Determination of the response distributions of cantilever beam under sinusoidal base excitation

Transcription

1 Journal of Physcs: Conference Seres OPEN ACCESS Determnaton of the response dstrbutons of cantlever beam under snusodal base exctaton To cte ths artcle: We Sun et al 13 J. Phys.: Conf. Ser Vew the artcle onlne for updates and enhancements. Related content - Dynamc analyss of hostng vscous dampng strng wth tme-varyng length P Zhang, J H Bao and C M Zhu - In-plane moton of wnd turbne blade under the combned acton of parametrc and forced exctatons L L, Y H L, H W Lv et al. - Dynamc analyss of slender launchng system connected by clamp band jont usng harmonc balance method Z Y Qn, S Z Yan and F L Chu Ths content was downloaded from IP address on 6/1/18 at 18:33

2 The 4th Symposum on the Mechancs of Slender Structures (MoSS13) Journal of Physcs: Conference Seres 448 (13) 11 IOP Publshng do:1.188/ /448/1/11 Determnaton of the response dstrbutons of cantlever beam under snusodal base exctaton We Sun, Yng Lu, Hu L and Dejun Pan School of Mechancal Engneerng & Automaton Northeastern Unversty, Shenyang 11819, Chna E-mal: wesun@me.neu.edu.cn Abstract. As a knd of base exctaton, shakng table s often used to test the dynamc characterstcs of structures. However, the predcton of response to base exctaton hasn t been solved effectvely, whch lmts the further research on the test and analyss method wth respect to base movement. Ths artcle s based on a cantlever beam and focuses on ts response predcton under snusodal base exctaton. By moment and force equlbrum equatons, an analytcal model s bult for ths cantlever beam, and then a method to predct dynamc response at base exctaton s proposed. Fnally, the method s used to solve the vbraton response dstrbutons of the cantlever beam at base exctaton. Correctness of ths method s also proved by comparng the result wth expermental data. 1. Introducton Base exctaton s an mportant way to test vbraton parameters of a structure. It s wdely used because of stable nput energy and beng close to real constraned condton [1-4]. Yoshkazu Arak[1] tested the response of an object placed on a vbraton solator by shaker. Francesco Pellcano[] tested the nonlnear parameters of cylndrcal shell by shaker wth a sne exctaton. However, the analyss of structure under base exctaton progressed slowly compared wth the experment mentoned above. Most scholars generally adopted FEM to analyze the dynamc characterstcs. For example, Lu[5] offered a method that smplfed base exctaton to dstrbuted force by means of equvalent loads and create a FEM model to solve the problem. Allahdadan[6] proposed a topology optmzaton scheme whch was used to reduce the vbraton response under the base exctaton by FEM. Some scholars[7-9] also were devoted to dong theory analyss for the structure under base exctaton. For example, on the bass of neglectng the mass of the beam, a composte structure at the base exctng was analyzed by Feng[7]. Kocaturk[8] analyzed the response of a vscoelastc support Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3. lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, journal ctaton and DOI. Publshed under lcence by IOP Publshng Ltd 1

