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A COMBINATION OF VECTORIAL FORM OF WAVE METHOD (VWM) WITH FOURIER EXPANSION SERIES IS PROPOSED AS A NEW VEHICLE FOR FREE AND FORCED VIBRATION ANALYSIS OF STEPPED CYLINDRICAL SHELLS WITH MULTIPLE INTERMEDIATE FLEXIBLE SUPPORTS. THE FLEXIBLE SUPPORTS CAN INCLUDE SPRINGS WITH ARBITRARY PROPERTIES IN THE POSSIBLE DIRECTIONS. BASED ON FLÃ¼GGE THIN SHELL THEORY AND VWM, THE REFLECTION, PROPAGATION AND TRANSMISSION MATRICES FOR A CIRCULAR CYLINDRICAL SHELL ARE DEFINED. FURTHERMORE, CONTIGUOUS VECTOR-MATRIX RELATIONSHIPS ARE ESTABLISHED FOR FREE AND FORCED VIBRATION ANALYSIS OF THE ISSUE INCLUDING ARBITRARY NUMBER OF THE DISCONTINUITIES IN THE SHELL THICKNESS, OR SHELL STEPS, AND INTERMEDIATE SUPPORTS. USING THESE VECTOR-MATRIX RELATIONS, THE EQUATIONS OF MOTION AS WELL AS THE SYSTEM CONTINUITY ARE WELL SATISFIED. DIMENSION OF THESE VECTORS AND MATRICES ARE COMPLETELY, INDEPENDENT OF THE NUMBER OF THE APPLIED INTERMEDIATE SUPPORTS AND GEOMETRICAL STEPS IN THE SHELL. HENCE, THE PRESENT APPROACH PROVIDES EXCELLENT COMPUTATIONAL ADVANTAGES AND MODELING FLEXIBILITY COMPARED TO THE CONVENTIONAL VIBRATION ANALYSIS METHODS AVAILABLE IN THE LITERATURE. THE RESULTS OF THE PRESENT STUDY ARE COMPARED WITH THE RESULTS AVAILABLE IN THE LITERATURE AS WELL AS THE RESULTS OF FINITE ELEMENT METHOD (FEM) AND FOUND IN EXCELLENCE AGREEMENT. FURTHERMORE, AS A CASE STUDY CASE, A CYLINDRICAL SHELL WITH THREE FLEXIBLE INTERMEDIATE SUPPORTS AND ALSO THREE GEOMETRICAL STEPS IS CONSIDERED. THE NATURAL FREQUENCY AND MODE SHAPES OF THE ISSUE ARE DERIVED, AND THE FORCED RESPONSES OF THE SHELL SUBJECT TO POINT LOAD EXCITATION ARE REPORTED.
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