FREE TRANSVERSE VIBRATIONS OF NON-UNIFORM BEAMS

Transcription

1 Please cite this aticle as: Izabela Zamosa Fee tansvese vibations of non-unifom beams Scientific Reseach of the Institute of Mathematics and Compute Science Volume 9 Issue pages 3-9. The website: Scientific Reseach of the Institute of Mathematics and Compute Science FREE TRANSVERSE VIBRATIONS OF NON-UNIFORM BEAMS Izabela Zamosa Institute of Mathematics Technical Univesity of Czestochowa Poland izabela.zamosa@im.pcz.pl Abstact. In this pape the Geen s function method fo the fee vibation poblem of nonunifom Benoulli-Eule beams is pesented. To find the Geen s function of the fouth ode diffeential opeato occuing at the beam s equation of motion the powe seies method is poposed. Intoduction The appoximate appoach fo the vibation poblem of non-unifom Benoulli Eule beams was pesented in [ ]. The authos of these publications used the Rayleigh-Ritz method fo beams with a polynomial coss section and moment of inetia with additional discete elements [] and fo diffeent bounday conditions [6]. An analytical solution of the fee vibation poblems of the non-unifom beams can only be found fo some special types of the beams. The closed fom solutions of the fee vibation poblem fo second and fouth ode polynomial paametes chaacteizing beams have been found in [3 ]. The Geen s function method was pesented in papes [-6].. Fomulation and solution of the poblem et s conside a non-unifom beam length with coss section aea A(x) and moment of inetia I(x) caying any numbe of discete elements. x Fig.. A setch of a non-unifom beam

2 I. Zamosa Accoding to the Benoulli-Eule theoy the following equation govens a fee vibation of the beam with attached discete elements (spings masses oscillatos) [5]: Y x t Y x t EI ( x) + ρ A( x) = F Y ( x t) x x t whee Y is the function of deflection ρ is the mass density of the beam mateial E is the modulus of elasticity and the fom of opeato F depends on the natue of the attached discete systems. Function Y satisfies homogeneous bounday conditions B Y ( x t) = Y ( x t ) () B = () Natual fequencies of the beam ae hamonic Y ( x t) equations () and () may be witten in the fom: d dx d Y x EI ( x) ρω A x Y x = Y x i t = Y x e ω theefoe F dx (3) B Y ( x) = Y ( x ) B = () x Y By intoducing non-dimensional coodinates and values: ξ = y = ρ Aω Ω = whee ω is the natual vibation of the beam we obtain equations EI (3)-() in the fom: d d y I Ω A y = F % y (5) B % y = y ξ = B % = (6) The solution to this poblem (5)-(6) can be obtained with the use of the Geen s function method and popeties of the Geen s function. If the Geen s function of d d the linea diffeential opeato = I A Ω is nown the solution to the poblem (5)-(6) may be witten as follows: ( ) ξ = y ξ = G ξ ζ F y ζ dζ (7)

3 Fee tansvese vibations of non unifom beam 5 Equation (7) is used in the analysis of the vibation of the beam. It yields to the fequency equation which is solved numeically with espect to the eigenfequencies. The eigenfunctions coesponding to the eigenfequencies ae deived by using equation (7).. Geen s function of the fouth ode diffeential opeato et s conside the following opeato []: d d d d p3 p + Ω p (8) The Geen s function of this opeato (8) satisfies the equation: ( ) G ξ ζ = δ ξ ζ (9) whee δ is the Diac delta function. Function G has the fom: ( ξ ζ ) ( ξ ζ ) ( ξ ζ ) ( ξ ζ ) G = G + G H () H is a unit step function G and G ae solutions to a homogeneous equation G also satisfies the following conditions: G ξ ζ = () G ( ξ ζ ) 3 ( ξ ζ ) ( ξ ζ ) ( ξ ζ ) G G G = = = = ξ ξ ξ p ζ ξ = ζ 3 ξ = ζ ξ = ζ ξ = ζ 3 () Assuming that p i p i = ξ fo i = 3 we ae looing fo the solution to =! V ξ = with the use of the powe seies method also as the sum v V = ξ. Substituting p i and V into a homogeneous equation leads to: =! P3 + + P + Ω P ξ = (3)! =

