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Complex Nanomaterials for Aerospace functions has been built for a group drawn to house technology and nanotechnology. Scientists and engineers from a number of NASA box facilities and the Jet Propulsion Laboratory, college of Puerto Rico, The Pennsylvania nation college, and INFN-Laboratori Nazionali di Frascati, Italy, have joined efforts to debate the functions of nanomaterials in sensors, surroundings revitalization in liveable area structures, existence help platforms, regenerative gasoline cells, lithium-ion batteries, powerful light-weight fabrics, nanoelectr.
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Limiting case 1, Ω —» T h e case of rotor speed a p p r o a c h i n g infinity implies that the centrifugal force stiffening term b e c o m e s significantly larger than the elastic stiffening t e r m . 22) E q u a t i o n ( 3 . 2 2 ) can b e r e c o g n i z e d as a form of the L e g e n d r e differential equation w h o s e solution is given by any of the L e g e n d r e p o l y n o m i a l s P (x). 23b) 2 T h e s e results define the vibration characteristics of w h a t a m o u n t s to a flexible cable with vanishingly small elastic stiffness.
This is especially important, as w e shall find, for reducing BASIC ANALYTICAL TECHNIQUES descriptions of c o n t i n u o u s s y s t e m s (which have an infinite of freedom) to a handful of normal modal variables. 23 number of degrees Basic proof of orthogonality. T h e proof offered for the orthogonality of natural m o d e s is based on the two-degree-of-freedom configuration defined earlier in F i g . 2 . 7 . H o w e v e r , the veracity of the orthogonality principle is not limited to this simple p r o b l e m , but can easily be generalized to other m o r e c o m p l i c a t e d , but linear, d y n a m i c s y s t e m s .
Often the d y n a m i c system is not rigidly fixed to a stationary point. In such a case a ' r i g i d - b o d y " m o d e appears that h a s zero frequency and involves zero spring deflections. F u r t h e r m o r e , for free vibrations the system then h a s a center of gravity that does not m o v e in space and hence b e c o m e s a system node point. C o n s i d e r the e x a m p l e given in F i g . 2 . 1 1 of t w o flywheels c o n n e c t e d dumbbell-like to each other by a flexible shaft: Let us find the natural frequency and node point for this system.