Computers and Mathematics with Applications

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1 Computers and Mathematcs wth Applcatons 56 ( Contents lsts avalable at ScenceDrect Computers and Mathematcs wth Applcatons journal homepage: An nnovatve egenvalue problem solver for free vbraton of Euler Bernoull beam by usng the Adoman decomposton method Hsn-Y La, Jung-Chang Hsu 2, Cha o-kuang Chen Department of Mechancal Engneerng, Natonal Cheng-Kung Unversty, No., Unversty Road, Tanan 70, Tawan, ROC a r t c l e n f o a b s t r a c t Artcle hstory: Receved 9 March 2008 Receved n revsed form 30 June 2008 Accepted 0 July 2008 Keywords: Unform Euler Bernoull beam Free vbraton Natural frequency Mode shape Adoman decomposton method Ths paper deals wth free vbraton problems of Euler Bernoull beam under varous supportng condtons. The technque we have used s based on applyng the Adoman decomposton method (ADM to our vbraton problems. Dong some smple mathematcal operatons on the method, we can obtan th natural frequences and mode shapes one at a tme. The computed results agree well wth those analytcal and numercal results gven n the lterature. These results ndcate that the present analyss s accurate, and provdes a unfed and systematc procedure whch s smple and more straghtforward than the other modal analyss Elsever Ltd. All rghts reserved.. Introducton The vbraton problems of unform Euler Bernoull beams can be solved by analytcal or approxmate approaches 4. Usng analytcal approaches, the closed form solutons of free vbraton under varous boundary condtons have been found n these lterature. Approxmate approaches such as the Raylegh Rtz method and the Galerkn method have been appled to calculate some lower natural frequences and mode shapes. However, t may be dffcult to determne hgher natural frequences and mode shapes on account of not choosng complete and correct admssble functons. Regster 5 derved a general expresson for the modal frequences and nvestgated the egenvalue for a beam wth symmetrc sprng boundary condtons. Wang 6 studed the dynamc analyss of generally supported beam usng Fourer seres. Yeh 7 studed the applcatons of dual MRM for determnng the natural frequences and natural modes of the Euler Bernoull beam usng the sngular value decomposton method. Recently, Km 8 studed the vbraton of unform beams wth generally restraned boundary condtons usng Fourer seres. Naguleswaran 9 obtaned an approxmate soluton to the transverse vbraton of the unform Euler Bernoull beam under lnearly varyng axal force. In ths study, a new computed approach called Adoman decomposton method (ADM s ntroduced to solve the vbraton problems. The concept of ADM was frst proposed by Adoman and was appled to solve lnear and nonlnear ntal/boundary-value problems n physcs 0. In recent years, a large amount of lterature developed concernng the ADM by applyng t to the applcatons n appled scences. For more detals about the method, see 5 and the references cted there. In ths paper the vbraton problems of unform beams wth varous boundary condtons are consdered. Usng the ADM, the governng dfferental equaton becomes a recursve algebrac equaton and boundary condtons become smple algebrac frequency equatons whch are sutable for symbolc computaton. Moreover, after some smple algebrac operatons on these frequency equatons, any th natural frequency and the closed form seres soluton of any th mode shape Correspondng author. Tel.: x6240; fax: E-mal addresses: hyla@mal.ncku.edu.tw (H.-Y. La, jchsu@mal.ksu.edu.tw (J.-C. Hsu, ckchen@mal.ncku.edu.tw (C.-K. Chen. Tel.: x6258; fax: Tel.: ; fax: /$ see front matter 2008 Elsever Ltd. All rghts reserved. do:0.06/j.camwa

