Beam vibrations: Discrete mass and stiffness models
|
|
- Tiffany Harvey
- 5 years ago
- Views:
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
Exercise 3 Identification of parameters of the vibrating system with one degree of freedom
Exercise 3 Identificatin f parameters f the vibrating system with ne degree f freedm Gal T determine the value f the damping cefficient, the stiffness cefficient and the amplitude f the vibratin excitatin
More information3D FE Modeling Simulation of Cold Rotary Forging with Double Symmetry Rolls X. H. Han 1, a, L. Hua 1, b, Y. M. Zhao 1, c
Materials Science Frum Online: 2009-08-31 ISSN: 1662-9752, Vls. 628-629, pp 623-628 di:10.4028/www.scientific.net/msf.628-629.623 2009 Trans Tech Publicatins, Switzerland 3D FE Mdeling Simulatin f Cld
More informationDetermining the Accuracy of Modal Parameter Estimation Methods
Determining the Accuracy f Mdal Parameter Estimatin Methds by Michael Lee Ph.D., P.E. & Mar Richardsn Ph.D. Structural Measurement Systems Milpitas, CA Abstract The mst cmmn type f mdal testing system
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed
More informationModeling the Nonlinear Rheological Behavior of Materials with a Hyper-Exponential Type Function
www.ccsenet.rg/mer Mechanical Engineering Research Vl. 1, N. 1; December 011 Mdeling the Nnlinear Rhelgical Behavir f Materials with a Hyper-Expnential Type Functin Marc Delphin Mnsia Département de Physique,
More informationSolution to HW14 Fall-2002
Slutin t HW14 Fall-2002 CJ5 10.CQ.003. REASONING AND SOLUTION Figures 10.11 and 10.14 shw the velcity and the acceleratin, respectively, the shadw a ball that underges unirm circular mtin. The shadw underges
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed
More informationSurface and Contact Stress
Surface and Cntact Stress The cncept f the frce is fundamental t mechanics and many imprtant prblems can be cast in terms f frces nly, fr example the prblems cnsidered in Chapter. Hwever, mre sphisticated
More informationDerailment Safety Evaluation by Analytic Equations
PAPER Derailment Safety Evaluatin by Analytic Equatins Hideyuki TAKAI General Manager, Track Technlgy Div. Hirnari MURAMATSU Assistant Senir Researcher, Track Gemetry & Maintenance, Track Technlgy Div.
More informationFree Vibrations of Catenary Risers with Internal Fluid
Prceeding Series f the Brazilian Sciety f Applied and Cmputatinal Mathematics, Vl. 4, N. 1, 216. Trabalh apresentad n DINCON, Natal - RN, 215. Prceeding Series f the Brazilian Sciety f Cmputatinal and
More informationFebruary 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA
February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA Mental Experiment regarding 1D randm walk Cnsider a cntainer f gas in thermal
More informationFall 2013 Physics 172 Recitation 3 Momentum and Springs
Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.
More informationLecture 7: Damped and Driven Oscillations
Lecture 7: Damped and Driven Oscillatins Last time, we fund fr underdamped scillatrs: βt x t = e A1 + A csω1t + i A1 A sinω1t A 1 and A are cmplex numbers, but ur answer must be real Implies that A 1 and
More information39th International Physics Olympiad - Hanoi - Vietnam Theoretical Problem No. 1 /Solution. Solution
39th Internatinal Physics Olympiad - Hani - Vietnam - 8 Theretical Prblem N. /Slutin Slutin. The structure f the mrtar.. Calculating the distance TG The vlume f water in the bucket is V = = 3 3 3 cm m.
More informationPhys101 Final Code: 1 Term: 132 Wednesday, May 21, 2014 Page: 1
Phys101 Final Cde: 1 Term: 1 Wednesday, May 1, 014 Page: 1 Q1. A car accelerates at.0 m/s alng a straight rad. It passes tw marks that are 0 m apart at times t = 4.0 s and t = 5.0 s. Find the car s velcity
More information20 Faraday s Law and Maxwell s Extension to Ampere s Law
Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law 20 Faraday s Law and Maxwell s Extensin t Ampere s Law Cnsider the case f a charged particle that is ming in the icinity f a ming bar magnet
More informationSections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.
Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage
More informationLecture 23: Lattice Models of Materials; Modeling Polymer Solutions
Lecture 23: 12.05.05 Lattice Mdels f Materials; Mdeling Plymer Slutins Tday: LAST TIME...2 The Bltzmann Factr and Partitin Functin: systems at cnstant temperature...2 A better mdel: The Debye slid...3
More informationBuilding to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.
Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define
More informationModule 4: General Formulation of Electric Circuit Theory
Mdule 4: General Frmulatin f Electric Circuit Thery 4. General Frmulatin f Electric Circuit Thery All electrmagnetic phenmena are described at a fundamental level by Maxwell's equatins and the assciated
More informationBootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >
Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);
More informationLead/Lag Compensator Frequency Domain Properties and Design Methods
Lectures 6 and 7 Lead/Lag Cmpensatr Frequency Dmain Prperties and Design Methds Definitin Cnsider the cmpensatr (ie cntrller Fr, it is called a lag cmpensatr s K Fr s, it is called a lead cmpensatr Ntatin
More informationKinematic transformation of mechanical behavior Neville Hogan
inematic transfrmatin f mechanical behavir Neville Hgan Generalized crdinates are fundamental If we assume that a linkage may accurately be described as a cllectin f linked rigid bdies, their generalized
More informationDispersion Ref Feynman Vol-I, Ch-31
Dispersin Ref Feynman Vl-I, Ch-31 n () = 1 + q N q /m 2 2 2 0 i ( b/m) We have learned that the index f refractin is nt just a simple number, but a quantity that varies with the frequency f the light.
More informationMethods for Determination of Mean Speckle Size in Simulated Speckle Pattern
0.478/msr-04-004 MEASUREMENT SCENCE REVEW, Vlume 4, N. 3, 04 Methds fr Determinatin f Mean Speckle Size in Simulated Speckle Pattern. Hamarvá, P. Šmíd, P. Hrváth, M. Hrabvský nstitute f Physics f the Academy
More informationCourse Stabilty of Structures
Curse Stabilty f Structures Lecture ntes 2015.03.06 abut 3D beams, sme preliminaries (1:st rder thery) Trsin, 1:st rder thery 3D beams 2:nd rder thery Trsinal buckling Cupled buckling mdes, eamples Numerical
More informationPressure And Entropy Variations Across The Weak Shock Wave Due To Viscosity Effects
Pressure And Entrpy Variatins Acrss The Weak Shck Wave Due T Viscsity Effects OSTAFA A. A. AHOUD Department f athematics Faculty f Science Benha University 13518 Benha EGYPT Abstract:-The nnlinear differential
More informationRelationship Between Amplifier Settling Time and Pole-Zero Placements for Second-Order Systems *
Relatinship Between Amplifier Settling Time and Ple-Zer Placements fr Secnd-Order Systems * Mark E. Schlarmann and Randall L. Geiger Iwa State University Electrical and Cmputer Engineering Department Ames,
More informationMath Foundations 20 Work Plan
Math Fundatins 20 Wrk Plan Units / Tpics 20.8 Demnstrate understanding f systems f linear inequalities in tw variables. Time Frame December 1-3 weeks 6-10 Majr Learning Indicatrs Identify situatins relevant
More informationROUNDING ERRORS IN BEAM-TRACKING CALCULATIONS
Particle Acceleratrs, 1986, Vl. 19, pp. 99-105 0031-2460/86/1904-0099/$15.00/0 1986 Grdn and Breach, Science Publishers, S.A. Printed in the United States f America ROUNDING ERRORS IN BEAM-TRACKING CALCULATIONS
More informationCESAR Science Case The differential rotation of the Sun and its Chromosphere. Introduction. Material that is necessary during the laboratory
Teacher s guide CESAR Science Case The differential rtatin f the Sun and its Chrmsphere Material that is necessary during the labratry CESAR Astrnmical wrd list CESAR Bklet CESAR Frmula sheet CESAR Student
More informationDYNAMIC MODELLING OF N-CARDAN TRANSMISSIONS WITH SHAFTS IN SPATIAL CONFIGURATION. Part II. THE ALGORITHM OF DYNAMIC MODELLING