3 The 4th Symposum on the Mechancs of Slender Structures (MoSS13) Journal of Physcs: Conference Seres 448 (13) 11 IOP Publshng do:1.188/ /448/1/11 cantlever beam under snusodal base exctaton by analytcal method, but gnored the dampng of the beam tself and only consdered the dampng and stffness of the support. The analyss results wll be naccurate f the vbraton characterstcs of the structure at base exctaton are not fully ntroduced. Therefore, a detaled analytcal analyss of the vbraton response of a cantlever beam under the base exctaton was done n ths paper. Frstly, the dynamc dfferental equaton of cantlever beam system was establshed consdered the base exctaton and the system dampng. Secondly, modal superposton method was used to solve the equaton and the response dstrbutons of cantlever beam under snusodal base exctaton were acqured. Fnally, the analyss method was demonstrated usng a study case and the valdty of the proposed method was verfed by the comparson of the analytcal soluton and the expermental results.. Analytcal model of cantlever beam under base exctaton The cantlever beam system under base exctaton s shown n fgure 1 and s extremely close to that of a constant secton Euler-Bernoull beam. Set E s the elastc modulus, I s the moment of nerta, m s the mass of per unt length, A s the cross-secton area and l s the length of the beam. The total dsplacement of one pont on the cantlever beam under the base exctaton s defned as: λ( x, t)= y( t) + ϕ( x, t) (1) where ϕ ( x, t) s the deflecton curve, yt () s the dsplacement of base. If the base s n snusodal movement, t can be defned as yt ()= Ysn( ω t), where ω s the radan frequency and Y s the dsplacement ampltude of the base movement. Fgure 1. Cantlever beam under base exctaton A small segment s taken from the beam and the force analyss s done whch are shown n fgure. In the fgure, V s the shear force on the vertcal secton, M s the bendng moment and c s velocty dampng coeffcent. (a) A small segment beam (b) Force analyss on the cross-secton Fgure. Force analyss

4 The 4th Symposum on the Mechancs of Slender Structures (MoSS13) Journal of Physcs: Conference Seres 448 (13) 11 IOP Publshng do:1.188/ /448/1/11 In ths study, two types of dstrbuted vscous dampng force are taken nto consderaton [1]: one λ s external dampng force whch s proportonal to velocty and represented as Dt ()= c, the t other s nternal dampng force whch s proportonal to deformaton velocty and expressed as ε σ r= cr t, where c r s the dampng coeffcent of deformaton velocty and ε s stran. The nternal dampng force can also produce addtonal bendng moment and shown n fgure 3. Fgure 3. Bendng moment produced from the nternal dampng force It s assumed that the stran on the cross-secton s lnear dstrbuton, and then the addtonal bendng moment produced by the nternal dampng force can be expressed as: ε M r= σ rzda= cr zda t () z Where z s the dstance between one pont and the neutral surface. From the materals mechancs, the equaton between stran and beam deflecton can be obtaned: z ε = z ϕ( x, t) x (3) Substtutng equaton (3) nto equaton (), the addtonal bendng moment becomes: ϕ( x, t) ϕ( x, t) ϕ( x, t) Mr= cr z zda= c d r z A= c ri t x x t x t z z (4) Then the total bendng moment of the small segment s: ϕ( x, t) ϕ( x, t) M = EI c ri x x t (5) From the moment balance of the whole small segment, can yeld: M x t x t M + x M V x+ c m x = x t t 1 λ(, ) λ(, ) d d (d ) (6) Neglectng the hgh order tem and referrng to the equaton (5), the equaton (6) can be smplfed as: M ϕ( xt, ) ϕ( xt, ) V = = EI c ri x x x x t (7) 3

5 The 4th Symposum on the Mechancs of Slender Structures (MoSS13) Journal of Physcs: Conference Seres 448 (13) 11 IOP Publshng do:1.188/ /448/1/11 Further, from the force balance of the whole small segment, can yeld: V λ( x, t) λ( x, t) V + dx V m + c dx= x t t (8) Substtutng the expresson of λ ( x, t) shown n equaton (1), the equaton (8) becomes: dx m + m + c + c dx= x t t t t V y() t ϕ( x,) t y() t ϕ( x,) t Further smplfcaton and organzaton can yeld: (9) V ϕ x t ϕ x t y t y t m c = m + c x t t t t (, ) (, ) ( ) ( ) At last, substtutng the expresson of V shown n equaton (7) nto the equaton (1) can yeld the movement equaton of the small segment under the base exctaton and shown as: 4 5 (,) x t (,) x t (,) x t (,) x t y() t y() t EI ϕ c 4 ri ϕ m ϕ c ϕ = m + c 4 x x t t t t t (1) (11) 3. Soluton of response dstrbutons of cantlever beam under base exctaton It should be noted from the equaton (1), f the deflecton curve ϕ( x, t) s known, the vbraton response of any pont on the cantlever beam can be determned. Here, the modal superposton method was adopted to solve the expresson of the deflecton curve ϕ( x, t). The every order natural frequency of the beam s set as ω, = 1,, 3,, and the relatve regular modal shape functon s Y ( x ), whch satsfes orthogonally condtons as follows: l my ( )d 1 x x = (1) Then the deflecton curve ϕ( x, t) can be expressed by the regular response η () t and the regular modal shape Y ( x) and shown as: ϕ( x, t)= Y( x) η ( t) (13) For a cantlever beam, the expresson of Y ( x ) s: snh kl sn kl Y( x) = D cosh kx cos kx (snh kx sn kx) cosh kl + cos kl Where D s a parameter related to the exctaton level, k s a parameter related to nature frequency. Correspondng to the frst fve order, the values of k are / L, / L, / L, / L, / L respectvely. Substtutng equaton (13) nto equaton (11) can yeld: = 1 (14) 4