2 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( can be obtaned. Fnally, three problems of unform beams are solved to verfy the accuracy and effcency of the present method. 2. The prncple of ADM In order to solve vbraton problems by the Adoman decomposton method (ADM the basc theory s stated n bref n ths secton. Consder the equaton Fy(x = g(x, where F represents a general nonlnear ordnary dfferental operator nvolvng both lnear and nonlnear parts, and g(x s a gven functon. The lnear terms n Fy are decomposed nto Ly + Ry, where L s an nvertble operator, whch s taken as the hghest-order dervatve and R s the remander of the lnear operator. Thus, Eq. ( can be wrtten as Ly + Ry + Ny = g(x, where Ny represents the nonlnear terms n Fy. Eq. (2 corresponds to an ntal-value problem or a boundary-value problem. Solvng for Ly, one can obtan y = Φ + L g L Ry L Ny, where Φ s an ntegraton constant, and LΦ = 0 s satsfed. Correspondng to an ntal-value problem, the operator L may be regarded as a defnte ntegraton from 0 to x. In order to solve Eq. (3 by the ADM, we decompose y nto the nfnte sum of seres y = y k, (4 and the nonlnear term Ny = f (y s decomposed as Ny = f (y = A k, where the A k are known as Adoman polynomals. Followng 0 4, Adoman polynomals can be derved as follows: A 0 = f (y 0, A = y f (y 0, A 2 = y 2 f (y 0 + y2 2! f (y 0, A 3 = y 3 f (y 0 + y y 2 f (y 0 + y3 3! f (y 0, and other polynomals can be generated n a smlar manner. Pluggng Eqs. (4 and (5 nto Eq. (3 gves y k = Φ + L g L R y k L A k. (6 Each term n seres (6 s gven by the recurrent relaton y 0 = Φ + L g, y k = L Ry k L A k, k. The ntal term y 0 was defned and the remander terms were determned by usng smple ntegratons. However, n practce all terms n seres (6 cannot be determned exactly, and the solutons can only be approxmated by a truncated seres n y k. 3. Usng the ADM to analyze the free vbraton problem of unform beam Consder a unform Euler Bernoull beam of fnte length l, the equaton of moton for lateral vbratons of a unform elastc beam gnorng shear deformaton and rotary nerta effects s EI 4 y(x, t 4 x + ρa 2 y(x, t 2 t = 0, 0 < x < l (8 ( (2 (3 (5 (7

3 3206 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( where y(x, t s the lateral deflecton at dstance x along the length of the beam and tme t, EI, ρ and A are the flexural rgdty, the mass per unt volume and the cross-sectonal area of the beam, respectvely. The beam subjected to the homogeneous boundary condtons s gven as 3 y(x, t 2 y(x, t y(x, t c r3 + c 3 r2 + c x 2 r + c r0 y(x, t = 0, r =, 2 (9 x x and d r3 3 y(x, t 3 x + d r2 2 y(x, t 2 x + d r y(x, t x + d r0 y(x, t x=0 x=l = 0. r =, 2. (0 The beam has 4 boundary condtons, two at the left end x = 0 and the other two at the rght end x = l. The 6 constants, c rj, d rj (r =, 2, j = 0,, 2, 3, depend on the gven types of the boundary condtons. For any mode of vbraton, the lateral deflecton y(x, t may be wrtten n the form y(x, t = Y(xh(t, where Y(x s the modal deflecton and h(t s a harmonc functon of tme t. If ω denotes the frequency of h(t, then ( 2 y(x, t 2 t = ω 2 Y(xh(t, and the egenvalue problem of Eq. (8 reduces to the dfferental equaton (2 EI d4 Y(x d 4 x ρaω 2 Y(x = 0, 0 < x < l whch must be satsfed throughout the doman 0 < x < l and whch s subject to the gven homogeneous boundary condtons at two ends x = 0 and x = l. Wthout loss of generalty, the followng dmensonless quanttes are ntroduced. X = x l, Y(x Y(X =, λ = ρaω2 l 4. (4 l EI The governng equaton (3 can be wrtten n the followng dmensonless form: d 4 Y(X λy(x = 0, 0 < X < (5 d 4 X and the boundary condtons of Eqs. (9 and (0 can be wrtten n the form: d 3 Y(X d 2 Y(X dy(x α r3 + α d 3 r2 + α X d 2 r + α r0 Y(X = 0, r =, 2 (6 X dx X=0 d 3 Y(X d 2 Y(X dy(x β r3 + β d 3 r2 + β X d 2 r + β r0 Y(X = 0, r =, 2 (7 X dx X= or α rj Y (j (0 = 0, r =, 2 (8 j=0 β rj Y (j ( = 0, r =, 2 (9 j=0 where the 6 constants, α rj, β rj (r =, 2, j = 0,, 2, 3, are dmensonless and Y (j (X denotes the jth-order dervatve wth respect to X, and sets Y(X = Y (0 (X. In Eq. (5, the lnear operator s set to be LY = Y (4 (X, and the nonlnear operator s NY = 0. Hence the deflecton Y(X can be solved by the ADM. From Eqs. (2 and (5 one can obtan Y(X = Φ + L λy(x, where L = dxdxdxdx. Now the decomposton Y(X = Y k(x can be put together wth Eq. (20 to yeld Y(X = Y k (X = Φ + λl Y k (X, (2 (3 (20