Fascicle f Management and Technlgical Engineering, Vlume VI (XVI), 7 DYNAMIC MODELLING OF N-CARDAN TRANSMISSIONS WITH SHAFTS IN SPATIAL CONFIGURATION. Part II. THE ALGORITHM OF DYNAMIC MODELLING Cdrua
More informationDead-beat controller design
J. Hetthéssy, A. Barta, R. Bars: Dead beat cntrller design Nvember, 4 Dead-beat cntrller design In sampled data cntrl systems the cntrller is realised by an intelligent device, typically by a PLC (Prgrammable
More informationAnalysis of Curved Bridges Crossing Fault Rupture Zones
Analysis f Curved Bridges Crssing Fault Rupture Znes R.K.Gel, B.Qu & O.Rdriguez Dept. f Civil and Envirnmental Engineering, Califrnia Plytechnic State University, San Luis Obisp, CA 93407, USA SUMMARY:
More informationElectric Current and Resistance
Electric Current and Resistance Electric Current Electric current is the rate f flw f charge thrugh sme regin f space The SI unit f current is the ampere (A) 1 A = 1 C / s The symbl fr electric current
More informationSupporting information
Electrnic Supplementary Material (ESI) fr Physical Chemistry Chemical Physics This jurnal is The wner Scieties 01 ydrgen perxide electrchemistry n platinum: twards understanding the xygen reductin reactin
More informationNGSS High School Physics Domain Model
NGSS High Schl Physics Dmain Mdel Mtin and Stability: Frces and Interactins HS-PS2-1: Students will be able t analyze data t supprt the claim that Newtn s secnd law f mtin describes the mathematical relatinship
More information(1.1) V which contains charges. If a charge density ρ, is defined as the limit of the ratio of the charge contained. 0, and if a force density f
1.0 Review f Electrmagnetic Field Thery Selected aspects f electrmagnetic thery are reviewed in this sectin, with emphasis n cncepts which are useful in understanding magnet design. Detailed, rigrus treatments
More informationFIELD QUALITY IN ACCELERATOR MAGNETS
FIELD QUALITY IN ACCELERATOR MAGNETS S. Russenschuck CERN, 1211 Geneva 23, Switzerland Abstract The field quality in the supercnducting magnets is expressed in terms f the cefficients f the Furier series
More informationKinetic Model Completeness
5.68J/10.652J Spring 2003 Lecture Ntes Tuesday April 15, 2003 Kinetic Mdel Cmpleteness We say a chemical kinetic mdel is cmplete fr a particular reactin cnditin when it cntains all the species and reactins
More informationEXPERIMENTAL STUDY ON DISCHARGE COEFFICIENT OF OUTFLOW OPENING FOR PREDICTING CROSS-VENTILATION FLOW RATE
EXPERIMENTAL STUD ON DISCHARGE COEFFICIENT OF OUTFLOW OPENING FOR PREDICTING CROSS-VENTILATION FLOW RATE Tmnbu Gt, Masaaki Ohba, Takashi Kurabuchi 2, Tmyuki End 3, shihik Akamine 4, and Tshihir Nnaka 2
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 11: Mdeling with systems f ODEs In Petre Department f IT, Ab Akademi http://www.users.ab.fi/ipetre/cmpmd/ Mdeling with differential equatins Mdeling strategy Fcus
More informationCHAPTER 8b Static Equilibrium Units
CHAPTER 8b Static Equilibrium Units The Cnditins fr Equilibrium Slving Statics Prblems Stability and Balance Elasticity; Stress and Strain The Cnditins fr Equilibrium An bject with frces acting n it, but
More informationEngineering Approach to Modelling Metal THz Structures
Terahertz Science and Technlgy, ISSN 1941-7411 Vl.4, N.1, March 11 Invited Paper ngineering Apprach t Mdelling Metal THz Structures Stepan Lucyszyn * and Yun Zhu Department f, Imperial Cllege Lndn, xhibitin
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More information(2) Even if such a value of k was possible, the neutrons multiply
CHANGE OF REACTOR Nuclear Thery - Curse 227 POWER WTH REACTVTY CHANGE n this lessn, we will cnsider hw neutrn density, neutrn flux and reactr pwer change when the multiplicatin factr, k, r the reactivity,
More informationLecture 5: Equilibrium and Oscillations
