POWER SECANT METHOD APPLIED TO NATURAL FREQUENCY EXTRACTION OF TIMOSHENKO BEAM STRUCTURES

Abstract

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MSO-LEVEL-TEXT:"\(%1\)"; MSO-LEVEL-TAB-STOP:36.0PT; MSO-LEVEL-NUMBER-POSITION:LEFT; TEXT-INDENT:-18.0PT;} @LIST L5 {MSO-LIST-ID:1087070885; MSO-LIST-TYPE:HYBRID; MSO-LIST-TEMPLATE-IDS:-1471508046 -612498340 68550681 68550683 68550671 68550681 68550683 68550671 68550681 68550683;} @LIST L5:LEVEL1 {MSO-LEVEL-TEXT:"\(%1\)"; MSO-LEVEL-TAB-STOP:36.0PT; MSO-LEVEL-NUMBER-POSITION:LEFT; TEXT-INDENT:-18.0PT;} @LIST L6 {MSO-LIST-ID:1296720610; MSO-LIST-TYPE:HYBRID; MSO-LIST-TEMPLATE-IDS:-1568785020 39191838 68550681 68550683 68550671 68550681 68550683 68550671 68550681 68550683;} @LIST L6:LEVEL1 {MSO-LEVEL-NUMBER-FORMAT:ALPHA-LOWER; MSO-LEVEL-TEXT:"\(%1\)"; MSO-LEVEL-TAB-STOP:36.0PT; MSO-LEVEL-NUMBER-POSITION:LEFT; TEXT-INDENT:-18.0PT;} @LIST L7 {MSO-LIST-ID:1713725830; MSO-LIST-TYPE:HYBRID; MSO-LIST-TEMPLATE-IDS:1361095152 68550681 68550681 68550683 68550671 68550681 68550683 68550671 68550681 68550683;} @LIST L7:LEVEL1 {MSO-LEVEL-NUMBER-FORMAT:ALPHA-LOWER; MSO-LEVEL-TAB-STOP:36.0PT; MSO-LEVEL-NUMBER-POSITION:LEFT; TEXT-INDENT:-18.0PT;} @LIST L8 {MSO-LIST-ID:1856193258; MSO-LIST-TYPE:HYBRID; MSO-LIST-TEMPLATE-IDS:2079874422 68550671 68550681 68550683 68550671 68550681 68550683 68550671 68550681 68550683;} @LIST L8:LEVEL1 {MSO-LEVEL-TAB-STOP:36.0PT; MSO-LEVEL-NUMBER-POSITION:LEFT; TEXT-INDENT:-18.0PT;} OL {MARGIN-BOTTOM:0CM;} UL {MARGIN-BOTTOM:0CM;} --> CONSIDERING THE CASE OF PLANE FRAMES, THIS WORK DEALS WITH THE ESTABLISHMENT OF A NEW FORMULATION FOR THE EXACT DYNAMIC STIFFNESS MATRIX (DSM) WHOSE DETERMINANT PRESENTS, UNIQUELY, ZEROS FOR THE NATURAL FREQUENCIES. IN THE ABSENCE OF POLES, IT IS POSSIBLE TO EMPLOY ITERATIVE PROCEDURE FOR THE EIGENVALUE EXTRACTION, AS WE PROPOSE HERE. APPLYING THE PROCEDURE TO THE MORE GENERAL CASE OF TIMOSHENKO BEAMS, WE INTRODUCE A NEW TECHNIQUE, CALLED “POWER DEFLATION”, THAT MAKES THE SECANT METHOD SUITABLE FOR TRANSCENDENTAL NONLINEAR EIGENPROBLEMS. FOR THE AUTOMATED COMPUTER CALCULATION, AND TO AVOID OVERFLOW OCCURRENCES THAT CAN COMPLETELY COMPROMISE THE OPERATION OF THE ITERATIVE PROCESS, LIMITING FREQUENCIES ARE FORMULATED AS WELL AS SCALING IS APPLIED TO THE EIGENVALUE PROBLEM. VARIOUS APPLICATION EXAMPLES ARE PRESENTED THAT PROVE THE OPERATION OF THE PROPOSED METHOD BY COMPARISON WITH THE RESULTS AVAILABLE IN THE LITERATURE.