Natural oscillation frequencies for arbitrary piping systems Janez Gale, Iztok Tiselj “Jožef Stefan” Institute Jamova cesta 39, SI-1000 Ljubljana, Slovenia
[email protected],
[email protected] ABSTRACT This paper concerns axial and lateral oscillations and oscillation frequencies of various empty (natural frequency of oscillation) and fluid filled (forced oscillation) piping systems. Forced oscillations of fluid filled piping systems and corresponding exchange of energy are denominated also Fluid-Structure-Interaction (FSI). Oscillations appear due to various external or internal impacts and are successfully described with eight-equation physical model for simulations of FSI during fast transients. The physical model is solved with characteristic upwind numerical method and is compiled into a computer code. Simulations were compared to the analytical solutions or solutions from the literature whenever was possible. Discussion on results and problems encountered is given. The proposed physical model gives accurate results, and it enables evaluation of natural frequency of arbitrarily loaded, arbitrarily shaped and arbitrarily supported piping systems. Piping systems are rarely empty, thus forced oscillations due to FSI effects were observed and simulated. Application of various fluids pointed out importance of the fluid’s compressibility on pipe’s axial oscillations. 1
INTRODUCTION
Sinusoidal natural frequency of axial oscillation of a mass hanging on a ‘massless’ spring is a trivial problem known to every student on technical study. This simple problem becomes quite complicated if one needs to consider also mass of the spring. Even more, if there is no hanging mass on the spring, what would be the trace of the oscillating ‘heavy’ spring’s endpoint due to initial disturbance? Note that the trace is not sinusoidal but sharp (saw pattern). Oscillations can be also lateral; an example is a cantilever (beam) with end mass. In practice, closed arbitrarily shaped and arbitrarily supported piping system, empty or filled with fluid, represents an oscillating beam. Valve, pump, end cap or measuring equipment could represent a mass clamped on that beam. The oscillations are initiated by removing a support, by adding load (accidental fall of equipment, pipe break), by start up of the pump, by earthquake, by other vibrating machinery, by valve closing, by steady state flow, by fluid transient inside the pipe, etc. The objective of this paper is to validate our eightequation physical model for evaluation of the natural frequency of oscillation of arbitrarily shaped, arbitrarily supported and arbitrarily loaded piping systems. Pipelines and piping systems are under consideration because they provide transport for a wide range of substances (water, chemicals, petrochemicals, etc.) and they fulfil safety functions (cooling systems in NPP). Pressure pulsations and mechanical vibrations affect performance and safety. Failure of piping systems can have disastrous effects, leading to 218.1
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injuries and fatalities as well as to substantial cost to industry and the environment. Symptoms include vibrations, noise and fatigue damage to piping systems, supports and machinery. Tijsseling [1] stated that FSI is not a widely recognized phenomenon and it is quite possible that it is responsible for a significant number of unexplained piping failures and other unacceptable behaviour. For example, failure due to fatigue could in fact be FSI induced; failure due to corrosion could again be partially attributed to FSI. Tijsseling cited Wylie [2] who stated that consideration of the FSI during piping system design phase is obliged for situations with high safety requirements that are mostly encountered in nuclear and chemical industry. Appropriate FSI analysis followed by appropriate design and definition of the optimal operating procedures is the best prevention against detrimental effects of the FSI. The FSI analyses may be useful also in post-accident analyses [3]. FSI analyses are very complicated, costly, time and processor consuming. There are no reliable and generally accepted criteria for estimation of importance of the FSI except the one proposed by Lavooij and Tijsseling [4]. They discussed natural frequencies of oscillation and, for the case of rapid valve closure in single elbow Tank-Pipe-Valve system, showed that FSI is important if the period of natural oscillation is smaller than period of the pressure wave in the fluid. The FSI can be intense if