Lecture 5: Equilibrium and Oscillatins Energy and Mtin Last time, we fund that fr a system with energy cnserved, v = ± E U m ( ) ( ) One result we see immediately is that there is n slutin fr velcity if
More informationPhysics 2B Chapter 23 Notes - Faraday s Law & Inductors Spring 2018
Michael Faraday lived in the Lndn area frm 1791 t 1867. He was 29 years ld when Hand Oersted, in 1820, accidentally discvered that electric current creates magnetic field. Thrugh empirical bservatin and
More informationQuantum Harmonic Oscillator, a computational approach
IOSR Jurnal f Applied Physics (IOSR-JAP) e-issn: 78-4861.Vlume 7, Issue 5 Ver. II (Sep. - Oct. 015), PP 33-38 www.isrjurnals Quantum Harmnic Oscillatr, a cmputatinal apprach Sarmistha Sahu, Maharani Lakshmi
More informationA mathematical model for complete stress-strain curve prediction of permeable concrete
A mathematical mdel fr cmplete stress-strain curve predictin f permeable cncrete M. K. Hussin Y. Zhuge F. Bullen W. P. Lkuge Faculty f Engineering and Surveying, University f Suthern Queensland, Twmba,
More informationLyapunov Stability Stability of Equilibrium Points
Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),
More informationLecture 17: Free Energy of Multi-phase Solutions at Equilibrium
Lecture 17: 11.07.05 Free Energy f Multi-phase Slutins at Equilibrium Tday: LAST TIME...2 FREE ENERGY DIAGRAMS OF MULTI-PHASE SOLUTIONS 1...3 The cmmn tangent cnstructin and the lever rule...3 Practical
More informationReview of the Roll-Damping, Measurements in the T-38 Wind Tunnel
Internatinal Jurnal f Scientific and Research Publicatins, Vlume 3, Issue 12, December 2013 1 Review f the Rll-Damping, Measurements in the T-38 Wind Tunnel Dušan Regdić *, Marija Samardžić **, Gjk Grubr
More informationQ1. A) 48 m/s B) 17 m/s C) 22 m/s D) 66 m/s E) 53 m/s. Ans: = 84.0 Q2.
Phys10 Final-133 Zer Versin Crdinatr: A.A.Naqvi Wednesday, August 13, 014 Page: 1 Q1. A string, f length 0.75 m and fixed at bth ends, is vibrating in its fundamental mde. The maximum transverse speed
More information205MPa and a modulus of elasticity E 207 GPa. The critical load 75kN. Gravity is vertically downward and the weight of link 3 is W3
ME 5 - Machine Design I Fall Semester 06 Name f Student: Lab Sectin Number: Final Exam. Open bk clsed ntes. Friday, December 6th, 06 ur name lab sectin number must be included in the spaces prvided at
More informationCHAPTER 4 DIAGNOSTICS FOR INFLUENTIAL OBSERVATIONS
CHAPTER 4 DIAGNOSTICS FOR INFLUENTIAL OBSERVATIONS 1 Influential bservatins are bservatins whse presence in the data can have a distrting effect n the parameter estimates and pssibly the entire analysis,
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 11 (3/11/04) Neutron Diffusion
.54 Neutrn Interactins and Applicatins (Spring 004) Chapter (3//04) Neutrn Diffusin References -- J. R. Lamarsh, Intrductin t Nuclear Reactr Thery (Addisn-Wesley, Reading, 966) T study neutrn diffusin
More informationENSC Discrete Time Systems. Project Outline. Semester
ENSC 49 - iscrete Time Systems Prject Outline Semester 006-1. Objectives The gal f the prject is t design a channel fading simulatr. Upn successful cmpletin f the prject, yu will reinfrce yur understanding
More informationFive Whys How To Do It Better
Five Whys Definitin. As explained in the previus article, we define rt cause as simply the uncvering f hw the current prblem came int being. Fr a simple causal chain, it is the entire chain. Fr a cmplex
More informationSpring Pendulum with Dry and Viscous Damping
Spring Pendulum with Dry and Viscus Damping Eugene I Butikv Saint Petersburg State University, Saint Petersburg, Russia Abstract Free and frced scillatins f a trsin spring pendulum damped by viscus and
More informationSuggested reading: Lackmann (2011), Sections
QG Thery and Applicatins: Apprximatins and Equatins Atms 5110 Synptic Dynamic Meterlgy I Instructr: Jim Steenburgh jim.steenburgh@utah.edu 801-581-8727 Suite 480/Office 488 INSCC Suggested reading: Lackmann
