Beam vibrations: Discrete mass and stiffness models

Transcription

1 Beam vibratins: Discrete mass and stiffness mdels Ana Cláudia Susa Neves Institut Superir Técnic, Universidade de Lisba, Prtugal May, 2015 Abstract In the present wrk the dynamic behavir f several beams with different supprt cnditins, frced r in free vibratin, is studied. Using the Discrete Element Methd (DEM), the expressins gverning the mtin f the blcks in which the beam is discretized are derived. A MATLAB prgram that calculates natural frequencies and mde shapes is develped; the results are then cmpared with the exact slutins, in rder t validate the mdels. The prgram als allws t simulate the evlutin f the dynamic systems in time, yielding displacements, velcities and acceleratins. The effect n the beam behavir due t the intrductin f ne r mre cracks is analyzed; cracks with different sizes and psitins are cnsidered. Three distinctive cases fr the studied mdels are cnsidered: nn-existence f cracks, permanently pen cracks r breathing cracks (cracks that pen and clse depending n the curvature sign). The btained results are shwn with the help f tables and graphics and, when pssible, cmpared with the exact slutins r with numerical r experimental results fund in literature. Key-wrds: vibratin f beams, rigid blcks, discrete stiffness, breathing crack. 1. Intrductin The methd applied in the develpment f the mdels presented in this paper is the Discrete Element Methd (DEM) (Neild et al., 2001). Using this methd ne can represent the beam as a discrete system f blcks (i.e. with a finite number f degrees f freedm) where the mass and the mment f inertia f each blck are lumped in its respective middle pint and where rtatinal and transverse springs, cnnecting adjacent blcks, simulate respectively the bending and shear distrtin. Thus, the beam can be seen as a sequence f rigid blcks linked by pairs f springs, as bserved in Figure 1. Figure 1. Beam mdel using the Discrete Element Methd. Once the mdel is defined, the differential equatins gverning its evlutin in time are established. Fr the particular case f a beam, these equatins invlve relative rtatins and displacements between blcks, as well as their derivates with respect t time. The dynamic behavir in time f several beam mdels is studied and simulated with the help f the DEM implemented in MATLAB. One f the mst imprtant criteria t btain gd results using the DEM is the prcess adpted fr the beam discretizatin. It is expectable that, as ne refines the mesh, the results tend twards the exact slutins. Hwever, the number f blcks shuld nt be indefinitely increased as that wuld lead t an increase f the time expended in the numerical calculatins. There shuld therefre exist a balance between the time and the effrt used in the calculatins and the aimed precisin fr the results. The easiness with which the DEM cnsiders cases where the beams are cracked shuld be nted. Assuming the existence f cracks between rigid blcks, the lcalized lss f stiffness (cincident with the crack psitin) is taken int cnsideratin by changing the stiffness cnstants f the springs cnnecting the blcks; the amunt f reductin f the stiffness will be dependent n the crack depth (Okamura et al., 1969). In Sectin 2 a cantilever beam mdel is studied and the system f rdinary differential equatins that gverns its mtin is btained. The actin f an external frce n the beam and the existence f a crack (which intrduces a lcal stiffness discntinuity 1

