- Email: [email protected]
Analysis of nonlinear shallow water waves in a tank by concentrated mass model
Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Analysis of nonlinear shallow water waves in a tank by concentrated mass model Satoshi Ishikawa a,n, Takahiro Kondou a, Kenichiro Matsuzaki b, Satoshi Yamamura c a b c
Department of Mechanical Engineering, Kyushu University, 744 Motooka Nishi-ku, Fukuoka, Japan Department of Mechanical Engineering, Kagoshima University, 1-21-40 Korimoto, Kagoshima, Japan OMRON Corporation, 211 Nishikusatsu, Kusatsu, Shiga, Japan
a r t i c l e i n f o
abstract
Article history: Received 13 October 2015 Received in revised form 12 February 2016 Accepted 16 February 2016 Handling Editor: L. G. Tham
The sloshing of liquid in a tank is an important engineering problem. For example, liquid storage tanks in industrial facilities can be damaged by earthquakes, and conversely liquid tanks, called tuned liquid damper, are often used as passive mechanical dampers. The water depth is less often than the horizontal length of the tank. In this case, shallow water wave theory can be applied, and the results indicate that the surface waveform in a shallow excited tank exhibits complex behavior caused by nonlinearity and dispersion of the liquid. This study aims to establish a practical analytical model for this phenomenon. A model is proposed that consists of masses, connecting nonlinear springs, connecting dampers, base support dampers, and base support springs. The characteristics of the connecting nonlinear springs are derived from the static and dynamic pressures. The advantages of the proposed model are that nonlinear dispersion is considered and that the problem of non-uniform water depth can be addressed. To confirm the validity of the model, numerical results obtained from the model are compared with theoretical values of the natural frequencies of rectangular and triangular tanks. Numerical results are also compared with experimental results for a rectangular tank. All computational results agree well with the theoretical and experimental results. Therefore, it is concluded that the proposed model is valid for the numerical analysis of nonlinear shallow water wave problems. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Vibration analysis method, Modeling, Shallow water wave, Sloshing, Concentrated mass model
1. Introduction The sloshing of liquid in a tank is an important engineering problem. For example, liquid storage tanks in industrial facilities can be damaged by earthquakes, as happened when the floating roof of a large oil storage tank was damaged in the 2003 Tokachi-oki earthquake, casing a fire to break out [1,2]. Conversely, liquid tanks, called tuned liquid dampers (TLDs), can be used as passive mechanical dampers to attenuate the vibration of tall buildings [3,4]. The water depth in these liquid tanks is often less than the horizontal length of the tank. This is the so-called shallow water condition. In this case, shallow water wave theory can be applied, and the results indicate that the surface waveform in a shallow excited tank exhibits complex behavior caused by nonlinearity and n Correspondence to: Department of Mechanical Engineering, Kyushu University, 744 Motooka Nishi‐ku, Fukuoka, Japan. Tel.: +81 92-802-3187; fax: +81 92-802-0001. E-mail address: [email protected] (S. Ishikawa).
http://dx.doi.org/10.1016/j.jsv.2016.02.029 0022-460X/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
dispersion of the liquid. One- and two-dimensional models are used for the numerical simulation of this phenomenon. Onedimensional models deal with only the horizontal equation of motion of a fluid, and two-dimensional models deal with twodimensional equations of a fluid in the horizontal and vertical directions. Regarding one-dimensional models, Chester [5] derived a theory that considers nonlinearity and linear dispersion and provided a steady-state one-dimensional solution for a shallow water wave in a horizontally excited tank. Chester and Bones [6] performed experiments with excitation near the resonance frequency and indicated the significance of nonlinearity and dispersion for a shallow water wave. Lepelletier [7] and Lepelletier and Raichlen [8] derived one-dimensional finite element methods from Boussinesq-type equations that consider linear dispersion. Shimizu and Hayama [9] derived one-dimensional fluid equations and simulated shallow water sloshing phenomena in a rectangular tank by using the difference scheme of the fluid equations. In this method, linear dispersion was modeled by changing the grid size. Therefore, the number of elements decreases as water depth increases, and the calculation is inaccurate when the water is deep under a shallow water condition. Sun et al. [3] and Fujino et al. [4] simulated nonlinear shallow water waves in a TLD by using Shimizu and Hayama’s method [9]. In these studies, the waveforms of the numerical results agree with the experimental results [6–9]. However, the slopes of the frequency responses do not correspond in the case of large amplitude, probably because these numerical methods consider only linear dispersion. The nonlinear dispersion needs to be considered in the case of large-amplitude waves or deep water even under shallow water conditions. Furthermore, the abovementioned one-dimensional calculation methods are complicated to derive. With regard to two-dimensional models, Faltinsen et al. [10] and Faltinsen and Timokha [11,12] derived multimodal methods based on a two-dimensional potential theory and simulated the sloshing phenomena in a rectangular tank under shallow and intermediate water depth conditions. Love and Tait [13] simulated sloshing phenomena in a TLD using Faltinsen et al.’s modal expansion techniques. However, the equations of this method are quite complicated. Antuono et al. [14,15] simulated the shallow water sloshing phenomena using a two-dimensional smoothed particle hydrodynamics (SPH) scheme. However, an SPH scheme is a particle method that requires considerable computation time. Nakayama [16] derived a boundary element method for shallow water waves in a TLD tank by using two-dimensional fluid equations. The computational cost of these two-dimensional methods is larger than that of one-dimensional methods. The numerical methods developed by Chester [5], Lepelletier [7,8], Shimizu and Hayama [9], and Faltinsen et al. [10–12] cannot be applied to non-uniform water depth problems because they assume a constant water depth. However, Deng and Tait [17] noted that a sloped-bottom tank is a more efficient TLD than a flat-bottom tank; therefore the non-uniform water depth problem is important. Moreover, numerical simulations conducted researchers other than Chester and Bones did not yield unstable solutions, and did not perform stability analysis. This study aims to establish a practical one-dimensional analytical model to analyze non-linear shallow water wave phenomena in a tank. This model consists of masses, connecting nonlinear springs, connecting dampers, base support dampers, and base support springs. The characteristics of the connecting nonlinear springs are derived from static and dynamic pressures, the latter of which also considers vertical motion of the fluid. The connecting nonlinear spring represents the nonlinear static restoring force and nonlinear dispersion. The base support damper and the base support spring are derived from the influence of shear stress from the bottom wall. The advantages of the proposed model are that nonlinear dispersion is considered, non-uniform water depth problems can be addressed, and model derivation is simple. Furthermore, the numerical simulation with the concentrated mass model is performed using the shooting method [18,19] to yield both stable and unstable solutions and to perform stability analysis. The shooting method is modified to apply the concentrated mass model. In this paper, a concentrated mass model is proposed. Next, we compare the linear natural frequencies in the concentrated mass model with the theoretical values for rectangular and triangular tanks to confirm the validity of the model for the non-uniform water depth problem. Then, an experiment on a rectangular tank is performed, and the experimental results are compared with the numerical results to confirm the model’s validity in a nonlinear region.
2. Concentrated mass model We consider one-dimensional nonlinear shallow water wave phenomena in a tank, as shown in Fig. 1. We consider the 2 condition that the amplitude of a water wave is too small to cause wave-breaking. When the Bond number Bo ¼ gl0 ρ=T s (g is the gravitational acceleration; l0 is the length of the tank; ρ is the density of the fluid; and T s is the surface tension) is larger than 104 , the effect of surface tension can be neglected [1]. We consider the case Bo 4104 and neglect the surface tension of the free surface.