4 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( where we have Y 0 (X = Φ = m=0 X m m! Y (m (0 = Y(0 + Y ( (0X + Y (2 (0 X 2 + Y (3 (0 X 3, (22 2! 3! as the ntal term of the decomposton, and Y k (X = λl Y k (X = λ x x x x Y k (XdXdXdXdX, k, (23 as the recurrence relaton of the decomposton. Consequently, all components of the decomposton can be dentfed and evaluated from Eq. (23. That s, by substtutng Eq. (22 nto Eq. (23, one can obtan and Y (X = λl Y 0 (X = λ Y 2 (X = λl Y (X = λ 2 m=0 Y k (X = λl Y k (X = λ k m=0 X 4+m (4 + m! Y (m (0, (24 X 8+m (8 + m! Y (m (0, (25 m=0 X 4k+m (4k + m! Y (m (0. (26 In practce, the soluton wll be the n-term approxmaton φ n (X = Y k (X = λ k X 4k+m (4k + m! Y (m (0 m=0 n X 4k+m = (4k + m! λk Y (m (0, (27 m=0 and from the convergence of ADM 5, φ n (X approaches Y(X as n. Y(X = lm n φ n (X = Y k (X. We can now form successve approxmants, φ n (X = n Y k(x, as n ncreases and the boundary condtons are also met. Thus φ (X = Y 0 (X, φ 2 (X = φ (X + Y (X, φ 3 (X = φ 2 (X + Y 2 (X serve as approxmate solutons wth ncreasng accuracy as n, and s also oblgated to, of course, satsfy the boundary condtons. By dfferentatng Eq. (27 wth respect to X, one can obtan φ n (X ( dφ n (X = = dx φ n (X (2 = d 2 φ n (X dx 2 = φ n (X (3 = d 3 φ n (X dx 3 = k= X 4k (4k! λk Y(0 + m=0 k= 2 m=0 k= X 4k+m 2 m= n (4k + m 2! λk Y (m (0 + X 4k+m 3 (4k + m 3! λk Y (m (0 + X 4k+m (4k + m! λk m=2 X 4k+m 2 (28 Y (m (0, (29 (4k + m 2! λk Y (m (0, (30 X 4k (4k! λk Y (3 (0. (3 Hence, at the left end X = 0, φ n (0 (j = Y (j (0, (j = 0,, 2, 3 are obtaned and at the rght end X =, φ n ( (j are functons of λ, n and Y (j (0, (j = 0,, 2, 3. By use of these results, the boundary condtons of Eqs. (8 and (9 can be wrtten as α rj Y (j (0 = 0, r =, 2 (32 j=0 j=0 f n rj (λy (j (0 = 0, r =, 2 (33