2 in the mdel) are als cnsidered. The crack is either assumed t be always pen r t behave like a breathing crack (a crack that pens and clses accrdingly t the sign f the curvature f the crss sectin). The time integratin f the equatins that gvern the mtin f a cracked cantilever beam is perfrmed by the Runge-Kutta methd and the time evlutin f the beam s dynamic respnse is presented and cmpared with experimental results. In Sectin 3, an analgus study is made fr a suspended cracked beam submitted t an scillatry external frce acting perpendicularly t the beam s suspensin plane and free f supprt cnditins in its plane f mtin. The last sectin is dedicated t the cnclusins that result frm the perfrmed numerical investigatins and t enumerate sme aspects that are wrth f future attentin. 2. Cantilever beam 2.1. Dynamics f a hmgeneus cantilever beam In this sectin a cantilever beam f length L with unifrm rectangular transverse sectin b h and mass density ρ (Figure 2) is cnsidered. The beam is decmpsed int N blcks, as illustrated in Figure 3 Figure 4. Discrete elements mdel f the cantilever beam where the stiffness is als discretized in the interfaces between blcks. In the current study the shear distrtin is neglected. S ne nly has the rtatinal springs between blcks. The system f gverning rdinary differential equatins is (Neves, 2015) (J LB A 1 m A T B T L) θ + A K A Tθ = 0, (1) a system f rdinary differential equatins in terms f the N 1 blck rtatins. The matrices (J + 1 LB 4 A 1 m A T B T L) and A K A T are symmetric. System (1) can als be written as θ = (J LB A 1 m A T B T L) A K A Tθ (2) and shall be cmplemented with a set f suitable initial cnditins θ (0) = θ 0, θ (0) = θ 0. (3) One can use system (2) t estimate the exact natural angular frequencies and the exact mde shapes. Assuming a slutin f the expnential type θ (t) = Θ e λt, (4) Figure 2. Hmgeneus cantilever beam with rectangular transverse sectin. which crrespnds t the discrete stiffness versin represented in Figure 4, where each pair f cnsecutive blcks is cnnected by a pair f springs (rtatinal and transverse). The first and last blcks have a length that is half the length f the intermediate blcks, the latter with length l n = L N 1, mass m n = ρbhl and mment f inertia J n = ρbhl n (h 2 + l 2 2 n ) arund an axis perpendicular t the plan f mtin. The first blck is cnsidered t be clamped. the fllwing eigenprblem is btained [λ 2 I + (J LB A 1 m A T B T L) A K A T] Θ = 0, (5) frm which it is pssible t calculate apprximatins fr frequencies and vibratin mde shapes. Cnsidering an external cncentrated varying frce acting n the cantilever beam tip and Rayleigh damping, the system f rdinary differential equatins (1) becmes (Neves, 2015) (J LB A 1 m A T B T L) θ + Cθ + A K A Tθ = F(t)l. (6) Figure 3. Scheme f the cantilever beam decmpsitin in blcks. It is als imprtant t cnsider the pssibility f existence f cracks in the span which will lead t a nnlinear behavir f the beam. The DEM enables t take int accunt the existence f ne r mre cracks with variable depth. The mdificatins are cnsidered in the stiffness matrix as an pen crack 2

3 will prduce a stiffness reductin in the sectin where it is situated. A cantilever beam with a breathing crack in its upper part, with height a and at a distance x c frm the clamped end is mdeled (Figure 5). As the crack is lcated in the upper part f the beam, when the curvature in that sectin is psitive the crack is assumed t be clsed (Figure 6) and there is n discntinuity in flexural stiffness; therwise, if the curvature in that sectin is negative the crack is pen (Figure 7) and the lcal stiffness changes. It is assumed that the crack thickness is negligible when cmpared t its height and that it des nt prpagate. The stiffness f the spring affected by the crack is determined based in (Okamura et al., 1969). If the crss sectin has a psitive curvature sign, the crack is cnsidered t be clsed and the stiffness f the crrespndent spring, like the ther springs, fllws the knwn expressin k c = EI (7) l where l stands fr the length f each blck r, equivalently, the length f influence f the n-th spring. But if the crss sectin has a negative curvature sign, the crack is cnsidered t be pen and the stiffness f the crrespndent spring is given by (Okamura et al., 1969) where 1 k c = l EI + 72 Ebh 2 F ( a h ) (8) F ( a h ) = 1.98 (a h ) ( a h ) ( a h ) ( a h ) ( a h ) ( a h )7 + (9) ( a h ) ( a h ) ( a h )10. Figure 5. Detail f the cantilever beam where the plan crack f height a and width b is lcated. Three mdels f the cantilever beam were cmputatinally implemented: Nnexistence f cracks (the beam has a linear behavir); An always pen crack (the beam als has a linear behavir); A breathing crack (the beam has a nnlinear behaviur); 2.2. Results Table 1 shws the gemetric and material prperties f the cantilever beam. Figure 6. Clsed crack, where θ l + θ r = M l EI. Table 1. Gemetric and material prperties f the cantilever beam. Gemetric prperties Figure 7. Open crack, where θ l + θ r = M ( l EI + 72 Ebh 2 F ( a h )). b [m] 0.06 h [m] 0.22 L [m] 8.0 Material prperties E [Pa] ρ [kg/m 3 ]

