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0593_C11_fm  Page 372  Monday, May 6, 2002  2:59 PM





                       372                                                 Dynamics of Mechanical Systems


                        By projecting the forces applied to T and the particles along the partial velocities at the
                       points of application, we obtain the generalized forces as:


                                                             ⋅
                                       F =  O x (  i O y ) ⋅  O  Mg j v + C  nv Q 1  +  C  nv Q 2
                                                                                ⋅
                                               +
                                                  j v −
                                                                      ⋅
                                                               G
                                                                        ˙
                                                      ˙
                                        x 1           x 1      x 1 ˙  1  2  x 1  2  2  x 1 ˙
                                                      [
                                                                                  2]
                                                                2 (
                                                 ⋅
                                           + C  nv  Q 3  + − mg j k x − ) n −  kx  n −  C n ⋅ v P 1
                                                           +
                                                                  x
                                                                                      ˙
                                              3  2  x 1 ˙          1  1    1 1  1     x 1
                                            [
                                                j k x − )
                                          +−  mg + (  3  x 2  n − (  2  − x 1  1  − C n 2 ] ⋅v P 2
                                                               k x − )n
                                                            1
                                                                                  ˙
                                                                            2
                                                                                 x 1
                                            [                            ]
                                          +−mgj  − kx n 1  − ( k x 3  − )n 1  − C n 2  ⋅v P 3
                                                               x
                                                                      3
                                                     3
                                                                2
                                                                            x 1 ˙
                                         = − ++++ mgcosθ        + kx  − kx  ++ 0
                                           00000
                                                                      2
                                                                            0
                                                                    2    1
                       or
                                                    F =  mgcosθ + kx −  2 kx                   (11.7.14)
                                                     x 1           2     1
                       and
                                      F =  O x (  i O y ) ⋅  O  Mg j v +  C  nv  Q 1  +  C  nv Q 2
                                                                                ⋅
                                                             ⋅
                                               +
                                                  j v −
                                                                      ⋅
                                                               G
                                                                                  ˙
                                                                        ˙
                                                               ˙
                                       x 2            x 2 ˙    x 2  1  2  x 2  2  2  x 2
                                                                                  2]
                                                      [
                                                                2 (
                                                           +
                                                 ⋅
                                           +  C  nv Q 3  + − mg j k x − ) n −  kx  n − C  n ⋅  v  P 1
                                                                  x
                                             3  2  x 2             1  1    1 1  1     x 2
                                                   ˙
                                                                                      ˙
                                            [
                                                j k x − )
                                          +−  mg + (  3  x 2  n − (  2  − x 1  1  − C n 2 ] ⋅v  P 2 ˙
                                                               k x − )n
                                                                            2
                                                            1
                                                                                 x 2
                                            [                            ]
                                          +−mgj  − kx n 1  − ( k x 3  − )n  1  − C n  2  ⋅v  P 3
                                                               x
                                                                2
                                                     3
                                                                      3
                                                                            x 2 ˙
                                         = − ++++ mgcosθ        + kx  + kx  − kx  + 0
                                           00000
                                                                           2
                                                                   3    1     2
                       or
                                                 Fx =  mgcosθ + kx + kx −  2 kx                (11.7.15)
                                                   2             3   1     2
                       and
                                      Fx = (  i O ) ⋅  O    j v +     ⋅  Q 1  +  ⋅  Q 2
                                               +
                                                  j v −
                                                             ⋅
                                                               G
                                        3   O x   y   x 3  Mg  x 3  C 1 nv ˙ x 3  C 2 nv ˙ x 3
                                                               ˙
                                                                               2
                                                      ˙
                                                                     2
                                                                                  2]
                                                                2 (
                                                                           n −
                                                            +
                                                 ⋅
                                           + C 3 nv ˙ Q 3 + − [  mg j k x − ) n −  kx 1 1  C 1 n ⋅  v ˙ P 1
                                                                   x
                                                2
                                                                      1
                                                                    1
                                                   x 3
                                                                                      x 3
                                           +− [  mg + (  3  x 2  n − (  2  − x 1  1  − C n 2 ] ⋅v ˙ P 2
                                                 j k x − )
                                                               k x − )n
                                                                            2
                                                             1
                                                                                  x 3
                                           +− [ mgj  − kx n 1  − ( k x 3  − )n  1  − C n  2 ] ⋅v ˙ P 3
                                                               x
                                                                2
                                                     3
                                                                       3
                                                                            x 3
                                         = − ++++++ mgcosθ            − kx 3  + kx 2
                                                                       2
                                           0000000
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