5 3208 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( where f n rj (λ are polynomals of λ correspondng to n. By solvng Eqs. (32 and (33 smultaneously for nontrval solutons Y (j (0, (j = 0,, 2, 3, the th estmated egenvalue λ n correspondng to n s obtaned, and n s decded by the followng equaton: λ n λ n ε, where λ n s the th estmated egenvalue correspondng to n, and ε s a preset small value. If Eq. (34 s satsfed, then s the th egenvalue λ. Substtutng λ nto Y (X, Y 2 (X,..., and Y k (X, we have λ n φ n n (X = Y (X, k where Y (X k s Y k(x whose λ s substtuted by λ, and φ n s the th egenfuncton correspondng to the th egenvalue λ. By normalzng Eq. (35, the th normalzed egenfuncton s defned as φ n (X = φ n (X 0 φn, (X 2 dx where φ n (X s the th mode shape functon of the beam correspondng to the th natural frequency ω n, ω n = λ n EI/ρAl 4. For the convenence of analyss, we defned the dmensonless natural frequency ω n, ω n = λ n. One can solve Eqs. (32 and (33 smultaneously to fnd the egenvalue λ, but t s complcated and tedous. Hence, the smplfed method s provded and t s to reduce the number of the equatons n Eqs. (32 and (33 by the known boundary condtons of the beam at the left end X = 0. One can choose any two quanttes of Y (j (0, (j = 0,, 2, 3, as the arbtrary constants, and the remanng two as the functons of these two arbtrary constants, then only two equatons n Eq. (33 can be easly solved for the nontrval solutons and the frequency equaton for egenvalue λ can be obtaned. Hence, let us dscuss the varous boundary condtons at the left end X = 0 and consder three cases as follows: Case : Free end at X = 0. The shear force and bendng moment are zero at X = 0, that s Y (2 (0 = 0, Y (3 (0 = 0, and Y(0, Y ( (0 are consdered as the arbtrary constants whch depend on the gven boundary condtons at the rght end X =. The n-term approxmaton n Eq. (27 can be wrtten as λ n φ n (X = X 4k (4k! λk Y(0 + Hence, Eq. (33 can be smplfed as X 4k+ (34 (35 (36 (4k +! λk Y ( (0. (37 f n n r0 (λy(0 + f r (λy ( (0 = 0, r =, 2. (38 For nontrval solutons Y(0, and Y ( (0, the frequency equaton s gven as f n n 0 (λ f (λ f n n 20 (λ f (λ = 0. 2 The th estmated egenvalue λ n correspondng to n s obtaned by Eq. (39, and n s decded by Eq. (34, by substtutng nto Eq. (38, one can obtan Y ( (0 = f n r0 (λn f n Y(0, r = or 2. (40 r (λn Furthermore, the th mode shape φ n φ n n (X = Y(0 (λ n k X 4k (4k! f correspondng to the th egenvalue λ n s obtaned by Eqs. (40 and (37 n r0 (λn X 4k+ f n r (λn (4k +! (39. (4 Furthermore, wthout loss of generalty, let us consder the free free beam, whose ends x = 0 and x = l are connected by translatonal sprngs and rotatonal sprngs as shown n Fg.. The boundary condtons at x = 0 and x = l are gven as EI d2 Y(x dy(x k d 2 RL = 0, (42 x dx x=0 EI d3 Y(x + k d 3 TL Y(x = 0, (43 x x=0

6 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( Fg.. An elastc Bernoull Euler beam wth rotatonal and translatonal boundary restrants. and EI d2 Y(x dy(x + k d 2 = 0, (44 x dx x=l EI d3 Y(x k d 3 TR Y(x = 0 (45 x x=l where k TL and k TR are the translatonal sprng constants, k RL and k are the rotatonal sprng constants. Eqs. (42 (45 can be expressed n the dmensonless form as and Y (2 (0 β RL Y ( (0 = 0, Y (3 (0 + β TL Y(0 = 0, Y (2 ( + β Y ( ( = 0, Y (3 ( β TR Y( = 0 (49 where β RL = k RL l/ei, β = k l/ei, and β TL = k TL l 3 /EI, β TR = k TR l 3 /EI. From the BCs of Eqs. (46 and (47, one can set Y (2 (0 = β RL Y ( (0; Y (3 (0 = β TL Y(0, (50 as the arbtrary constants. Hence Eq. (27 can be wrtten as X φ n 4k (X = (4k! β 4k+3 TLX X 4k+ λ k Y(0 + (4k + 3! (4k +! + β 4k+2 RLX λ k Y ( (0. (5 and λ n From the BCs of Eqs. (48 and (49, we have φ n ( (2 + β φ n ( ( = 0, φ n ( (3 β TR φ n ( = 0. By substtutng Eq. (5 nto Eqs. (52 and (53, and usng Eq. (38, one can obtan f 0 (λ = β TR β TRβ RL + λ k 2! (4k 2! + β RL (4k! β TR (4k +! β TRβ RL, k= f (λ = β + β RL + β β RL + λ k (4k! + β + β RL + β β RL, (4k! (4k +! f 20 (λ = β TR β TL + β TLβ TR 3! f 2 (λ = β TL β TLβ 2! k= + λ k k= + λ k k= (4k 3! β TR + β TL (4k! + β TRβ TL, (4k + 3! (4k 2! + β (4k! β TL (4k +! β TLβ By substtutng Eqs. (54 and (55 nto Eq. (39, the frequency equaton s obtaned. Then the th estmated egenvalue correspondng to n s obtaned by Eq. (39, and n s decded by Eq. (34. Hence one can obtan Y ( (0 = f n 0 (λn f n (λn Y(0.. (46 (47 (48 (52 (53 (54 (55 (56