the frequency of the dynamic load is close to the natural frequency of the pipe (resonance). Natural frequency of oscillation is therefore an important indicator how feasible and intense FSI could be. The problem that arises here lays in the lack of procedures for evaluation of natural oscillation properties for arbitrarily shaped and supported heavy piping systems with or without additional mass. Arscott [5] claims that simple axial oscillation of light springs carrying loads in various combinations has a well-established place in all mechanics textbooks, but none of the standard works appear to deal thoroughly with the oscillation of a spring (pipe) that’s mass is not negligible. He gave solution of a problem for the unloaded spring (pipe) oscillation around equilibrium due to arbitrary excitation mechanism. However, the approach presented in this study is very general and enables consideration either arbitrarily distributed point masses only, either arbitrarily heavy piping system only, either both. Applied physical model is eight-equation system of linear first order PDEs that enable junction and Poisson coupling of fluid and structure. The thermodynamic state is approximated by constant state properties of the fluid (constant density, bulk modulus, temperature, speed of sound, etc.). The model is solved with high resolution characteristic upwind numerical method, that is based on Godunov’s methods. The physical model and numerical method are described in references [6] and [7]. The present study points out that the model that enables full FSI simulations enables also study of natural oscillations of arbitrary piping systems. 2
STRUCTURAL OSCILLATIONS
The frequency of a wave f, also oscillation or motion, is a rate at which the motion repeats. The frequency f is given in terms of the angular frequency ω or in terms of the period T (period is a time at which the oscillation is repeated) by: f =
ω 2π
or
f =
1 T
(1)
The natural frequency fn is a property of the structure and it is defined as frequency of oscillation of the structure in case, that the load is rapidly removed from the structure. Every natural oscillatory motion is damped because it is not possible to eliminate energy losses due
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to non-conservation forces like friction, viscosity, resistance etc. The fundamental equation of structural dynamics also known as equation of damped simple harmonic motion reads: ∂ 2w ∂w +β + ω 2w = 0 2 ∂t ∂t
(2)
where w is displacement (axial or lateral), β is the damping constant, and ω is the angular frequency. The equation is analytically solvable, but only for some certain simplified combinations of the piping structure and load. Harris and Crede [8] discussed natural frequencies prominently in their Shock and vibration handbook. Table 1 shows a compilation of frequently used expressions for the natural frequencies of common (simple) piping systems, where m stands for concentrated mass or load, mb stands for total mass of the pipe and mu stands for pipe mass per unit of length. The cross-section shape in considered cases is not relevant, therefore the equations in Table 1 hold also for empty piping systems. The presence of a fluid inside the pipe influences the natural frequency in a way that is not trivially predictable and differs from case to case. Table 1: Angular natural frequency for various piping systems [5, 8]. Description Fixed-free cantilever pipe (axial) Fixed-free cantilever pipe (lateral)
m
mb,mu L
L
mb,mu
m b,m u L/2
SE mL
ωn =
SE ( m + 0.333mb ) L
ωn =
3EI mL3
ωn =
3EI ( m + 0.23mb ) L3
ωn = 3.52
EI mu L4
ωn = 4
3EI mL3
48EI ( m + 0.5mb ) L3
ωn = 9.87
EI mu L4
ωn = 8
3EI mL3
m
L/2
Fixed-fixed pipe (lateral)
ωn = m
mb,mu
Pinned-pinned pipe (lateral)
3
Angular natural frequency Massless pipe Massive pipe Massive pipe with load with load without load
Geometry
L/2 m
L/2
ωn = ωn =
ωn =
1 4L
E
ρt
196EI EI 3 ω n = 22.4 4 m 0.375 m L + m ( b) uL
AXIAL OSCILLATIONS OF EMPTY PIPE
The problem of axially oscillating empty pipe was studied on a simple straight cantilever piping system schematically presented in Figure 1. The system can move in axial direction. At time zero, steady axial force Fx = 10 kN starts to act on a pipe’s endpoint and the pipe’s endpoint starts to oscillate around new equilibrium position. Note, that there is a difference between force and mass (load) i.e. force affects only amplitude of the oscillation, while mass affects amplitude and frequency. Thus, the empty pipe oscillates with (unloaded) natural frequency regardless the size of the force Fx. Arscott [5] gave exact solution of the problem of axial oscillation without damping for the unloaded heavy spring. The solution is valid also for axially oscillating empty piping systems without added load. The point at position s along the pipe oscillates with amplitude a around equilibrium due to some initial disturbance in accordance with equation: w x ( s, t )=
