CHAPTER 10 TRANSVERSE VIBRATIONS-VI: FINITE ELEMENT ANALYSIS OF ROTORS WITH GYROSCOPIC EFFECTS

Transcription

1 CHAPTER TRANSVERSE VIBRATIONS-VI: FINITE ELEMENT ANALYSIS OF ROTORS WITH GYROSCOPIC EFFECTS Previous, groscopic effects on a rotor with a singe disc were discussed in great detai b using the quasi-static dnamic anases with the hep of the anatica approach. For muti-dof sstems with groscopic effect of thin discs was described whie discussing the transfer matri method. There we have not considered groscopic effects due to thick feibe shafts (i.e., the shaft with distributed mass stiffness properties). In previous chapter, we deat with anasis of simpe rotor with the anatica (i.e., the continuous) numerica (finite eement) approaches without considering the important effects ike shear deformation groscopic effects. In the present chapter, the anasis woud be etended to initia a singe disc of a simpe rotor. Then it wi be etended for more genera rotor sstems with the hep of a powerfu anasis too of the finite eement method. The Timoshenko beam theor woud be used for the deveopment of governing equation of the continuous sstem anasis. The finite eement formuation wi be deveoped for the spinning Timoshenko beam which incudes higher effects ike the rotar inertia, shear deformation groscopic effects. The eigen vaue probem woud be deveoped through the state space form of the governing equation. With the hep of eampes the etraction of moda parameters from the specia eigen vaue probem wi be epained. The stard Campbe diagram for various cases are obtained, which shows the natura whir frequenc variation with the spin speed of the shaft for asnchronous whir. This diagram can be used to obtain the critica speeds of such rotor sstems.. Rotor Sstems with a Singe Disc In the present section a finite eement anasis for an overhang rotor with groscopic effects as shown in Figure. woud be iustrated. However, the procedure then can be etended to other boundar conditions (Jeffocott rotor, rotor with intermediate support, etc.). The shaft has been modeed with consistent mass stiffness matri. However, groscopic coupe due to disc is on considered for the shaft it is negected. The shaft is treated as feibe massess. Figure. A cantiever rotor sstem

2 593 Now since we woud be considering the groscopic coupe, hence we need to consider both pane motion simutaneous. Let us mode the rotor as singe eement with two nodes. The eementa equations of motion in z- pane as shown in Figure.(a) (without groscopic effects) can be written as S u u m ϕ 4 6 EI ϕ M z m = u 6 u S with 3 m Id ϕ sm 4 ϕ M z sm 4 m = ρ A 4 + m (.) The eementa equations of motion in -z pane as shown in Figure.(b) (without groscopic effect) can be written as S v v m ϕ 4 6 EI ϕ M z m = v 6 v S 3 m Id ϕ sm 4 ϕ M z sm 4 + m (.) (a) Eement in z- pane Figure. A rotor eement (b) Eement in -z pane In chapter 5 groscopic coupe effect of disc aone was described it is given as (pease note that the disc is at node )

3 594 u I ω I p ϕ p ϕ v (.3) It can be eped in the foowing form to accommodate both the nodes of the shaft eement u ϕ v ϕ ω u I p ϕ v I p ϕ (.4) It coud be seen that the groscopic effect eads to couping of motions in -z z- panes. In equations (.) (.) now groscopic coupe effect (equation (.4)) woud be added equations of motion of the eement, in both panes, can be written as u ϕ m v 56 + m m Id 4 + m ϕ m v 56 + m ϕ Id sm 4 + m ϕ u

4 595 ω + v I p u 6 6 ϕ 4 6 ϕ v 6 6 v ϕ 4 6 EI ϕ 3 u 6 u I p ϕ 4 ϕ ϕ u 6 v sm 4 ϕ S _ M S M = + S M S M z z z z (.5) The above formuation coud aternative be obtained b a more forma approach b using the additiona kinetic energ term of the disc deveoped in Chapter 5 due to groscopic moments in finite eement formuations of Euer-Bernoui beams as described in Chapter 9, which wi ead to the simiar resuts. This wi be iustrated whie deveoping the finite eement formuation with the Timoshenko beam mode (in which the rotar groscopic effects of shafts wi aso be considered) in subsequent sections. For the present iustration of an overhang rotor, boundar conditions are given as u = v = ϕ = ϕ = M = S = M = S = (.6) z z After appication of boundar conditions (.6), on eimination of he first four rows coumns equations of motion (.5) reduces to m 56 + m I u u 6 u d 4 + m ϕ I ϕ 4 p EI ϕ m ω + 3 m v = v 6 v 56 + m ϕ I p ϕ sm 4 I ϕ d sm 4 m (.7) which is equation of motions of the foowing stard form

5 596 [ M ]{ η} ω[ G]{ η} + [ K ]{ η} = { } (.8) For the formuation of stard form of the eigen vaue probem, equation (.8) need to be transformed to the state space form as discussed in Chapter 5. The foowing eampe wi iustrate the procedure of obtaining whir natura frequencies for a tpica case. For the asnchronous whir the eigen vaue probem of the form as equation (.8) is not a stard one. For obtaining the stard form of the eigen vaue, equation (.8) has to epressed in the state space form. To iustrate the transformation, et us consider the foowing simpe eampe. Eampe.: Obtain the state space form of the foowing stard governing equation for vibrations of a spring-mass-damper sstem. m + c + k = (a) Soution: Let us epress the veocit as = v (b) From equation (a), we get c k v = v (c) m m Equations (b) (c) which can be combined as = v k / m c / m v (d) which can be written in the stard state space form, as with { h} = [ D]{ h} (e)

6 597 = = v = k / m c / m v { h } ; [ D] ; { h} Answer Now coming back to conversion of equation (.8) to the state space form, et { } = { v} (.9) So that equation (.8) takes the foowing form { v} = ω [ M ] [ G]{ v} [ M ] [ K ]{ } (.) Combining above two equations, we have { h } = [ D]{ h} (.) with [ D] = [ ] M [ K ] ω[ M ] [ G] ; { h} { } { v} = (.) Let us assume the soution of equation (.) of the foowing form { } { } h = h e υ (.3) t where ν is the whir frequenc. On substituting equation (.3) into equation (.), we get { h } [ D]{ h } ν = (.4) which is a stard eigen vaue probem. Eigen vaues of equation (.4) appear as pure imaginar conjugate pairs with magnitudes equa to natura whir frequencies. For the case when groscopic effect is not present, the eigen vaue wi be of the form ± jα ± jα. However, with groscopic effect it takes the form ± jα + ± jα + where α > α > α the correspond to forward backward whirs, respective. The detaied iustration of the soution of eigen vaue probem wi be

