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1、CHAPTER 17. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION171 INTRODUCTION172 BEAM FLEXURE: ELEMENTARY CASE结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 17. PARTIAL DIFFERENTIAL EQUATIONS OF MOTIONsumming moments about point Aequilibrium relationship for vertical forces结构动力学结构动力学dyanmicsofstructuresch1CHAPTER
2、17. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION173 BEAM FLEXURE: INCLUDING AXIALFORCE EFFECTS结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 17. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION173 BEAM FLEXURE: INCLUDING AXIALFORCE EFFECTSmomentequilibrium equation now becomesRotation of beam axis结构动力学结构动力学dyanmicsof
3、structuresch1CHAPTER 17. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION174 BEAM FLEXURE: INCLUDING VISCOUS DAMPINGMaterial dampingDistributed damping结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 17. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION176 AXIAL DEFORMATIONS: UNDAMPED结构动力学结构动力学dyanmicsofstructuresch1CHAPTER
4、 17. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATIONFirst, constant EI and m181 BEAM FLEXURE: ELEMENTARY CASEone form of solution of this equation can be obtained easily by separation of variables using结构动力学结构动力学dyanmicsofstruc
5、turesch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATION结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATIONThese real constants must be evaluated so as to satisfy the known boundary conditions (displacement, slope, moment, or shear) at the ends of the beam. Taking this ac
6、tion, any three of the four constants can be expressed in terms of the fourth and an expression (called the frequency equation) can be obtained from which the frequency parameter a is determined.Example E181. Simple Beamfrom which one obtains结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF U
7、NDAMPED FREE VIBRATION结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATION结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATIONExample E182. Cantilever Beam结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATION结构动力学结构动力
8、学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATION结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATION结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATIONExample E183. Cantilever Beam with Rigid Mass at Free Endthe translat
9、ional and rotary inertial force components结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATIONforce and moment equilibrium of the rigid mass requires that boundary conditions182 BEAM FLEXURE: INCLUDING AXIALFORCE EFFECTS结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSI
10、S OF UNDAMPED FREE VIBRATIONIn fact, it is evident that when the axial force P equals zero, so that g = 0, then 结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATION183 BEAM FLEXURE: WITH DISTRIBUTED ELASTIC SUPPORTSeparating variables as beforegiving two independent equat
11、ions结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATION184 BEAM FLEXURE: ORTHOGONALITY OF VIBRATION MODE SHAPES结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATIONThe second orthogonality condition,For a nonuniform beam, the equation of motion
12、 in free vibrations结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATION185 FREE VIBRATIONS IN AXIAL DEFORMATIONUsing the solution结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATIONExample E185. Cantilever Bar结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATION结构动力学结构动力学dyanmicsofstructuresch1CHAPTER 18. ANALYSIS OF UNDAMPED FREE VIBRATION结构动力学结构动力学dyanmicsofstructuresch1