Fig. 1. Coordinate system of water wave.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3
The fluid is modeled as a concentrated mass model that consists of masses, nonlinear connecting springs, connecting dampers, base support dampers, and base support springs, as shown in Fig. 2. We consider that the fluid is incompressible and is divided into N trapezoidal elements. The mass is concentrated at the nodal points. The density of fluid is ρ, length of element i ð ¼ 1; ⋯; NÞ in equilibrium state is li , the depth of water at nodal point i is hi , and the horizontal displacement at nodal point i is xi . The water level in element i is ηi . The width of the tank is taken as the unit length. Considering the mass of fluid in the shaded area in the equilibrium state in Fig. 3(a), the mass of each mass point, mi , can be written as mi ¼
ρ 8
li ðhi 1 þ 3hi Þ þ li þ 1 ð3hi þ hi þ 1 Þ
(1)
When the mass points at both ends of element i move, we assume the water level changes so that the volume of the element is conserved, as shown in Fig. 3(b). Then, using the displacements xi 1 and xi , ηi is given implicitly by
ηi ¼
hi xi þ hi 1 xi 1 xi xi 1 þ li
(2)
2.1. Nonlinear connecting spring We derive a nonlinear spring from the hydrostatic and hydrodynamic pressures when the water level changes. The vertical equation of motion of a small element at vertical coordinate y (shown in red in Fig. 3b) in element i is given by ∂vi ðyÞ 1 ∂pi ðyÞ ¼ g ∂t ρ ∂y
(3)
where vi ðyÞ is the velocity in the vertical direction, pi ðyÞ is the gauge pressure and g is the gravitational acceleration. The boundary conditions at the water surface and bottom face become ) vi ðηi Þ ¼ η_ i (4) vi ð hi Þ ¼ 0 where ″ U ″ ¼ d=dt. Assuming vi ðyÞ is a linear function of y, and using boundary conditions (4), the velocity becomes vi ð yÞ ¼ η_ i
y þ hi
ηi þ hi
(5)
Fig. 2. Concentrated mass model.
Fig. 3. Elements of concentrated mass mode.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
4
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Differentiating this equation with respect to t and neglecting the terms of η_ 2i , we obtain ∂vð yÞ y þhi ¼ η€ i ∂t ηi þhi
(6)
Substituting Eq. (6) into Eq. (3), integrating from y ¼ y to ηi , and considering pðηi Þ ¼ 0, the pressure in element i becomes pi ðyÞ ¼ ρgðηi yÞ þ ρη€ i
ðηi yÞðηi þ 2hi þ yÞ 2ðηi þ hi Þ
(7)
The first term on the right-hand side describes the static pressure and the second term describes the dynamic pressure. Integrating this equation, the force acting on mass point i from the pressure in element i becomes Z ηi 1 1 pi ðyÞdy ¼ ρ g þ η€ i ðηi þ hi Þ2 (8) f i ðhi ; ηi Þ ¼ 2 3 hi The force in the equilibrium state (ηi ¼ η€ i ¼ 0) is Z f i ðhi ; 0Þ ¼
0 hi
1 2 pðyÞdy ¼ ρghi 2
(9)
Differentiating Eq. (2), η€ i becomes
η€ i ¼
2hi lðx_ i x_ i 1 Þ2 ðxi xi 1 þlÞ
3
hi lðx€ i x€ i 1 Þ 2
ðxi xi 1 þ lÞ
(10)
From (Eqs. (8)–10), the force variation is df i ðhi ; ηi Þ ¼ f i ðhi ; ηi Þ f ðhi ; 0Þ ¼ αi þ βi γ i
1 2
αi ¼ ρg ηi 2 þ 2ηi hi 1 3
βi ¼ ρ ηi þhi γi ¼
(12)
2 2hi li ðx_ i x_ i 1 Þ2 ðxi xi 1 þ li Þ
(11)
3
2 1 ρhi li ηi þ hi ðx€ i x€ i 1 Þ 2 3 ðxi xi 1 þ li Þ
(13)
(14)
The restoring force, df i , consists of three terms: αi is derived from the static pressure, and it describes the nonlinear static restoring force due to gravity; and β i and γ i are derived from the dynamic pressure due to the vertical velocity or acceleration, and they describe the nonlinear dispersion. Linearizing (Eqs. (12)–14) using linearized ηi in Eq. (2), we can obtain
αi ¼
ρghi ðhi xi hi 1 xi 1 Þ li
βi ¼ 0 γi ¼
ρh3i ðx€ i x€ i 1 Þ 3li
(15) (16) (17)
αi in Eq. (15) describes the linear static restoring force, and γ i in Eq. (17) describes the linear dispersion.
Fig. 4. Control volume and velocity distribution.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
5
2.2. Base support damper and base support spring We derive a base support spring and base support damper from the shear stress acting from the bottom face when there is a velocity as shown in Fig. 4. We assume that the flow is laminar, bottom face is horizontal, and flow velocity distribution, pffiffiffiffiffiffiffiffi
~ jωt j ¼ 1 is given outside the boundary layer. The u ¼ ue is ignored. The control volume is the shape influence of gravity
of the thin plate with thickness dy in the colored area ~l i ¼ ðli þ li þ 1 Þ=2 represented by nodal point i in Fig. 4. We define μ as the viscosity coefficient and uðyÞ as the velocity in the horizontal direction in the control volume. The equation of motion of the control volume is given by
ρ~l i
∂u ~ ∂2 u ¼ liμ 2 ∂t ∂y
(18)
We assume that the velocity is given by uð y; tÞ ¼ YðyÞejωt Substituting Eq. (19) into Eq. (18), we obtain
(19)
9 Yð yÞ ¼ Aeλy þ Be λy = qffiffiffiffi ; λ ¼ jνω
(20)
~ jωt , the where ν ¼ μ=ρ and A and B are constant. Considering this equation and boundary conditions uð0Þ ¼ 0 and uð1Þ ¼ ue velocity becomes
uð yÞ ¼ u~ 1 e λy ejωt (21) From this equation, the shear stress from the bottom face is ∂u τ0 ¼ μ ¼ u~ μλejωt ∂y y ¼ 0
(22)
If we assume x_ i ¼ ui ejωt , the damping force, f i , and restoring force, f i , of the equivalent concentrated model in Fig. 5 are bc
bk
f i ¼ cbi ui ejωt bc
(23)
jk ui ejωt b
bk
fi ¼
(24)
ω
Replacing the shear stress of Eq. (22) with the damping force of Eq. (23) and the restoring force of Eq. (24), the damping b coefficient of the base support damper, cbi , and the constant of the base support spring, ki , are written as h i (25) cbi ¼ Re μ~l i λ h i b ki ¼ ωIm μ~l i λ where λ is
λ¼ σ
qffiffiffiffi
9 =
jω
ν ¼ ð1 þ jÞσ pffiffiffiffi ; ¼ 2ων
(26)
(27)
From Eqs. (25), (26), and (27) and ~l i ¼ ðli þli þ 1 Þ=2, the damping coefficient of the base support damper and the constant of the base support spring are written as cbi ¼
μσ ðli þ li þ 1 Þ 2
(28)
xi m
cb
kb
Fig. 5. Equivalent mass-damper-spring model.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
b
ki ¼
ωμσ ðli þ li þ 1 Þ
(29)
2
We also assume that (Eqs. (28) and 29) are satisfied when the bottom face is inclined. The damping coefficient of the base support damper and the constant of the base support spring in (Eqs. (28) and 29) are functions of ω. However, a superharmonic component of the exciting angular frequency Ω is generated because this analysis model is a nonlinear system. Here, the angular frequency is set as ω ¼ Ω because the component of Ω is dominant. 2.3. Connecting damper We assume that damping is generated by the relative velocity between the mass points, and we model the damping with a connecting damper. The damping coefficient of the connecting damper, cci , for element i is assumed to be cci ¼
ζ
(30)
li
Here, ζ is a damping parameter. This damper includes the effects of the side wall, surface tension, turbulence, and bulk damping [1]. However, the value of ζ is determined experimentally in this paper. 2.4. Equation of motion Finally, the equation of motion of mass point i is written as mi x€ i ¼ αi αi þ 1 þ β i β i þ 1 þ γ i γ i þ 1 þ cciþ 1 ðx_ i þ 1 x_ i Þ cci ðx_ i x_ i 1 Þ cbi x_ i ki xi þ f i ðtÞ b
e
(31)
e f i ðtÞ
where is an external force acting on the mass point i. The nonlinearity of this model is derived from the nonlinear static restoring force α from the static pressure, and the nonlinear dispersion β, γ from the dynamic pressure. Arranging the equations in a matrix gives equations in the form _ tÞ MðxÞ x€ ¼ Fðx; x;
(32)
_ tÞ is a function of x, x, _ and time t. The where x is the displacement vector at each point; MðxÞ is the mass matrix; and Fðx; x; mass matrix consists of mi in Eq. (1) and components derived from γ i in Eq. (14), because γ i includes the accelerations of mass points.
