Page 274 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)
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PIPES CONVEYING FLUID: LINEAR DYNAMICS I1              255







                              40



                              30
                           w
                                                        A  .
                              20                            , , \ A ‘
                                                                    . . . . . . .
                                                                    -V-
                                                                    I...   *...
                                I








             Figure 4.31  Experimental  boundaries  of  the  first-mode  principal  and  fundamental  parametric
             resonances  compared  with  theory,  for  a  clamped-clamped  pipe  with  B = 0.202,  y  = 14.8,
             cr = 4.57 x   (T  = 0 and  uo = 4.55. For  primary  resonance:  -,   theory;  -A-,  experiment.
                  For secondary resonance:  . . . . . ., theory; -v-, experiment (Pai‘doussis & Issid  1976).


             for 0 < p < 0.5 and a frequency range usually spanning the first three eigenfrequencies
             of  the system.
               If  the  flow  velocity  was  close  to  the  critical  value  mentioned  in  the  foregoing,  the
             following  observations  were  made.  At  low  frequencies,  a  secondary  parametric  reso-
             nance  in  the second mode of  the system was observed, which was difficult to pin-point
             for  the  reasons  already  given.  With  increasing  frequency,  the  amplitude  of  oscillation
             increased, to  a maximum of  typically  five diameters,  then decreased and finally ceased.
             The system remained  stable with  increasing  frequency up to a certain value, where the
             principal primary instability was encountered, also in the second mode of the system, with
             the pulsation  frequency equal to  twice the oscillation frequency;  the onset of  this reso-
             nance was as easily pin-pointed as it was violent. With increasing frequency, a maximum
             amplitude of  20 pipe diameters was reached, then subsided and ceased.
               Certain  variations  to  the  foregoing  behaviour  should  be  noted.  In  some  cases,  at
             low  frequencies  and  high  amplitudes  of  pulsation,  quasiperiodic  motion  in  a  combina-
             tion  resonance  region  was  encountered,  involving  the  first  and  higher  modes.  In  some
             cases,  following the principal primary instability, either a combination resonance region
             or  a  region  involving  the  superposition  of  more  than  one  parametric  resonance  was
             observed,  where  the  pipe  seemed  to  be  oscillating  about  a  quasi-stationary  deflected
             shape, thus  displaying  clearly  nonlinear behaviour;  these regions  were  difficult to deci-
             pher, and the mode shape transitions seemed to be gradual and difficult to pin-point. In yet
             other cases, the  stable region between  second-mode secondary and primary  instabilities
             disappeared.  In  some cases, parametric  resonances  associated with the third mode were
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