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258                        DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES





                                         NACELLE






                                                         Blade tip -
                                                         deflected position
                                                Q
                                                           Blade tip -
                                                           undeflected position
                    Blade section
                    with maximum twist
                                                β
                                             P
                                                           δ 11
                                      δ 12

                                                    Weak principal axis

          Figure 5.23 Deflection of Tip Due to Flapwise Bending of Twisted Blade (Viewed Along
          Blade Axis)
          ä 12 =ä 11 approximates to 0.13, with the result that blade first mode flapwise oscilla-
          tions will result in some relatively small simultaneous edgewise inertia loadings.
          These will not excite significant edgewise oscillations, because the edgewise first
          mode natural frequency is typically about double the flapwise one.
            It can be seen from the above that the effects of interaction between flapwise and
          edgewise oscillations are generally minor, so they will not be considered further.
            Blades will also be subject to torsional vibrations. However, these can generally
          be ignored, because both the exciting loads are small, and the high torsional
          stiffness of a typical hollow blade places the torsional natural frequencies well
          above the exciting frequencies.
            Finally, in the case of a blade hinged at the root, the whole blade will experience
          oscillations involving rigid body rotation about the hinge. This phenomenon is
          considered in Section 5.8.8.



          5.8.2 Mode shapes and frequencies

          The mode shape and frequency of the first mode can be derived by an iterative
          technique called the Stodola method after its originator. Briefly, this consists of
          assuming a plausible mode shape, calculating the inertia loads associated with it for
          an arbitrary frequency of 1 rad=s, and then computing the beam deflected profile
          resulting from these inertia loads. This profile is then normalized, typically by
          dividing the deflections by the tip deflection, to obtain the input mode shape for the
          second iteration. The process is repeated until the mode shape converges, and the
          first mode natural frequency is calculated from the formula:
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