3. Comparison of natural frequencies in tank To confirm the validity of the proposed model in a linear region for non-uniform water depth, we compare the natural frequency in the concentrated mass model with the theoretical values of the natural frequency in rectangular and triangular tanks, as shown in Fig. 6. The natural frequency in the rectangular tank from the linear shallow wave theory is given by [20] r pffiffiffiffiffiffiffiffi fr ¼ gh0 (33) 2L where r is the order of the natural frequency. The natural frequency in the triangular tank is given by [20] qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 βr gh0 fr ¼ 2π l0
(34)
where β r is shown in Table 1 for each order r. Because there is no dispersion in the linear shallow wave theory, we set βi ¼ γ i ¼ 0 in the concentrated mass model. The natural frequency is calculated by the eigenvalue analysis of the concentrated mass model with the mass in Eq. (1) and the linear restoring force df i ¼ αi in Eq. (15). The number of elements is N ¼ 100. The mass points at both ends are considered to be fixed, and the displacements of these points are set to 0. The dimensions are L ¼ 2l0 ¼ 1:0 m and h0 ¼ 0:1 m. Table 2 shows a comparison between the theoretical values obtained using Eq. (33) and the numerical results for the natural frequencies from the first to the fourth order for a rectangular tank. Table 3 shows the theoretical values obtained
h0
h0
L Rectangular tank
l0
l0
Triangular tank
Fig. 6. Rectangular and triangular tanks. (a) Rectangular tank (b) Triangular tank.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
7
Table 1 Values of βr Order r
1
2
3
4
βr
1.446
3.671
7.618
12.31
Table 2 Natural frequencies of rectangular tank. Order r
1
2
3
4
Theoretical value from Eq. (33) Numerical results
0.495 Hz 0.495 Hz
0.990 Hz 0.990 Hz
1.485 Hz 1.484 Hz
1.980 Hz 1.979 Hz
Order r
1
2
3
4
Theoretical value from Eq. (34) Numerical results
0.379 Hz 0.379 Hz
0.604 Hz 0.604 Hz
0.870 Hz 0.870 Hz
1.105 Hz 1.106 Hz
Table 3 Natural frequencies of triangular tank.
Level sensor
Laser displacement sensor
L
H
Vibration exciter
h0
Linear guide Fig. 7. Experimental apparatus.
using Eq. (34) and the numerical results in the case of a triangular tank. The numerical results agree well with the theoretical results in both cases. Therefore, the mass and linear spring in the concentrated mass model are valid in a linear region, and this model can be used for non-uniform water depth problems such as those in the triangular tank.
4. Comparison with experimental results 4.1. Experimental apparatus To confirm the validity of the proposed model, we performed sloshing experiments in a rectangular tank, as shown in Fig. 7. The cuboid tank consists of acrylic boards with thickness 10 mm. The dimensions are L ¼ 1000 mm, H ¼ 400 mm and width ¼100 mm. The cuboid tank is set on a linear guide, and is excited using a vibration exciter. The displacement of the tank is measured using a laser displacement meter (LK-G80, Keyence), and a capacitance type water level gauge (WTS01A0505, CS Tokki) is installed at a point 50 mm from the left wall. We performed experiments in four configurations, cases 1–4. The water depth, h0 , and the amplitude of the displacement of the excitation, Δ, in each case are shown in Table 4. All cases satisfy the shallow water wave condition (water depth is less than one-fifth of the wavelength). 4.2. Analysis method We simulate the sloshing phenomena with the concentrated mass model and obtain a steady periodical solution by using the shooting method (see Appendix A). Numerical integration in the method is performed by the Runge–Kutta method, with time interval being 1/1024 of a cycle. The number of partitions is 50. We simulate two conditions, nonlinear dispersion and linear dispersion. For nonlinear dispersion, we use βi and γ i in (Eqs. (13) and 14). For linear dispersion, we use the linearized βi and γ i in (Eqs. (16) and 17). The nonlinear static restoring force αi in Eq. (12) is used in both conditions. ζ n in Table 4 is the damping parameter, ζ , from Eq. (30) used for nonlinear dispersion and ζ l is ζ used for linear dispersion. The other parameters for the simulation are shown in Table 5. The boundary conditions at both ends are fixed: that is, the displacements Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
Table 4 Parameter values for each case.
case case case case
1 2 3 4
h0 ½mm
Δ ½mm
ζ n ½N U s
ζ l ½N U s
50 50 50 150
1.7 2.25 3.5 1.5
0.1 0.06 0.07 0.51
0.07 0.04 0.04 0.49
Table 5 Parameter values. L ½m
1.0
μ ½Pa U s
1:52 10 3
ρ kg=m3
g m=s2
: Experimental results : Nonlinear dispersion, stable : Nonlinear dispersion, unstable
0.35
0.4
Numerical results
40
η [mm]
20
0 0.3
: Linear dispersion, stable : Linear dispersion, unstable
40
η [mm]
η [mm]
40
1000 9.81
20
0 0.3
Frequency [Hz] Case 1
0.35
20
0 0.3
0.4
0.35
Frequency [Hz]
Frequency [Hz]
Case 2
Case 3
0.4
Fig. 8. Frequency response (h0 ¼ 50 mm). I
I
are zero. The inertial force, f , acts on all mass points when the tank is excited. f is written as f ðtÞ ¼ mΩ Δ sin Ωt 2
I
(35)
4.3. Experimental and numerical results Fig. 8 shows the frequency responses around the first-order resonance for cases 1–3. The linear natural frequency is 0.35 Hz. The vertical axis shows the total amplitude of the water level at the measured point and the horizontal axis shows the excitation frequency. The red circles show the experimental results, black lines show the numerical results for nonlinear dispersion, and blue lines show the numerical results for linear dispersion. The solid lines show stable solutions, and dashed lines show unstable solutions. There are some peaks around the first-order resonance in each case. In case 1, the difference between the numerical results for nonlinear dispersion and linear dispersion is small. However, the difference is large in case 3 when the water level amplitude is large. The results for nonlinear dispersion in case 3 are closer to the experimental results than those for linear dispersion. The difference is significant in case 4, as shown in Fig. 11, when the water level amplitude, ηi , and water depth, h0 , are large. The slopes of the frequency response curve are completely different for nonlinear dispersion and linear dispersion. Therefore, the nonlinearity of the dispersion is necessary when the water level amplitude or water depth is large. Fig. 9 shows the time series waveforms in case 3 at 0.340 Hz, 0.358 Hz, and 0.386 Hz; these are the frequencies around the three peaks. At 0.386 Hz, there is one pulse per cycle. At 0.358 Hz and 0.340 Hz, there are two and three pulses per cycle, respectively. Fig. 10 shows the transition of surface deformation in case 3 at 0.340 Hz, 0.358 Hz, and 0.386 Hz. This figure describes the spatial distribution of the water level at the time, as given by the following equation: t ¼ t0 þ
2nπ 8Ω
ðn ¼ 0; 1; 2; ⋯; 7Þ
(36)
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Numerical results
Experimental results 40
η [mm]
η [mm]
40
0.340 Hz
20 0 0
2
4 Time [s]
6
η [mm]
η [mm]
20 0 2
4 Time [s]
6
0
2
4 Time [s]
6
8
0
2
4 Time [s]
6
8
0
2
4 Time [s]
6
8
20 0
8
40
η [mm]
40
η [mm]
0
40
0
0.386 Hz
20
8
40
0.358 Hz
9
20 0 0
2
4 Time [s]
6
20
8
0
Fig. 9. Time series waveform (case 3, nonlinear dispersion).
Fig. 10. Transition of surface deformation (case 3, nonlinear dispersion).
[mm]
150
100
50
0 0.5
0.6 Frequency [Hz]
0.7
Fig. 11. Frequency response (case 4, h0 ¼ 150 mm).
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
120 80 40 0 −40
Numerical results [mm]
[mm]
10
0
2
4 Time [s]
6
8
120 80 40 0 −40 0
Experimental results
2
4 Time [s]
6
8
6
8
Fig. 12. Time series waveform (case 4, nonlinear dispersion, 0.61 Hz).
1
100
0.5
50
0 0
0 2
4 Time [s]
6
8
0
2
4 Time [s]
Fig. 13. Energy dispersion rate (case 3, nonlinear dispersion, 0.61 Hz).
