Weakly nonlinear sloshing in a truncated circular conical tank

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1 IOP PUBLISHING (3pp) FLUID DYNAMICS RESEARCH doi:.88/ /45/5/555 Weakly nonlinear sloshing in a truncated circular conical tank I P Gavrilyuk M Hermann I A Lukovsky 3 O V Solodun 3 and A N Timokha 34 University of Cooperative Education Eisenach D-9987 Germany Friedrich-Schiller-Universitaet Jena D-7745 Jena Germany 3 Institute of Mathematics National Academy of Sciences of Ukraine Kiev 6 UKraine 4 Department of Marine Technology Norwegian University of Science and Technology NO-749 Trondheim Norway alexander.timokha@ntnu.no Received 3 March 3 in final form 7 August 3 Published 5 September 3 Online at stacks.iop.org/fdr/45/555 Communicated by M Funakoshi Abstract Sloshing of an ideal incompressible liquid in a rigid truncated (tapered) conical tank is considered when the tank performs small-magnitude oscillatory motions with the forcing frequency close to the lowest natural sloshing frequency. The multimodal method the non-conformal mapping technique and the Moiseev type asymptotics are employed to derive a finite-dimensional system of weakly nonlinear ordinary differential (modal) equations. This modal system is a generalization of that by Gavrilyuk et al 5 Fluid Dyn. Res Using the derived modal equations we classify the resonant steady-state wave regimes occurring due to horizontal harmonic tank excitations. The frequency ranges are detected where the planar and/or swirling steady-state sloshing are stable as well as a range in which all steadystate wave regimes are not stable and irregular (chaotic) liquid motions occur is established. The results on the frequency ranges are qualitatively supported by experiments by Matta E PhD Thesis Politecnico di Torino Torino. (Some figures may appear in colour only in the online journal). Introduction The multimodal method is a rather popular analytically approximate approach to the nonlinear liquid sloshing problem. The method makes it possible to replace in a rigorous mathematical way the original free-boundary problem by a low-dimensional system of nonlinear ordinary differential equations (modal equations) and thereby it facilitates analytical studies of the 3 The Japan Society of Fluid Mechanics and IOP Publishing Ltd Printed in the UK /3/555+3$33.

2 contained liquid dynamics and associated hydrodynamic loads. Examples are reviewed in the books by Lukovsky (99) and Faltinsen and Timokha (9) as well as in Ikeda and Ibrahim (5) Ikeda et al () Takahara and Kimura () and Lukovsky et al (). Along with the aforementioned low-dimensional modal systems providing analytical studies the literature contains computationally oriented versions of the multimodal method. The latter versions deal with multi-dimensional modal systems of complex structure and relatively large dimension. Normally they are used for simulating the transient sloshing. The computationally oriented modal equations are well represented by the fully nonlinear Perko s systems (see Moore and Perko 964 Perko 969 La Rocca et al ) and weakly nonlinear adaptive multimodal systems appearing in the papers by Faltinsen et al (6 ) Limarchenko (7) Love et al () Love and Tait ( ). The nonlinear multimodal method was originally proposed for tanks with vertical walls at the free surface. Using the non-conformal mapping technique by Lukovsky (975) makes it possible to generalize the method for tanks with non-vertical walls. However practical examples of the generalization are rare and almost fully represented by Lukovsky and Timokha () Gavrilyuk et al (5) Limarchenko (7) and Faltinsen and Timokha (3). A reason is that the multimodal method is rather sensitive to errors in satisfying the volume (mass) conservation condition and therefore it is desirable to have analytically approximate natural sloshing modes which exactly satisfy the Laplace equation and the zero Neumann condition on the wetted tank walls. The required analytically approximate natural sloshing modes have been constructed for a non-truncated circular conical tank (Gavrilyuk et al 5) and recently for a truncated circular conical tank (Gavrilyuk et al 8). Bearing in mind that applications normally deal with truncated conical shapes the constructed modes will be used in this paper to derive a seven-dimensional asymptotic nonlinear modal system of the Moiseev type which is in fact a generalization of that by Gavrilyuk et al (5). In section we give differential and variational formulations of the problem. Applying the multimodal method combined with the non-conformal mapping technique yields a fully nonlinear infinite-dimensional (modal) system of nonlinear ordinary differential equations coupling the generalized coordinates and velocities. These equations are known for upright tanks as Perko-type modal equations (Moore and Perko 964 Perko 969 La Rocca et al ). Section 3 shortly outlines results by Gavrilyuk et al (8) on the analytically approximate natural sloshing modes which are used in derivations of the Moiseev-type (Narimanov Moiseev) asymptotic modal equations. In section 4 the latter equations are presented in an explicit form. Because derivation of these equations is a tedious analytical procedure with cumbersome formulae involved the required technical details are reported in appendix A. Practically oriented readers do not need to follow computations of the appendix but alternatively may take the numerical non-dimensional hydrodynamic coefficients at nonlinear terms tabled for certain realistic tank proportions and liquid fillings. The hydrodynamic coefficients at the linear terms can be found in Gavrilyuk et al (8 ). The derived Moiseev type asymptotic modal equations are used to classify the steadystate resonant sloshing occurring due to a small-amplitude harmonic (horizontal or angular) tank excitation. The forcing frequency is close to the lowest natural sloshing frequency. In section 5 we construct an approximate time-periodic solution of the nonlinear modal equations which describes the steady-state wave regimes implying planar and swirling motions. Based on this solution we study the possibility of secondary resonances. In contrast to Gavrilyuk et al (5) these resonances depend on the two input parameters which can be interpreted as the semi-apex angle and the non-dimensional liquid depth. When a secondary resonance occurs the Moiseev-type modal equations may be inapplicable. The first Lyapunov