7 598 made in net eampe, when the method of finite eement wi be appied to rotors with groscopic effects. Eampe. Obtain the forward backward snchronous transverse critica speeds for a genera motion of a rotor as shown in Figure.3. The rotor is assumed to be fied supported at one end. Take mass of the thin disc m = kg with the radius of 3. cm. The shaft is assumed to be massess its ength, L, diameter, d, are. m. m, respective. Take shaft Young s moduus E =. N/m. Using the finite eement method considering the mass of the shaft with materia densit ρ = 78 kg/m 3 obtain first two forward backward snchronous bending critica speeds b drawing the Campbe diagram. Figure.3 Soution: On taking singe eement for the present probem, we have the foowing data: m = kg, I p 4 = mr = kg-m, 4.5 E =. N/m, ρ = 78 kg/m 3, I p 4 = mr = kg-m, 4.5 = L =. m, d =.5 m, A = d =. = 7.854, π π I = d =. = m 4, π 4 π ρ A = =.97 kg, EI = =.885 N/m 3 3 From equation (.7), we have u u.7 ϕ.45 ϕ 3 ω v v sm.7 ϕ.45 ϕ

8 u 5.6 ϕ + = v sm.6 ϕ (a) Hence, equation (a) has the same form as that of equation (.8). The state space form can be written as with { h } = [ D]{ h} (b) [ D] [ ] [ I ] [ ] M [ K ] ω[ M ] [ G] = ; { h} { } { } = ; { } u ϕ = v ϕ ; { } u ϕ = (c) v ϕ [ M ] = , (d) sm.7 [ G] = 3.45, [ K] = sm.6 (e) To compare eigen vaues eigen vectors without with groscopic effects the resuts from both the anasis are provided. Tabe. ists eigen vaues eigen vectors corresponding to one eement eigen anasis with rad/s ( ~ rpm, i.e. ver sow speed) of rotor speed (i.e., negigibe sma groscopic effects). In eigen vaues the rea part is reated with the damping the imaginar part is reated with the natura whir frequenc of the sstem. It can be observed that the imaginar part of eigen vaue in seria numbers to 4 are a the same. are compe conjugate. Thus, the whir frequenc from seria number is positive the whir natura frequenc from seria number is eact same with a negative sign. The negative frequenc has no phsica significance, hence, can be omitted. Without groscopic effect eigen vaue from seria numbers 3 are same, eigen vaues from seria number 4 is compe conjugate of 3. Hence, actua there is on one natura whir frequenc (i.e., 9.68 rad/s see Tabe.) from the first four eigen vaues. Simiar, from seria numbers 5 to 8 there is another natura whir frequenc (i.e., rad/s see Tabe.). Hence, without groscopic effect case with one eement we coud be abe to get on two natura frequencies

9 6 (refer Tabe.3). Eigen vectors are aso appearing in compe conjugate (see Tabe.). It shoud be noted that with zero speed the dnamic matri [D] becomes singuar, since a the diagona eements are zero with ω =. Hence, it wi be worthwhie to write the eigen vaue probem as ([ M ] [ K ] λ[ I ]){ } { } = (f) which is a famiiar form as discussed in detai in Chapters 7 9 for respective torsiona transverse vibrations. The eigen vaue eigen vector for matri [D], for eampe, at 5 rad/s, are provided in Tabe.. It can be observed that these eigen vaues are aso pure imaginar occurring in compe conjugate (e.g., seria numbers:, 3 4, 5 6, 7 8). Seria numbers to 4 beong to the first natura mode (as in the case of without groscopic effects these are a same, however, now 3 are not same). Eigen vaues 4 are compe conjugate of eigen vaues 3, respective; hence on eigen vaues 3 need to be considered (for the same reason as given for the case of without groscopic effects). Seria numbers 3 beong to the backward forward whirs, respective (which are beow above the first natura frequenc without groscopic coupe, i.e rad/s). Simiar seria numbers 5 7 beong to the backward forward whirs, respective (which are beow above the second natura frequenc without groscopic coupe, i.e rad/s); seria numbers 6 8 are eact same as 5 7 with a negative sign (since rea part in a eigen vaues are anwa zero). Eigen vectors corresponding to these eigen vaues are aso isted in Tabe.. It can be observed that the are appearing as compe number. However, on cose observation, it can be seen that either rea part is zero or imaginar part is zero (e.g., for the eigen vaue, the rea part of st nd eigen vector components are zero, whereas, for the imaginar part of 3 rd 4 th eigen vector components are zero. When the rea part is zero the phase is 9 when the rea part is zero the phase is. Tabe. Eigen vaue eigen vectors without groscopic coupe (i.e., at speed of rad/s) S.N. Eigen vaue Eigen vectors j j.36 +.j j. -.36j.+.933j.-.79j.933+.j.79+.j

10 j j.36 +.j j.+.36j.-.933j.-+.79j.933+.j.79+.j j j j j. -.36j.-.933j.-.79j.933+.j.79+.j j j j j. +.36j.+.933j.+.79j.933+.j.79+.j j j j j j j.77 +.j j j j j j j j j.77 +.j j. +.77j j j j j j j.77 +.j j j j j j j j j.77 +.j j. +.77j 6

11 6 Tabe. Eigen vaue eigen vectors with groscopic effects at 5 rad/s S.N. Eigen vaue Eigen vectors j j. +.j j -. -.j j j.3 +.j.77 +.j j j. -.j j -. +.j j j.3 -.j.77+.j j j j j.4 -.j j.77 +.j j. +.77j j j j j.4 +.j j.77+.j j. -.77j j j j. +.5j. +.33j j j j j j j j. -.5j. -.33j j j j j j j j.5 -.j.4 -.j.88 +.j.76 +.j

12 j. +.76j j j j.5 +.j.4 +.j.88 -.j.76 +.j. -.88j. -.76j Figs..4 (a), (b) show Campbe diagrams for the present probem with eements, respective. Tabe.3 ists the critica speeds up to the third mode with without groscopic coupe. It shoud be noted that the critica speed without groscopic coupe is awas in between the corresponding mode forward backward whir critica speeds, which indicate the spitting of natura frequenc in the case of groscopic effect. It can be observed that the spit in the whir natura frequenc is distinct at higher speeds at higher mode numbers. Fig.5(a) (b) show mode shapes for inear anguar dispacements, respective. Tabe.3 Critica speeds with different number of eements Mode no. Critica speed without groscopic effects (rad/s) Critica speed with groscopic effects (rad/s) for eement for eements eement eements Backward whir Forward whir Backward whir Forward whir * * * Critica speed coud not be obtained because estimated natura whir frequenc with one eement never intersected the ω = ω ine. This is due to the fact that the FEM awas gives over estimation nf of the natura frequencies error in second natura is epected to be more with singe eement, hence variation of the natura whir frequenc was diverging nature. * With a singe eement, it is possibe to get critica speeds on up to the second mode.

