Page 168 - Marks Calculation for Machine Design
P. 168

P1: Rakesh
                          January 4, 2005
                 Brown˙C03
        Brown.cls
                  150
                            U.S. Customary 14:16  STRENGTH OF MACHINES  SI/Metric
                  solution                           solution
                  Step 1. Convert the units on the speed of  Step 1. Convert the units on the speed of
                  rotation (ω) from rpm to (rad/s)   rotation (ω) from rpm to (rad/s)
                              rev  2πrad  1 min                 rev  2π rad  1 min
                      ω = 1,000  ×     ×                ω = 1,000  ×      ×
                             min   rev   60 s                   min   rev   60 s
                         (1,000)(2 π) rad                   (1,000)(2 π) rad
                       =                                  =
                             60    s                           60    s
                       = 105 (rad/s)                      = 105 (rad/s)
                  Step 2. Using Eq. (3.43) calculate the quantity  Step 2. Using Eq. (3.43) calculate the quantity
                  (σ o ) as                          (σ o ) as
                                                             2 2
                          2 2
                    σ o = ρω r o                       σ o = ρω r  o
                                          2                                 2
                             slug     rad    2                   kg     rad     2

                       =  15.2    105     (3ft)          =  7,850    105    (1m)
                              ft 3    s 2                       m 3      s 2
                        "           #  slug · ft ft        "           #  kg · m m
                                 2
                                                                    2
                       = (15.2)(105) (3) 2               = (7,850)(105) (1) 2
                                       s 2  ft 3                          s 2  m 3
                                lb                                 N
                       = 1,508,220  = 9,975 lb/in 2      = 86,546,250
                                ft 2                               m 2
                       = 10.0 kpsi                       = 86.5MPa
                  Step 3. Calculate the maximum tangential  Step 3. Calculate the maximum tangential
                  stress (σ t ) at the inside radius (r i ), where the  stress (σ t ) at the inside radius (r i ), where the
                  radial (σ r ) is zero, using Eq. (3.44).  radial (σ r ) is zero, using Eq. (3.44).
                                          2  
                              2
                             3 + ν   1 − ν r                   3 + ν   1 − ν r
                      σ t max  = σ o  1 +  i           σ t max  = σ o  1 +  i
                               4     3 + ν r o 2                4      3 + ν r 2 o
                                 3 + 0.3                             3 + 0.3
                          = (10 kpsi)                      = (86.5MPa)
                                   4                                   4
                                          2                                   2
                                 1 − 0.3 (1in)                     1 − 0.3 (0.025 m)
                           × 1 +                             × 1 +
                                 3 + 0.3 (36 in) 2                 3 + 0.3  (1m) 2
                                  3.3                                3.3
                          = (10 kpsi)                      = (86.5MPa)
                                  4                                  4
                                 0.7  1                            0.7 0.000625

                           × 1 +                             × 1 +
                                 3.3 1296                          3.3  1
                          = (10 kpsi)(0.825)               = (86.5MPa)(0.825)
                           × (1 + 0.00016)                   × (1 + 0.00013)
                          = 8.25 kpsi                      = 71.4MPa
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