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,CHAPTER17.PARTIALDIFFERENTIALEQUATIONSOFMOTION,171INTRODUCTION,172BEAMFLEXURE:ELEMENTARYCASE,CHAPTER17.PARTIALDIFFERENTIALEQUATIONSOFMOTION,summingmomentsaboutpointA,equilibriumrelationshipforverticalforces,CHAPTER17.PARTIALDIFFERENTIALEQUATIONSOFMOTION,173BEAMFLEXURE:INCLUDINGAXIALFORCEEFFECTS,CHAPTER17.PARTIALDIFFERENTIALEQUATIONSOFMOTION,173BEAMFLEXURE:INCLUDINGAXIALFORCEEFFECTS,momentequilibriumequationnowbecomes,Rotationofbeamaxis,CHAPTER17.PARTIALDIFFERENTIALEQUATIONSOFMOTION,174BEAMFLEXURE:INCLUDINGVISCOUSDAMPING,Materialdamping,Distributeddamping,CHAPTER17.PARTIALDIFFERENTIALEQUATIONSOFMOTION,176AXIALDEFORMATIONS:UNDAMPED,CHAPTER17.PARTIALDIFFERENTIALEQUATIONSOFMOTION,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,First,constantEIandm,181BEAMFLEXURE:ELEMENTARYCASE,oneformofsolutionofthisequationcanbeobtainedeasilybyseparationofvariablesusing,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,Theserealconstantsmustbeevaluatedsoastosatisfytheknownboundaryconditions(displacement,slope,moment,orshear)attheendsofthebeam.Takingthisaction,anythreeofthefourconstantscanbeexpressedintermsofthefourthandanexpression(calledthefrequencyequation)canbeobtainedfromwhichthefrequencyparameteraisdetermined.,ExampleE181.SimpleBeam,fromwhichoneobtains,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,ExampleE182.CantileverBeam,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,ExampleE183.CantileverBeamwithRigidMassatFreeEnd,thetranslationalandrotaryinertialforcecomponents,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,forceandmomentequilibriumoftherigidmassrequiresthatboundaryconditions,182BEAMFLEXURE:INCLUDINGAXIALFORCEEFFECTS,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,Infact,itisevidentthatwhentheaxialforcePequalszero,sothatg=0,then,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,183BEAMFLEXURE:WITHDISTRIBUTEDELASTICSUPPORT,Separatingvariablesasbefore,givingtwoindependentequations,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,184BEAMFLEXURE:ORTHOGONALITYOFVIBRATIONMODESHAPES,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,Thesecondorthogonalitycondition,Foranonuniformbeam,theequationofmotioninfreevibrations,CHAPTER18.ANALYSISOFUNDAMPEDFREEVIBRATION,185FREEVIBRATIONSINAXIALDEFORMATION,Usingthesolution,CHAPTER18.ANALYSISOFUND
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