3 Figure. Sketch of the tapered conical container and adopted notations. method is implemented in section 7 to distinguish stable and unstable steady-state wave regimes. We draw the response curves and detect the frequency ranges where the steadystate regimes are stable. The response curves are qualitatively similar to those reported by Gavrilyuk et al (5) for a non-truncated conical tank. Along with the stability ranges for planar and swirling a frequency interval is indicated where irregular (chaotic) swirling may happen. The results on the frequency ranges are qualitatively supported by the model tests conducted by Matta ().. Statement of the problem We consider a rigid truncated (tapered) conical tank of the semi-apex angle θ. The tank performs small-magnitude oscillatory motions with six degrees of freedom. These degrees are associated with translatory tank velocity v (t) = ( η η η 3 ) and the angular tank motions which could be defined by the instant angular velocity ω(t) = ( η 4 η 5 η 6 ). The tank is partially filled by an ideal incompressible liquid performing an irrotational flow. The liquid motions as well as v (t) and ω(t) are considered in the tank-fixed coordinate system Oxyz whose origin O is superposed with the artificial cone vertex so that the Ox-axis coincides with the symmetry axis (figure ). The gravity acceleration vector g has the opposite direction to O x... Free-boundary problem When introducing the absolute velocity potential (x y z t) defined in the Oxyz-coordinate system and function ζ(x y z t) implicitly defining the free surface (t) : ζ(x y z t) = the free-boundary sloshing problem can be written in the form (see chapter in Faltinsen and Timokha 9) = r Q(t) () ν = v ν + ω (r ν) r S(t) () 3

