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Extra info for Innovations in Intelligent Machines, Volume 5: Computational Intelligence in Control Systems Engineering (Studies in Computational Intelligence, Volume 561)
These amplitudes cannot be straightforwardly added to obtain the total amplitude if damping is present, because the phase angles φm are Transverse Vibrations of Strings different for each mode. 94b) the summations, of course, extending for all m. Thus, the total amplitude W may be regarded as the vector sum of the amplitudes of the separate displacement components at point x = x0. 94a) shows that the in-phase components Am sinαmx0 are added up separately, and so are the out-of-phase components Bm sinαmx0 and the total resultant amplitude is found by the same procedure used to obtain the magnitude of any vector.
An external, distributed force (p, having dimensions of force per unit length) is also shown, acting normal (perpendicular) to the string. 1) where ρ = ρ(x) is the mass density per unit length of string. , they are first-order differentials). The remaining one contains (ds)2, 13 14 Chapter Two which is of higher order, and can therefore be discarded with no error. 4) is a nonlinear partial differential equation, with T depending on w. Indeed, if one wanted to consider large amplitude vibrations of the string, it could be useful.
89a) show that at every natural frequency (ωm), the mth mode would be strongly excited if damping were small. This large amplitude response, which would occur at each natural frequency, is called a resonance. Thus, as in any 41 42 Chapter Two continuous system, an infinite number of resonances would exist. However, if some damping is present (as in the case of any real system), then, the amplitudes at the resonant frequencies will typically decrease as m is increased. 89a) in the case when the exciting frequency exactly equals one of the natural frequencies.