Representation o Coriolis orces and simulation procedures or moving luid-conveying pipes Jörg Gebhardt*, Frank Kassubek** *ABB Corporate Research Center Germany, Department or Automation Device Technologies, Wallstadter Straße 59, 6856 Ladenburg, Germany **ABB Schweiz AG, Corporate Research, Department or Inormation Technologies, CH-5405 Baden-Dättwil, Switzerland Abstract: For inclusion o inertia eects o moving luid columns or strings into structural mechanics simulations, a combined inite-element modelling scheme is proposed in ABAQUS. The concept has been realized in the orm o user subroutines (UEL) o beam type. These can be coupled to existing standard (e.g., Euler-Bernoulli or Timoshenko) elements. The proposed Coriolis elements can be applied or, e.g., prediction o loads rom high-speed lows and phase shits o oscillating luid-conveying pipe systems with arbitrary complex designs. The modelling scheme also allows a lexible coupling to shell or solid inite-element models. Keywords: inite elements, simulation, vibration, inertial orce, Coriolis orce, luid-structureinteraction, multi-physics, mass low sensors. 1. Introduction In this short note inclusion o inertia eects or luid conveying pipes into inite element structural mechanics simulations is discussed. Fluid structure interaction is a complex and very interesting ield. Here we discuss a rather simple system. Consider a (possibly curved) pipe with some lateral delection w(x,t) conveying a luid. The luid motion will inluence the pipe delection in two ways: irstly, i the luid moves on a curved path, a centriugal orce will occur. Secondly, as the luid at each position x along the pipe has approximately a transverse velocity corresponding to the velocity o the pipe (neglecting compressibility eects here), the low along the tube axis will lead to the transport o momentum, i the velocities w t (x,t) along the tube axis are not constant. Both orces are inertial orces, proportional to the mass density per length o the luid. They are also o technical importance: so-called Coriolis low meters use the latter orce or a very precise measurement o mass low. They have been termed Coriolis meters, as the second inertial orce described above is the Coriolis orce in the case o a curved or straight luid- 009 SIMULIA Customer Conerence 1
conveying pipe with position-dependent transverse velocities. This is most evident in the case o pipe oscillations which are locally perpendicular to the osculating plane o a curved pipe. Let us now discuss the eect quantitatively. For a straight tube illed with a luid which may move with velocity v in axial direction, lateral delection w(x,t) is described by the dierential equation (e.g.: Garnett et al., 1993; Raszillier,1991) A A w EIw Fw A v w A vw 0. tt xxxx xx Here, and are the densities o the tube material and luid, respectively. E is the Young s modulus o the tube material, I the tube cross section s moment o inertia. A and A are cross section areas o pipe wall and luid column, respectively. F is the orce deining axial pre-stress. The equation o motion is easily derived rom the Langrangian L o the tube and L o the luid xx tx A 0 EI L T V w wt A v L T V wt v wx xx F, w x, which are deined as the dierence T - V o kinetic and potential energy. First and second row in the vector notation give axial and transversal velocity component, respectively. In the ollowing, we show that the term A w tt can be described realistically with acoustic elements in Abaqus. Furthermore, the centriugal orce term A v w xx and the Coriolis term A v w tx are modelled by two Abaqus user subroutines o type UEL (Dassault Systémes, 008a, ch. 7.16, and Dassault Systémes, 008b, ch 1.1.).. Acoustic elements or luid inertia The most straightorward method or inclusion o luid mass inertia in vibration calculations o a pipe is adding nonstructural mass to the walls o the pipe. This method, as it turns out, gives suiciently accurate results or pure bending modes. For modes including rotational motion about the pipe s longitudinal axis, however, in most cases the luid almost does not take part in the rotation. The inluence o luid inertia on pipe motion, i modelled by nonstructural mass attached to the tube walls, is overestimated in such cases. A realistic model o luid inertia or luids with moderate viscosity should give the correct result or bending motion without adding inertia to axial rotation. We have tested (Gebhardt, 008) the suitability o Abaqus acoustic elements or this kind o modelling in the case o standard titanium tubes which are illed with luids o variable density. 009 SIMULIA Customer Conerence