4 ν = v ν + ω (r ν) ζ/ t r (t) (3) ζ + t (v + ω r) + U = r (t) (4) dq = V l = const (5) Q(t) where ν is the outer normal vector Q(t) is the liquid domain S(t) = S (t) S is the wetted tank surface (S is the tank bottom and S (t) is the wetted tank walls) r = (x y z) is the radius vector U = r g is the gravity potential (g is the gravity acceleration vector). Equality (5) expresses the liquid volume V l conservation which is in addition the necessary solvability condition of the Neumann boundary problem () (3). The free-boundary problem () (5) needs initial conditions determining the instant free-surface pattern and the normal velocity on (t) at t = t i.e. ζ(x y z t ) = ζ (x y z) / ν (t ) = (x y z) (t ) where ζ (x y z) and (x y z) (t ) are the two known functions. For the steady-state wave solutions the periodicity condition should be adopted... The Bateman Luke formulation Instead of dealing with the free-boundary problem () (5) the multimodal method normally employs the Bateman Luke variational formulation whose equivalence to the original free surface problem is for instance proved in section.5.3. by Faltinsen and Timokha (9) and in chapter by Lukovsky and Timokha (995). According to this formulation the solution ( and ζ ) coincides with the extrema points of the action t ( ) A(ζ ) = [p p ] dx dy dz dt (6) t Q(t) for arbitrary t and t (t < t ) subject to the variations satisfying δ t t = δζ t t =. (7) The pressure field p(x y z t) can be determined by using the Bernoulli equation rewritten in the non-inertial coordinate system Oxyz as follows: t + (v + ω r) + U = p p (8) ρ where p is the ullage pressure and ρ is the liquid density..3. General modal equations The Bateman Luke variational formulation was extensively used by many authors to derive nonlinear modal equations for upright tanks (the tanks having vertical walls at the free surface) when the single-valued presentation of (t): ζ = x f (y z t) = is possible. The derivation assumed that f is expanded in a Fourier series with the time-dependent coefficients {β N (t)} playing the role of the generalized coordinates. For non-vertical walls the Fourier representation is impossible and therefore we have to introduce the generalized 4

5 coordinates implicitly ζ = ζ(x y z; {β N (t)}) (9) subject to the volume conservation condition (5) considered as a holonomic constraint. We introduce the modal representation of the velocity potential (x y z t) = v r + ω (x y z {β N (t)}) + F N (t)ϕ N (x y z) () where {ϕ N } is a complete set of linearly independent harmonic functions defined in Q(t) for any admissible instant liquid shapes Ω = ( 3 ) are the Stokes Joukowski potentials which are parametrically dependent functions of β N as being the solution of the Neumann boundary value problem i = in Q(t) ν = yν z zν y N= ν = zν x xν z 3 ν = xν y yν x on (t) S(t). () Here ν i are the projections of the outer normal on the coordinate axes and {F N (t)} play the role of the generalized velocities. The Fourier-type solution () should keep the volume (mass) conservation that requires {ϕ N } to exactly satisfy the Laplace equation and the zero- Neumann boundary condition on the wetted tank surface. Because ζ and are independent variables in the Bateman Luke formulation the generalized coordinates {β N } and velocities {F N } are also independent and due to (7) satisfy the condition δf N t=t t = δβ N t=t t =. Substituting () into (6) and varying F N (Faltinsen and Timokha (9) chapter 5) leads to the kinematic modal equations da N dt K A N β K β K = K A N K F K for all N. () Following the derivations in (Faltinsen and Timokha (9) pp 3 3) leads to the dynamic modal equations K A K β N Ḟ K + + ω KL A K L β N ( lω β i l ωt β i F k F L + (ω v g) l β i ω J β i ) ( lωt + ω d β i dt l ωt β i ω ) = for all N. (3) The modal equations () and (3) are formulated with respect to the aforementioned generalized coordinates and velocities where A N = ϕ N dq A N K = ( ϕ N ϕ K ) dq l = Q(t) Q(t) l kω = ρ x dq l = Q(t) Q(t) k dq l kωt = ρ Q(t) y dq l 3 = Q(t) k t Q(t) dq J i j = ρ z dq (4) S(t)+ (t) i j t dq 5

6 r r Figure. The physical and transformed meridional cross-sections. Ji j = J ji i j k = 3 are implicitly defined nonlinear functions of generalized coordinates β N (the time evolution of Q(t) is fully determined by (9)). Here J is the inertia tensor of the contained liquid l/v l is the dynamic liquid mass center but the vectors l ω and l ωt have no a clear physical interpretation. 3. Analytically approximate natural sloshing modes Normally the functional set {ϕ N } in () is associated with the natural sloshing modes which are the eigenfunctions of the spectral boundary problem ϕ ϕ = r Q ν = r S ϕ ν = κϕ r ϕ ds = (5) ν formulated in the hydrostatic (mean) liquid domain Q bounded by the hydrostatic free surface and the mean wetted tank surface S. A mathematical inconsistency is that the natural sloshing modes are defined in the unperturbed domain Q but to make the integrals (4) mathematically correct the multimodal method requires the eigensolution of (5) which is expandable over into the ullage domain. Another important limitation is that the modal solution should be as precise as to satisfy the mass (volume) conservation and therefore the functional set {ϕ N } must be harmonic functions satisfying the zero-neumann condition on the wetted tank surface. To get an explicit definition of (9) we employ the non-conformal mapping technique by introducing the curvilinear coordinate system Ox x x 3 x = x y = x x cos x 3 z = x x sin x 3 (6) (x 3 = η is the angular coordinate) transforming the conical domain to the circular cylindrical shape as demonstrated in figure for the meridional cross-section of the static (mean) liquid domain in the physical G and transformed G planes. Considering the eigensolution of (5) in the curvilinear coordinate system 6 ϕ(x x x 3 ) = ψ m (x x ) sin m x 3 cos m x 3 m =... (7)

