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264                        DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES


            It is seen that the damping ratio for the second mode is under half that for the
          first.



          5.8.5 Response to deterministic loads—step-by-step dynamic
                 analysis

          As set out in Section 5.8.1, blade dynamic response to time varying loading is best
          analysed in terms of the separate excitation of each blade mode of vibration, for
          which, under the assumptions of unstalled flow and mass-proportional aerody-
          namic damping , the governing equation is
                                                  ð R
                                  _
                          €
                                           2
                        m i f i (t) þ c i f i (t) þ m i ø f i (t) ¼  ì i (r)q(r, t)dr ¼ Q i (t)  (5:70)
                          f
                                  f
                                           i
                                                   0
          where f i (t) and ì i (r) are the tip displacement and mode shape for the ith mode
          respectively. Starting with the initial tip displacement, velocity and acceleration
          arbitrarily set at zero, this equation can be used to derive values for these quantities
          at successive time steps over a complete blade revolution by numerical integration.
          The procedure is then repeated for several more revolutions until the cyclic blade
          response to the periodic loading becomes sensibly invariant from one revolution to
          the next.


          Linear acceleration method

          The precise form of the equations linking the tip displacement, velocity and
          acceleration at the end of a time step to those at the beginning depends on how the
          acceleration is assumed to vary over the time step. Newmark has classified
          alternative assumptions in terms of a parameter â which measures the relative
          weightings placed on the initial and final accelerations in deriving the final
          displacement The simplest assumption is that the acceleration takes a constant
          value equal to the average of the initial and final values (â ¼ 1=4). Clough and
          Penzien (1993), however, recommend that the acceleration is assumed to vary
          linearly between the initial and final values, as this will be a closer approximation
          to the actual variation. Step-by-step integration with this assumption is known as
          either the linear acceleration method or the Newmark â ¼ 1=6 method.
            Expressions for the tip displacement, velocity and acceleration at the end of the
                                     €
                              _
          first time step – f i1 , f i1 and f i1 respectively – are derived in terms of the initial
                                     f
                              f
                      _
                             €
                             f
                      f
          values – f i0 , f i0 and f i0 – as follows. The acceleration at time t during the time step
          of total duration h is

                                                 €    €
                                                      f
                                     €     €     f f i1   f i0  t              (5:81)
                                     f f i (t) ¼ f i0 þ
                                           f
                                                    h
          This can be integrated to give the velocity at the end of the time step as
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