4 Natural Frequencies Table 2 shws the exact natural frequencies f the cantilever beam tgether with the nes btained thrugh the DEM (apprximated frequencies), fr an increasing number f blcks N = 24, 48, 96, 192, 384. Figure 8 represents the rati between the first five frequencies calculated with the prgram develped with MATLAB and their hmlgus exact natural frequencies, as a functin f the number f blcks N mentined in Table 1. Table 2. Exact and apprximated natural frequencies f the cantilever beam. blcks were needed, which means the frequency cnvergence is much slwer fr the cantilever beam. This bvius difference in the cnvergence rates is prbably due t the fact that the mde shapes f a cantilever beam are nt the simple circular trignmetric functins as fr the simply supprted beam Mde shapes f the cantilever beam withut cracks Figures 9 and 10 represent the first five exact nrmalized mde shapes and the first five mde shapes btained using the DEM fr a beam discretizatin f 96 blcks. Exact frequencies (p e ) (rads 1 ) Apprximated frequencies (p a ) (rads 1 ) N = 24 N = 48 N = 96 N = 192 N = ,10 17,35 17,72 17,91 18,01 18,05 113,45 108,73 111,01 112,17 112,75 113,04 317,67 304,30 310,46 313,65 315,27 316,08 622,51 595,65 607,28 613,40 616,54 618, ,06 982, , , , , , , , , , ,45 Figure 9. First, secnd and third mde shapes f the cantilever beam (96 blcks). : 1 st exact mde; : 1 st calculated mde; : 2 nd exact mde; : 2 nd calculated mde; : 3 rd exact mde; : 3 rd calculated mde. Figure 8. The first five frequencies f the cantilever beam as a functin f the number f blcks (N) : p 1a /p 1e -----: p 2a /p 2e -----: p 3a /p 3e -----: p 4a /p 4e -----: p 5a /p 5e. Frm the bservatin f Table 1 and Figure 8 ne can cnclude that 384 blcks are required t btain an estimatin f the first three exact frequencies with an errr lwer than 0.5% fr the cantilever beam, while fr the simply supprted beam in (Neves, 2015), t btain the same limitatin f errr, nly 12 Figure 10. Furth and fifth mde shapes f the cantilever beam (96 blcks). : 4 th exact mde; : 4 th calculated mde; : 5 th exact mde; : 5 th calculated mde. 4

5 Even withut an excellent precisin, with a discretizatin f 96 blcks it is already pssible t replicate well enugh the first three mde shapes. Obviusly, by increasing the number f blcks mre precise mde shapes wuld be btained; but the big imprvement achieved fr the first five frequencies and mdes due t the transitin frm a lw number f blcks t 96 blcks is hardly repeatable if the number f blcks is increased frm 96 t mre: the apprximatin imprvement is nt maintained with an indefinite refinement Dynamic evlutin f the scillatin f the cracked cantilever beam In this sectin the numerical results btained using DEM are cmpared with the results fund in the reference (Lutridis et al., 2005) where the nly situatin cnsidered is that f ne breathing crack. With the purpse f studying the nnlinear respnse f the cracked cantilever beam, a Plexiglas mdel with the characteristics fund in Table 3 is cnsidered (Lutridis et al., 2005). Figure 11. Experimental mdel f the cantilever: breathing crack a = 0.3h (40 blcks). Acceleratin at the free end; damping factr ξ = 15%. DEM curve; Experimental values (Lutridis et al., 2005). Table 3. Gemetric and material prperties f the cantilever beam (Lutridis et al., 2005). Gemetric prperties b [m] 0.02 h [m] 0.02 L [m] Material prperties E [Pa] ρ [kg/m 3 ] 1200 A transverse crack is assumed at a distance l f = 70 mm frm