∞
8a
∑ π2
n =0
⎛ πt πs ⎞ ⎛ sin ⎜ ( 2n + 1) ⎟ cos ⎜⎜ ( 2n + 1) 2L 2L ⎠ ( 2n + 1) ⎝ ⎝
( −1)n
2
E ⎞ ⎟ ρt ⎠⎟
(3)
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Figure 2 shows oscillation of the endpoint obtained with three different approaches: (i) with our code where pipe is divided into N = 640 computational nodes, (ii) with structural dynamics code Abaqus/Explicit based on Finite Element Method, and (iii) with Arscott’s analytical solution. It is obvious that there are no differences. Figure 2 shows that axial oscillations of the real cantilever are not sinusoidal but sharp with ‘saw’ trace. The simulated frequency of oscillation is fsim = 626 Hz which is very close to the natural frequency of the unloaded cantilever f = 625.8 Hz. As already mentioned, every natural oscillatory motion is damped because it is not possible to eliminate energy losses due to non-conservation forces. Thus, oscillations of the cantilever piping system slowly become sinusoidal as time evolves because of damping. Several mechanisms introduce numerical error into simulation. The effect is known as numerical dissipation. Numerical dissipation essentially emulates natural damping. Figure 3 shows, that one of the mechanisms that affect numerical dissipation is number of computational volumes. It is evident that denser grid reduces numerical dissipation. Figure 4 shows oscillations of the endpoint for case with maximized numerical damping. Comparison of the simulation to the analytical solution of equation for damped simple harmonic motion (2) shows that numerical dissipation obeys the analytical law of damping. Oscillations for the case in Figure 4 are damped where damped angular natural frequency ωd = 3933.3 rad/s, damping constant β = 100.9 rad/s, and amplitude a = 0.1337 mm. 0.35
Endpoint oscillations [mm]
x Fx
St E L 2
St = 5.938E-4 m , E = 200 GPa,
N = 640
Abaqus/Explicit
0.3 0.25 Detail
0.2 0.33
0.15 0.325
0.1 0.32
0.05
ρt = 7980 kg/m3, L = 2 m, S = 5.938E-4 m2
Analytical
0 0
0.5
1
1.5
2
2.36
2.5
2.38
2.4
3
2.42
2.44
3.5
4
Time [ms]
Fig. 1: Geometry of the fluid-filled pipe loaded with axial force Fx. N = 10
N = 40
N = 160
N = 10
N = 640
0.25 0.2 0.15 0.1 0.05 0 0
ae-βtsin(ω t-θ) d
0.25
0.3
Endpoint oscillations [mm]
Endpoint displacement [mm]
0.35
Fig. 2: Axial oscillations of the endpoint: comparison between our code, Abaqus and analytical solution.
0.5
1
1.5
2
Time [ms]
2.5
3
3.5
4
Fig. 3: Axial oscillations of the endpoint: influence of grid refinement.
0.2
0.15
0.1
0.05
0 0
0.002 0.004 0.006 0.008
0.01
Time [s]
0.012 0.014 0.016
0.018
0.02
Fig. 4: Axial oscillations of the endpoint: comparison to damped sinusoidal oscillation.
As natural damping, also numerical dissipation finally stops any movement of the piping system. From basic equations of statics of structures, it is possible to evaluate pipe’s endpoint equilibrium extension due to axial force. Analytically evaluated extension is wx = 0.168 mm, which is 0.004 mm (2.3 %) more than simulated wx,sim = 0.164 mm. Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, Sept. 10-13, 2007
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4
AXIAL OSCILLATIONS OF FLUID-FILLED PIPE
The considered transient in a closed fluid-filled cantilever piping system is an extension of the empty pipe transient presented in previous section. Figure 5 schematically shows geometry and properties of the pipe, and Table 2 collects important thermodynamic state properties for all applied fluids (air, water, methanol and dichlorodifluoromethane – R12). Fluid thermodynamic properties are constant during simulation and are valid for fluid at room pressure (p = 1 bar, except R12: p = 7 bar) and room temperature (T = 293 K). The cantilever pipe at rest is instantaneously loaded in axial direction with axial force Fx = 10 kN. The transient (and equilibrium) are controlled with properties of the both, the pipe and the fluid. x
Table 2: Properties of fluids at T = 293 K and p = 1 (7) bar Fluid ρ [kg/m3] K [Pa] c [m/s]
Fx
St E L 2
St = 5.938E-4 m , E = 200 GPa, ρt = 7980 kg/m3, L = 2 m, S = 5.938E-4 m2
Fig. 5: Geometry of the fluid-filled pipe loaded with axial force Fx.