13 64 Natura whir frequenc, ωnf (rad/s) F B ω = ω nf 5 5 B Shaft spin speed, ω (rad/s) F Figure.4(a) Campbe diagram (with eement) Natura whir frequenc, ωnf (rad/s) F 3F 3B Shaft spin speed, ω (rad/s) ω = ω nf B Figure.4(b) Campbe diagram (with eements)

14 65. Reative inear dispacements Shaft ength Figure.5(a) Normaized inear dispacement mode shapes at ω = rad/s (forward); F F () - first mode shape ( ω nf = 9.43 rad/s), () - second mode shape( ω 647. nf = rad/s) Reative anguar dispacements Shaft ength Figure.5(b): Normaized anguar dispacement mode shapes at ω = rad/s (forward); F F () - first mode shape( ω nf = 9.43 rad/s), () - second mode shape ( ω 647. nf = rad/s)

15 66. TIMOSHENKO BEAM THEORY In previous section we discussed the groscopic effect on a disc aone in a sender feibe beam. The cross-sectiona dimensions of the beam were considered to be sma in comparison with the ength. In this case the rotation of a beam eement is equa to sope of the eastic ine of the shaft. When we have to investigate higher frequenc modes an infinitesima eement woud have appreciabe amount of rotar inertia. When the beam is thick in that case the cross-sectiona dimensions of the beam are considered to be comparabe in comparison with the ength the shear effect becomes predominant. The Euer-Bernoui beam theor is based on the assumption that pane cross-sections remain pane perpendicuar to the ongitudina ais after bending. This assumption impies that a shear strains are zero. When the normait assumption is not used, i.e., pane sections remain pane but not necessari to the ongitudina ais after deformation, the transverse shear strain ε z is not zero. Therefore, the rotation of a transverse norma pane about -ais is not equa to - dv/dz. Beam theor based on these reaed assumptions is caed a shear deformation beam theor, most common known as the Timoshenko beam theor. Consider a short stubb beam as shown in Figure.6(a) before deformation. Let L be the ength of the shaft, A be the cross sectiona area, E be the Young s moduus, ρ be the mass densit of the shaft materia. Assume that the deformation of the beam is pure due to the shear that a vertica eement before deformation remains vertica after deformation moves b distance v s (subscript s to represents the pure shear) in the transverse direction as shown in Figure.6(b). For the pure shear case aso there wi be no couping in deformations in the two transverse directions (i.e., in the directions). Since there is no couping of motion in two transverse directions on negecting dispacement in the aia direction, the dispacement fied is given b u = u = v ( z, t) u = (.5) s z Line eements tangentia to the eastic ine of the beam undergo a rotation β(z,t) due to the shear as shown in Figure.6(c). Engineering strains from dispacement fieds of equation (.5) are ε = v (.6) z s ε =, ε =, ε =, ε =, ε = (.7) zz z z where prime ( ) represents derivative with respect to z.

16 67 Figure.6(a) A beam before defection Figure.6(b) The beam after the shear defection Figure.6(c) Rotation due to shear

17 68 The stress fied consists of on one component τ = Gv (.8) z s where, G is the shear moduus of the beam materia. Equation (.8) shows that the shear stress across the cross section of the beam is uniform independent of the coordinate. However, if we introduce epicit dependenc of stress fied on another independent variabe,, then anasis becomes unnecessari compicated. In actua case this is not true therefore a shear correction factor k sc is appied to equation (.8), rather than making the theor more compicated b having one more independent parameter in the anasis. Figure.7 Tpica cross sections of the shaft Severa definitions of the shear correction factors are found in iteratures. Present investigation uses the one given b Cowper (966) (mentioned aso b Shames Dm, 5). These are given for some cases as (Figure.7) with r k sc k sc = r r i ( + ν ) 6 = 7 + 6ν = 6( + ν )( + r ) ( 7 + 6ν )( + r ) + ( + ν ) r for a circuar cinder (.) for a hoow cinder (.) where, ν is the Poisson s ratio, r i r o are the inner outer radii of the cross section of shaft as shown in Figure.7. Thus considering the shear correction factor, equation (.8) can be written as

18 69 τ = k Gv (.) z sc s In genera, the beam is not subjected to the pure shear on. The significant component of the deformation arises due to the bending moment as in the case of Euer Raeigh (as compared to Timoshenko beam mode it considers on the rotar inertia effect without shear deformation effects) beams. Combining the effect of shear considered above with the Euer beam theor (Chapter 9), the foowing dispacement fied for a point P(,, z) as shown in Figure.8 is assumed u = ; u v( z, t) = u ϕ ( z, t) { v β ( z, t) } z = = (.) with v v v = s + b, v vs vb β ϕ = + = + (.3) (a) Shear (b) Bending (Euer) Figure.8 A dispacement fied of the Timoshenko beam The tota sope of the beam, v, consists of two parts, one due to the bending, which is ϕ (z,t) other due to on the shear, which is β(z,t). The aia dispacement of a point at a distance from center ine is on due to the bending sope that the shear force has no contribution to this. So there are two independent variabes in this probem. One independent variabe is the tota dispacement of the point, v(z,t) other is the sope due to the bending ϕ (z,t). The engineering strain fied from equation (.) is ε u z z zz = = ϕ ; ε = + = ( v ϕ ) z u z u z (.4)

19 6 ε = ; ε = ; ε = ; ε = (.5) z The corresponding stress fied with the shear correction factor is given as σ = Eϕ ; τ = k AG( v ϕ ) (.6) zz z sc σ = ; σ = ; τ = τ = (.7) z The strain (conservative) energ is L { ( ) ϕ sc ϕ } U = EI + k AG v dz (.8) The kinetic energ is L { ρ ρ ϕ } T = Av + I dz (.9) If f(z,t) is the distributed force for transverse oads then the work done (non-conservative) b eterna forces can be written as Wnc L = f ( z, t) vdz (.3) The eementa equation of motion boundar conditions can be obtained from Hamiton s principe, as foows t t ( ) δ T U + δwnc dt = (.3) Substituting equation (.8), (.9) (.3), into equation (.3), we get { } t L L L δ { } ( ) ρ ρ ϕ ϕ sc ϕ δ t Av + I dz EI + k AG v dz + f ( z, t) vdz dt = (.3)

20 6 On operating the variation operator, from equation (.3) we get { ρav δ ( v ) + ρi ϕ δ ( ϕ )} dz { EI ϕ δ ( ϕ ) + kscag ( v ϕ ) δ ( v ϕ )} dz + f ( z, t) δvdz dt = (.33) t L L L t On changing the order of variation differentiation in equation (.33), we get t L t ρ Av ( v) I ( ) dz dt t δ ρ ϕ t δϕ + + t L t L EI ϕ ( δϕ ) k AG( v ϕ ) δ v δϕ dz dt f ( z, t) δ + + vdzdt = (.34) sc t t On performing integration b parts of terms, which has both the differentia variationa operators (i.e., first four terms), in equation (.34), we get L t t t L L t L Av v dz Av v dzdt + I dz I dzdt + { ρ ( δ )} { ρ ( δ )} { ρ ϕ ( δϕ )} { ρ ϕ ( δϕ )} t t t t L L t t L t t L [ ϕ ( δϕ )] { ϕ ( δϕ )} { sc ( ϕ )( δ )} sc ( ϕ )( δ ) EI dt + EI dzdt k AG v v dt + k AG v v dzdt t t t t t L t L + { kscag ( v ϕ )( δϕ )} dzdt + f ( z, t) δvdzdt = (.35) t t The first third terms of equation (.35) wi vanish, since variations are not defined in tempora domain. Remaining terms can be rearranged in the foowing form t L { ρ Av + ksc AG( v ϕ ) f ( z, t) } δ vdzdt { ( ) } t L t t t L t [ ϕ δϕ ] { sc ( ϕ )( δ )} EI ( ) dt k AG v v dt = t t ρi ϕ EI ϕ ksc AG v ϕ dzδϕ dt (.36) L