the clamped end f the cantilever, in its upper part, with a height a = m (30% f the beam s height). An exciting frce is applied experimentally by a 15 mm diameter vice cil with a 3 g mass. The cil was place in the field f a permanent magnet and was excited by an scillatr B&K type 2010 using a sinus signal. An accelermeter was munted n the free end f the beam t pick up the vibratinal respnse. It is cnsidered that, after the set-up f the experimenting material, the beam is subjected t a harmnic frce with amplitude F máx = N (btained frm (3 g) m/s 2 ). The exciting frequency is f F = 44 Hz, apprximately half f the first natural frequency (f 1 = 91 Hz). Rayleigh damping (Clugh, Penzien, 1993) is cnsidered and the damping cefficients fr the tw first frequencies are ξ 1 = ξ 2 = Figure 12. Experimental mdel f the cantilever: breathing crack a = 0.3h (80 blcks). Acceleratin at the free end; damping factr ξ = 15%. DEM curve; Experimental values (Lutridis et al., 2005). Figures 11 and 12 represent the acceleratin f the free end f the cantilever beam calculated with the DEM, cnsidering the existence f a breathing crack and a mesh f 40 and 80 blcks, respectively; the results fund in the article (Lutridis et al., 2005) are als represented in the same figures. Frm the bservatin f Figures 11 and 12 ne can see that the refinement frm 40 t 80 blcks des nt imprve significantly the results. The apprximatin t the experimental results is quite gd: the irregularity f the acceleratin near zer, due t the change f the crack state, is well reprduced. 5

6 3. Suspended beam 3.1. Dynamics f a hmgeneus suspended beam The dynamics f a suspended beam as shwn in Figure 13 is nw studied. The beam is suspended by its ends and is actuated by a frce in the directin perpendicular t the gravity acceleratin, transversely t the beam s axis. Under the actin f the frce, the beam mtin is free f supprt cnditins. A breathing crack is riented transversely and it pens and clses accrding t the curvature sign in the crack sectin. As shwn in Figure 14, the beam is discretized in cntiguus blcks cnnected by rtatinal springs (shear distrtin is again neglected) and the frce F(t) acts in the mass center f the k-th blck. [ the fllwing eigenprblem is btained λ 2 (J LB A 1 m A T B T L) + D T K D λ 2 ( 1 2 LB A 1 m A T a 1 ) 0 = { } 0 λ 2 ( 2J 1 l 1 a 1 T ) 2K 1 l 1 a 2 T λ 2 m 1 ] Θ { } = Y 1 frm which is pssible t calculate apprximatins fr frequencies and vibratin mde shapes Results (12) Table 4 shws the gemetric and material prperties f the suspended beam, which cincide with thse in reference (Saavedra, Cuitiñ, 2001). Table 4. Gemetric and material prperties f the suspended beam. Gemetric prperties b [m] h [m] L [m] 0.9 Material prperties Figure 13. Mdel f the cracked suspended beam acted by a frce transverse t the suspensin plan. E [Pa] ρ [kg/m 3 ] Natural frequencies Table 5 shws the exact natural frequencies f the suspended beam tgether with the nes btained thrugh the DEM (apprximated frequencies), fr an increasing number f blcks N = 8, 12, 24, 48, 96. Table 5. Exact and apprximated natural frequencies f the suspended beam. Figure 14. Discrete elements mdel f the suspended beam acted by a frce transverse t the suspensin plan. The system f rdinary differential equatins that gverns the mtin f the mdel is (Neves, 2015) J + 1 LB 4 A 1 m A T B T 1 L LB 2 A 1 m A T a 1 θ [ 2J ] { } = 1 a l 1 T m 1 1 y 1 D T 1 K D 0 θ F(t)LB = [ 2K 1 a l 2 T 0 ] { 2 A 1 e k 1 } + { }. 1 y 1 0 (10) Assuming F(t) = 0 and a slutin f the expnential type θ (t) = Θ e λt and y 1 (t) = Y 1 e λt (11) Exact frequencies (p e ) (rads 1 ) Apprximated frequencies (p a ) (rads 1 ) N = 8 N = 12 N = 24 N = 48 N = ,07 785,27 785,34 785,36 785,36 785, , , , , , , , , , , , , , , , , , , , , , , , ,10 Figure 15 represents the rati between the first five frequencies calculated with the prgram develped with MATLAB and the first five hmlgus exact natural frequencies, as a functin f the number f blcks N. 6