7 makes it possible to separate the spatial variables (x x ) and x 3 so that one yields the following m-family of spectral boundary problems: p ψ m x + q ψ m + s ψ m x x x + d ψ m m cψ m = in G (8) x s ψ m x + q ψ m x = on L (9) p ψ m x + q ψ m x = κ m pψ m on L () p ψ m x + q ψ m x = on L () ψ m (x ) < m =... () x ψ x dx = (3) where G = {(x x ) : x x x x x } p = x x q = x x s = x (x + ) d = + x c = /x and L L and L are defined in figure. The natural sloshing frequencies are σ mn = gκmn g κ mn = (4) r where κ mn = r κ mn are the non-dimensional eigenvalues normalized by the mean free surface radius r. The lowest natural sloshing frequency is associated with κ. Dependences of the non-dimensional spectral parameters κ mn on the lower-to-upper radii r = r /r (see figure ) are extensively discussed by Gavrilyuk et al (8). By using the Trefftz method Gavrilyuk et al (8) constructed the required analytically approximate Trefftz solution of (8) (3) which exactly satisfies (8) (9) and () q ψ m = ψ mn (x x ) = q a (m) nk w(m) k + k= l= ā (m) nl w (m) l (5) where w (m) k (x x ) = N (m) k x ν mk T ν (m) mk (x ) w (m) k (x x ) = N (m) k x ν mk ν mk (x ) with T ν (m) mk (x ) and T ν (m) mk (x ) expressed via the associate Legendre polynomials of the first kind P ν (m) (µ) as follows: T ν (m) mk (x ) = ( + x ) ν mk P ν (m) mk T (m) + x ν mk (x ) = ( + x ) ν mk P ν (m) mk. + x The numbers ν mk are the roots of P ν (m) (cos θ)/ θ θ=θ = and N (m) k and N (m) k are the normalizing multipliers introduced to satisfy the condition w (m) k L = w (m) L k L = L where implies the mean square-root norm on L L. 7 T (m)

8 4. Weakly nonlinear modal equations 4.. Modal solution We consider (9) in the x x x 3 -coordinates i.e. ζ = ζ(x x x 3 {β mi }) and postulate it as ζ = x f (x x 3 t) = x f (x x 3 {p mi } {r mi }) = x x β (t) (p mi (t) cos(mx 3 ) + r mi (t) sin(mx 3 )) f mi (x ) (6) m= i= where x is the distance between the origin and the mean free surface (see figure ) and f mi (x ) = σ mi g ψ mi(x x ) (7) defines the radial natural surface profiles and σ mi are the natural sloshing frequencies by (4). Satisfying the volume conservation condition (5) makes β (t) a function of other generalized coordinates β = G({p mi } {r mi }). The modal representation of the velocity potential () takes the form (x x x 3 t) = v r + ω + m= i= ( ) P mi (t) cos(mx 3 ) + R mi (t) sin(mx 3 ) ψ mi (x x ). According to (6) and (8) integrals (4) are fully determined by the generalized coordinates {p mi } and {r mi } in which the capital indices should be replaced by the complex indices (mi cos) and (mi sin) so that for instance when N = (mi cos) A N = A (micos) = π r f (x x 3 {p mi }{r mi }) π x x x ψ mi (x x ) cos(mx 3 ) dx dx dx 3. (8) 4.. The Moiseev asymptotics Henceforth we assume that the problem is scaled by the mean free surface radius so that all geometric parameters and generalized coordinates are non-dimensional and of course the circle has the unit radius. The ratio between the bottom and free-surface radii is denoted by r = r /r. We adopt the Moiseev asymptotics (Narimanov 957 Moiseev 958 Lukovsky et al ) for the introduced generalized coordinates and velocities. This asymptotics holds true for resonant tank excitations with the mean forcing frequency close to the lowest natural sloshing frequency and the secondary resonances are neglected. The Moiseev asymptotics has been widely used in the papers on the multimodal method (Faltinsen et al Gavrilyuk et al 5 7 Ikeda and Ibrahim 5 Lukovsky et al Takahara and Kimura Faltinsen and Timokha 3) as well as in other semi-analytical approaches to the nonlinear sloshing problem with a finite liquid depth (Ockendon and Ockendon 973 Bridges Waterhouse 994 Ockendon et al 996). The Moiseev asymptotics suggests that the non-dimensional forcing magnitude is of the order ɛ. For axisymmetric tanks this causes the two primary excited lowest modes differing only by the π/-azimuthal angle and associated with the non-dimensional generalized coordinates p and r to dominate. These are of the order O(ɛ /3 ). A simple 8