Oscillations of the remote end [mm]
Pressure at remote end [MPa]
0.6 0.4 0.2 0 -0.2 -0.4 Air Water Methanol R12
-0.6 -0.8 -1 0
0.5
1
1.5
2
Water Methanol R12* Air
2.5
3
3.5
4
4.5
5
Time [ms]
Fig. 6: Pressure history in various fluids near the remote end of the cantilever pipe.
998.2 791.1 1330 1.204
2.009E9 0.987E9 0.379E9 1.418E5
1418,9 1116,7 533,6 343,2
Air Water Methanol R12
0.3 0.25 0.2 0.15 0.1 0.05 0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time [ms]
Fig. 7: Oscillation history of the remote end of the cantilever pipe.
Figure 6 shows relative pressure history near the remote end for the first 5 milliseconds of the transient for all fluids and Figure 7 shows pipe’s extension history at the same position and time period. Figures 6 and 7 and comparison of equilibrium extension wx and fluid pressure p between simulation and analytical values in Table 3 show that: • • •
The energy form the loaded pipe is transported through FSI coupling into the fluid. The pressure rise/drop is increased and amplitude of endpoint oscillation is reduced. The inertia effects of the mass of the fluid (density) are not the most important and are not trivially related to FSI. Proof: the heaviest fluid (R12) is not the most influential. Casadei [9] recommended FSI analysis if the fluid is incompressible. Figures 6 and 7 show that intensity of the FSI directly depends on stiffness i.e. bulk modulus. The effects of the FSI are most evident and intense in low compressible water-filled piping system and less evident in a very compressible air-filled piping system. The transient depends on superposition of several properties related to the fluid and the pipe like speed of sound, geometry, density, and characteristic period of the transient, but compressibility is the most important.
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•
Less compressible fluids exhibit higher level of numerical oscillation damping. This effect is not evident from Figures 6 and 7. Equilibrium extension is smaller and equilibrium fluid pressure change due to change in internal volume of the fluid (axial force) is larger in less compressible fluid. Comparison of simulated equilibrium to analytical solution exhibits high accuracy and reliability of the applied mathematical model and numerical technique.
• •
Table 3: Equilibrium extension and pressure Simulation
Fluid
Water Methanol R12 Air
5
Analytical
Relative difference
wx [mm]
p [Pa]
wx [mm]
p [Pa]
Δwx [%]
Δp [%]
0,16635 0,16691 0,16709 0,16715
-56942 -30280 -12105 -4,5
0,16658 0,16704 0,16736 0,16758
-56864 -30405 -12323 -4,8
-0,14 -0,08 -0,16 -0,25
0,14 -0,41 -1,77 -5,77
LATERAL OSCILLATIONS OF EMPTY PIPE
Stokey in a handbook of Harris and Crede [8] gave analytically equations for frequencies of lateral oscillation based on exact method, lumped parameter solution and solution by Rayleigh’s method for numerous combinations of vibrating beams, plates, masses and supports. Derivation of these analytical solutions is usually based on assumption of symmetric load and symmetric supports, which in turn actually enables analytical solution. As a consequence, tables in literature that collects natural lateral frequencies refer to idealized symmetric cases with centralized masses and are thus useless for real structures. Simulation of cases in this section points out that our eight-equation model gives an elegant numerical way to estimate frequency of arbitrarily loaded and asymmetrically supported piping systems. m = 100 N at L/3 Pinned supports
m = 100 N at L/2 Pinned supports
m1
m
m
-0.4 -0.6 -0.8
0.01
0.02
0.03
0.04
0.05
Time [s]
0.06
0.07
0.08
0.09
Fig. 8: Geometry and lateral oscillation of the midpoint for beam with centric mass and equivalent mass.