6 The 4th Symposum on the Mechancs of Slender Structures (MoSS13) Journal of Physcs: Conference Seres 448 (13) 11 IOP Publshng do:1.188/ /448/1/11 d d EI Y( x) η ( t) + c I Y( x) η ( t) + my( x) η ( t) + cy( x) η ( t) r 4 = 1 dx = 1 dx = 1 = 1 yt () yt () = m + c t t Utlzng the orthogonally condtons, equaton (15) s ntegrated along the longtudnal drecton of the beam and can become: 4 4 l η d Y( ) l d ( ) l x Y x () t EI Y( )d () ( ) ( )d () ( )d 4 4 () x x t c d ri cy x Y x x t my x x F t x η dx η = (16) = l yt () yt () Where F() t = m c Y( )d + x x t t Assume that the two dampng coeffcent cand c r mentoned above are proportonal to mand E, and express as c= αm c = β E. Then equaton (16) can be smplfed as:, r ( t) + ( + ) ( t) + ( t) = F ( t) l yt () yt () Accordngly, F() t = m αm Y( )d + x x t t. Furthermore, the -order modal dampng rato (15) η α βω η ω η (17) ζ can be expressed as: α + βω α βω ζ = = + (18) ω ω Thus the coeffcentα, β can be determned from the modal dampng rato and nature frequency, the soluton formulas are: ( ) ω ω ζ ω ζ ω α = ω ω1 (19) ( ) ζ ω ζ ω β = ω 1 1 ω1 () Where ζ 1, ζ, ω 1, ω can be obtaned from the experment. Combnng equaton (17) and equaton (18) yelds the fnal equaton: ( t) + ( t) + ( t) = F ( t) η ζ ωη ω η (1) The soluton of equaton (1) can be easly obtaned by reference to that of a sngle-degree-of-freedom system and shown as: 5

7 The 4th Symposum on the Mechancs of Slender Structures (MoSS13) Journal of Physcs: Conference Seres 448 (13) 11 IOP Publshng do:1.188/ /448/1/11 Where 1 t ζω t η ζωη ζωτ η() t e + = snω t+ η cos ω t+ F( τ) e sn ω ( t τ)dτ ω ω () ω = ω 1 ζ (3) as: l ( ) η = mϕ x, Ydx (4) l ( ) η = m ϕ x, Ydx (5) Fnally, the total dsplacement of the cantlever beam under snusodal base exctaton s expressed λ( x, t) y t ϕ( x, t)= y t Y( x) η ( t) (6) = () + () + From the equaton (6), the response dstrbutons of cantlever beam under base exctaton can be determned. 4. Study case 4.1. Problem descrpton In ths study, a T-6Al-4V beam whch s n cantlever state s chosen as study object. The shakng table s taken as base exctaton devce and the acceleraton sensor of lght mass s taken as the vbraton test devce. Then the experment system s constructed and shown n fgure 4 and the relatve expermental equpments are lsted n table 1 detaledly. = 1 shakng table acceleraton sensor cantlever beam Fgure 4. Experment system 6