More informationMODULE FOUR. This module addresses functions. SC Academic Elementary Algebra Standards:
MODULE FOUR This mdule addresses functins SC Academic Standards: EA-3.1 Classify a relatinship as being either a functin r nt a functin when given data as a table, set f rdered pairs, r graph. EA-3.2 Use
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 2: Mdeling change. In Petre Department f IT, Åb Akademi http://users.ab.fi/ipetre/cmpmd/ Cntent f the lecture Basic paradigm f mdeling change Examples Linear dynamical
More informationBASD HIGH SCHOOL FORMAL LAB REPORT
BASD HIGH SCHOOL FORMAL LAB REPORT *WARNING: After an explanatin f what t include in each sectin, there is an example f hw the sectin might lk using a sample experiment Keep in mind, the sample lab used
More informationKinetics of Particles. Chapter 3
Kinetics f Particles Chapter 3 1 Kinetics f Particles It is the study f the relatins existing between the frces acting n bdy, the mass f the bdy, and the mtin f the bdy. It is the study f the relatin between
More information, which yields. where z1. and z2
The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin
More informationELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA. December 4, PLP No. 322
ELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA by J. C. SPROTT December 4, 1969 PLP N. 3 These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated. They are fr private
More informationLab 11 LRC Circuits, Damped Forced Harmonic Motion
Physics 6 ab ab 11 ircuits, Damped Frced Harmnic Mtin What Yu Need T Knw: The Physics OK this is basically a recap f what yu ve dne s far with circuits and circuits. Nw we get t put everything tgether
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/ Cntent Stichimetric matrix Calculating the mass cnservatin relatins
More informationSOLUTION OF THREE-CONSTRAINT ENTROPY-BASED VELOCITY DISTRIBUTION
SOLUTION OF THREECONSTRAINT ENTROPYBASED VELOCITY DISTRIBUTION By D. E. Barbe,' J. F. Cruise, 2 and V. P. Singh, 3 Members, ASCE ABSTRACT: A twdimensinal velcity prfile based upn the principle f maximum
More information37 Maxwell s Equations
37 Maxwell s quatins In this chapter, the plan is t summarize much f what we knw abut electricity and magnetism in a manner similar t the way in which James Clerk Maxwell summarized what was knwn abut
More informationSimple Models of Foundation-Soil Interactions
IACSIT Internatinal Jurnal f Engineering and Technlgy, Vl. 5, N. 5, Octber 013 Simple Mdels f Fundatin-Sil Interactins Shi-Shuenn Chen and Jun-Yang Shi Abstract This study aims t develp a series f simplified
More informationAP Physics Kinematic Wrap Up
AP Physics Kinematic Wrap Up S what d yu need t knw abut this mtin in tw-dimensin stuff t get a gd scre n the ld AP Physics Test? First ff, here are the equatins that yu ll have t wrk with: v v at x x
More informationGuaranteeing Reliability with Vibration Simulation and Testing. Dr. Nathan Blattau
Guaranteeing Reliability with Vibratin Simulatin and Testing Dr. Nathan Blattau . Nathan Blattau, Ph.D. - Senir Vice President Has been invlved in the packaging and reliability f electrnic equipment fr
More informationYeu-Sheng Paul Shiue, Ph.D 薛宇盛 Professor and Chair Mechanical Engineering Department Christian Brothers University 650 East Parkway South Memphis, TN
Yeu-Sheng Paul Shiue, Ph.D 薛宇盛 Prfessr and Chair Mechanical Engineering Department Christian Brthers University 650 East Parkway Suth Memphis, TN 38104 Office: (901) 321-3424 Rm: N-110 Fax : (901) 321-3402
More informationLab 1 The Scientific Method
INTRODUCTION The fllwing labratry exercise is designed t give yu, the student, an pprtunity t explre unknwn systems, r universes, and hypthesize pssible rules which may gvern the behavir within them. Scientific
More informationo o IMPORTANT REMINDERS Reports will be graded largely on their ability to clearly communicate results and important conclusions.