7 320 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( Furthermore, the th mode shape φ n { φ n (X = Y(0 (λ n k X 4k Fg. 2. The clamped beam wth rotatonal and translatonal boundary constrants at the rght end. correspondng to the th egenvalue λ n s obtaned from Eqs. (5 and (56. 4k+3 n } 0 (λn X 4k+ 4k+2. (57 (4k! β TLX (4k + 3! f f n (λn (4k +! + β RLX Case 2: Clamped at X = 0. The deflecton and slope are zero at X = 0. By usng the ADM n ths case, we set Y(0 = 0, Y ( (0 = 0, and Y (2 (0, Y (3 (0 are consdered as the arbtrary constants whch depend on the gven boundary condtons at the rght end X =. The n-term approxmaton n Eq. (27 can be wrtten as λ n φ n (X = X 4k+2 λk Y (2 (0 + Hence, Eq. (33 can be smplfed as X 4k+3 (4k + 3! λk Y (3 (0. (58 f n r2 (λy (2 (0 + f n r3 (λy (3 (0 = 0, r =, 2. (59 For nontrval solutons Y (2 (0, and Y (3 (0, the frequency equaton s gven as f n n 2 (λ f (λ 3 f n n 22 (λ f (λ = The th estmated egenvalue λ n correspondng to n s obtaned by Eq. (60, and n s decded by Eq. (34, by substtutng nto Eq. (59, one can obtan Y (3 (0 = f n r2 (λn f n r3 (λn Y (2 (0, r = or 2. (6 By followng the same procedure as Case, one can obtan the th estmated egenvalue λ n egenfuncton φ n (X = Y (2 (0 φ n (λ n k X 4k+2 f n r2 (λn f n r3 (λn X 4k+3 (4k + 3! (60 from Eq. (60 and the th. (62 Furthermore, let us consder the clamped-free beam, whose rght end x = l s connected by translatonal sprng and rotatonal sprng as shown n Fg. 2. The boundary condtons at x = l are gven n Eqs. (44 and (45. By substtutng Eq. (58 nto Eqs. (6 and (62, and usng Eq. (59, one can obtan f 2 (λ = λ k (4k! + β (4k +!, f 3 (λ = λ k (4k +! + β + λ k f 22 (λ = β TR 2! f 23 (λ = k= λ k (4k! (4k! β TR β TR. (4k + 3!, (63 (64

8 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( λ n By substtutng Eqs. (63 and (64 nto Eq. (58, the closed seres form of frequency equaton s gven as λ k (4k! + β λ k (4k +! (4k +! + β β TR + λ k 2! (4k! β TR λ k (4k! β = 0. (65 TR (4k + 3! k= The th estmated egenvalue λ n correspondng to n s obtaned by Eq. (65, and n s decded by Eq. (34, by substtutng nto Eq. (59, one can obtan Y (3 (0 = n n (λ n (λ n k k (4k! + β (4k+! (4k+! + Furthermore, the th egenfuncton φ n (X = Y (2 (0 (λ n φ n k Y (2 (0. (66 β (4k+2! correspondng to the th egenvalue λ n s obtaned by Eqs. (66 and (60. n (λ n X 4k+2 j (4j! + β (4j+! j=0 X 4k+3 n. (67 (λ n j (4j+! + β (4k + 3! (4j+2! In fact, Eq. (67 s the same as Eq. (57 when β RL, and β TL. j=0 Case 3: Hnged at X = 0. The deflecton and bendng moment are zero at X = 0, that s Y(0 = 0, Y (2 (0 = 0, and Y ( (0, Y (3 (0 are consdered as the arbtrary constants whch depend on the gven boundary condtons at the rght end X =. The n-term approxmaton n Eq. (27 can be wrtten as φ n (X = X 4k+ (4k +! λk Y ( (0 + Hence, Eq. (33 can be smplfed as X 4k+3 (4k + 3! λk Y (3 (0. (68 f n r (λy ( (0 + f n r3 (λy (3 (0 = 0, r =, 2. (69 For nontrval solutons Y ( (0, and Y (3 (0, the frequency equaton s gven as f n n (λ f 3 (λ f n n 2 (λ f (λ = By followng the same procedure as n Case, one can obtan the th estmated egenvalue λ n from Eq. (70 and Y (3 (0 = f n r (λn f n r3 (λn Y ( (0, r = or 2. (7 By substtutng Eq. (7 nto Eq. (68, one can obtan the th egenfuncton φ n φ n n (X = Y ( (0 (λ n k X 4k+ (4k +! f n r (λn X 4k+3 f n. (72 r3 (λn (4k + 3! Smlarly, let us consder the hnged-free beam, whose rght end x = l s connected by translatonal sprng and rotatonal sprng as shown n Fg. 3. The boundary condtons at x = l are gven as Eqs. (44 and (45. By substtutng Eq. (68 nto Eqs. (48 and (49, and usng Eq. (69, one can obtan f (λ = β + λ k (4k! + β (4k! k= f 3 (λ = λ k (4k +! + β (70 (73