Calculations with Abaqus showed that, indeed, a luid-column model built rom acoustic elements can accurately account or inertia in bending modes. Deviations rom a Timoshenko beam model, which is used as reerence, are less than 0.4 % or the modes considered here. On the other hand, dierent rom the model with nonstructural mass, representation o the luid as acoustic medium in Abaqus eectively decouples rotations o luid and tube around the tube axis. In the latter case tube rotation requency remains independent rom luid illing, whereas an eective-mass ormulation strongly aects torsional mode requencies. For simulation o a inite boundary layer in cases o higher luid viscosity, both approaches may be combined. 3. User element or centriugal orces For a general bent pipe illed with a moving luid, centriugal orces have two consequences. First, they cause static loads on all sections o the tube which have non-vanishing curvature in the base state. Second, they have to be considered as solution-dependent orces on those tube sections which show a curvature ater delection. An Abaqus user element can account or both o these eects. User subroutine UEL provides, in particular, the variables coords(n) and u(n) or base state node coordinates and delections, respectively. Looking at the discretized system o equations or the node delections u in static and dynamic analyses M u tt C u K u F, t (where M, C and K denote mass, damping and stiness matrix, respectively) we note that the centriugal load gives additions to the stiness matrix and to the right-hand side vector o external orces. The UEL is written as a three-node element as depicted in Figure 1. ext 009 SIMULIA Customer Conerence 3
Figure 1. Geometry o the centriugal orce element. In its simplest orm, suiciently accurate or many applications, the element acts on the corresponding nine translational degrees o reedom, and its addition K UEL to the stiness matrix reads. The contribution to F ext, which in Abaqus is accessible as the variable rhs(n), is deined analogously.the contribution to the right-hand side vector has, e.g., successully been tested (Gebhardt, 008) or a rotating ring structure, reproducing the correct tangential principal stress v, where and v denote the ring material s density and tangential velocity, respectively. 4 009 SIMULIA Customer Conerence
Correctness o the stiness contribution has been checked through the requency decrease o the lowest eigenmode o a standard stainless steel pipe illed with water. Results have been compared to Euler-Bernoulli theory or the ground mode shit with pinned-pinned boundary conditions, where an analytical ormula can be derived and used as reerence. Numerical data show good agreement with theory even or rather extreme luid velocities, where the model is shited considerably towards the instability o the physical system itsel. 4. User elements or Coriolis orces The user element geometry is given im Figure. Figure. Geometry o the Coriolis orce element. The Coriolis orce on an element is given by F A v v v, C luid luid luid 1 where the bracket with subscript denotes the components perpendicular to the element s longitudinal axis. The orce has been coded as rhs vector in an Abaqus user subroutine o type UEL. The UEL has been designed or use in the *Dynamic, direct procedure. This procedure has been chosen because it oers, besides the ability to call user elements, the opportunity to speciy heuristic damping or the complete model via *Damping, alpha=, beta=. 009 SIMULIA Customer Conerence 5
For veriication o the element ormulation, we chose a running steel wire (with Young s modulus E = 00 GPa, and density = 7940 kg/m 3 ) o length m and cross-section radius 1 mm. It has been deined between the end points x1 1m with pinned-pinned boundary conditions. Damping has been speciied by alpha=0 and beta=0.01. This produces quality actors Q i or the systems i th eigenmode with angle requency i as ollows: Q i 1 So, since the system s lowest eigenrequency 0 is about 6. Hz, the quality actor Q 0 assumes a value o about 17. In order to get stable steady state solutions or a situation where mainly the ground state is driven by a periodic orce, thereore, we have to integrate the system rom arbitrary starting conditions or a time T Q convergenc e 0. 0 i. The system is driven periodically at the resonance requency 0 by a lateral orce (F = 10-4 N) in the middle o the tube. The steady state under inluence o Coriolis orces can be expected to be seen ater 0 30 s integration time. Testing o the element, now, has been done by the study o induced relative phase shits between two given points on the wire (here chosen as m ). x R L 0. 56 Analytical investigation o the system (Kassubek, 004) leads to closed expressions or the expected phase unction along the wire in irst order perturbation theory. Reducing the dynamics, additionally, to a two-level-system o the ground mode and irst harmonic, we get: 3 v x) arctan 3 L ( luid 0 sin x 1 01 L The phase shits extracted rom Abaqus calculations are compared to analytical results in Figure 3. The analytical exact results have been obtained by imposing the boundary conditions on the general solution o the Euler-Bernoulli equation at given requency (with low), leading to nontrivial solutions only or a discrete spectrum. The inite element result is compared to the lowest eigenmode. 6 009 SIMULIA Customer Conerence