21 6 Variations δ v δϕ in spatia domain are arbitrar, this ieds the differentia equations of motion as sc ( ϕ ) ρ Av k AG v = f (.37) ( ) ρi ϕ EI ϕ k AG v ϕ = (.38) sc boundar conditions as sc L ( ϕ ) δ v k AG v = EI ϕ δϕ = (.39) L Equations of motions (.37) (.38) can be combined to a singe equation. On rearranging equation (.37) for free vibrations, we get ρ ϕ = v v (.4) k G sc which gives ρ ρ ρ ϕ = v v ; ϕ = v v ϕ = v v (.4) k G k G k G sc sc sc On differentiating equation (.38) with respect to z, we get ( ) EI ϕ ρi ϕ + k AG v ϕ = (.4) sc On substituting equations (.4) (.4) into equation (.4), we get ρ ρ ρ EI v v ρi v v + ksc AG v v v = kscg kscg kscg which gives, the Timoshenko beam equation for free vibrations as v v E v ρ I v EI + ρ A ρi = 4 z t kscg t z kscg t (.43)

22 63 It shoud be noted that due to the rotar inertia the shear deformation in the Timoshenko beam, two etra terms (the third fourth terms in the eft h side of the above equation) are appearing in the equation of motion as compared to the Euer-Bernoui beam, in which on the first two terms in the eft h side appears. The third fourth terms containing k sc are reated to the shear effect, whereas without it is reated to the rotar inertia (i.e., ρ I v ). It shoud be noted that simiar equation of motion coud be deveoped for z- pane of the foowing form u u E u ρ I u EI + ρ A ρi = 4 z t kscg t z kscg t (.44) where u is the inear transverse dispacement in the -ais direction, I is the moment of inertia of the cross section about the -ais. Eact Soution: Equation (.43) is soved for a simp supported end condition of a shaft. For obtaining the cosed form soution of equation (.43) for the specified boundar conditions, it can be simpified b using a genera soution of the foowing form nzπ v( z, t) = An sin sinωnf t L (.45) On substituting equation (.45) into equation (.43), we get 4 4 ρ L 4 ρ AL ρl E 4 ωnf ( nπ ) ωnf ( nπ ) = EkscG EI E kscg (.46) On soving equation (.46) for ω nf, we get with b + b 4ac ωnf = (.47) a 4 ρ AL ρ E b = + + EI E kscg ( nπ ) ; a 4 ρ L Ek G = ; ( ) 4 sc c = nπ (.48)

23 64 The above eact soution wi be used for comparing natura frequencies for a simp supported end condition of a shaft with the finite eement method as a benchmark soution for the convergence stud. On the simiar ines for other boundar conditions epressions of natura frequencies coud be attempted..3 Finite Eement Formuations of the Timoshenko Beam For the finite eement anasis, we need to discretise the beam into number of eements as shown in Fig..9. Consider a finite eement of the shaft of ength in an eementa co-ordinate sstem --z. Since the two orthogona transverse pane motions are uncouped, the deformation of the eement is initia considered in the -z pane with two nodes as shown in Figure.. The motion in z- pane coud be anased in the simiar ines. Let v be the noda inear transverse dispacement of the shaft eement. Let v = dv dz be the tota sope of the beam, which consists of two parts, one due to the bending, which is ϕ other due to the shear, which is β is given b ( v ϕ ). From the dispacement fied of equation (.), we know that the aia dispacement of a point P(,, z) at a distance from the centre ine is on due to the bending sope that the shear force has no contribution to this. In previous section we observed that there are two independent variabes in this probem (i.e., v ϕ ). Figure.9 Discretization of a beam into number of eements Figure. A tpica beam eement in -z pane

24 65 From equations (.37) (.38) without considering the work done b the eterna forces, equations of motion are sc ( ) ρ Av k AG v ϕ = (.49) ( ) EI ϕ ρi ϕ + k AG v ϕ = (.5) sc In the finite eement mode, the continuous dispacement fied can be approimated in terms of descretised generaized-dispacements of eement nodes. In the present stud, each eement in singe pane (e.g., -z) has two nodes each node has two generaised dispacements (one inear the other rotationa). Therefore, dispacements coud be obtained within the eement b using appropriate shape functions to be derived in the subsequent subsection..3. Weak Formuations of the Timoshenko Beam Eement for the Static Case Let the static case be considered first b dropping time derivative terms. Equations of motion (.49) (.5) coud be written as beow whie boundar conditions remains the same sc ( ) k AG v ϕ = (.5) EI ( ϕ ) ϕ + k AG v = (.5) sc On assuming approimate soution of the foowing form, we have { } ( ne ) ( e v ) ( z) = S( z) u (.53) { } ( ne ) ( e ϕ ) ( z) = T( z) u (.54) On substituting approimate soutions of equations (.53) (.54) in equations of motion (.5) (.5), the residue of each equation is given b ( ) R = k AG v ϕ (.55) ( e) ( e) ( e) sc ( ϕ ) R = EIϕ + k AG v (.56) ( e) ( e) ( e) ( e) sc

25 66 Empoing the Gaerkin principe, one has { } ( e S R ) dz = (.57) { } ( e T R ) dz = (.58) Using equations (.55) (.56) into equations (.57) (.58), we get ( e) ( e) { S} ksc AG ( v ϕ ) dz = (.59) ( ) ( e) ( e) ( e) { } ϕ sc ( ϕ ) T EI + k AG v dz = (.6) On performing integration b parts, it gives ( e) ( e) ( e) ( e) { } sc ( ϕ ) { } sc ( ϕ ) S k AG v S k AG v dz = (.6) ( e) ( e) ( e) ( e) { T} EIϕ { T } EIϕ dz { T} kscagv dz { T} kscagϕ + dz = (.6) On ooking into the competeness compatibiit requirements, we need the compatibiit requirement up to to ( e) v ( e ) ( e) v ϕ (i.e., up to the first derivative) the competeness requirement up ( e) ϕ (i.e., up to the vaue of variabes itsef). Hence, according to these requirements a inear interpoation function woud serve the purpose. The variabes ( e) v ϕ do not have same the same phsica units; the can be interpoated, in genera, with different degrees of interpoation. ( e) ( e) As we know for thin beams we have ϕ = v, hence a inear in ( e) This wi make the bending energ ( ) ( e ) ( e) v impies a constant in ϕ. EI ϕ dz to zero. This numerica probem is known as shear ocking (since it originates from the shear effect). To overcome this, severa aternative methods have been deveoped in the iterature (Redd, 3). ( e )