9 Associated with p and r Associated with p Associated with p and r Associated with p 3 and r 3 Figure 3. Wave patterns associated with the generalized coordinates included into our nonlinear modal analysis. Except for p these patterns appear twice differing by π/-azimuthal rotation. The drawings for θ = 3 and the non-dimensional ratio of the lower (bottom) and upper (the mean free surface) radii is r =.5. trigonometric analysis by the angular coordinate leads to the following asymptotic relations for the generalized coordinates and velocities: P σ R σ p r = O(ɛ /3 ) P n σ n R n σ n P n σ n p n r n p n = O(ɛ /3 ) P 3n σ 3n R 3n σ 3n P (n+) σ (n+) R (n+) σ (n+) p 3n r 3n p (n+) r (n+) = O(ɛ) n. (9) Remaining non-dimensional generalized coordinates and velocities are of the order o(ɛ) and can be neglected within the framework of the Moiseev asymptotics Finite-dimensional asymptotic modal equations Derivation of asymptotic modal systems based on the Moiseev asymptotics (9) implies neglecting the o(ɛ)-order terms in the modal equations () and (3). As a consequence we arrive at an infinite-dimensional system of nonlinear ordinary differential equations with respect to the generalized coordinates and velocities (9). Examples of such infinitedimensional systems are given by Lukovsky et al () and Faltinsen and Timokha (3). Other existing asymptotic analytically oriented modal equations e.g. in Lukovsky (99) Gavrilyuk et al (5) involve two dominant r and p and three second-order generalized coordinates and velocities associated with p p and r. Faltinsen and Timokha (9) showed that these five-dimensional nonlinear modal equations enable an accurate approximation of the steady-state sloshing due to resonant excitations of the lowest natural modes. This means that the weakly nonlinear modal equations of the Moiseev type do not require to include a large set of generalized coordinates of the second and third order. A physical reason for that is that the major of kinematic energy is normally accumulated by the natural sloshing modes possessing the lower natural sloshing frequencies. For an upright circular cylindrical tank it was enough to account for three second-order generalized coordinates associated with p p and r in addition to the two dominant generalized coordinates r and p. Based on this fact we include in our modal analysis the aforementioned five lowest modes associated with and in addition the two third-order generalized coordinates p 3 and r 3. The wave patterns of the adopted natural sloshing modes are shown in figure 3. Technical derivation details for the seven-dimensional Moiseev-type modal system are outlined in appendix A. For brevity the generalized coordinates and velocities are 9