9 322 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( Fg. 3. A pnned beam wth rotatonal and translatonal boundary constrants at the rght end. f 2 (λ = β TR + λ k (4k 2! β TR (4k +! f 23 (λ = k= λ k (4k! β TR. (4k + 3! (74 λ n By substtutng Eqs. (73 and (74 nto Eq. (70, the closed seres form of frequency equaton s gven as β + λ k (4k! + β λ k (4k! (4k +! + β k= n β TR + λ k (4k 2! β TR λ k (4k +! (4k! β = 0. (75 TR (4k + 3! k= The th estmated egenvalue λ n correspondng to n s obtaned by Eq. (75, and n s decded by Eq. (34, by substtutng nto Eq. (7, one can obtan β + n (λ n k Y (3 k= (0 = n (λ n k (4k! + β (4k! (4k+! + Furthermore, the th mode shape φ n (X = Y (2 (0 (λ n φ n k Y (2 (0. (76 β (4k+2! correspondng to the th egenvalue λ n s obtaned by Eqs. (76 and (72. β + n (λ n X 4k+2 j (4j! + β (4j! j= X 4k+3 n. (77 (λ n j (4j+! + β (4k + 3! (4j+2! j=0 In fact, Eq. (77 s the same as Eq. (55 when β RL = 0, and β TL. Hence, by usng the method of ADM, we can easly solve the vbraton problem wth varous boundary condtons. The proposed method s very effcent wth the ad of symbolc computaton. 4. Verfcatons and examples In order to demonstrate the feasblty and the effcency of ADM n ths paper, the prevous three cases are dscussed as follows. By usng the results of the prevous cases, one can obtan the natural frequences and mode shapes of the beam wth varous boundary condtons at both ends. The computed results are compared wth the analytcal and numercal results n the lterature. 4.. A clamped-free beam wth elastc sprng restrants at x = l Let us consder the clamped-free beam, whose rght end x = l s connected by translatonal sprng and rotatonal sprng as shown n Fg. 2. Followng the dervaton of the prevous Case 2, frst we assume β =, β TR =. By solvng Eq. (65 and takng real root for λ, Table for n = 8 s obtaned. From ths table, we can obtan any egenvalue one at a tme. The larger the approxmate term s, more egenvalues one can fnd. From Eq. (34 and Table, we have λ 5 λ 4 ε = (78 Thus, the frst egenvalue, dmensonless natural frequency and natural frequency correspondng to n = 5 can be obtaned as

10 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( Table Results of the th egenvalues λ n for n = 8 approxmate terms n λ n λ n 2 λ n 3 λ n 4 λ n 5 λ n (β = k l/ei =, β TR = k TR l 3 /EI =. Fg. 4. The convergence of the frst, second and thrd natural frequences (ω = 4.685, ω 2 = , ω 3 = λ = λ 5 = , (79 ω = ω 5 = λ 5 = (80 λ 5 ω = ω 5 EI = l 2 ρa = EI ρal. (8 4 By substtutng λ 5 nto Eq. (67 and normalzng t by Eq. (36, the frst mode shape functon s gven as φ 5 = X X X X X X X X X X 9. (82 By usng the gven analytcal method 4, the frst dmensonless natural frequency and mode shape functon can be obtaned as ω a = 4.685, φ a = sn(2.4907X snh(2.4907x cos(2.4907X cosh(2.4907x (84 where ω a and φ a are analytcal solutons of the frst natural frequency and mode shape functon, respectvely. From Eqs. (80 and (83, we deduce that ω 5 = ω a = The results obtaned by usng Eq. (82 are compared closely wth the analytcal results obtaned from Eq. (84. Followng the same procedure as shown above, the other natural frequences and mode shapes can be obtaned. In Fgs. 4 and 5, as the approxmate term number n ncreases, the natural frequences (83