7 Frm the bservatin f Table 5 and Figure 15 ne can cnclude that t btain an errr lwer than 0.5% in the estimatin f the first three exact frequencies f the suspended beam 12 blcks can be used. T btain an errr lwer than 1% in the estimatin f the first five frequencies at least 24 blcks are needed. Figure 17. Furth and fifth mde shapes f the suspended beam (31blcks). : 4 th exact mde; : 4 th calculated mde; : 5 th exact mde; : 5 th calculated mde. Figure 15. The first five frequencies f the suspended beam as a functin f the number f blcks (N) : p 1a /p 1e -----: p 2a /p 2e -----: p 3a /p 3e -----: p 4a /p 4e -----: p 5a /p 5e Mdes shapes f the suspended beam Figures 16 and 17 represent the first five exact nrmalized mde shapes and the first five mde shapes btained using the DEM, als nrmalized, fr a beam discretizatin f 31 blcks. Frm the bservatin f Figure 16 ne can see that 31 blcks reprduce well the first three mde shapes; the same cnclusin is taken frm Figure 17 fr the furth and fifth mde shapes. This accuracy is in agreement with the fast cnvergence f the frequencies fr a lw number f blcks, as shwn in Table Dynamic evlutin f the cracked suspended beam In this sectin the numerical results btained using the DEM are cmpared with the results f the reference (Saavedra, Cuitiñ, 2001). Table 4 shws the gemetric and material prperties f the simulated suspended beam. A transverse crack is cnsidered at a distance l f = 585 mm frm the left end f the beam and with a height a = m (40% f the beam s height). The exciting frce, applied at a distance f 270 mm frm the left end, has an amplitude F máx = 10N and a frequency f F = 62 Hz, half f the first natural frequency (f 1 = 124 Hz). Rayleigh damping is cnsidered and the damping cefficients fr the tw first frequencies may have tw different values: ξ 1 = ξ 2 = 0.1 and ξ 1 = ξ 2 = 0.2. Figure 16. First, secnd and third mde shapes f the suspended beam (31 blcks). : 1 st exact mde; : 1 st calculated mde; : 2 nd exact mde; : 2 nd calculated mde; : 3 rd exact mde; : 3 rd calculated mde. The decisin t discretize the beam in 31 blcks is justified by the fact that it is then pssible t lcate the crack, the frce and the accelermeter exactly at the psitins (see Figure 18) mentined in (Saavedra, Cuitiñ, 2001). 7

8 Figure 18. Crack, frce and respnse evaluatin psitins. Figures 19 and 20 represent the acceleratin f sectin A (at a distance f 810 mm frm the left end f the beam) calculated with the DEM, cnsidering the existence f a breathing crack, a mesh f 31 blcks and a damping factr ξ = 0.1 r ξ = 0.2 respectively; the results fund in the article (Saavedra, Cuitiñ, 2001) are als represented in the same Figures. Figure 19. Acceleratin at sectin A f the suspended beam (ξ = 1%): breathing crack a = 0.4h (31 blcks). DEM curve; Experimental values (Saavedra, Cuitiñ, 2001). One verifies that the apprximatins btained with the DEM, especially cnsidering 2% f damping, are quiet acceptable and are, in general, better than the apprximatins btained in the mdels described in the articles (Saavedra, Cuitiñ, 2001) and (Sinha & Friswell, 2002).A shift between the respnse f the mdel and the experimental respnse is bserved. Hwever, the steady state respnse frm the mdel has a frequency equal t the frequency f the exciting frce. The shift between the respnses can be due t the fact that in the experimental situatin gravity is present which may prduce variatins in the crack state. 4. Cnclusins 4.1. Cntributins The crack detectin in structures is a very imprtant issue and mre practical and less nerus detectin methds are cntinuusly investigated; this tpic is cmmn t many engineering branches such as Civil, Mechanical and Aernautical. This wrk intends t cntribute t the characterizatin f the vibratins that the existence f cracks f different characteristics causes in the free r frced dynamic respnse f sme structures. Thus, frm the data cllected frm accelermeters strategically placed in the beam the existence f cracks can be detected and their depth and lcatin may be assessed. In the curse f