10 denoted as follows: p = p r = r p = p r = r p = p r 3 = r 3 p 3 = p 3 P = P R = R P = P R = R P = P R 3 = R 3 P 3 = P 3. The result is the following system of ordinary differential equations coupling the nondimensional generalized coordinates: p + σ p + d 8 (ṗ + ṙ ) + d ( p p + r r ) + σ g (p + r ) = (3) r + σ r + d r ( p p + r r + ṗ + ṙ ) + d (p ( r p p r ) + ṗ (ṙ p ṗ r )) +d 3 ( p r r p + ṗ ṙ ṗ ṙ ) + d 4 ( r p p r ) + d 5 ( r p + ṙ ṗ ) + d 6 p r +σ (g p r + g (p r p r ) + g 3 (p + r )r ) + ( v 3 + gθ ) = (3) p + σ p + d p ( p p + r r + ṗ + ṙ ) + d (r ( p r r p ) + ṙ (ṗ r ṙ p )) + d 3 ( p p + r r + ṗ ṗ + ṙ ṙ ) + d 4 ( p p + r r ) + d 5 ( p p + ṗ ṗ ) + d 6 p p + σ (g p p + g (p p + r r ) + g 3 (p + r )p ) + ( v gθ 3 ) = (3) r + σ r + d 7 ṗ ṙ + d 9 ( p r + r p ) + σ g 4 p r = (33) p + σ p + d 7 (ṗ ṙ ) + d 9( p p r r ) + σ g 4(p r ) = (34) r 3 + σ 3 r 3 + d ( r (p r ) + p p r ) + d (r (ṗ ṙ ) + ṗ ṙ p ) + d 3 ( p r + r p ) + d 4 ( p r + r p ) + d 5 (ṗ ṙ + ṗ ṙ ) + σ 3 (g 5(p r + p r ) + g 6 r (3p r )) = (35) p 3 + σ 3 p 3 + d ( p (p r ) r p r ) + d (p (ṗ ṙ ) ṗ ṙ r ) + d 3 ( p p r r ) + d 4 ( p p r r ) + d 5 (ṗ ṗ ṙ ṙ ) + σ 3 (g 5(p p r r ) + g 6 p (p 3r )) =. (36) Here the non-dimensional hydrodynamic coefficients are functions of the mean liquid domain parameters; the corresponding formulae for them are given in appendix A. The natural sloshing frequencies σ i = σ i are defined by (4) where κ m are the corresponding nondimensional eigenvalues whose numerical values (as well as those for ) can be found in Gavrilyuk et al (8 ) Non-dimensional hydrodynamic coefficients Whereas r the tank becomes non-truncated and as expected the non-dimensional hydrodynamic coefficients tend to the numerical values by Gavrilyuk et al (5). For another limit case θ the tank tends to the upright circular cylindrical shape and modal equations (3) (36) should transform to the corresponding seven modal equations taken from the infinite-dimensional modal system by Lukovsky et al (). Figures 4 and 5 illustrate how the non-dimensional hydrodynamic coefficients d i and g j depend on < θ < 45 for the fixed non-dimensional liquid depth h =. The limit values

11 Figure 4. Coefficients d i i =... 6 d 8 d and g i i = 3 as functions of θ for the non-dimensional depth h = Figure 5. Coefficients d 7 d 9 d i i =... 5 and g i i = as functions of θ for the non-dimensional depth h =. on the vertical axis (θ = ) coincide with the coefficients in the front of the corresponding nonlinear terms computed for an upright circular cylindrical tank which were calculated by using the exact natural sloshing modes (Lukovsky et al ). The limit values are marked by d i. The modal equations (3) (36) contain the hydrodynamic coefficients g j which are not zero only for tanks with non-vertical walls. The graphs confirm that the limit numerical values g j tend to zero when the semi-apex angle tends to zero. Tables 3 present the numerical non-dimensional hydrodynamic coefficients d i and g j (m = 3 i =... 5 j =... 6) for three semi-angles but κ m and can be found in Gavrilyuk et al (8 ). The hydrodynamic coefficients of the modal equations can be rewritten in the dimensional form using the formulae d i = { r d i for i = 9 r d i for i = { r g i for i = 4 5 ḡ i = r g i for i = 3 6. (37) 5. The time-periodic solution of the modal equations We consider forced steady-state resonant liquid sloshing occurring due to harmonic translatory tank excitations. For brevity the excitations are assumed along the Oz-axis in notations of figure implying that η i = i 3 and η 3 = Hcos(σ t). Our task consists of