I
where t 0 is the time when the inertial force, f , in Eq. (35) is zero. The horizontal axis shows the distance from the left side wall. At 0.386 Hz, one solitary wave moves back and forth in the tank. At 0.358 Hz and 0.340 Hz, two and three solitary waves move in this manner, respectively. Fig. 12 shows the waveform in case 4 at 0.61 Hz. The numerical calculations use nonlinear dispersion. In this case, there is no solitary wave, and the waveform in the tank is near the first-order mode shape. The waveforms from the model agree well with the experimental results. Therefore, the proposed model is valid for the numerical analysis of nonlinear shallow water wave problems. Fig. 13 shows the energy dispersion rates of dampers. The left panel shows the time-series energy dispersion rate of the base support damper at mass point 1 given the following equation: Eb ¼ cb1 x_ 1 U x_ 1
(37)
The right panel of the figure shows the energy dispersion rate of the connecting support damper at the measurement point (element 2) given the following equation: Ec ¼ cc2 ðx_ 2 x_ 1 Þ U ðx_ 2 x_ 1 Þ
(38)
The scales of the vertical axis are different. The dispersion rates become large when the solitary wave passes through. The energy dispersion rate of the connecting damper is larger than that of the base support damper because the damping coefficient of the connecting damper that includes surface tension, turbulence, and bulk damping (cc2 ¼ 0:35 N s/m) is larger than that of the base support damper that considers the shear stress from the bottom face (cb1 ¼ 2:72 10 3 N s/m).
5. Conclusion To analyze nonlinear shallow water wave phenomena in a tank, we propose a concentrated mass model that represents the nonlinear static restoring force and nonlinear dispersion. A comparison of the natural frequencies obtained from the concentrated mass model and the theoretical value of rectangular and triangular tanks shows good agreement. This shows that the proposed model is also valid for the non-uniform water depth problem. In addition, an experiment on a rectangular tank has been performed, and the experimental results have been compared with the numerical results obtained from the model. The numerical results agree well with the experimental results both qualitatively and quantitatively for phenomena such as the solitary waves. The comparison also shows that the nonlinearity of dispersion is necessary when the water level amplitude or water depth is large. In summary, the proposed model is suitable for the numerical analysis of nonlinear shallow water waves in a tank. Future tasks include the modeling of viscous damping and the non-uniform water depth problems, including experimental verification.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
11
Appendix A Modification of shooting method for the proposed model The shooting method is an accurate and powerful numerical method based on numerical integration to find a steady periodic solution directly [18,19]. The equation of motion in Eq. (32) is transformed into the first-order differential equation " # " #) I 0 x_ x _y ¼ V 1 ðyÞ U Yðy; tÞ y ¼ ; VðyÞ ¼ ; Yðy; tÞ ¼ (A.1) _ tÞ 0 MðxÞ Fðx; x; x_ where I is the unit matrix and 0 is the zero matrix. The following equation should hold so that the solution of Eq. (A.1) is a periodic solution of period T. yð0; y0 Þ ¼ y0 ¼ yðT; y0 Þ
(A.2)
Here, yðt; y Þ is the solution of Eq. (A.1) at time t when y is the initial value of y at t ¼ 0. The shooting method finds the initial value y0 that satisfies Eq. (A.2) using Newton method. In this method, the variational equation of Eq. (A.1) is necessary. The components of VðyÞ are variable in the case of the proposed model, so the variational equation becomes ∂Yðy; tÞ Wðy; tÞ ξð t Þ (A.3) ξ_ ðt Þ ¼ V 1 yðt; y0 Þ ∂y y ¼ yðt;y0 Þ 0
0
where ξðtÞ is the variation of yðt; y0 Þ, and W is the matrix given by h ∂V ∂V ∂V W ¼ ∂y1 y_ ∂y2 y_ ⋯ ∂yN y_ 0 " # 0 ∂V _ ði ¼ 1; ⋯; NÞ y ¼ ∂M ∂yi € ∂x x
⋯
i9 0 > > > = > > > ;
i
(A.4)
where yi is the i-th component of vector y. Assuming y0k is the k-th approximate initial value of the iterative calculation and δy0k is the correction amount, the condition that the corrected initial value y0k þ δy0k satisfies Eq. (A.2) becomes ΦðTÞ I δy0k ¼ y0k yðT; y0k Þ (A.5) y0k þ 1 ¼ y0k þ δy0k
(A.6)
where ΦðtÞ is the state transition matrix about ξðtÞ. ΦðTÞ is given by h i9 ΦðTÞ ¼ ξ1 ðTÞ ⋯ ξn ðTÞ =
ξi ð0Þ ¼ ei
;
(A.7)
where ei is the unit vector whose i-th element is 1. Each column of ΦðTÞ, ξi ðTÞ, is calculated by solving Eq. (A.3) using numerical integration. We repeat the iterative calculation of Eqs. (A.5) and (A.6) until ‖δy0k ‖ becomes small. Stability analysis is performed by calculating the characteristic multipliers, that is, eigenvalues of ΦðTÞ. When all eigenvalues, λi , are less than 1, the solution is stable. If even one eigenvalue is more than 1, the solution is unstable.
References [1] O.M. Faltinsen, A.N. Timokha, Sloshing, Cambridge University Press, New York, 2009. [2] K. Hatayama, Lessons from the 2003 Tokachi-oki Japan, earthquake for prediction of long-period strong ground motions and sloshing damage to oil storage tanks, J. Seismol. 12 (2) (2008) 255–263. [3] L. Sun, M., Y. Fujino, B.M. Pacheco, M. Isobe, Nonlinear waves and dynamic pressures in rectangular tuned liquid damper (TLD), Proceedings of the Japan Society of Civil Engineers 410 (1989) 81–92. [4] Y. Fujino, L. Sun, M., B.M. Pacheco, P. Chaiseri, Tuned liquid damper (TLD) for suppressing horizontal motion of structures, Journal of Engineering Mechanics 118 (10) (1992) 2017–2030. [5] W. Chester, Resonant oscillations of water wave I Theory, Proceedings of the Royal Society of London 306 (1968) 5–22. [6] W. Chester, J.A. Bones, Resonant oscillations of water wave II experiment, Proceedings of the Royal Society of London 306 (1968) 23–39. [7] T.G. Lepelletier, Tsunami- Harbor oscillations induced by nonlinear transient long waves (PhD thesis), California Institute of Technology, 1981. [8] T.G. Lepelletier, F. Raichlen, Nonlinear oscillations in rectangular tanks, Journal of Engineering Mechanics 114 (1) (1988) 1–23. [9] T. Shimizu, S. Hayama, Nonlinear response of sloshing based on the shallow water wave theory, JSME International Journal 30 (263) (1987) 806–813. [10] O.M. Faltinsen, O.F. Rognebakke, I.A. Lukovsky, A.N. Timokha, Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth, Journal of Fluid Mechanics 407 (2000) 201–234. [11] O.M. Faltinsen, A.N. Timokha, An adaptive multimodal approach to nonlinear sloshing in a rectangular tank, Journal of Fluid Mechanics 432 (2001) 167–200. [12] O.M. Faltinsen, A.N. Timokha, Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth, Journal of Fluid Mechanics 470 (2002) 319–357. [13] J.S. Love, M.J. Tait, Nonlinear simulation of a tuned liquid damper with damping screens using a modal expansion technique, Journal of Fluids and Structures 26 (2010) 1058–1077. [14] M. Antuono, A. Colagrossi, S. Marrone, C. Lugni, Propagation of gravity waves through an SPH scheme with numerical diffusive terms, Computer Physics Communications 182 (2011) 866–877. [15] M. Antuono, A. Colagrossi, S. Marrone, C. Lugni, Two dimensional modal method for shallow-water sloshing in rectangular basins, Journal of Fluid Mechanics 700 (2012) 419–440.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
12
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
[16] W. Nakayama, Boundary Element Method Applied to the Analysis of shallow liquid sloshing in moving tanks, JSME International Journal, Series C 39 (4) (1996) 800–807. [17] X. Deng, M.J. Tait, Theoretical modeling of TLD with different tank geometries using linear long wave theory, Journal of Vibration and Acoustics 131 (4) (2009) 0140141–01401410. [18] T.J. Aprille Jr., T.N. Trick, A Computer algorithm to determine the steady-state response of nonlinear oscillators, IEEE Transactions on Circuit Theory 19 (4) (1972) 356–360. [19] H. Tamura, K. Matuszaki, Numerical scheme and program for the solution and stability analysis of a steady periodic vibration problem, JSME International Journal 39 (3) (1996) 456–463. [20] H. Lamb, Hydrodynamics, Cambridge University Press, 1945.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Analysis of nonlinear shallow water waves in a tank by concentrated mass model Satoshi Ishikawa a,n, Takahiro Kondou a, Kenichiro Matsuzaki b, Satoshi Yamamura c a b c
