Moon, F.C.

Wauer, Jörg

Moon, F. C.

KITopen

52 International Symposium on the Vibrations of Continuous Systems, July, 2007On Bending-Torsional Flutter of a Cantilever with Tip Fluid JetJo¨rg Wauer and Francis C. MoonInstitut fu¨r Technische Mechanik, Universita¨t Karlsruhe (TH), Germanyand Cornell University, Ithaca, NY, U.S.AIntroductionFrom the 1950s to the 1970s, stability problems of structural members subjected to follower forcesattracted very much attention. Meanwhile it seems to be clear (see [4], for instance) that many ofsuch problems are academic and structural members subjected to ﬂuid ﬂow loading belong to the fewrepresentatives of non-conservative mechanical systems of practical relevance.A special type of this problem class is a cantilevered bar subjected to a transverse follower forceof ﬂuid jet, which during the early 1970s (see [5,8], for example) was studied in many details. Theinvestigations were based on linear models taking into consideration some pre-deformation due tothe time-independent loading or on a simpliﬁed geometrically non-linear theory. In addition, thetreatment was focused on a special arrangement producing the follower force of ﬂuid jet. Since i)a consistent geometrically nonlinear theory of elasticity for such problems is available now (see [6,9]where the academic problem of a cantilever subjected to follower tip moment was treated) and ii)other speciﬁcations of tip jets are interesting to be discussed, the described problem is re-consideredhere.In a ﬁrst step, a suﬃciently consistent formulation of the governing boundary value problemis presented, where two diﬀerent mechanisms of ﬂuid jet loads are discussed. The correspondingvariational equations are derived next where a non-dimensional notation is introduced. This non-dimensional formulation is the key to recognizing the important parameters inﬂuencing the stabilitybehavior and the ﬂutter load, in particular qualitatively but also quantitatively.Physical ModelConsider a slender beam of length L and mass per unit length µS = ρA (ρ mass density, A cross-sectional area) with narrow rectangular cross-section (thickness h height H) so that the smaller ofthe bending stiﬀnesses EI2 and the torsional stiﬀness GIT are much smaller than the other bendingstiﬀness EI1, see Fig 1a. All data are assumed to be constant. One end of the beam is rigidly ﬁxedand there is introduced a Cartesian reference frame {OeXeY eZ} with origin O coinciding with thecentroid S(Z = 0) of the cross-sectional area, where the unit base vectors eX ,eY correspond withthe symmetry axes of that cross-section and eZ is directed along the non-deformed bar axis. Due tothe deformations (displacements u, v, w and a torsional angle ϕ), the centroid S(Z) of a general cross-section located at the position ZeZ displaces to s with a corresponding changed orientation denotedby the body-ﬁxed reference frame {oexeyez} with origin o coinciding with location s of the deformedcross-sectional centroid, where ez, ey correspond with the symmetry axes of the actual cross-section(which itself remains undeformed) and ez is the outside normal of that cross-section.Two layouts of follower ﬂuid jet load are considered, see Fig. 1b and c. The ﬁrst one is that intro-a) b) c)Figure 1. Bar Model, Coordinate Systems and Layouts of Fluid Jet Load.duced by [5] and [8]. In this case, the bar has a uniform circular bore of certain suitable diameter alongFirst published in: EVA-STAR (Elektronisches Volltextarchiv – Scientific Articles Repository) http://digbib.ubka.uni-karlsruhe.de/volltexte/1000011253 53International Symposium on the Vibrations of Continuous Systems, July, 2007the Z-axis (in un-deformed state), through which an incompressible ﬂuid (e. g., water) at constantspeed U0 is ﬂowing. A nozzle is connected at the free end of the beam with the hole such that theﬂuid leaves the system in the deformed state at the centroid s(Z = L) in the form of a jet followingthe end cross-section in its actual state shooting into the ey(Z = L)-direction.The second speciﬁcation is geared to that one suggested by Como [2] and Wohlhart [10] here alsoat the tip end. To make their academic follower force practicable, a slender rigid attachment of massM and length H with a central hole as a model of a jet engine is appropriately ﬁxed at the endcross-section along ey(Z = L) where air is coming in with speed U0 at Y = −H/2 and exhaust gas isleaving with larger speed U1 > U0 (neglecting the fuel mass rate against the air ﬂow rate) at Y = H/2so that a transverse follower propulsion in the −ey-direction results.FormulationThe governing boundary value problems are derived based on the assumptions as follows: The ﬂuidis inviscid. Rotary inertia and shear deformation of the beam are neglected as well as gravity eﬀects.Then, Hamilton’s principleδ t1t0(T − V ) dt+ t1t0Wvirt dt = 0will be applied where T is the kinetic energy of the open system to be considered, V is the corre-sponding potential energy and Wvirt contains the virtual work of all non-conservative forces and thecontributions of mass transport over the open boundaries.Independently from the two follower force realizations, there are the energy contributionsTS = 12 