WuhanUniversityof Technology18.1 Beam flexure: elementary caseExampleE182.CantileverBeamThefreevibrationanalysisofthesimplebeaminthepreviousexamplewasnotdifficultbecauseitsmodeshapesweredefinedbyonlyonetermintheshapefunctionexpressionofEq.(1815).Toprovideamorerepresentativeexampleoftheanalysisprocedurerequiringallfourterms,considerthecantileverbeamshowninFig.E182a.Itsfourboundaryconditionsto be satisfied are+中(x)El,m=constantsT18-11
18-11 Wuhan University of Technology Example E182. Cantilever Beam The freevibration analysis of the simple beam in the previous example was not difficult because its mode shapes were defined by only one term in the shapefunction expression of Eq. (1815). To provide a more representative example of the analysis procedure requiring all four terms, consider the cantilever beam shown in Fig. E182a. Its four boundary conditions to be satisfied are 18.1 Beam flexure: elementary case
Wuhan University of Technology-18.1 Beam flexure: elementary caseΦ(0) = 0Φ(0) = 0M(L)=EI0"(L)= 0V(L) =EIo"(L)=0SubstitutingEg.(1815)and itsderivativeexpressions intotheseequationsgives(0)=(A1cos0+A2sin0+A3 cosh0+A4sinh0)=0'(0)=a(-A1 sin0+A2 cos0+A3 sinh0+A4 cosh0)= 0o"(L)=a(-ArcosaL-A2 sinaL+A3coshaL+A4sinhaL)=0o"(L) =a (A1 sinaL -A2 cosaL + A3 sinhaL + A4 coshaL)= 018-12
18-12 Wuhan University of Technology 18.1 Beam flexure: elementary case Substituting Eq. (1815) and its derivative expressions into these equations gives