26 67.3. Derivation of Shape Functions Now on assuming that ( e) v is a cubic ponomia, then ϕ shoud be of the same order as ( e ) ( e) v that is therefore a quadratic. These are eact shape functions in the static anasis. The inter-eement compatibiit requires that v, ϕ, v assumed as ϕ must be continuous. Therefore, ( e) v ϕ can be ( e ) ( ) v ( ξ, η) = v ξ + v η + aξη + b ξ η ξη (.63) ( e ) ϕ ξ η = ϕ ξ + ϕ η + ξη (.64) ( e ) (, ) c with ξ = z z η = (.65) where a, b c are unknown interpoation coefficients, ξ η are caed natura coordinates. A four boundar conditions of the eement in Fig.. are satisfied as v ( e) z= = v ; v ( e) = v ; ( e) = ϕ = ϕ z= z ϕ ϕ ( e) = (.66) z= because ξη ( ξ η ξη ) vanish at boundaries of the eement. Therefore a, b, c are assumed to be interna degrees of freedom (DOFs), which wi be eiminated subsequent. Because of these DOFs we coud be abe to make the shape function of the required degree of ponomia (i.e. a cubic for ( e) v a quadratic for ϕ ( e) ) without vioating desired boundar conditions. It coud be observed that without interna DOFs, the assumed shape function woud be inear for both ( e) v ϕ ( e). The presence of ξη ( ξ η ξη ) is necessar in order to compete the ponomia of the required degree to avoid the shear ocking. Equation (.66) can be written as ( e) v ( ξ, η) = ξ η ξη ( ξ η ξη ) v ϕ v ϕ a b c T { u} ( ne) = S (.67) ( e ϕ ) ( ξ, η) = ξ η ξη v ϕ v ϕ a b c T with { u} ( ne) = T (.68)

27 68 S = S (.69) z= = z= T = T = = = (.7) z z where, { u } ( ne) is the noda dispacement vector. On substituting equations (.67)-(.7) in equations (.6) (.6), we get sc z= ( ne) ( ne) ksc AG{ S } S dz{ u} ksc AG{ S } T dz{ u} = (.7) k AG ( v ϕ ) ksc AG ( v ϕ ) z= ( ne) ( ne) ( ne) EI { T } T dz{ u} ksc AG{ T} S dz{ u} + ksc AG{ T} T dz{ u} = EI ϕ z = (.7) EI ϕ z= On combining equations (.7) (.7), the stiffness reationship coud be obtained as with [ ] { } ( ne) K u { f } { } ( ne) = [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] (.73) = + + (.74) [ ] { } K = ksc AG S S dz = ksc AG (3 ) (5 ) (.75)

28 69 [ ] { } K = ksc AG S T dz 6 6 = ksc AG (.76) [ ] { } K3 = T EI T dz 3 6 = EI (.77) [ ] { } K4 = kscag T S dz 6 = ksc AG (.78) [ ] { } K5 = kscag T T dz 3 6 = ksc AG (.79) where [ K ] k k k k k 6 k k EI k k EI k k k k k k k 6 k k EI k k EI k k = k k 6 6 k k 5 3 k k k k k k EI (.8)

29 6 k = k AG (.8) sc Equation (.73) has a vector { u } ( ne), which contain three interna DOFs that need to be eiminated at this stage with the hep of the static condensation scheme. Defining the eterna DOFs (four in numbers) of the eement as masters interna DOFs (three in numbers) of the eement as saves, the stiffness matri in equation (.73) is subdivided as with [ k ] [ ] [ ] [ k ] [ k ] { u} { } { } ( ne) k k ( ne) f = EI = 3 ksc sm 3 (.5Φ + ) ( 6 + ) 3 (.5Φ + ) ; [ k ] (.8) ksc AG = 6 3 5Φ + 6 T [ k ] [ k ] = = ksc AG ; { u} ( ne) { } ( ne d ) a = b c EI kscga ( ne) ϕ ϕ T ( ne) T Φ = ; { d} = v v { f } = S M z S M z S = k AG ( v ϕ ) ; = sc z M = EI ϕ = ; z z S = k AG ( v ϕ ) ; = sc z M = EI ϕ. = z z where Φ is caed the shear parameter, S is the shear force, M z is the bending moment. Equation (.8) can be written as a ne [ k ]{ d} + [ k ] b = { f } ( ) ( ne) c a ( ne) [ k ]{ d} + [ k ] b = { } c (.83) The second set of equation (.83) gives

30 6 with a b = k k c [ ] [ ] v ϕ v ϕ T v ϕ µ µ = µ µ (.84) v 6µ 6µ 3µ 3µ ϕ µ = ( + Φ) (.85) On substituting equation (.84) into the first set of equation (.83), we get ( ) { } [ ]{ } [ ] [ ] [ ]{ } ( ne) ( ne) ( ne) k d + k k k d = f (.86) which can be can be written as with [ ]{ d} ( ne) { f } ( ne) k = (.87) [ k] [ k ] [ k ][ k ] [ k ] = (.88) where [ k ] is the condensed stiffness matri. Now interna DOFs coud be eiminated from the shape assumed functions aso. On substituting a, b, c obtained from equation (.84) into equation (.67), we get T ( e) ( ξ, η) = ξ η ξη ( ξ η ξη ) ϕ ϕ v v v a b c v a ϕ = ξ η + ξη ( ξ η ξη ) b v c ϕ v v ϕ µ µ ϕ = ξ η + ξη ( ξ η ξη ) µ µ v v ϕ 6µ 6µ 3µ 3µ ϕ which coud be simpified as

31 6 = { } (ne) ( e) v Nv Nϕ N 3v N4ϕ = N d (.89) with Ni ( z) = i ( z) + Φ i ( z) + Φ [ α β ] EI ; kscga Φ = ; i =,, 3, 4. 3 α = 3η + η ; β = η ; 3 = ( + ) α η η η ; β = η η ; ( ) / 3 α3 = 3η η β ; 3 = η ; 3 4 = ( + ) α η η z ; β = η + η ; 4 ( ) / ( ne) d = v ϕ v ϕ η = { } T where { } ( ne) d is the noda dispacement vector N is the transationa shape function vector. The α ( z) functions are associated with the bending deformation the β ( z) functions are due to i the shear deformation of a Timoshenko beam. i Simiar, on substituting a, b c obtained from equation (.84) in equation (.68), we get with ( e) 3 4 { } ϕ = M v + M ϕ + M v + M ϕ = M d (.9) M i ( z) = i ( z) + Φ i ( z) + Φ [ ε δ ] ( ne) EI ; kscga Φ = ; i =,, 3, 4 ε = η η δ = ; (6 6 ) / ; ε = η + η δ = η; 4 3 ; ε = η η δ 3 = ; 3 (6 6 ) / ; ε = η η δ ; = η ; z η =