BASD High Schl Frmal Lab Reprt GENERAL INFORMATION 12 pt Times New Rman fnt Duble-spaced, if required by yur teacher 1 inch margins n all sides (tp, bttm, left, and right) Always write in third persn (avid
More informationGeneral Chemistry II, Unit II: Study Guide (part 1)
General Chemistry II, Unit II: Study Guide (part 1) CDS Chapter 21: Reactin Equilibrium in the Gas Phase General Chemistry II Unit II Part 1 1 Intrductin Sme chemical reactins have a significant amunt
More informationOn Huntsberger Type Shrinkage Estimator for the Mean of Normal Distribution ABSTRACT INTRODUCTION
Malaysian Jurnal f Mathematical Sciences 4(): 7-4 () On Huntsberger Type Shrinkage Estimatr fr the Mean f Nrmal Distributin Department f Mathematical and Physical Sciences, University f Nizwa, Sultanate
More information^YawataR&D Laboratory, Nippon Steel Corporation, Tobata, Kitakyushu, Japan
Detectin f fatigue crack initiatin frm a ntch under a randm lad C. Makabe," S. Nishida^C. Urashima,' H. Kaneshir* "Department f Mechanical Systems Engineering, University f the Ryukyus, Nishihara, kinawa,
More informationOF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION
U. S. FOREST SERVICE RESEARCH PAPER FPL 50 DECEMBER U. S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY OF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION
More informationEffects of piezo-viscous dependency on squeeze film between circular plates: Couple Stress fluid model
Turkish Jurnal f Science & Technlgy Vlume 9(1), 97-103, 014 Effects f piez-viscus dependency n squeeze film between circular plates: Cuple Stress fluid mdel Abstract U. P. SINGH Ansal Technical Campus,
More informationMath Foundations 10 Work Plan
Math Fundatins 10 Wrk Plan Units / Tpics 10.1 Demnstrate understanding f factrs f whle numbers by: Prime factrs Greatest Cmmn Factrs (GCF) Least Cmmn Multiple (LCM) Principal square rt Cube rt Time Frame
More informationRevision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax
.7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical
More informationNUMERICAL SIMULATION OF CHLORIDE DIFFUSION IN REINFORCED CONCRETE STRUCTURES WITH CRACKS
VIII Internatinal Cnference n Fracture Mechanics f Cnete and Cnete Structures FraMCS-8 J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang (Eds) NUMERICAL SIMULATION OF CHLORIDE DIFFUSION IN REINFORCED
More informationDesign and Simulation of Dc-Dc Voltage Converters Using Matlab/Simulink
American Jurnal f Engineering Research (AJER) 016 American Jurnal f Engineering Research (AJER) e-issn: 30-0847 p-issn : 30-0936 Vlume-5, Issue-, pp-9-36 www.ajer.rg Research Paper Open Access Design and
More informationSTUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE AFDELING MACHINEBOUW - DROOGBAK la - AMSTERDAM
REPORT N. 31 M APRIL 1960 STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE AFDELING MACHINEBOUW - DROOGBAK la - AMSTERDAM NETHERLANDS RESEARCH CENTRE T.N.O. FOR SHIPBUILDING AND NAVIGATION ENGINEERING
More informationSoliton-Effect Optical Pulse Compression in Bulk Media with χ (3) Nonlinearity. 1 Introduction
Nnlinear Analysis: Mdelling and Cntrl, Vilnius, IMI,, N 5 Lithuanian Assciatin f Nnlinear Analysts, Slitn-Effect Optical Pulse Cmpressin in Bulk Media with χ (3) Nnlinearity Received: 9.7. Accepted: 11.1.