Department of Mechanical Engineering, Kyushu University, 744 Motooka Nishi-ku, Fukuoka, Japan Department of Mechanical Engineering, Kagoshima University, 1-21-40 Korimoto, Kagoshima, Japan OMRON Corporation, 211 Nishikusatsu, Kusatsu, Shiga, Japan
a r t i c l e i n f o
abstract
Article history: Received 13 October 2015 Received in revised form 12 February 2016 Accepted 16 February 2016 Handling Editor: L. G. Tham
The sloshing of liquid in a tank is an important engineering problem. For example, liquid storage tanks in industrial facilities can be damaged by earthquakes, and conversely liquid tanks, called tuned liquid damper, are often used as passive mechanical dampers. The water depth is less often than the horizontal length of the tank. In this case, shallow water wave theory can be applied, and the results indicate that the surface waveform in a shallow excited tank exhibits complex behavior caused by nonlinearity and dispersion of the liquid. This study aims to establish a practical analytical model for this phenomenon. A model is proposed that consists of masses, connecting nonlinear springs, connecting dampers, base support dampers, and base support springs. The characteristics of the connecting nonlinear springs are derived from the static and dynamic pressures. The advantages of the proposed model are that nonlinear dispersion is considered and that the problem of non-uniform water depth can be addressed. To confirm the validity of the model, numerical results obtained from the model are compared with theoretical values of the natural frequencies of rectangular and triangular tanks. Numerical results are also compared with experimental results for a rectangular tank. All computational results agree well with the theoretical and experimental results. Therefore, it is concluded that the proposed model is valid for the numerical analysis of nonlinear shallow water wave problems. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Vibration analysis method, Modeling, Shallow water wave, Sloshing, Concentrated mass model
1. Introduction The sloshing of liquid in a tank is an important engineering problem. For example, liquid storage tanks in industrial facilities can be damaged by earthquakes, as happened when the floating roof of a large oil storage tank was damaged in the 2003 Tokachi-oki earthquake, casing a fire to break out [1,2]. Conversely, liquid tanks, called tuned liquid dampers (TLDs), can be used as passive mechanical dampers to attenuate the vibration of tall buildings [3,4]. The water depth in these liquid tanks is often less than the horizontal length of the tank. This is the so-called shallow water condition. In this case, shallow water wave theory can be applied, and the results indicate that the surface waveform in a shallow excited tank exhibits complex behavior caused by nonlinearity and n Correspondence to: Department of Mechanical Engineering, Kyushu University, 744 Motooka Nishi‐ku, Fukuoka, Japan. Tel.: +81 92-802-3187; fax: +81 92-802-0001. E-mail address: [email protected] (S. Ishikawa).
http://dx.doi.org/10.1016/j.jsv.2016.02.029 0022-460X/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
dispersion of the liquid. One- and two-dimensional models are used for the numerical simulation of this phenomenon. Onedimensional models deal with only the horizontal equation of motion of a fluid, and two-dimensional models deal with twodimensional equations of a fluid in the horizontal and vertical directions. Regarding one-dimensional models, Chester [5] derived a theory that considers nonlinearity and linear dispersion and provided a steady-state one-dimensional solution for a shallow water wave in a horizontally excited tank. Chester and Bones [6] performed experiments with excitation near the resonance frequency and indicated the significance of nonlinearity and dispersion for a shallow water wave. Lepelletier [7] and Lepelletier and Raichlen [8] derived one-dimensional finite element methods from Boussinesq-type equations that consider linear dispersion. Shimizu and Hayama [9] derived one-dimensional fluid equations and simulated shallow water sloshing phenomena in a rectangular tank by using the difference scheme of the fluid equations. In this method, linear dispersion was modeled by changing the grid size. Therefore, the number of elements decreases as water depth increases, and the calculation is inaccurate when the water is deep under a shallow water condition. Sun et al. [3] and Fujino et al. [4] simulated nonlinear shallow water waves in a TLD by using Shimizu and Hayama’s method [9]. In these studies, the waveforms of the numerical results agree with the experimental results [6–9]. However, the slopes of the frequency responses do not correspond in the case of large amplitude, probably because these numerical methods consider only linear dispersion. The nonlinear dispersion needs to be considered in the case of large-amplitude waves or deep water even under shallow water conditions. Furthermore, the abovementioned one-dimensional calculation methods are complicated to derive. With regard to two-dimensional models, Faltinsen et al. [10] and Faltinsen and Timokha [11,12] derived multimodal methods based on a two-dimensional potential theory and simulated the sloshing phenomena in a rectangular tank under shallow and intermediate water depth conditions. Love and Tait [13] simulated sloshing phenomena in a TLD using Faltinsen et al.’s modal expansion techniques. However, the equations of this method are quite complicated. Antuono et al. [14,15] simulated the shallow water sloshing phenomena using a two-dimensional smoothed particle hydrodynamics (SPH) scheme. However, an SPH scheme is a particle method that requires considerable computation time. Nakayama [16] derived a boundary element method for shallow water waves in a TLD tank by using two-dimensional fluid equations. The computational cost of these two-dimensional methods is larger than that of one-dimensional methods. The numerical methods developed by Chester [5], Lepelletier [7,8], Shimizu and Hayama [9], and Faltinsen et al. [10–12] cannot be applied to non-uniform water depth problems because they assume a constant water depth. However, Deng and Tait [17] noted that a sloped-bottom tank is a more efficient TLD than a flat-bottom tank; therefore the non-uniform water depth problem is important. Moreover, numerical simulations conducted researchers other than Chester and Bones did not yield unstable solutions, and did not perform stability analysis. This study aims to establish a practical one-dimensional analytical model to analyze non-linear shallow water wave phenomena in a tank. This model consists of masses, connecting nonlinear springs, connecting dampers, base support dampers, and base support springs. The characteristics of the connecting nonlinear springs are derived from static and dynamic pressures, the latter of which also considers vertical motion of the fluid. The connecting nonlinear spring represents the nonlinear static restoring force and nonlinear dispersion. The base support damper and the base support spring are derived from the influence of shear stress from the bottom wall. The advantages of the proposed model are that nonlinear dispersion is considered, non-uniform water depth problems can be addressed, and model derivation is simple. Furthermore, the numerical simulation with the concentrated mass model is performed using the shooting method [18,19] to yield both stable and unstable solutions and to perform stability analysis. The shooting method is modified to apply the concentrated mass model. In this paper, a concentrated mass model is proposed. Next, we compare the linear natural frequencies in the concentrated mass model with the theoretical values for rectangular and triangular tanks to confirm the validity of the model for the non-uniform water depth problem. Then, an experiment on a rectangular tank is performed, and the experimental results are compared with the numerical results to confirm the model’s validity in a nonlinear region.
2. Concentrated mass model We consider one-dimensional nonlinear shallow water wave phenomena in a tank, as shown in Fig. 1. We consider the 2 condition that the amplitude of a water wave is too small to cause wave-breaking. When the Bond number Bo ¼ gl0 ρ=T s (g is the gravitational acceleration; l0 is the length of the tank; ρ is the density of the fluid; and T s is the surface tension) is larger than 104 , the effect of surface tension can be neglected [1]. We consider the case Bo 4104 and neglect the surface tension of the free surface.