32 63 where M is the rotationa shape function vector. Functions, ε ( z), are associated with the bending deformation functions, δ ( z), are due to the shear deformation of a Timoshenko beam. i i.3.3 Weak Formuation of the Timoshenko Beam Eement for the Dnamic Case For dnamic case the shape functions derived in the previous section wi sti be vaid coud take the foowing form (, ) = ( ) ( ) + ( ) ( ) + ( ) ( ) + ( ) ( ) = N( z) { d( t) } ( ne) (.9) ( e) v z t N z v t N z ϕ t N 3 z v t N4 z ϕ t ( e) 3 4 { } ϕ ( z, t) = M ( z) v ( t) + M ( z) ϕ ( t) + M ( z) v ( t) + M ( z) ϕ ( t) = M ( z) d( t) (.9) ( ne) On substituting equations (.9) (.9) in equations of motion (.49) (.5), residues of each equations of motion are given b ( ϕ ) R = ρ Av k AG v (.93) ( e) ( e) ( e) ( e) sc ( ) R = EI ϕ ρi ϕ + k AG v ϕ (.94) ( e) ( e) ( e) ( e) ( e) sc Using the Gaerkin method to minimize the residue, one has ( e) { } N R dz = (.95) { } ( e M R ) dz = (.96) On substituting of equations (.93) (.94) in equations (.95) (.96) the weak formuation can be obtained as ρ A{ N} N{ d } dz + k AG{ N } N { d} ksc AG ( v ϕ ) z= dz ksc AG{ N } M { d} dz = ksc AG ( v ϕ ) z= (.97) ( ne) ( ne) ( ne) sc

33 64 EI ϕ EI ϕ z= (.98) ( ne) ( ne) ( ne) ( ne) z ρi { M} M { d = } dz + EI { M } M { d} dz ksc AG{ M} N { d} dz + ksc AG{ M} M{ d} dz = Equations (.97) (.98) can be written as with [ M ]{ d} [ K ]{ d} [ K ]{ d} k AG ( v ϕ ) sc z= ( ne) ( ne) ( ne) + = (.99) ksc AG ( v ϕ ) z= ne [ M ]{ d} [ K ]{ d} [ K ]{ d} [ K ]{ d} ( ) ( ne) ( ne) ( ne) z= EI ϕ + + = (.) EI ϕ z= 4(7 Φ + 47 Φ+ 78) sm. M = A N N dz ρa (35 Φ + 77Φ+ 44) (7Φ + 4Φ+ 8) = 84( +Φ) 4(35 Φ + 63Φ+ 7) (35 Φ + 63Φ+ 6) (7 Φ + 47 Φ+ 78) (35 Φ + 63Φ+ 6) (7 Φ + 4Φ+ 6) (35 Φ + 77Φ+ 44) (7 Φ + 4Φ+ 8) [ ] ρ { } [ ] ρ { } M = I M M dz 36 sm ρi 3(5Φ ) (Φ + 5Φ + 4) = 3( ) 36 3(5Φ ) 36 + Φ 3(5Φ ) (5Φ 5Φ ) 3(5Φ ) (Φ + 5Φ + 4) [ ] sc { } K = k AG N N dz [ ] sc { } = (5Φ + Φ + 6) sm ksc AG 6 (5Φ + Φ + 8) + Φ Φ + Φ + Φ + Φ + 6 (5Φ + Φ + ) 6 (5Φ + Φ + 8) 6( ) (5 6) 6 (5 6) (5Φ + 6) sm K = k AG N M dz k 6 (5 8) sc AG Φ + Φ + = 6( + Φ) (5Φ + 6) 6(5Φ + 5Φ ) (5Φ + 6) 6 (5Φ + Φ + ) 6 (5Φ + Φ + 8) [ 3 ] { } K = M EI M dz sm EI 6 ( Φ + Φ + 4) = 3 ( + Φ) 6 6 ( Φ + Φ ) 6 ( Φ + Φ + 4)

34 65 [ 4 ] { } K = M ksc AG N dz (5Φ + 6) sm k 6(5 5 ) (5 8) sc AG Φ + Φ Φ + Φ + = 6( ) (5Φ + 6) 6 (5Φ + 6) + Φ 6(5Φ + 5Φ ) (5Φ + Φ + ) 6(5Φ + 5Φ ) (5Φ + Φ + 8) [ K5 ] { } = M ksc AG M dz 36 sm k 3(5 ) ( 5 4) sc AG Φ Φ + Φ + = 3( ) 36 3(5Φ ) 36 + Φ 3(5Φ ) (5Φ 5Φ ) 3(5Φ ) (Φ + 5Φ + 4) EI with Φ = k GA sc On combining equations (.99) (.), we get [ M ]{ d} + [ K ]{ d} = { f } ( ne ) ( ne ) ( ne ) (.) with [ M ] = [ M ] + [ M ] = [ M ] + [ M ] T R [ M T ] = [ M T ] + Φ[ M ] [ ] T + Φ M T [ M R ] = [ M R ] + Φ[ M ] [ ] R + Φ M R [ K ] = [ K ] [ K ] + [ K ] [ K ] + [ K ] = [ K ] + Φ [ K ] where { f } ( ne) is generaized force vector, the mass matri [ M ] consists of the transationa mass matri [ M T ] the rotationa mass matri [ M R ]. Detais of the mass matri stiffness matrices are given in Appendi.. Now, through eampes the effect of rotar inertia shear deformation woud be iustrated. Eampe. Obtain natura frequenc parameters defined b a non-dimensiona form as 4 4 ( ρal ω EI ) ω = of a uniform, non-rotating simp supported Timoshenko beam for the first n n / four modes. The senderness parameter R (=r/l) is to be varied from. (thin beam) to. (thick beam). Show the comparison for different number of eements with the eact anatica formua. ω n

35 66 Soution: Since in previous chapters the usua finite eement procedures have been deat in detai (i.e., regarding the eementa equations, assemb procedures, appication of boundar conditions, eigen vaue probem formuations, etc.), hence here those detais for the present probem is omitted. The uniform, non-rotating simp supported Timoshenko beam for obtaining natura frequencies, is first discretised into five numbers of eements. From the eigen vaue formuation of the present probem first four owest natura frequencies obtained are compared with the pubished work (Ku 998) b the cassica cosed form soution given b Shames Dm, 5. Ku (998) has considered C continuit (i.e., compatibiit up to inear dispacements) for the Timoshenko beam mode whereas mode represented here considers C continuit (i.e., the compatibiit up to the inear anguar dispacements). Comparisons of non-dimensiona natura frequencies for first four modes are presented in tabuar form in Tabes.4-.7, respective. In pubished work of Ku, 998, on first two natura frequencies are avaiabe. The present stud etracts natura frequencies in first four modes. These have been compared with cassica cosed form soutions. These studies are aso conducted b discretising the beam into the three seven number of eements in order to make convergence comparisons are summaries in Tabes R Tabe.4 Natura frequenc parameters ω n of a uniform, non-rotating simp supported Timoshenko beam for the first mode. Present work Ku (998) Shames *(3) (5) (7) (3) (5) (7) Dm (5) *Number in the bracket indicates number of eements considered. R Tabe.5 Natura frequenc parameters ω n of a uniform, non- rotating simp supported Timoshenko beam for the second mode. Present work Ku (998) Shames (3) (5) (7) (3) (5) (7) Dm (5)