More informationChapter 14. Nanoscale Resolution in the Near and Far Field Intensity Profile of Optical Dipole Radiation
Chapter 4 Nanscale Reslutin in the Near and Far Field Intensity Prfile f Optical Diple Radiatin Xin Li * and Henk F. Arnldus Mississippi State University * xl@msstate.edu hfa@msstate.edu Jie Shu Rice University
More informationNUROP CONGRESS PAPER CHINESE PINYIN TO CHINESE CHARACTER CONVERSION
NUROP Chinese Pinyin T Chinese Character Cnversin NUROP CONGRESS PAPER CHINESE PINYIN TO CHINESE CHARACTER CONVERSION CHIA LI SHI 1 AND LUA KIM TENG 2 Schl f Cmputing, Natinal University f Singapre 3 Science
More informationComparison of two variable parameter Muskingum methods
Extreme Hydrlgical Events: Precipitatin, Flds and Drughts (Prceedings f the Ykhama Sympsium, July 1993). IAHS Publ. n. 213, 1993. 129 Cmparisn f tw variable parameter Muskingum methds M. PERUMAL Department
More informationEquilibrium of Stress
Equilibrium f Stress Cnsider tw perpendicular planes passing thrugh a pint p. The stress cmpnents acting n these planes are as shwn in ig. 3.4.1a. These stresses are usuall shwn tgether acting n a small
More informationFundamental Concepts in Structural Plasticity
Lecture Fundamental Cncepts in Structural Plasticit Prblem -: Stress ield cnditin Cnsider the plane stress ield cnditin in the principal crdinate sstem, a) Calculate the maximum difference between the
More informationImage Processing Adam Finkelstein & Tim Weyrich Princeton University
Syllabus I. Image prcessing II. Mdeling Cmputer Animatin III. Rendering Rendering IV. Animatin (Michael Bstck, CS426, Fall99) Image Prcessing Adam Finkelstein & Tim Weyrich Princetn University (Rusty Cleman,
More informationLecture 6: Phase Space and Damped Oscillations
Lecture 6: Phase Space and Damped Oscillatins Oscillatins in Multiple Dimensins The preius discussin was fine fr scillatin in a single dimensin In general, thugh, we want t deal with the situatin where:
More informationCHAPTER 3 INEQUALITIES. Copyright -The Institute of Chartered Accountants of India
CHAPTER 3 INEQUALITIES Cpyright -The Institute f Chartered Accuntants f India INEQUALITIES LEARNING OBJECTIVES One f the widely used decisin making prblems, nwadays, is t decide n the ptimal mix f scarce
More informationES201 - Examination 2 Winter Adams and Richards NAME BOX NUMBER
ES201 - Examinatin 2 Winter 2003-2004 Adams and Richards NAME BOX NUMBER Please Circle One : Richards (Perid 4) ES201-01 Adams (Perid 4) ES201-02 Adams (Perid 6) ES201-03 Prblem 1 ( 12 ) Prblem 2 ( 24
More information(Communicated at the meeting of January )
Physics. - Establishment f an Abslute Scale fr the herm-electric Frce. By G. BOR ELlUS. W. H. KEESOM. C. H. JOHANSSON and J. O. LND E. Supplement N0. 69b t the Cmmunicatins frm the Physical Labratry at
More informationA PLETHORA OF MULTI-PULSED SOLUTIONS FOR A BOUSSINESQ SYSTEM. Department of Mathematics, Penn State University University Park, PA16802, USA.
A PLETHORA OF MULTI-PULSED SOLUTIONS FOR A BOUSSINESQ SYSTEM MIN CHEN Department f Mathematics, Penn State University University Park, PA68, USA. Abstract. This paper studies traveling-wave slutins f the
More information[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )
(Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well
More information