L0µS(u2,t + v2,t + k2Sϕ2,t) dZ, k2S =I1 + I2AandVS = 12 L0EI2u2,ZZ + EI1(−v,ZZ + ϕu,ZZ)2 +GITϕ2,Z dZof the structural member together with its virtual work termWvirt,S = −de L0µS(u,tδu+ v,tδv + k2Sϕ,tδϕ) dZ− di L0(EI2u,ZZtδu,ZZ + EI1v,ZZtδv,ZZ +GITϕ,Ztδϕ,Z) dZ (1)characterizing external and internal damping in a linear formulation which is suﬃcient.Two alternative ﬂuid ﬂow contributions to kinetic energy and virtual work representing the distinctfollower force concepts have to be added. In case 1 where ﬂuid ﬂows through the central hole of thebar and leaves at the nozzle following the end cross-section into ey(L)-direction (here in such a formthat the ﬂuid speed within the nozzle remains unchanged), there is an additional kinetic energyT (1)F =12 L0µFu2,t + v2,t + 2U0(u,tu,Z + v,tv,Z) dZand a work termW (1)virt,F = −µFU0(u,t − U0ϕ)δu+ (v,t + U0)δvZ=L.If the shortening of the bar were to be taken into consideration, a supplement would occur that wouldnot signiﬁcantly modify the stability analysis.In case 2 (assuming that the ﬂuid mass within the attached engine is negligible compared to M)the kinetic energy of the jet engineT (2) = 12M(u2,t + v2,t) +MH212 ϕ2,tZ=Land the contribution of the ﬂuid jetW (2)virt,F = µFU0(u,t − U0ϕ)δu+ (v,t + U0)δv− U1(u,t − U1ϕ)δu+ (v,t + U1)δvZ=L54 International Symposium on the Vibrations of Continuous Systems, July, 2007have to be added. Also in this case, the shortening of the bar end (together with the inclination anglev,Z(Z = L)) can easily be taken into account.It is straightforward now to evaluate Hamilton’s principle for both cases to get the respectivegoverning boundary value problems.It will be mentioned that, for a consistent post buckling analysis or the examination of interactingoscillations in both lateral directions of the beam, a geometrically nonlinear theory of elasticity tothe cubic order (see [1], for instance) or a nonlinear elastica theory (see [3,7], for instance) should beapplied.Evaluation and ResultsIn a ﬁrst step, the steady deformation state is determined which is for both follower force realizationsa pure time-independent bending about the stiﬀer of the two main inertia lateral axes without torsion:u0(Z, t), ϕ0(Z, t) ≡ 0, v0(Z, t) = v0(Z) whereEI1v0,ZZ = −µFU20 (L− Z) (case 1) or EI1v0,ZZ = −µF(U21 − U20 )(L− Z) (case 2).Taking now solutions of the formu(Z, t) = u0 +∆u(Z, t), v(Z, t) = v0 +∆v(Z, t), ϕ(Z, t) = ϕ0 +∆ϕ(Z, t)and substituting them into the governing nonlinear boundary value problems, we get – linearizingin the ∆-quantities – the variational equations as the starting point for the stability analysis. Asexpected the coupled boundary value problem in ∆u and ∆ϕ constitute the lateral buckling problemwhile the other decoupled one in ∆v describes simple damped oscillations.Introducing non-dimensional variables and parameters, it becomes obvious that besides two char-acteristic damping coeﬃcients De and Di the stiﬀness ratio GIT /(EI2) together with the slenderness(kS/L)2 may drastically inﬂuence the eigenvalues as a function of the load parameter µFU20L2/√GITEI2(case 1) or µF(U21 −U20 )L2/√GITEI2 (case 2). The stability behavior will be discussed in detail where,in particular, some qualitative results not given in [5,8] will be presented.Acknowledgement The ﬁrst co-author is grateful to Professor F.C. Moon who invited him for asabbatical stay at Cornell University, NY, U.S.A. where some of the work originated.References1. Botz, M., Zur Dynamik von Mehrko¨rpersystemen mit elastischen Balken, Diss. TU Darmstadt, Ger-many, 1992.2. Como, M., Lateral Buckling of a Cantilever Subjected to a Transverse Follower Force, Int. J. SolidsStructures, Vol. 2, 515–523, 1966.3. Cusumano, J. P., and Moon, F.C., Chaotic Non-planar Vibrations of the Thin Elastica, Part I and II,J. Sound Vibration, Vol. 179, 185–208 and 209–226, 1995.4. Elishakoﬀ, I., Controversy Associated with the So-called “Follower Forces”: Critical Overview, AppliedMechanics Reviews, Vol. 58, 117–142, 2005.5. Kalmbach, C. F., Dowell, E.H., and Moon, F.C., The Suppression of a Dynamic Instability of an ElasticBody Using Feedback Control, Int. J. Solids Structures, Vol. 10, 361–381, 1974.6. Meyer, A., Tan, H., and Wauer, J., Die Wirkung a¨ußerer und innerer Da¨mpfung auf die Stabilita¨t desPrandtlschen Kragtra¨gers unter mitgehender Momentenbelastung, Ing.-Archiv, Vol. 54, 205–219, 1984.7. Plaut, R.H., Postbuckling and Vibration of End-supported Elastica Pipes Conveying Fluid and ColumnsUnder Follower Loads, J. Sound Vibration, Vol. 289, 264–277, 2006.8. Prasad, S.N., and Herrmann, G., Stability of a Cantilevered Bar Subjected to a Transverse FollowerForce of Fluid Jet, Ing.-Archiv, Vol. 39, 341–356, 1970.9. Wauer, J., Der Prandtlsche Kragtra¨ger unter konservativer und nichtkonservativer Momentenbelastung,ZAMP, Vol. 29, 334–340, 1978.10. Wohlhart, K., Dynamische Kippstabilita¨t eines Plattenstreifens unter Folgelast, Z. Flugwissenschaft,Vol. 19, 291–298, 1971.

On Bending-Torsional Flutter of a Cantilever with Tip Fluid Jet

https://publikationen.bibliothek.kit.edu/1000011253/783631

On Bending-Torsional Flutter of a Cantilever with Tip Fluid Jet

Abstract

Similar works

Full text

Available Versions

KITopen