36 67 Tabe.6 Natura frequenc parameters ω n of a uniform, non-rotating simp supported Timoshenko beam for the third mode. R Present work Shames Dm (3) (5) (7) () (5) (5) Tabe.7 Natura frequenc parametersω n of a uniform, non-rotating simp supported Timoshenko beam for the fourth mode. R Present work Shames Dm (3) (5) (7) () (5) (5) Due to the shear effect, defection of the beam increases which reduces the effective stiffness. So the non-dimensionaised natura frequenc decreases as the diameter of the beam is increased (i.e. with the senderness ratio). This trend coud be observed in a four modes. The accurac of the present soution is we within 5% for a cases. B FEM we over-predict with coarser discretisation it coud be seen that as the number of eements are increased from 3 to 7 for first two modes from 3 to 5 for the third fourth modes the natura frequenc parameter is decreasing. Especia converge (not much improvement) coud be observed in Tabed.6.7 between the number of eements to 5. The difference in the cosed form soution of Shames Dm (5) the FEM resuts coud be due to the approimate nature of the eigen function ( sin n π z L cosed-form soution). chosen in the Eampe.3 A tpica simp supported rotor disc sstem as shown in Figure. is to be anazed for obtaining the whir natura frequencies to show the appication of the finite eement method. Phsica properties of the rotor sstem are given as: the diameter of shaft is.m, the ength of shaft is 3.5 m, the Young s moduus of materia of shaft is.8 N/m, the mass densit of the

37 68 shaft materia is 783 kg/m 3, the Poisson s ratio is.3, the number of rigid discs are 4, the mass of each rigid disc is 6.3 kg. Soution: The rotor is discretised into the seven fourteen eements, respective, as shown in Figures... In the case of seven eements, two identica rigid bearings are ocated at node numbers two seven; four rigid discs are ocated at node numbers three, four, five si. In case of fourteen eement member, two identica rigid bearings are ocated at node numbers three thirteen, rigid discs are ocated at node numbers five, seven, nine eeven. The shaft is assumed uniform aong the span. Rigid discs are considered as point masses these point masses are added to mass matri corresponding to ocations of rigid discs inear acceerations. The assembed mass stiffness matrices are obtained b usua method boundar conditions are appied to get the reduced form dnamic matri. Natura whir frequencies are obtained b soving eigen vaue probem. Natura whir frequencies are obtained for 7 eements 4 eements mode are given in Tabe.8. Resuts show that good convergence has aread occurred with 7 eements mode. Figure. Rotor bearing sstem with rigid disks (7-eements of.5 m ength each) Figure. Rotor bearing sstem with rigid disks (4-eements of.5 m ength each)

38 69 For the present case R is ess than. so effect of shear is ess but as rotar inertia is aso considered, natura whir frequencies of the rotor bearing sstem decreases as compared to Euer-Bernoui case. This decrease is.3%,.8%, % 3% for the first, second third fourth modes, respective. The difference is higher at higher modes since the rotar inertia becomes predominant at higher frequencies. It shoud be noted that in the present probem, the rotor is assumed to be non-rotating because of this groscopic coupe is absent. The case of rotating shaft with the rotar inertia, shear deformation groscopic coupe woud be considered in the net section. Tabe.8 Natura whir frequencies of rotor bearing sstem with rigid disks (For non-rotating Timoshenko beam case) Natura whir frequencies without Natura whir frequencies with considering shear rotar inertia shear rotar inertia effects Mode No. effects (rad/sec) (rad/sec) For 7 eements For 4 eements For 7 eements For 4 eements Whiring of Timoshenko Shafts In the present section the anasis of whiring of spinning Timoshenko shaft is presented which incudes higher effects ike the rotar inertia, the shear effect the groscopic effect. A finite eement formuation incuding these effects is presented b using the consistent mass matri approach. Because of the groscopic effect, two perpendicuar transverse motions are now couped most important natura frequencies of the rotor sstem depend upon the anguar speed of the shaft. It eads to the forward backward whirs phenomena, to which we have aread described in detai for singe mass rotor sstem in Chapter 5. The finite eement formuation in accordance with rea co-ordinate sstem with groscopic effect ieds the groscopic matri as skew smmetric a other matrices as smmetric. The eigen vaue probem (without damping) gives eigen vaues as pure imaginar eigen vectors as compe. The finite eement anasis of the Timoshenko rotating beam are compared with resuts obtained b Weaver et a. (99) Esheman Eubanks (969), who determined critica speeds from approimate reations..4. Equations of Motion of a Spinning Timoshenko Shaft The anasis of the previous section coud be etended for the present case with incusion of groscopic effects. However, this woud ead to couping of motion in two orthogona panes. The

39 63 transation of the cross section centreine negecting aia motion is given b two dispacements u(z,t) v(,t) in the direction respective, which consists of contribution due to both the bending deformation contribution due to the shear deformation. u( z, t) = u ( z, t) + u ( z, t) v( z, t) = v ( z, t) + v ( z, t) (.) b s b s The rotation of the cross section is described b the rotation anges ϕ ( z, t) ϕ ( z, t) about aes, respective; which are associated with bending deformation of the eement as vb (, ) ( z, t ϕ ) z t = z ub (, ) ( z, t ϕ ) z t = z (.3) On differentiating equations (.), we get ub ( z, t) u ( z, t) = + u s ( z, t) z vb ( z, t) v ( z, t) = + v s ( z, t) z (.4) On noting equations (.3), equations (.4) take the form u ( z, t) = ϕ ( z, t) + u ( z, t) v ( z, t) = ϕ ( z, t) + v ( z, t) s s or u ( z, t) = u ( z, t) ϕ ( z, t) v ( z, t) = v ( z, t) + ϕ ( z, t) (.5) s s where u s v s are shear strains. For the differentia shaft eement ocated at z, the potentia energ of an eement of ength,, can be epressed as {( ϕ ) ( ) } ϕ { } U = EI k AG u v dz nc sc s s Where the first term is the eastic bending shear deformation energ. On noting equation (.5), we get {( ϕ ) ( ϕ ) } ( ϕ ) ( ϕ ) { } nc = sc (.6) U EI k AG u v dz

40 63 The kinetic energ of a shaft eement rotating at a constant speed, ω, incuding the transationa rotationa forms is given b { ( ) ( ) ( )} ρ ρ ϕ ϕ Pω ϕϕ ϕ ϕ Pω T = A u + v + I + + I dz + I dz (.7) In equation (.7) various kinetic energ terms are contributed as foows: the first term is due to inear motions, the second term is due to titing motions, the third term is due to groscopic coupes the ast term is due to spinning of the rotor. In Chapter 5 derivation of terms reated to groscopic moments have been described in detai, whie discussing the energ method. Anguar momentum in - z pane z- pane are opposite in direction have same magnitude. The quantit ϕϕ is a constant quantit in the conservative sstem, so it gives d( ϕϕ ) = ; so that ϕϕ = ϕ ϕ (.8) dt Noting equation (.8), equation (.7) becomes { ( ) ( ) } ρ ρ ϕ ϕ Pωϕ ϕ Pω T = A u + v + I + I dz + I dz (.9) If f ( z, t ) f ( z, t) are distributed transverse forces in directions, respective; then the work done b eterna forces is ( ) W = f u + f v dz (.) nc from the Hamiton s principe, we have t t { δ δ } ( T U ) + Wnc dt = (.) On substituting equation (.6), (.9), (.) into equation (.), we get t δ { ( ) ( ) ρ + + ρ ϕ + ϕ Pωϕ ϕ + Pω } t A u v I I I dz dt

