or basic load rating Pi, lbf, is the constant axial load that a group of ball screw assemblies can endure for a rated life of 1 million inches of screw travel. The rated life is the length of travel that 90 percent of a group of assemblies will complete or exceed before any signs of fatigue failure appear. The catalog ratings, developed from laboratory test results, therefore involve the effects of hertzian contact stresses, manufacturing processes, and surface fatigue failure. The catalog selection process requires choosing the appropriate combination of screw diameter, ball diameter, and lead, so that the axial load F will be sufficiently less than the basic static thrust capacity or the basic load rating for the rated axial travel life. For a different operating travel life of X inches, the modified basic load rating PiX, lbf, is obtained from PiX = Pi 10 6 X
(13.26)
An equivalent load rating P can be obtained for applications involving loads P1, P2 , P3 , . . . , Pn that occur for C1, C2 , C3 , . . . , Cn percent of the life, respectively: P=
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