8 The 4th Symposum on the Mechancs of Slender Structures (MoSS13) Journal of Physcs: Conference Seres 448 (13) 11 IOP Publshng do:1.188/ /448/1/11 No. Table 1. The expermental equpments n the base exctaton test. Name 1 LMS SCADAS moble front-end PCB modal hammer 3 KINGDESIGN EM-1F shaker 4 B&K4517 acceleraton sensor of lght mass 5 Hgh-performance notebook computers The materal parameters of the beam have already been tested n the related research. The Young s 3 modulus s 11.3Gpa, Posson's rato s.31, and the densty s 44kg / m. The fnshed dmensons of the beams are 5mm mm 1.5mm and a 5mm secton to be mounted n the clampng devce. 4.. Soluton of the response of Cantlever Beam by analytcal method Durng the analytcal analyss, the base exctaton s set as dsplacement harmonc exctaton wth frequences 39 Hz and 51 Hz respectvely and the level of base exctaton need be set as consstency wth the experment. Because the level of base exctaton s depcted by acceleraton n the real experment, the acceleraton should be converted nto the dsplacement by the formula Y = a/ ω, where a s the acceleraton ampltude. In ths study, the acceleraton exctaton ampltudes are set as.3 g,.8 g,1.6g respectvely. Correspondng to these acceleraton ampltudes, the dsplacement ampltudes whch are used for the analytcal analyss are lsted n table. 7 Table. Dsplacement exctaton ampltude ( 1 m). Exctng frequency Exctng level of shakng table.3g.8g 1.6g The dampng parameter can be calculated by the modal dampng ratos acqured n the subsequent experment, and the obtaned α s.518 and β s The natural frequences of the cantlever beam are calculated frstly and are lsted n table 3.Then equaton (6) s appled to compute the dsplacement response under base exctaton wth dfferent exctng levels and frequences. Here,two locaton on the beam are chosen as response testng ponts whch exst the dstances of.1m and.15m from the clamp area. In order to compare wth the experment, t s requred that the dsplacement responses obtaned by analytcal soluton be transformed nto the acceleraton responses. Fgure 5 s an example about the response of x =.15 m locaton. In ths calculaton, the exctaton ampltude s 1.6g and exctaton frequency s 51Hz. From the fgure 5, the response ampltude can be obtaned and the value s 13.5m/s, Smlar wth the example, the responses of the beam under dfferent exctaton frequency and levels can be computed for the two locatons and the relatve calculaton results are lsted n from table 4 to table 7. 7

9 The 4th Symposum on the Mechancs of Slender Structures (MoSS13) Journal of Physcs: Conference Seres 448 (13) 11 IOP Publshng do:1.188/ /448/1/11 Table 3. The natural frequences by analytcal calculaton and measurement. No Calculated value A Measured value B Dfference A B / B Acceleraton(m/S a(m/s ) ) Tme t(s) (S) Fgure 5. Steady- state response of the cantlever beam Table 4. Response of the beam when exctaton frequency s 39Hz and x=.1m/ m/s. Exctng level.3g.8g 1.6g Calculated value A Measured value B Dfference A B / B(%) Table 5. Response of the beam when exctaton frequency s 51Hz and x=.1m/ m/s. Exctng level.3g.8g 1.6g Calculated value A Measured value B Dfference A B / B(%) Table 6. Response of the beam when exctaton frequency s 39Hz and x=.15m/ m/s. Exctng level.3g.8g 1.6g Calculated value A Measured value B Dfference A B / B(%)