12 Table. Non-dimensional hydrodynamic coefficients d i (i =... 8) computed within the five significant figures. r d d d 3 d 4 d 5 d 6 d 7 d 8 θ = θ = θ = finding a time-periodic solution of (3) (36) implying the steady-state wave regimes. The lowest-order generalized coordinates r (t) and p (t) are presented by the Fourier series r (t) = (A k cos(kσ t) + A k sin(kσ t)) p (t) = (B k cos(kσ t) + B k sin(kσ t)) k= where according to the Moiseev asymptotics the leading asymptotic contribution is associated with the first harmonics i.e. r (t) = A cos σ t + A sin σ t + o(ɛ /3 ); p (t) = B cos σ t + B sin σ t + o(ɛ /3 ) (38) and A A B B = O(ɛ /3 ) ɛ = H. As follows from substituting (38) into the modal equations (3) (33) and (34) the generalized coordinates p (t) r (t) and p (t) are functions of the dominant amplitude k=

13 Table. Non-dimensional hydrodynamic coefficients d i (i = ) computed within the five significant figures. r d 9 d d d d 3 d 4 d 5 θ = θ = θ = parameters A A B and B i.e. p (t) = ( A + A + B + B ) o () ( A A + B B ) o () cos σ t (A A + B B ) o () sin σ t + o ( ɛ /3) (39) r (t) = (A B + A B ) o () (A B A B ) o () cos σ t (A B + A B ) o () sin σ t + o(ɛ /3 ) (4) p (t) = ( A + A B ) B o () + ( A A B + ) B o () cos σ t + (A A B B ) o () sin σ t + o(ɛ /3 ). (4) 3

14 Table 3. Non-dimensional hydrodynamic coefficients g j ( j =... 6) computed within the five significant figures. r g g g g 3 g 4 g 5 g 6 θ = θ = θ = Analogously one can find r 3 (t) = (A (A + A 3B B ) A B B )o () 3 cos σ t + (A (A + A B 3B ) A B B )o () 3 sin σ t + (A (A 3A 3B + 3B ) + 6A B B )o (3) 3 cos 3σ t +(A (3A A 3B + 3B ) 6A B B )o (3) 3 sin 3σ t + o (ɛ) (4) p 3 (t) = (B (3A + A B B ) + A A B )o () 3 cos σ t + (B (A + 3A B B ) +A A B )o () 3 sin σ t + (B (3A 3A B + 3B ) 6A A B )o (3) 3 cos 3σ t +(B (3A 3A 3B + B ) + 6A A B )o (3) 3 sin 3σ t + o (ɛ) (43) where coefficients om k are defined in appendix A. 4

15 Substituting (38) (39) (4) into the modal equations (3) and (3) and gathering the lowest-order terms at the first harmonics lead to the system of algebraic equations (m (A + A + B ) + m B )A + m 3 A B B + ( σ )A = H (m (A + A + B ) + m B )A + m 3 A B B + ( σ )A = (m (A + B + B ) + m A )B + m 3 A A B + ( σ )B (44) = (m (A + B + B ) + m A )B + m 3 A A B + ( σ )B = with respect to the dominant amplitude parameters where coefficients m i depend on hydrodynamic coefficients of the modal equations; see formulae (B.) in appendix A. The algebraic system (44) is similar to those by Gavrilyuk et al (5) where we showed that its solvability condition consists of A = B = and therefore there are only two nonzero amplitude parameters which can be found from the system m A 3 + m A B + ( σ ) A = H m B 3 + m A B + ( σ ) B = (45) whose solution obviously depends on m i and in turn on the non-dimensional ratio of the bottom and free surface radii r σ (r ) and θ (m i = m i ( σ r θ )). As shown by Gavrilyuk et al (5) one can distinguish two types of solutions of (45) and the corresponding steady-state wave regimes. The first solution type A B = implies the so-called planar waves. The second solution type A B leads to the so-called swirling. The planar waves are described by the asymptotic solution p = r = p 3 = r = A cos σ t + o(ɛ) p = A o() + A o() cos σ t + o(ɛ ) r 3 = A 3 o() 3 cos σ t + A 3 o(3) 3 cos 3σ t + o(ɛ 3 ) p = A o() A o() cos σ t + o(ɛ ) (46) where the amplitude parameter A is the root of the cubic equation Swirling implies m A 3 + ( σ ) A H =. (47) r (t) = A cos σ t + o(ɛ) p (t) = B sin σ t + o(ɛ) r (t) = A B o () sin σ t + o(ɛ ) p (t) = ( A + ) B o () ( A B) o () cos σ t + o(ɛ ) p (t) = ( A B ) o () + ( A + B ) o () cos σ t + o(ɛ ) r 3 (t) = A (A B )o() 3 cos σ t + A (A + 3B )o(3) 3 cos 3σ t + o(ɛ 3 ) p 3 (t) = B (A B )o() 3 sin σ t + B (3A + B )o(3) 3 sin 3σ t + o(ɛ 3 ) where the amplitude parameters A and B are computed from the equations (48) m 6 A 3 + m 5( σ )A H = B = (m A ( σ ))/m > (49) m 5 = m 3 /m and m 6 = m 4 m 5. The latter inequality in (49) is the solvability condition. 5