0.1
Centric mass Eccentric mass
0
Right endpoint displacement [mm]
0 -0.2
-0.2 -0.4 -0.6 -0.8 -1 0
m3
L
0.1
0.2 Mass Force
Midpoint displacement [mm]
Midpoint displacement [mm]
0.2
m2
s
L
L
-1 0
m = {10, 5, 30} kg at s = {0.2, 1.2, 1.8} m Pinned supports at s = {0.4, 1.4} m
0.01
0.02
0.03
0.04
0.05
Time [s]
0.06
0.07
0.08
0.09
Fig. 9: Geometry and lateral oscillation of the midpoint for beam with eccentric mass is compared to centric mass case.
0.1
0.05
0
-0.05
-0.1
-0.15 0
0.05
0.1
0.15
0.2
0.25
Time [s]
Fig. 10: Geometry and right endpoint oscillation for arbitrary beam with eccentric mass.
Figures 8 and 9 show two cases with simple pinned-pinned beam with centric (Lm = L/2) and eccentric (Lm = L/3) mass. All beams in this section are empty and have the same properties: cross-section area St = 5.938E-4 m2, moment of inertia (second moment of area) It = Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, Sept. 10-13, 2007
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2.952E-5 m4, Young’s elasticity modulus E = 200 GPa, density ρt = 7980 kg/m3 and length L = 2 m. The transient is induced at time zero, when load instantaneously burdens the structure and
the structure starts to oscillate. Figure 8 shows oscillation of the midpoint of a pinned-pinned beam (or pipe) loaded with centric force Fy = 100 N, and comparison to the case with mass that is equivalent to the force Fy. Again, external forces in FSI simulations are needed to excite the structure, but they do not influence natural frequency of the unloaded structure since forces have no inertia. Analytical natural frequency of lateral oscillations of a pinned-pinned unloaded beam is f = 43.5 Hz, while simulation gives fsim = 43.7 Hz . Similar accuracy was obtained for the case with centric mass: analytics gives fcm = 24.5 Hz, and simulation gives fsim,cm = 24.8 Hz. The natural frequency of oscillation with centric mass is decreased in comparison to unloaded beam. Additional results show that other variables like shear forces or bending momentum are the same for both cases. Figure 9 shows comparison of oscillations of a beam with centric and eccentric mass. The amplitude and bending of the midpoint are reduced while frequency is increased from fsim,cm = 24.8 Hz to fsim,ecm = 28.5 Hz for case with eccentric mass. Figure 10 shows the case with arbitrary supports and masses, where oscillation of any point along the beam is a complicated function of load, beam’s stiffness and supports. Figure 10 shows oscillations of the right endpoint at s = L. Fy1
Fy2 q
Fy
c1
Fy
c L
c2
L
L 0.05
-2 -3 Initial state Analytical solution Simulation
-4 0
0.2
0.4
0.6
0.1
-0.05 -0.1 -0.15 -0.2 -0.25
Initial state Analytical solution Simulation
-0.3 0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
Length [m]
1
1.2
1.4
1.8
2
0
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
50
-80
50
0
-50
-100 0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-100 0
2
Analytical solution Simulation 0.2
0.4
0.6
Length [m]
0.8
1
1.2
1.4
1.6
1.8
0
-50
-100 2
0
100
0.8
1
1.2
1.4
1.6
1.8
2
1.4
1.6
1.8
2
10
0.8
1
1.2
Length [m]
1.4
1.6
1.8
2
50
Moment [Nm]
Moment [Nm]
-150
0.6
0.6
0
-100
0.4
0.4
Length [m]
Analytical solution Simulation -50
0.2
Analytical solution Simulation 0.2
Length [m]
0
Moment [Nm]
0.2
Length [m]
Shear force [N]
-60
-200 0
Initial state Analytical solution Simulation
-0.1 -0.15
100
-40
-120 0
1.6
0 -0.05
Analytical solution Simulation
Shear force [N]
Shear force [N]
0.8
0.05
Length [m]
0 -20
Bending [mm]
-1
-5
0.15
0
Bending [mm]
Bending [mm]
0
0
-50
-100 0
0.4
0.6
0.8
1
1.2
Length [m]
1.4
1.6
1.8
-20 -30 -40
Analytical solution Simulation 0.2
-10
2
-50 0
Analytical solution Simulation 0.2
0.4
0.6
0.8
1
1.2
Length [m]
Fig. 11: Bending, shear force and Fig. 12: Bending, shear force and Fig. 13: Bending, shear force and moment in cantilever due to static moment in beam due to static moment in cantilever beam due to force at endpoint. forces and gravity. static force at endpoint.