4 6 I. Zamosa whee P = p v j= j i i j + i j i = 3. We may wite this equation (3) as follows: Ω = + + p3 jv + j p jv + j p jv j j = j j= j j= j () Fom () we can deteminate values of unnown coefficients v + ( =...) by means of factos v v v v 3: + + v p v p v p v = 3 j + j + j + j Ω j j p3 j= j j= j j= j (5) Equation () has fou linea independent solutions: V v = ξ = 3 (6) =! To find them we must assume that functions V satisfy conditons: i d V i ξ = = δ i = 3; = 3 (7) i+ whee δ i+ is the Konece delta function. Because of wite (7) in the fom i d V i ξ = i = v we can v i = δ i + i = 3; = 3 (8) With the use of fomula (8) and equation () othe coefficients of function V ae calculated: + + v p v p v p v = 3 j + j + j + j Ω j j p3 j= j j= j j= j fom: = 3 ; =... (9) V ξ = has the Finally a geneal solution to the homogeneous equation = = ξ () V C V C = = =! whee v ae expessed by (8) and (9). v

5 Fee tansvese vibations of non unifom beam 7 To calculate a paticula solution G ( ξ ζ ) H ( ξ ζ ) to equation (9) it s necessay to have in mind condition () whee p ( ζ ) obtain the function G (ξζ) as follows: = ( ξ ζ ) ( ζ ) 3 p 3 = ζ. Using them we =! G = C V () ( ) W whee C ( ζ ) = W = det ij p3 ( ζ ) W w ζ W det i j= 3 wij ζ i= 3 j i d Vj ( ζ ) j = 3 and w ( ζ ) = ij dζ i = fo In esult the Geen s function G(ξζ) of the diffeential opeato can be pesented in the fom: ( ξ ζ ) ( ξ ζ W = + ) p ( ζ ) W ξ () = = G C V H V 3 Unnown constants C ae calculated with the use of bounday conditions (6). Fo example the system of equations ( i C ) V + = = ( ) W ( i+ C ) + V = i = (3) = p ( ζ ) W 3 gives the coefficients fo a fee-fee beam: C ( ζ ) = b a3 b3 a b a3 M ζ C ( ζ ) = c a3 c3 a c a3 M ζ + C3 ( ζ ) = c a c a c a M ζ + C ( ζ ) = c a3 c a3 c3 a M ζ ()

6 8 I. Zamosa Fo a clamped-fee beam the bounday conditions ae as follows: i CV = = ( ) W ( i+ C ) + V = i = (5) = p ( ζ ) W 3 and the coefficients C ae: C ( ζ ) = c3 d c d3 c d3 N ζ C ( ζ ) = c3 d c d3 c d3 N ζ C3 ( ζ ) = c d c d c d N ζ C ( ζ ) = c d3 c d3 c3 d N ζ (6) Intoduced functions and values occuing in () (6) ae given as: a = V V V V " "' "' " ij i j i j d = V V V V ' ' ij i j i j " "' = ( ζ ) + ( ζ ) ( ζ ) ( ζ ) b B V A V i i i A( ζ ) = W V W + " ζ ( ζ ) ζ = and denominatos M N: c = B V A V " "' i i i B = W V W ( ζ ) + "' ( ζ ) = ( ζ ) 3 ( ζ ) a a a a a a M = p a a + a a a a 3 3 ( ζ ) 3 ( ζ ) d a d a d a N = p d a + d a d a

7 Fee tansvese vibations of non unifom beam 9 Conclusions The pesented solution fo the fee vibation poblem may be used fo numeical calculations. The pesented method may be used to analyze stepped non-unifom beams with additional discete elements. Refeences [] Auciello N.M. Mauizi M.J. On the natual vibations of tapeed beams with attached inetia elements Jounal of Sound and Vibation (3) [] Zhou D. Cheung Y.K. The fee vibation of a type of tapeed beams Compute Methods in Applied Mechanics and Engineeing 88(-3). [3] Elishaoff I. Becquet R. Closed-fom solutions fo natual fequency fo inhomogeneous beams with one sliding suppot and the othe pinned Jounal of Sound and Vibation 38(3) [] Zamojsa I. Analiza dgań własnych jednowymiaowych uładów dysetno-ciągłych. Zastosowanie metody funcji Geena paca dotosa 6. [5] Kula S. Dynamiczne funcje Geena w analizie dgań własnych ciągłych i dysetno-ciągłych uładów mechanicznych Monogafie 6 Wydawnictwo Politechnii Częstochowsiej Częstochowa 999. [6] Kula S. Zamojsa I. Application of the Geen s function method in fee vibation analysis of non-unifom beams Scientific Reseach of the Institute of Mathematics and Compute Science 5() 87-9.