Fig. 1. Coordinate system of water wave.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3
The fluid is modeled as a concentrated mass model that consists of masses, nonlinear connecting springs, connecting dampers, base support dampers, and base support springs, as shown in Fig. 2. We consider that the fluid is incompressible and is divided into N trapezoidal elements. The mass is concentrated at the nodal points. The density of fluid is ρ, length of element i ð ¼ 1; ⋯; NÞ in equilibrium state is li , the depth of water at nodal point i is hi , and the horizontal displacement at nodal point i is xi . The water level in element i is ηi . The width of the tank is taken as the unit length. Considering the mass of fluid in the shaded area in the equilibrium state in Fig. 3(a), the mass of each mass point, mi , can be written as mi ¼
ρ 8
li ðhi 1 þ 3hi Þ þ li þ 1 ð3hi þ hi þ 1 Þ
(1)
When the mass points at both ends of element i move, we assume the water level changes so that the volume of the element is conserved, as shown in Fig. 3(b). Then, using the displacements xi 1 and xi , ηi is given implicitly by
ηi ¼
hi xi þ hi 1 xi 1 xi xi 1 þ li
(2)
2.1. Nonlinear connecting spring We derive a nonlinear spring from the hydrostatic and hydrodynamic pressures when the water level changes. The vertical equation of motion of a small element at vertical coordinate y (shown in red in Fig. 3b) in element i is given by ∂vi ðyÞ 1 ∂pi ðyÞ ¼ g ∂t ρ ∂y
(3)
where vi ðyÞ is the velocity in the vertical direction, pi ðyÞ is the gauge pressure and g is the gravitational acceleration. The boundary conditions at the water surface and bottom face become ) vi ðηi Þ ¼ η_ i (4) vi ð hi Þ ¼ 0 where ″ U ″ ¼ d=dt. Assuming vi ðyÞ is a linear function of y, and using boundary conditions (4), the velocity becomes vi ð yÞ ¼ η_ i
y þ hi
ηi þ hi
(5)
Fig. 2. Concentrated mass model.
Fig. 3. Elements of concentrated mass mode.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
4
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Differentiating this equation with respect to t and neglecting the terms of η_ 2i , we obtain ∂vð yÞ y þhi ¼ η€ i ∂t ηi þhi
(6)
Substituting Eq. (6) into Eq. (3), integrating from y ¼ y to ηi , and considering pðηi Þ ¼ 0, the pressure in element i becomes pi ðyÞ ¼ ρgðηi yÞ þ ρη€ i
ðηi yÞðηi þ 2hi þ yÞ 2ðηi þ hi Þ
(7)
The first term on the right-hand side describes the static pressure and the second term describes the dynamic pressure. Integrating this equation, the force acting on mass point i from the pressure in element i becomes Z ηi 1 1 pi ðyÞdy ¼ ρ g þ η€ i ðηi þ hi Þ2 (8) f i ðhi ; ηi Þ ¼ 2 3 hi The force in the equilibrium state (ηi ¼ η€ i ¼ 0) is Z f i ðhi ; 0Þ ¼
0 hi
1 2 pðyÞdy ¼ ρghi 2
(9)
Differentiating Eq. (2), η€ i becomes
η€ i ¼
2hi lðx_ i x_ i 1 Þ2 ðxi xi 1 þlÞ
3
hi lðx€ i x€ i 1 Þ 2
ðxi xi 1 þ lÞ
(10)
From (Eqs. (8)–10), the force variation is df i ðhi ; ηi Þ ¼ f i ðhi ; ηi Þ f ðhi ; 0Þ ¼ αi þ βi γ i
1 2
αi ¼ ρg ηi 2 þ 2ηi hi 1 3
βi ¼ ρ ηi þhi γi ¼
(12)
2 2hi li ðx_ i x_ i 1 Þ2 ðxi xi 1 þ li Þ
(11)
3
2 1 ρhi li ηi þ hi ðx€ i x€ i 1 Þ 2 3 ðxi xi 1 þ li Þ
(13)
(14)
The restoring force, df i , consists of three terms: αi is derived from the static pressure, and it describes the nonlinear static restoring force due to gravity; and β i and γ i are derived from the dynamic pressure due to the vertical velocity or acceleration, and they describe the nonlinear dispersion. Linearizing (Eqs. (12)–14) using linearized ηi in Eq. (2), we can obtain
αi ¼
ρghi ðhi xi hi 1 xi 1 Þ li
βi ¼ 0 γi ¼
ρh3i ðx€ i x€ i 1 Þ 3li
(15) (16) (17)
αi in Eq. (15) describes the linear static restoring force, and γ i in Eq. (17) describes the linear dispersion.
Fig. 4. Control volume and velocity distribution.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
5
2.2. Base support damper and base support spring We derive a base support spring and base support damper from the shear stress acting from the bottom face when there is a velocity as shown in Fig. 4. We assume that the flow is laminar, bottom face is horizontal, and flow velocity distribution, pffiffiffiffiffiffiffiffi
~ jωt j ¼ 1 is given outside the boundary layer. The u ¼ ue is ignored. The control volume is the shape influence of gravity
of the thin plate with thickness dy in the colored area ~l i ¼ ðli þ li þ 1 Þ=2 represented by nodal point i in Fig. 4. We define μ as the viscosity coefficient and uðyÞ as the velocity in the horizontal direction in the control volume. The equation of motion of the control volume is given by
ρ~l i
∂u ~ ∂2 u ¼ liμ 2 ∂t ∂y
(18)
We assume that the velocity is given by uð y; tÞ ¼ YðyÞejωt Substituting Eq. (19) into Eq. (18), we obtain
(19)
9 Yð yÞ ¼ Aeλy þ Be λy = qffiffiffiffi ; λ ¼ jνω
(20)
~ jωt , the where ν ¼ μ=ρ and A and B are constant. Considering this equation and boundary conditions uð0Þ ¼ 0 and uð1Þ ¼ ue velocity becomes
uð yÞ ¼ u~ 1 e λy ejωt (21) From this equation, the shear stress from the bottom face is ∂u τ0 ¼ μ ¼ u~ μλejωt ∂y y ¼ 0
(22)
If we assume x_ i ¼ ui ejωt , the damping force, f i , and restoring force, f i , of the equivalent concentrated model in Fig. 5 are bc
bk
f i ¼ cbi ui ejωt bc
(23)
jk ui ejωt b
bk
fi ¼
(24)
ω
Replacing the shear stress of Eq. (22) with the damping force of Eq. (23) and the restoring force of Eq. (24), the damping b coefficient of the base support damper, cbi , and the constant of the base support spring, ki , are written as h i (25) cbi ¼ Re μ~l i λ h i b ki ¼ ωIm μ~l i λ where λ is
λ¼ σ
qffiffiffiffi
9 =
jω
ν ¼ ð1 þ jÞσ pffiffiffiffi ; ¼ 2ων
(26)
(27)
From Eqs. (25), (26), and (27) and ~l i ¼ ðli þli þ 1 Þ=2, the damping coefficient of the base support damper and the constant of the base support spring are written as cbi ¼
μσ ðli þ li þ 1 Þ 2
(28)
xi m
cb
kb
Fig. 5. Equivalent mass-damper-spring model.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
b
ki ¼
ωμσ ðli þ li þ 1 Þ
(29)
2
We also assume that (Eqs. (28) and 29) are satisfied when the bottom face is inclined. The damping coefficient of the base support damper and the constant of the base support spring in (Eqs. (28) and 29) are functions of ω. However, a superharmonic component of the exciting angular frequency Ω is generated because this analysis model is a nonlinear system. Here, the angular frequency is set as ω ¼ Ω because the component of Ω is dominant. 2.3. Connecting damper We assume that damping is generated by the relative velocity between the mass points, and we model the damping with a connecting damper. The damping coefficient of the connecting damper, cci , for element i is assumed to be cci ¼
ζ
(30)
li
Here, ζ is a damping parameter. This damper includes the effects of the side wall, surface tension, turbulence, and bulk damping [1]. However, the value of ζ is determined experimentally in this paper. 2.4. Equation of motion Finally, the equation of motion of mass point i is written as mi x€ i ¼ αi αi þ 1 þ β i β i þ 1 þ γ i γ i þ 1 þ cciþ 1 ðx_ i þ 1 x_ i Þ cci ðx_ i x_ i 1 Þ cbi x_ i ki xi þ f i ðtÞ b
e
(31)
e f i ðtÞ
where is an external force acting on the mass point i. The nonlinearity of this model is derived from the nonlinear static restoring force α from the static pressure, and the nonlinear dispersion β, γ from the dynamic pressure. Arranging the equations in a matrix gives equations in the form _ tÞ MðxÞ x€ ¼ Fðx; x;
(32)
_ tÞ is a function of x, x, _ and time t. The where x is the displacement vector at each point; MðxÞ is the mass matrix; and Fðx; x; mass matrix consists of mi in Eq. (1) and components derived from γ i in Eq. (14), because γ i includes the accelerations of mass points.