Figure 3. Veriication o the Coriolis orce element. The phase dierence between the points x L R =± 0.56 m is given, as a unction o dimensionless-scaled low velocity, or the Abaqus calculation ( numerical ), or irst order perturbation theory ( pt theory ) and or the analytical solution ( exact ). As can be seen, agreement is exceptionally good. Remaining small deviations may be caused mainly by damping inluences and the limitations o irst order perturbation theory with truncated spectrum. It should be mentioned that the exact solution takes also in account the centriugal orce, which leads to the slight non-linearity or large lows. We have veriied that it perectly agrees with the inite element solutions i the centriugal terms in the exact solution are neglected. A unit dimensionless low, as it is deined here, means 1/ o the critical low where system instability sets in. The tumbling wave unction caused by Coriolis orces can be viewed as real part o a complex solution i 0 w( t, x) a( x) i b( x) e t o the equation o motion. For illustration, in Figure 4 we have plotted or our system analytical solutions or a(x) and b(x) in the special case o zero drive and zero damping. In this case, a(x) is a symmetrical unction in x, whereas b(x) is anti-symmetrical. 009 SIMULIA Customer Conerence 7
Figure 4: Eigenunction vs position (in x/l). Let: Real part o the basic unction (dimensionless velocity vl/(a/ei) 1/ =0.1 ). Top right: imaginary part. 5. Conclusions We have seen that Abaqus may serve as a very lexible tool or modelling inertia o systems like pipes with luid low, conveyor belts or chain drives. It has been shown that Abaqus UELs can be applied or modelling Coriolis orces, which is slightly more intricate than modelling other inertia orces because it contributes to the damping structure rather than to the stiness or mass matrices. A drawback in currently available Abaqus versions: It is at least inconvenient, i not impossible, to model additions to the damping matrix directly by user subroutines, since the latter are currently not yet supported by the relevant procedures *Steady dynamics,direct, *Steady state dynamics,subspace and *Complex requency. A request or enhancement on this topic has been issued. In this note, a practical workaround has been presented: Using *Dynamic, direct, the vibrating system can be driven and integrated in time until the steady state is reached. O course, this procedure is computationally expensive or larger systems and in cases where damping is small. For this reason, and or convenient postprocessing o phase and damping data in vibrational systems, it would stll be very desirable to include the Coriolis eect into procedures or steadystate calculation or complex eigensolvers. Nevertheless, user elements as proposed in this note, together with advanced strategies or coupling and contact to general structures, give remarkable opportunities to study dynamical eects e.g. o pipe boundary conditions (Kassubek, 004) or o time-dependent low conditions (see, e.g. Gebhardt, 004; Gebhardt, 006). 8 009 SIMULIA Customer Conerence
6. Reerences 1. R.B. Garnett C.P. Stack and G.E. Pawlas, A Finite Element or thevibration Analysis o a Fluid-Conveying Timoshenko Beam. American Institute o AeroNautics and Astronautics, Report No. AIAA-93-155-CP, 1993.. H. Raszillier and F. Durst. Coriolis-eect in mass metering. Archive o Applied Mechanics, 61:19 14, 1991. 3. Gebhardt, J., Modelling inertia in luid-conveying pipes, Abaqus User Conerence, Bad Homburg, Germany, 008. 4. Kassubek, F., Use o perturbation theory or Coriolis low meters, ABB internal report, 003. 5. Dassault Systèmes, Abaqus Analysis User s Manual V. 6.8, 008a. 6. Dassault Systèmes, Abaqus User Subroutines Reerence Manual V. 6.8, 008b. 7. Kassubek, F., Gebhardt, J., Friedrichs, R., Keller, S., Inluence o boundary conditions on Coriolis low meter measurement, in: VDI-Berichte 189 Sensoren und Meßsysteme, Ludwigsburg, Germany, 004. 8. Gebhardt, J., Kassubek, F., Time scales in vibrational systems, in: VDI-Berichte 189 Sensoren und Meßsysteme, Ludwigsburg, Germany, 004. 9. Gebhardt, J., Modelling and perturbation theory o dynamical eects in a Coriolis low sensor, 13.ITG/GMA-Fachtagung Sensoren und Meßsysteme, Freiburg, Germany, 006. 009 SIMULIA Customer Conerence 9