《结构动力学》课程教学课件（讲稿）18 Distributed-parameter systems（Analysis of undamped free vibration）

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

Wuhan University of Technology18.1 Beam flexure: elementary caseS3,4 =±aS1,2=±iaIncorporatingeachoftheserootsintoEq.(1811)separatelyandaddingtheresultingfourterms,oneobtainsthecompletesolutionΦ(r) =G1 exp(iar) + G2 exp(-iar) + G3 exp(ar) +G4 exp(-ar)o(r)=Aicosar+A2sinar+A3coshar+A4sinhaawhere A,Az,A3, and Ay are real constants which can be expressed in terms ofthe components of Gi, G2, G3, and G4. These real constants must be evaluatedsoastosatisfytheknownboundaryconditions(displacement,slope,momentorshear)attheendsofthebeam.18-6

18-6 Wuhan University of Technology 18.1 Beam flexure: elementary case Incorporating each of these roots into Eq. (1811) separately and adding the resulting four terms, one obtains the complete solution where A1, A 2, A 3, and A 4 are real constants which can be expressed in terms of the components of G1, G 2, G 3, and G 4. These real constants must be evaluated so as to satisfy the known boundary conditions (displacement, slope, moment, or shear) at the ends of the beam

Wuhan University of Technology18.1 Beam flexure: elementary caseto(x)xEI,m=constantsL(a)E(01=元mL401(t) =sin 元xLEI02=4元mL42元x02(x)=sinLEI03=9元mL3元(x)=sinL(b)18-7

18-7 Wuhan University of Technology 18.1 Beam flexure: elementary case

Wuhan University of Technology-18.1 Beam flexure: elementary caseExampleE181.SimpleBeamConsideringtheuniformsimplebeamshowninFigE181a,itsfourknownboundaryconditionsareb(0) = 0M(0) = EI @"(0) = 0Φ(L) = 0M(L)= EI"(L) =0Makinguseof Eq.(1815)and itssecondpartial derivativewith respecttox,Eqs.(a) can be written asΦ(O)=A1 cos0+A2 sin0+A3 cosh0+A4 sinh0= 0@"(0) = α2 (-A1 cos 0 - A2 sin0+ A3 cosh 0 + A4 sinh 0) = 018-8

18-8 Wuhan University of Technology Example E181. Simple Beam Considering the uniform simple beam shown in Fig. E181a, its four known boundary conditions are 18.1 Beam flexure: elementary case Making use of Eq. (1815) and its second partial derivative with respect to x, Eqs. (a) can be written as

Wuhan University of Technology-18.1 Beam flexure: elementary case(A1 +A3) = 0(-A1 + A3) = 0A1 = A3 = 0.Similarly, Eqs. (b) can be written in the formo(L)=A2 sinaL+A4 sinhaL=0d"(L)= a2 (-A2 sinaL + A4 sinhaL) = 02A4sinhaL=0(r) = A2 sinac18-9

18-9 Wuhan University of Technology 18.1 Beam flexure: elementary case Similarly, Eqs. (b) can be written in the form

Wuhan Universityof Technology18.1 Beam flexure: elementary caseA2 = 0. Φ(L) = 0sinaL=0which is the systemfrequency equation:it requiresthata=/Ln=0.1.2,..SubstitutingthisexpressionintoEg.(188)andtakingthesquarerootofbothsidesyieldthefrequencyexpressionEIWn=n2?㎡L4nTdn(α) = A2 sinn=1.2aL18-10

18-10 Wuhan University of Technology 18.1 Beam flexure: elementary case which is the system frequency equation: it requires that Substituting this expression into Eq. (188) and taking the square root of both sides yield the frequency expression