3. Comparison of natural frequencies in tank To confirm the validity of the proposed model in a linear region for non-uniform water depth, we compare the natural frequency in the concentrated mass model with the theoretical values of the natural frequency in rectangular and triangular tanks, as shown in Fig. 6. The natural frequency in the rectangular tank from the linear shallow wave theory is given by [20] r pffiffiffiffiffiffiffiffi fr ¼ gh0 (33) 2L where r is the order of the natural frequency. The natural frequency in the triangular tank is given by [20] qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 βr gh0 fr ¼ 2π l0
(34)
where β r is shown in Table 1 for each order r. Because there is no dispersion in the linear shallow wave theory, we set βi ¼ γ i ¼ 0 in the concentrated mass model. The natural frequency is calculated by the eigenvalue analysis of the concentrated mass model with the mass in Eq. (1) and the linear restoring force df i ¼ αi in Eq. (15). The number of elements is N ¼ 100. The mass points at both ends are considered to be fixed, and the displacements of these points are set to 0. The dimensions are L ¼ 2l0 ¼ 1:0 m and h0 ¼ 0:1 m. Table 2 shows a comparison between the theoretical values obtained using Eq. (33) and the numerical results for the natural frequencies from the first to the fourth order for a rectangular tank. Table 3 shows the theoretical values obtained
h0
h0
L Rectangular tank
l0
l0
Triangular tank
Fig. 6. Rectangular and triangular tanks. (a) Rectangular tank (b) Triangular tank.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
7
Table 1 Values of βr Order r
1
2
3
4
βr
1.446
3.671
7.618
12.31
Table 2 Natural frequencies of rectangular tank. Order r
1
2
3
4
Theoretical value from Eq. (33) Numerical results
0.495 Hz 0.495 Hz
0.990 Hz 0.990 Hz
1.485 Hz 1.484 Hz
1.980 Hz 1.979 Hz
Order r
1
2
3
4
Theoretical value from Eq. (34) Numerical results
0.379 Hz 0.379 Hz
0.604 Hz 0.604 Hz
0.870 Hz 0.870 Hz
1.105 Hz 1.106 Hz
Table 3 Natural frequencies of triangular tank.
Level sensor
Laser displacement sensor
L
H
Vibration exciter
h0
Linear guide Fig. 7. Experimental apparatus.
using Eq. (34) and the numerical results in the case of a triangular tank. The numerical results agree well with the theoretical results in both cases. Therefore, the mass and linear spring in the concentrated mass model are valid in a linear region, and this model can be used for non-uniform water depth problems such as those in the triangular tank.
4. Comparison with experimental results 4.1. Experimental apparatus To confirm the validity of the proposed model, we performed sloshing experiments in a rectangular tank, as shown in Fig. 7. The cuboid tank consists of acrylic boards with thickness 10 mm. The dimensions are L ¼ 1000 mm, H ¼ 400 mm and width ¼100 mm. The cuboid tank is set on a linear guide, and is excited using a vibration exciter. The displacement of the tank is measured using a laser displacement meter (LK-G80, Keyence), and a capacitance type water level gauge (WTS01A0505, CS Tokki) is installed at a point 50 mm from the left wall. We performed experiments in four configurations, cases 1–4. The water depth, h0 , and the amplitude of the displacement of the excitation, Δ, in each case are shown in Table 4. All cases satisfy the shallow water wave condition (water depth is less than one-fifth of the wavelength). 4.2. Analysis method We simulate the sloshing phenomena with the concentrated mass model and obtain a steady periodical solution by using the shooting method (see Appendix A). Numerical integration in the method is performed by the Runge–Kutta method, with time interval being 1/1024 of a cycle. The number of partitions is 50. We simulate two conditions, nonlinear dispersion and linear dispersion. For nonlinear dispersion, we use βi and γ i in (Eqs. (13) and 14). For linear dispersion, we use the linearized βi and γ i in (Eqs. (16) and 17). The nonlinear static restoring force αi in Eq. (12) is used in both conditions. ζ n in Table 4 is the damping parameter, ζ , from Eq. (30) used for nonlinear dispersion and ζ l is ζ used for linear dispersion. The other parameters for the simulation are shown in Table 5. The boundary conditions at both ends are fixed: that is, the displacements Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
Table 4 Parameter values for each case.
case case case case
1 2 3 4
h0 ½mm
Δ ½mm
ζ n ½N U s
ζ l ½N U s
50 50 50 150
1.7 2.25 3.5 1.5
0.1 0.06 0.07 0.51
0.07 0.04 0.04 0.49
Table 5 Parameter values. L ½m
1.0
μ ½Pa U s
1:52 10 3
ρ kg=m3
g m=s2
: Experimental results : Nonlinear dispersion, stable : Nonlinear dispersion, unstable
0.35
0.4
Numerical results
40
η [mm]
20
0 0.3
: Linear dispersion, stable : Linear dispersion, unstable
40
η [mm]
η [mm]
40
1000 9.81
20
0 0.3
Frequency [Hz] Case 1
0.35
20
0 0.3
0.4
0.35
Frequency [Hz]
Frequency [Hz]
Case 2
Case 3
0.4
Fig. 8. Frequency response (h0 ¼ 50 mm). I
I
are zero. The inertial force, f , acts on all mass points when the tank is excited. f is written as f ðtÞ ¼ mΩ Δ sin Ωt 2
I
(35)
4.3. Experimental and numerical results Fig. 8 shows the frequency responses around the first-order resonance for cases 1–3. The linear natural frequency is 0.35 Hz. The vertical axis shows the total amplitude of the water level at the measured point and the horizontal axis shows the excitation frequency. The red circles show the experimental results, black lines show the numerical results for nonlinear dispersion, and blue lines show the numerical results for linear dispersion. The solid lines show stable solutions, and dashed lines show unstable solutions. There are some peaks around the first-order resonance in each case. In case 1, the difference between the numerical results for nonlinear dispersion and linear dispersion is small. However, the difference is large in case 3 when the water level amplitude is large. The results for nonlinear dispersion in case 3 are closer to the experimental results than those for linear dispersion. The difference is significant in case 4, as shown in Fig. 11, when the water level amplitude, ηi , and water depth, h0 , are large. The slopes of the frequency response curve are completely different for nonlinear dispersion and linear dispersion. Therefore, the nonlinearity of the dispersion is necessary when the water level amplitude or water depth is large. Fig. 9 shows the time series waveforms in case 3 at 0.340 Hz, 0.358 Hz, and 0.386 Hz; these are the frequencies around the three peaks. At 0.386 Hz, there is one pulse per cycle. At 0.358 Hz and 0.340 Hz, there are two and three pulses per cycle, respectively. Fig. 10 shows the transition of surface deformation in case 3 at 0.340 Hz, 0.358 Hz, and 0.386 Hz. This figure describes the spatial distribution of the water level at the time, as given by the following equation: t ¼ t0 þ
2nπ 8Ω
ðn ¼ 0; 1; 2; ⋯; 7Þ
(36)
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Numerical results
Experimental results 40
η [mm]
η [mm]
40
0.340 Hz
20 0 0
2
4 Time [s]
6
η [mm]
η [mm]
20 0 2
4 Time [s]
6
0
2
4 Time [s]
6
8
0
2
4 Time [s]
6
8
0
2
4 Time [s]
6
8
20 0
8
40
η [mm]
40
η [mm]
0
40
0
0.386 Hz
20
8
40
0.358 Hz
9
20 0 0
2
4 Time [s]
6
20
8
0
Fig. 9. Time series waveform (case 3, nonlinear dispersion).
Fig. 10. Transition of surface deformation (case 3, nonlinear dispersion).
[mm]
150
100
50
0 0.5
0.6 Frequency [Hz]
0.7
Fig. 11. Frequency response (case 4, h0 ¼ 150 mm).
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
120 80 40 0 −40
Numerical results [mm]
[mm]
10
0
2
4 Time [s]
6
8
120 80 40 0 −40 0
Experimental results
2
4 Time [s]
6
8
6
8
Fig. 12. Time series waveform (case 4, nonlinear dispersion, 0.61 Hz).
1
100
0.5
50
0 0
0 2
4 Time [s]
6
8
0
2
4 Time [s]
Fig. 13. Energy dispersion rate (case 3, nonlinear dispersion, 0.61 Hz).