this wrk a prgram in MATLAB envirnment that allws the applicatin f the Discrete Elements Methd (DEM) t the analysis f the dynamic behavir f sme structures is develped. Taking as a starting pint the mdel f a simply supprted beam, fr which expressins were btained in (Neild et al., 2001), the expressins fr the remaining mdels were derived. The DEM is a valid and simple methd t simulate the behaviur f cracked beams. Based n the presented tables and figures and als n the results presented in (Neves, 2015), the fllwing cnclusins culd be inferred: Figure 20. Acceleratin at sectin A f the suspended beam (ξ = 2%): breathing crack a = 0.4h (31 blcks). DEM curve; Experimental values (Saavedra, Cuitiñ, 2001). The DEM leads t gd apprximatins f the natural frequencies and mde shapes; The apprximatins are better fr the lwer frequencies and mde shapes, fr a given number f blcks used in the discretizatin, thugh in sme cases the cnvergence is nt mntnus; The DEM adapts well in the situatins where cracks exist and the prgramatin in MATLAB is easily mdified; The results btained with DEM give gd apprximatins f the experimental results; 8

9 The results btained with DEM in situatins where experimental results are nt available cme ut as expected, namely: The stiffness f a cracked beam is lwer than the stiffness f an uncracked beam and that cnditin is reflected in the reductin f the natural frequencies f the cracked beam and in its free dynamic respnse; The frequencies are mre sensitive t the existence f cracks when these are lcated in regins f higher curvature f the crrespnding vibratin mdes; The natural frequencies f a beam where there is a breathing crack have intermediate values between thse f an uncracked beam and a beam with an always pen crack; The existence f breathing cracks lcated in regins f larger curvature yields mre irregular respnses when the crack state changes (when the crack ges frm being pen t clsed r vice-versa); Fr a cracked beam, the vibratin amplitudes in a frced vibratin increase cmpared t the case f an uncracked beam, accrding t the verified reductin f stiffness Future develpments One imprtant aspect that wuld be interesting t develp in future wrks is the cnsideratin f the shear distrtin (which was ignred in the numerical simulatins f the presented mdels). Taking it int accunt, the derivatin f the expressins that rule the blcks mtins becmes mre cmplex but the achieved results will be mre accurate, especially when beams have small slenderness ratis L/h. cracked beams using instantaneus frequency. NDT&E internatinal,, Neild, S., McFadden, P., Williams, M. (2001). A discrete mdel f a vibrating beam using a time-stepping apprach. Jurnal f Sund and Vibratin, 239(1), Neves, C. (2015). Vibrações de vigas: Mdels de massa e rigidez discretas. MSc Thesis in Civil Engineering, Institut Superir Técnic, Lisba. Okamura, H., Liu, H., Chu, C.-S., Liebwitz, H. (1969). A cracked clumn under cmpressin. Engineering Fracture Mechnics, 1, Orhan, S. (2007). Analysis f free and frced vibratin f a cracked cantilever beam. NDT&E Internatinal, Ra, S. (2004). Mechanical Vibratins. Pearsn Educatin Inc., Prentice Hall. Saavedra, P., Cuitiñ, L. (2001). Crack detectin and vibratin behaviur f cracked beams. Cmputers and Structures, 79, Saeedi, K., Bhat, R. (2011). Clustered natural frequencies in multi-span. Shck and Vibratin, 18, Sinha, J., Friswell, M. (2002). Simulatin f the dynamic respnse f a cracked beam. Cmputers and Structures, 80, It wuld als be interesting t study a larger number f pssibilities fr the crack lcalizatin and quantity f cracks, as a way t deepen the knwledge abut hw the crack lcalizatin mdifies the dynamic behavir f a beam. The analysis culd als be extended t mre cmplex structures, such as cntinuus beams with tw r mre spans r even frames. References Clugh, R., Penzien, J. (1993). Dynamics f Structures. McGraw-Hill Internatinal Editin (Civil Engineering Series). Lutridis, S., Duka, E., Hadjilentiadis, L. (2005). Frced vibratin behaviur and crack detectin f 9