11 324 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( Fg. 5. The convergence of the fourth, ffth, and sxth natural frequences (ω 4 = , ω 5 = , ω 6 = Fg. 6. The frst, second and thrd mode shape functons. Table 2 The th dmensonless natural frequency ω of a clamped beam, as shown n Fg. 2, for n = 8 approxmate terms β TR ω 6 ω 9 2 ω 3 ω 4 4 ω 6 5 ω (β = k l/ei = 0. ω ω 6 converge to 4.685, , , , , and very quckly one by one wthout mssng any frequency. Those complete natural frequences whch lead to correspondng mode shapes correctly are shown n Fgs. 6 and 7. Fnally, the dmensonless natural frequences ω ω 6 for the clamped-free beam wth translatonal sprng boundary condton and no rotatonal sprng boundary condton (β = 0 are provded n Table 2. These values are obtaned from Eq. (65 usng the 8 approxmate terms. From ths table, the larger the sprng constant β TR s, the larger the natural frequences

12 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( Fg. 7. The fourth, ffth, and sxth mode shape functons. Table 3 Results of the th egenvalues λ n for n = 8 approxmate terms n λ n λ n 2 λ n 3 λ n 4 λ n 5 λ n (β = k l/ei = 0, β TR = k TR l 3 /EI = 25. ω ω 6 are, and the beam s consdered as a clamped-free beam when β TR β TR. 0, and as a clamped-pnned beam when 4.2. A pnned-free beam wth elastc sprng restrants at x = l We consder the beam as shown n Fg. 3. By usng the results of the prevous Case 3, frst let β = 0, β TR = 25, by solvng Eq. (75 and takng real root for λ, Table 3 for n = 8 s obtaned. From ths table and Eq. (34, let ε = 0.000, we can obtan 6 natural frequences one at a tme. In Fgs. 8 and 9, the dmensonless natural frequences ω ω 6 converge to , , 5.000, , , and very quckly one by one wthout mssng any frequency and the results obtaned by usng Eq. (75 are compared closely wth the analytcal results obtaned from 4. From Eq. (77, the 6 mode shape functons can be obtaned for n = 8, whch are shown n Fgs. 0 and. The frst mode shape functon n the present method and analytcal method 4, respectvely, are gven as φ 6 = X X X X X X X X X X X X 23 (85 φ a = sn( x snh( x. (86 Fnally, the dmensonless natural frequences ω ω 6 for the pnned-free beam wth translatonal sprng boundary condton and no rotatonal sprng boundary condton (β = 0 are provded n Table 4. These values are obtaned from

13 326 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( Fg. 8. The convergence of the frst, second and thrd natural frequences (ω = 4.685, ω 2 = , ω 3 = Fg. 9. The convergence of the fourth, ffth, and sxth natural frequences (ω 4 = , ω 5 = , ω 6 = Fg. 0. The frst, second, and thrd mode shape functons. Eq. (75 usng the 8 approxmate terms. From ths table, the beam s consdered as a pnned-free beam when β TR 0, and as a pnned-pnned beam when β TR.

14 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( Fg.. The fourth, ffth, and sxth mode shape functons. Table 4 The th dmensonless natural frequency ω of a pnned beam, as shown n Fg. 3, for n = 8 approxmate terms β TR ω 6 ω 9 2 ω 3 ω 3 4 ω 6 5 ω (β = k l/ei = 0. Table 5 The frst dmensonless natural frequency ω of a free free beam, as shown n Fg., for n = 8 approxmate terms β TL β RL (β = β RL, β TR = β TL A free free beam wth rotatonal and translatonal restrants at both ends Let us consder the free free beam as shown n Fg.. By followng the dervaton of the prevous Case, one can solve Eqs. (54, (55 and (39 for varous values β, β TR, β RL, and β TL. In ths paper, we assume β = β RL, β TR = β TL and the frst three dmensonless natural frequences ω ω 3 are provded n Tables 5 7 for n = 8. Fgs. 2 4 are the threedmensonal plots of Tables 5 7, respectvely. The beam s consdered as a free free beam when β RL 0, β TL 0, a clamped clamped beam when β RL, β TL. a pnned-pnned beam when β RL 0, β TL. The present results show qute a good agreement wth analytcal 4 and the frst two dmensonless natural frequences ω, ω 2 can be