41 63 {( ) ( ) } ( sc ) ( ) { } t δ EI ϕ ϕ k AG u ϕ v ϕ dz dt t t + + = t δ ( f u f v) dz dt (.) On operating the variation operator in equation (.), we get t { ρ δ ( ) + ρ δ ( ) + ρ ϕδ ( ϕ ) + ρ ϕ δ ( ϕ ) Pωδ ( ϕ ) ϕ Pωϕ δ ( ϕ )} t Au u Av v I I I I dzdt t { EIϕ δ ( ϕ ) + EIϕ δ ( ϕ ) + kscag ( u ϕ ) δ ( u ϕ ) + kscag ( v + ϕ ) δ ( v + ϕ )} dzdt t t { f δu f δv} dzdt t + + = (.3) On changing the order of variation differentiation in equation (.3), we get t ρau ( δu) + ρav ( δv) + ρiϕ ( δϕ ) + ρiϕ ( δϕ ) I Pω ( δϕ ) ϕ I Pωϕ δ ( ϕ ) dzdt + t t t t t t t EIϕ ( δϕ ) + EIϕ ( δϕ ) + kscag ( u ϕ ) ( δu) δϕ + kscag ( v + ϕ ) ( δv) + δϕ dzdt z z z z t t { f δu f δv} dzdt t + + = (.4) On performing integration b parts of terms, which has both the differentia variationa operators, in equation (.4), we get t t t t Avu u dz Au u dzdt Av v dz Av v dzdt + { ρ ( δ )} { ρ ( δ )} + { ρ ( δ )} { ρ ( δ )} t t t t t t t t + { ρi ϕ ( δϕ )} dz { ρi ϕ ( δϕ )} dzdt + { ρi ϕ ( δϕ )} dz { ρi ϕ ( δϕ )} dzdt + t t t t

42 633 t t IPωϕδ ( ϕ ) dz + { IPωϕ δ ( ϕ ) IPωϕ δ ( ϕ )} dzdt + t t t t t t L ϕ ( δϕ ) { ϕ ( δϕ )} { ϕ ( δϕ )} { ϕ ( δϕ )} EI dt + EI dzdt EI dt + EI dzdt t t t t t t t { k AG ( u ϕ )( δu) } dt + { k AG( u ϕ )( δu) } dzdt + { k AG( u ϕ )( δϕ )} dzdt sc sc sc t t t t t t { ( sc ϕ )( δ )} { ( sc ϕ )( δ )} { sc ( ϕ )( δϕ )} k AG v + v dt + k AG v + v dzdt k AG v + dzdt t t t t + ( fδu + f δ v) dzdt = (.5) t The first, third, fifth, seventh ninth terms of equation (.5) wi vanish, since variations are not defined in tempora domain. Remaining terms can be rearranged in the foowing form + + t t { ρau kscag ( u ϕ ) f } δudzdt { + ( ) sc )} t t ρav k AG v ϕ f δvdzdt t { ρi ϕ EIϕ kscag ( u + ϕ ) I Pωϕ } dzδϕ dt t t { ρi ϕ EIϕ ksc AG ( v ϕ ) + I Pωϕ } dzδϕ dt t t t t t { } ( )( ) [ ] ( )( ) { } EIϕ ( δϕ ) dt EIϕ ( δϕ ) dt k AG u ϕ δu + dt k AG v ϕ δ v dt = sc sc t t t t (.6)

43 634 Variations δ u, δϕ, δ v δϕ in spatia domain are arbitrar, this ieds the differentia equations of motion as ρau k AG( u + ϕ ) = f ; EIϕ + k AG( u + ϕ ) ρi ϕ + I ωϕ = ; sc sc P ρav k AG( v ϕ ) = f ; EIϕ + k AG( v ϕ ) ρi ϕ I ωϕ =. sc sc P (.7) boundar (geometrica natura) conditions are k AG( u ϕ ) δu =, EIϕ δϕ =, sc k AG( v + ϕ ) δ v =, sc EIϕ δϕ = (.8) It shoud be noted that now a four equations of motion (.7) are couped due to groscopic coupe terms need to be soved simutaneous. In the net subsection the finite eement formuation of governing equations are presented..4. Finite Eement Formuation Now the finite eement formuation for governing equations (.7) wi be deveoped b using the Gaerkin s method. As compared to previous section finite eement formuations, here two pane governing equations need to be considered simutaneous. In the finite eement mode, the continuous dispacement fied can be approimated in terms of generaised dispacements of the eement nodes. In the present finite eement mode (see Figures.3.4), each eement has two nodes each node have four generaized dispacements (two inear other two rotationa). The inear dispacement within the eement can be obtained b using appropriate shape functions are defined as { η } ( ) u v( z, t) =N ( z) { η( t) } ( ne) v u( z, t) =N ( z) ( t) ne (.9) ( ne) { η( t) } u v ϕ ϕ u v ϕ ϕ = T

44 635 Nu = N N N3 N4 Nv = N N N3 N4 Figure.3 A tpica beam eement in -z pane Figure.4 A tpica rotor eement in z- pane In the matri form, it coud be combined as with, [ Nt ( z) ] u( z, t) = v( z, t) [ ]{ } ( ne N ( z) η( t) ) t Nu N N N3 N4 = = N v N N N3 N4 (.) where N i,(i =,, 3, 4) is caed the transationa shape function matri are same as derived in previous section, which are given in equation (.89), { ( t) } ( ne) η is noda dispacement vector

45 636 anguar dispacement (it shoud be carefu noted that in previous section one pane motion was considered according in equation (.89) the order of stacking of dispacements are different, which can be chosen as per convenience, however, the mass stiffness matrices might take different form that shoud be taken care of) within the eement can be obtained b { } ( ne = η ) { } ( ne ϕ (, ) ( ) η( ) ) ϕ ( z, t) M ( z) ( t) ϕ z t = M z t (.) ϕ N M M M M ϕ = 3 4 N ϕ =M M M 3 M 4 In the matri form ϕ ( z, t) = ϕ ( z, t) [ ]{ } ( ne M ( z) η ) t where, [ M ( ) ] t M M M M M ϕ 3 4 = = M M M M 3 M 4 ϕ?? is caed the rotationa shape function matri. M i, i =,, 3, 4 are given in equation (.9)..4.3 The Weak Form Finite Eement Formuations Since in the previous section, aread the weak form finite eement formuation without groscopic coupe effect has been performed in great detai, hence, for the present case the same is described brief. Negecting the work done b eterna forces substituting equations (.9) (.) in equations of motion (.7), residues can be given as R = ρau k AG( u + ϕ ) ( e) ( e) ( e) ( e) sc R = ρav k AG( v ϕ ) ( e) ( e) ( e) ( e) sc