Additional examples are presented in Figures 11, 12, and 13 where beams have the same geometric properties as beams studied previously. Beams have loads at various positions: beam’s length is L = 2 m, c = c2 = 1.6 m, and c1 = 0.6 m. The load, i.e. external lateral force for cases in Figs. 11, and 13 is equal to Fy = 100 N and the beam in Fig. 12 is burdened with two eccentric lateral forces, Fy1 = 20 N and Fy2 = 50 N, and distributed force due to gravity q = 46.51 N/m. The natural frequency of oscillation of the (unloaded) cantilever in Fig. 11 is fsim Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, Sept. 10-13, 2007
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= 15.6 Hz, which is close to analytical solution f = 16.0 Hz, the natural frequency of the midpoint of the (unloaded) beam in Fig. 12 oscillates with frequency fsim = 44.1 Hz, while analytical frequency of oscillation is f = 43.5 Hz. Each movement is damped until equilibrium
is reached. Figures 11, 12, and 13 show equilibrium bending, shear force and bending momentum and all simulated variables (blue dots) perfectly match with analytical solution depicted with red line. 6
CONCLUSIONS
Axial and lateral oscillations and oscillation frequencies of various empty (natural frequency of oscillation) and fluid filled (forced oscillation) piping systems were successfully simulated and discussed. Analysis of a problem of forced oscillations of fluid filled piping systems was performed and the results showed that compressibility of the fluid indicates how feasible and intense is FSI during transient conditions. The analysis was conducted by application of several fluids with various thermodynamic state properties. Several piping systems were simulated and compared to the analytical results with great accuracy. The main conclusion is that applied eight-equation physical model for simulations of FSI during fast transients is a reliable and effective tool for estimation of natural frequencies of arbitrarily shaped, arbitrarily supported and arbitrarily loaded piping systems, that can be fluid-filled or empty. ACKNOWLEDGMENTS The Ministry of Higher Education, Science and Technology of the Republic of Slovenia and Reactor Engineering Division of Jožef Stefan Institute supported this work. REFERENCES [1] Tijsseling, A. S., 1993, Fluid.structure interaction in case of waterhammer with cavitation, Ph.D thesis, Delft University of Technology, Faculty of Civil Engineering. [2] Tijsseling, A. S., 1996, “Fluid-Structure Interaction in Liquid-Filled Pipe Systems: A Review“, Journal of Fluids and Structures, 10, pp. 109-146. [3] Obradović, P., 1990, “Fluid-Structure Interactions: an Accident which has Demonstrated the Necessity for FSI Analysis”, Transactions of the 15th IAHR Symposium on Hydraulic Machinery and Cavitation, Paper J2. [4] Lavooij, C. S. W., Tijsseling, A. S., 1991, “Fluid-Structure Interaction in Liquid-Filled Piping Systems“, Journal of Fluids and Structures, 5, pp. 573-595. [5] Arscott, F. M., 1955, The oscillation of a heavy spring, The Mathematical Gazette, 39 328: 126-131. [6] Gale, J., Tiselj, I., 2006, Eight Equation Model for Arbitrary Shaped Pipe Conveying Fluid, Proc. of Int. Conf.: Nuclear Energy for New Europe 2006. [7] Gale, J., Tiselj, I., 2008, Godunov's method for simulations of fluid-structure interaction in piping systems, ASME Journal of Pressure Vessel Technology (final manuscript approved). [8] Harris, C. M., Crede, C. E., 1976, Shock and wibration handbook, McGraw-Hill book company, 2nd edition. [9] Casadei, F., Halleux, J. P., Sala, A., Chille, F., 2001, “Transient Fluid-Structure Interaction Algorithms for Large Industrial Applications“, Computer Methods in Applied Mechanics and Engineering, 190, pp. 3081-3110.
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