Wuhan University of TechnologyChapter18Analysis of undamped free vibration18-1
18-1 Wuhan University of Technology Chapter 18 Analysis of undamped free vibration
Wuhan University of TechnologyContents18.1Beamflexure:elementary case18.2 Beam flexure: including axial-force effects18.3Beamflexure:with distributedelastic support18.4 Beam flexure: orthogonality of vibration mode shapes18.5Freevibrationsinaxialdeformation18.6 Orthogonality of axial vibration modes18-2
18-2 Wuhan University of Technology 18.1 Beam flexure: elementary case 18.2 Beam flexure: including axial-force effects 18.3 Beam flexure: with distributed elastic support 18.4 Beam flexure: orthogonality of vibration mode shapes 18.5 Free vibrations in axial deformation 18.6 Orthogonality of axial vibration modes Contents
Wuhan Universityof Technology18.1 Beam flexure: elementary caseFirst,letusconsidertheelementarycasepresentedinSection172withEI(x)and m(x) set equal to constants EI and m, respectively. As shown by Eq. (177)thefreevibration equationof motionforthissystemis04v(r,t)a2v(c,t)EI0m0r4Ot2min(r,t)+Ei(,t)=0v(c,t)=o(c)Y(t)ms() (t) + 晋 (a) (t) = 018-3
18-3 Wuhan University of Technology 18.1 Beam flexure: elementary case First, let us consider the elementary case presented in Section 172 with EI(x) and m(x) set equal to constants EI and , respectively. As shown by Eq. (177), the freevibration equation of motion for this system is
Wuhan Universityof Technology-18.1 Beam flexure: elementary casemY(t)si(c)EIY(t)o(a)Becausethefirstterminthisequationisafunctionofxonlyandthesecondterm is a function of t only, the entire equation can be satisfied for arbitraryvaluesofxandtonlyif eachtermisaconstant inaccordancewithY(t)msiv(r)2EIY(t)d(r)wherethesingleconstantinvolvedisdesignatedintheforma4forlatermathematical convenience.ThiseguationyieldstwoordinarydifferentialequationsY(t) +w2Y(t) =0gi(a)-a40(c)=0a4EIi.e.EIm18-4
18-4 Wuhan University of Technology 18.1 Beam flexure: elementary case Because the first term in this equation is a function of x only and the second term is a function of t only, the entire equation can be satisfied for arbitrary values of x and t only if each term is a constant in accordance with where the single constant involved is designated in the form a 4 for later mathematical convenience. This equation yields two ordinary differential equations
Wuhan University of Technology-18.1 Beam flexure: elementary caseThe first of these [Eq. (187a)] is the familiar freevibration expression for anundampedSDOFsystemhavingthesolution[seeEq.(231)]Y(t)=Acoswt+B sinwtinwhichconstantsAandBdependupontheinitialdisplacementandvelocityconditions, i.e.,Y(0)Y(t) = Y (O) cos wt +sinwtwThesecond equationcanbesolvedintheusualwaybyintroducingasolutionoftheformd(c) = G exp(sr)(s4 - a4) G exp(sr) = 018-5
18-5 Wuhan University of Technology 18.1 Beam flexure: elementary case The first of these [Eq. (187a)] is the familiar freevibration expression for an undamped SDOF system having the solution [see Eq. (231)] in which constants A and B depend upon the initial displacement and velocity conditions, i.e., The second equation can be solved in the usual way by introducing a solution of the form