FINITE TRANSFORM AND FINITE DIFFERENCE SOLUTIONS TO THE RAYLEIGH EQUATION

THE RAYLEIGH EQUATION FOR LATERAL VIBRATION OF BEAMS IS SOLVED FOR BOTH THE UNDAMPED AND LIGHTLY-DAMPED CASES. THE LOADING IS COMPLETELY GENERAL, AS ARE THE INITIAL CONDITIONS. THE EQUATION IS SOLVED EXACTLY, BY MEANS OF A FINITE TRANSFORM BASED ON STURM-LIOUVILLE THEORY, AND APPROXIMATELY, THROUGH THE USE OF CENTRAL DIFFERENCE OPERATORS. THE TWO SOLUTIONS ARE COMPARED THROUGH NINE NUMERICAL EXAMPLES. THE RESULTS OF EACH EXAMPLE ARE PRESENTED GRAPHICALLY AS WELL AS IN TABULAR FORM. THE FINITE TRANSFORM APPROACH, LEADING TO A RAPIDLY CONVERGENT SERIES SOLUTION, IS SHOWN TO BE VERY USEFUL. IT DOES, HOWEVER, BECOME LIMITED IF THE LOADING FUNCTION AND/OR THE INITIAL CONDITIONS ARE NOT EASILY EXPRESSIBLE IN A MATHEMATICAL FORM WHICH WILL PERMIT INTEGRATION OF THE NECESSARY EXPRESSIONS. FOR TRACTABLE INTEGRALS THIS APPROACH WILL QUICKLY PRODUCE RESULTS OF ANY DESIRABLE ACCURACY. THE FINITE DIFFERENCE TECHNIQUE REQUIRES A RATHER LARGE COMPUTER FOR MOST PRACTICAL PROBLEMS. ITS USE, NEVERTHELESS, MAY BE NECESSARY WHEN EITHER THE LOADING FUNCTION AND/OR THE INITIAL CONDITIONS ARE SUCH THAT THE CALCULATIONS IN THE TRANSFORM APPROACH BECOME EITHER UNWIDELY OR IMPOSSIBLE TO COMPLETE. WHEN THIS OCCURS, THE FINITE DIFFERENCE RECURRENCE FORMULA MAY BE UTILIZED TO OBTAIN EXCELLENT PREDICTIONS OF MAXIMUM DEFLECTIONS. THIS FORMULA MUST BE USED JUDICIOUSLY, HOWEVER, AS CURVATURES AND INTERMEDIATE DEFLECTIONS MAY BE SIGNIFICANTLY IN ERROR. /AUTHOR/

  • Supplemental Notes:
    • TECH REPT 75
  • Corporate Authors:

    New Mexico State University, Las Cruces

    Department of Civil Engineering, P.O. Box 30001
    Las Cruces, NM  USA  88003-8001
  • Authors:
    • Endebrock, E G
    • Durrant, S O
  • Publication Date: 0

Media Info

  • Features: Figures; References; Tables;
  • Pagination: 82 p.

Subject/Index Terms

Filing Info

  • Accession Number: 00209693
  • Record Type: Publication
  • Files: TRIS
  • Created Date: Aug 20 1972 12:00AM