I
where t 0 is the time when the inertial force, f , in Eq. (35) is zero. The horizontal axis shows the distance from the left side wall. At 0.386 Hz, one solitary wave moves back and forth in the tank. At 0.358 Hz and 0.340 Hz, two and three solitary waves move in this manner, respectively. Fig. 12 shows the waveform in case 4 at 0.61 Hz. The numerical calculations use nonlinear dispersion. In this case, there is no solitary wave, and the waveform in the tank is near the first-order mode shape. The waveforms from the model agree well with the experimental results. Therefore, the proposed model is valid for the numerical analysis of nonlinear shallow water wave problems. Fig. 13 shows the energy dispersion rates of dampers. The left panel shows the time-series energy dispersion rate of the base support damper at mass point 1 given the following equation: Eb ¼ cb1 x_ 1 U x_ 1
(37)
The right panel of the figure shows the energy dispersion rate of the connecting support damper at the measurement point (element 2) given the following equation: Ec ¼ cc2 ðx_ 2 x_ 1 Þ U ðx_ 2 x_ 1 Þ
(38)
The scales of the vertical axis are different. The dispersion rates become large when the solitary wave passes through. The energy dispersion rate of the connecting damper is larger than that of the base support damper because the damping coefficient of the connecting damper that includes surface tension, turbulence, and bulk damping (cc2 ¼ 0:35 N s/m) is larger than that of the base support damper that considers the shear stress from the bottom face (cb1 ¼ 2:72 10 3 N s/m).
5. Conclusion To analyze nonlinear shallow water wave phenomena in a tank, we propose a concentrated mass model that represents the nonlinear static restoring force and nonlinear dispersion. A comparison of the natural frequencies obtained from the concentrated mass model and the theoretical value of rectangular and triangular tanks shows good agreement. This shows that the proposed model is also valid for the non-uniform water depth problem. In addition, an experiment on a rectangular tank has been performed, and the experimental results have been compared with the numerical results obtained from the model. The numerical results agree well with the experimental results both qualitatively and quantitatively for phenomena such as the solitary waves. The comparison also shows that the nonlinearity of dispersion is necessary when the water level amplitude or water depth is large. In summary, the proposed model is suitable for the numerical analysis of nonlinear shallow water waves in a tank. Future tasks include the modeling of viscous damping and the non-uniform water depth problems, including experimental verification.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
11
Appendix A Modification of shooting method for the proposed model The shooting method is an accurate and powerful numerical method based on numerical integration to find a steady periodic solution directly [18,19]. The equation of motion in Eq. (32) is transformed into the first-order differential equation " # " #) I 0 x_ x _y ¼ V 1 ðyÞ U Yðy; tÞ y ¼ ; VðyÞ ¼ ; Yðy; tÞ ¼ (A.1) _ tÞ 0 MðxÞ Fðx; x; x_ where I is the unit matrix and 0 is the zero matrix. The following equation should hold so that the solution of Eq. (A.1) is a periodic solution of period T. yð0; y0 Þ ¼ y0 ¼ yðT; y0 Þ
(A.2)
Here, yðt; y Þ is the solution of Eq. (A.1) at time t when y is the initial value of y at t ¼ 0. The shooting method finds the initial value y0 that satisfies Eq. (A.2) using Newton method. In this method, the variational equation of Eq. (A.1) is necessary. The components of VðyÞ are variable in the case of the proposed model, so the variational equation becomes ∂Yðy; tÞ Wðy; tÞ ξð t Þ (A.3) ξ_ ðt Þ ¼ V 1 yðt; y0 Þ ∂y y ¼ yðt;y0 Þ 0
0
where ξðtÞ is the variation of yðt; y0 Þ, and W is the matrix given by h ∂V ∂V ∂V W ¼ ∂y1 y_ ∂y2 y_ ⋯ ∂yN y_ 0 " # 0 ∂V _ ði ¼ 1; ⋯; NÞ y ¼ ∂M ∂yi € ∂x x
⋯
i9 0 > > > = > > > ;
i
(A.4)
where yi is the i-th component of vector y. Assuming y0k is the k-th approximate initial value of the iterative calculation and δy0k is the correction amount, the condition that the corrected initial value y0k þ δy0k satisfies Eq. (A.2) becomes ΦðTÞ I δy0k ¼ y0k yðT; y0k Þ (A.5) y0k þ 1 ¼ y0k þ δy0k
(A.6)
where ΦðtÞ is the state transition matrix about ξðtÞ. ΦðTÞ is given by h i9 ΦðTÞ ¼ ξ1 ðTÞ ⋯ ξn ðTÞ =
ξi ð0Þ ¼ ei
;
(A.7)
where ei is the unit vector whose i-th element is 1. Each column of ΦðTÞ, ξi ðTÞ, is calculated by solving Eq. (A.3) using numerical integration. We repeat the iterative calculation of Eqs. (A.5) and (A.6) until ‖δy0k ‖ becomes small. Stability analysis is performed by calculating the characteristic multipliers, that is, eigenvalues of ΦðTÞ. When all eigenvalues, λi , are less than 1, the solution is stable. If even one eigenvalue is more than 1, the solution is unstable.
References [1] O.M. Faltinsen, A.N. Timokha, Sloshing, Cambridge University Press, New York, 2009. [2] K. Hatayama, Lessons from the 2003 Tokachi-oki Japan, earthquake for prediction of long-period strong ground motions and sloshing damage to oil storage tanks, J. Seismol. 12 (2) (2008) 255–263. [3] L. Sun, M., Y. Fujino, B.M. Pacheco, M. Isobe, Nonlinear waves and dynamic pressures in rectangular tuned liquid damper (TLD), Proceedings of the Japan Society of Civil Engineers 410 (1989) 81–92. [4] Y. Fujino, L. Sun, M., B.M. Pacheco, P. Chaiseri, Tuned liquid damper (TLD) for suppressing horizontal motion of structures, Journal of Engineering Mechanics 118 (10) (1992) 2017–2030. [5] W. Chester, Resonant oscillations of water wave I Theory, Proceedings of the Royal Society of London 306 (1968) 5–22. [6] W. Chester, J.A. Bones, Resonant oscillations of water wave II experiment, Proceedings of the Royal Society of London 306 (1968) 23–39. [7] T.G. Lepelletier, Tsunami- Harbor oscillations induced by nonlinear transient long waves (PhD thesis), California Institute of Technology, 1981. [8] T.G. Lepelletier, F. Raichlen, Nonlinear oscillations in rectangular tanks, Journal of Engineering Mechanics 114 (1) (1988) 1–23. [9] T. Shimizu, S. Hayama, Nonlinear response of sloshing based on the shallow water wave theory, JSME International Journal 30 (263) (1987) 806–813. [10] O.M. Faltinsen, O.F. Rognebakke, I.A. Lukovsky, A.N. Timokha, Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth, Journal of Fluid Mechanics 407 (2000) 201–234. [11] O.M. Faltinsen, A.N. Timokha, An adaptive multimodal approach to nonlinear sloshing in a rectangular tank, Journal of Fluid Mechanics 432 (2001) 167–200. [12] O.M. Faltinsen, A.N. Timokha, Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth, Journal of Fluid Mechanics 470 (2002) 319–357. [13] J.S. Love, M.J. Tait, Nonlinear simulation of a tuned liquid damper with damping screens using a modal expansion technique, Journal of Fluids and Structures 26 (2010) 1058–1077. [14] M. Antuono, A. Colagrossi, S. Marrone, C. Lugni, Propagation of gravity waves through an SPH scheme with numerical diffusive terms, Computer Physics Communications 182 (2011) 866–877. [15] M. Antuono, A. Colagrossi, S. Marrone, C. Lugni, Two dimensional modal method for shallow-water sloshing in rectangular basins, Journal of Fluid Mechanics 700 (2012) 419–440.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i
12
S. Ishikawa et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
[16] W. Nakayama, Boundary Element Method Applied to the Analysis of shallow liquid sloshing in moving tanks, JSME International Journal, Series C 39 (4) (1996) 800–807. [17] X. Deng, M.J. Tait, Theoretical modeling of TLD with different tank geometries using linear long wave theory, Journal of Vibration and Acoustics 131 (4) (2009) 0140141–01401410. [18] T.J. Aprille Jr., T.N. Trick, A Computer algorithm to determine the steady-state response of nonlinear oscillators, IEEE Transactions on Circuit Theory 19 (4) (1972) 356–360. [19] H. Tamura, K. Matuszaki, Numerical scheme and program for the solution and stability analysis of a steady periodic vibration problem, JSME International Journal 39 (3) (1996) 456–463. [20] H. Lamb, Hydrodynamics, Cambridge University Press, 1945.
Please cite this article as: S. Ishikawa, et al., Analysis of nonlinear shallow water waves in a tank by concentrated mass model, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.029i