10 The 4th Symposum on the Mechancs of Slender Structures (MoSS13) Journal of Physcs: Conference Seres 448 (13) 11 IOP Publshng do:1.188/ /448/1/11 Table 7. Response of the beam when exctaton frequency s 51Hz and x=.15m/ m/s. Exctng level.3g.8g 1.6g Calculated value A Measured value B Dfference A B / B(%) Response test of the Cantlever Beam. Correspondng to the analytcal analyss, two measurng ponts are arranged n the cantlever beam. Frstly, the natural frequences and modal dampng ratos are acheved by the hammer exctng method. The natural frequences obtaned by experment are also lsted n table 3 and the modal dampng ratos are lsted n table 8. Furthermore, the base exctaton s done at the frequences and levels mentoned n analytcal analyss to get the responses of two desgnated ponts. Table 8. The modal dampng ratos of the beam obtaned by experment Smlar wth analytcal analyss, the response of when the exctaton frequency s 51Hz and the locaton s x =.15 m also s taken as an example of the experment and shown n fgure 6. Acceleraton (m/s ) Tme(S) Frequency(Hz) Fgure 6. 3D waterfall chart of the steady-state response of the cantlever beam From the fgure 6, the response ampltude can be obtaned and the value s 1.9m/s whch s almost consstent wth the analytc calculaton results. The other experment results are also lsted n from table 4 to table The Comparson between Experment and Analyss From the comparson n the table 4 to table 7, t can be known that there s only a lttle dfference between the experment and analyss. In partcular, the response at x =.15m, the maxmum dfference between the experment and analyss s smaller than 7% for the dfferent exctng frequences, whch 9

11 The 4th Symposum on the Mechancs of Slender Structures (MoSS13) Journal of Physcs: Conference Seres 448 (13) 11 IOP Publshng do:1.188/ /448/1/11 can be thought consstent of both. Then, the correctness of the proposed method about the response analyss of cantlever beam under base exctaton s verfed. At the locaton of x =.1 m, the dfference between the analyss and experment s much bgger than the results of x =.15m. The bgger dfference may be come from the naccurate poston of vbraton pckup pont. To verfy ths concluson, the poston of vbraton pckup pont s modfed as x =.96m and analytcal calculaton s done agan. The relatve results are lsted n table.9. Table 9. Response of the beam when exctaton frequency s 39Hz. Exctng level.3g.8g 1.6g Calculated value A(x=.96m) Measured value B Dfference A B / B(%) It has been shown n table 9, changng the value of x n the vcnty of x =.1, the dfference between the calculaton results and experment become smaller. Ths shows that the dfference between experment and analytcal analyss at the locaton of x =.1s ndeed from the naccuracy of dstance. 5. Concluson In ths paper the response calculaton of cantlever beam under the base exctaton was studed. On the bass of consderng the characterstc of base exctaton and the system dampng of cantlever beam, the movement equaton of the cantlever beam was derved by moment and force equlbrum. Furthermore, the modal superposton method was adopted to solve the vbraton response dstrbutons of the cantlever beam at base exctaton. At last, a T-6Al-4V beam whch s n cantlever state was chosen as study object and the relevant experment system was set up. The response dstrbutons of cantlever beam under base exctaton were computed and compared wth the experment results. The two results are consstent, and then the correctness of the proposed method s verfed. The proposed method can promote the understandng of dynamcs mechansm of structure under base exctaton and provde a reference for the dynamc analyss of smlar structure under base exctaton. 6. References [1] Arak Y, Kawabata S, Asa T and Masu T 11 J. Engneerng Structures [] Pellcano F 11 J. Solds and Structures [3] Han Q, Wang J and L Q 11 J. Sound and Vbraton [4] Sarank M, Lenor D and Jézéquel L 1 J. Computers and Structures [5] Lu J T, Du P A and Huang M 9 J. Journal of Unversty of Electroncal Scence and Technology of Chna [6] Allahdadan S, Boroomand B and Barekaten A R 11 J. Fnte Elements n Analyss and Desgn

12 The 4th Symposum on the Mechancs of Slender Structures (MoSS13) Journal of Physcs: Conference Seres 448 (13) 11 IOP Publshng do:1.188/ /448/1/11 [7] Feng A H, Zhu X D and Lan X J 11 J. Internatonal Journal of Non-Lnear Mechancs [8] Kocatur kt 4 J. Sound and Vbraton, [9] Kumar V, Mller J K and Rhoads J F 11 J. Sound and Vbraton [1] Clough R W and Penzen 11 Dynamcs of Structures(Berkeley: Computers & Structures) 11