15 328 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( Table 6 The second dmensonless natural frequency ω 2 of a free free beam, as shown n Fg., for n = 8 approxmate terms β TL β RL (β = β RL, β TR = β TL. Table 7 The thrd dmensonless natural frequency ω 3 of a free free beam, as shown n Fg., for n = 8 approxmate terms β TL β RL (β = β RL, β TR = β TL. Fg. 2. Plot of the frst dmensonless natural frequency (ω for a free free beam wth symmetrc sprng boundary condtons (β = β RL, β TR = β TL, β RL = k RL l/ei, β TL = k TL l 3 /EI. obtaned and are lsted n Tables 8 and 9. From these tables, one can fnd that the calculated results n the study compared wth the results of other lterature are n close agreement. 5. Concluson Ths paper presents an effectve method to solve vbraton problems wth varous elastcally supported condtons. By usng the proposed method, any th natural frequency and mode shape functon can be obtaned one at a tme. The larger the approxmate term n s gvng, more natural frequency can be found at the same tme. The computed results are

16 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( Fg. 3. Plot of the second dmensonless natural frequency (ω 2 for a free free beam wth symmetrc sprng boundary condtons (β = β RL, β TR = β TL, β RL = k RL l/ei, β TL = k TL l 3 /EI. Fg. 4. Plot of the thrd dmensonless natural frequency (ω 3 for a free free beam wth symmetrc sprng boundary condtons (β = β RL, β TR = β TL, β RL = k RL l/ei, β TL = k TL l 3 /EI. Table 8 The frst dmensonless natural frequency ω of a free free beam, as shown n Fg., for n = 8 approxmate terms β TL β RL = 0 2 β RL = β RL = 0 2 β RL = 0 6 Present n = 8 (I (II Present n = 8 (I (II Present n = 8 (I (II Present n = 8 (I (II (β = β RL, β TR = β TL. (I Regster s results 5. (II Km s results 8. compared closely wth the results obtaned by usng other analytcal and numercal methods. Ths study provdes a unfed and systematc procedure whch s seemngly smpler and more straghtforward than the other methods. On the bass of these results, the frequency expressons derved n the present paper can be used n the desgn of beams wth varous supportng condtons.

17 3220 H.-Y. La et al. / Computers and Mathematcs wth Applcatons 56 ( Table 9 The second dmensonless natural frequency ω 2 of a free free beam, as shown n Fg., for n = 8 approxmate terms β TL β RL = 0 2 β RL = β RL = 0 2 β RL = 0 6 Present n = 8 (I (II Present n = 8 (I (II Present n = 8 (I (II Present n = 8 (I (II (β = β RL, β TR = β TL. (I Regster s results 5. (II Km s results 8. References Leonard Merovtch, Fundamentals of Vbratons, Internatonal Edton, McGraw-Hll, Andrew Dmarogonas, Vbraton for Engneers, 2nd ed., Prentce-Hall, Inc., W. Weaver, S.P. Tmoshenko, D.H. Young, Vbraton Problems n Engneerng, 5th ed., John Wley & Sons, Inc., Wllam T. Thomson, Theory of Vbraton wth Applcatons, 2nd ed., A.H. Regster, A note on the vbratons of generally restraned, end-loaded beams, Journal of Sound and Vbraton 72 (4 ( J.T.S. Wang, C.C. Ln, Dynamc analyss of generally supported beams usng Fourer seres, Journal of Sound and Vbraton 96 (3 ( W. Yeh, J.T. Chen, C.M. Chang, Applcatons of dual MRM for determnng the natural frequences and natural modes of an Euler Bernoull beam usng the sngular value decomposton method, Engneerng Analyss wth Boundary Elements 23 ( H.K. Km, M.S. Km, Vbraton of beams wth generally restraned boundary condtons usng Fourer seres, Journal of Sound and Vbraton 245 (5 ( S. Naguleswaran, Transverse vbraton of an unform Euler Bernoull beam under lnearly varyng axal force, Journal of Sound and Vbraton 275 ( G. Adoman, Solvng Fronter Problems of Physcs: The Decomposton Method, Kluwer Academc Publshers, 994. A.M. Wazwaz, S.M. El-Sayed, A new modfcaton of the Adoman decomposton method for lnear and nonlnear operators, Appled Mathematcs and Computaton 22 ( A.M. Wazwaz, A new algorthm for calculatng Adoman polynomals for nonlnear polynomals, Appled Mathematcs and Computaton ( A.M. Wazwaz, The modfed decomposton method and Pade approxmants for solvng the Thomas Ferm equaton, Appled Mathematcs and Computaton 05 ( A.M. Wazwaz, A relable modfcaton of Adoman s decomposton method, Appled Mathematcs and Computaton 02 ( M.M. Hossen, H. Nasabzadeh, On the convergence of Adoman decomposton method, Appled Mathematcs and Computaton 82 (