16 r i.8 i.7 i.6 3 i.5 4 i i n r i.8.8 i.7 3 i i i n r i i i 3 i 4 i r i n i 3 i 3 i 33 i 34 i 35 i 3n Figure 6. The graphs of r = r (i mn ) (r is the ratio of the bottom and free surface radii) for the semi-apex angle θ = 3. The secondary resonance is expected at r = and Secondary resonances When constructing the time-periodic solution we assumed that the forcing frequency σ is close to the lowest natural sloshing frequency σ i.e. σ σ. (5) The constructed solution is valid if and only if coefficients in front of the polynomial terms by the amplitude parameters are of the order O(). However these coefficients become large when σ is close to one from the natural sloshing frequencies σ i and σ i i or alternatively when 3σ tends to one from the natural sloshing frequencies σ 3i i and σ i i. This closeness is associated with the so-called secondary resonances. The necessary condition of the secondary resonance consists of satisfying the relations σ σ n σ σ n 3σ σ 3n 3σ σ (n+) n (5) together with (5). The secondary resonance is not avoidable with the strong equalities in (5) and (5). To analyze the secondary resonances with strong equalities in (5) and (5) we plot in figures 6 8 the graphs of i mn (θ r ) as functions of the non-dimensional parameter r (r is 6

17 r.797 i i i 3 i 4 i 5 r i n i n i i 3 i 4 i r i i i 3 i 4 i r i n i 3 i 3 i 33 i 34 i 35 i 3n Figure 7. The graphs of r = r (i mn ) (r is the ratio of the bottom and free surface radii) for the semi-apex angle θ = 45. The secondary resonance is expected at r = and.7. the ratio of the bottom and free surface radii) with a fixed value of the semi-apex angle i n (θ r ) = σ n = κn i n (θ r ) = σ n = κn σ κ σ κ i 3n (θ r ) = σ 3n = κ3n 3σ 3 κ i (n+) (θ r ) = σ (n+) = κ(n+) n. 3σ 3 κ The functions i mn = i mn (θ r ) do not depend on the forcing frequency σ and one can see that the condition (5) i mn = (53) for certain indices m and n is equivalent to the strong equality in the corresponding m n- equation of (5) and (5) simultaneously. The case r = corresponds to the V-shaped conical tank but the limit r implies the shallow water condition. The calculations were done for the three semi-apex angles θ = 3 45 and 6. The strong equality i = happens for r =.896 implying that the first axisymmetric mode is subject of the secondary resonance for larger r and the double harmonics σ can then be amplified. As for the triple harmonics 3σ it can occur for the modes (3) (4) (3) and (33). So for r =.65 the modes (33) are subject to the secondary resonance but the modes (3) is resonantly excited at r =.835. Finally the modes (3) are exposed to the secondary resonance at r =.86 and the modes (4) at r = The strong secondary 7

18 .9 r i i i 3 i 4 i 5 i n r i i 3 i 4 i 5 i n r i i i 3 i 4 i 5 r i n i 3 i 3 i 33 i 34 i 35.. i 3n Figure 8. The graphs of r = r (i mn ) (r is the ratio of of the bottom and free surface radii) for the semi-apex angle θ = 6. The secondary resonance is expected as r =.67 and.396. resonances for the semi-apex angle θ = 3 are not expected for the non-dimensional ratio r.5. As follows from figure 7 the secondary resonances also exist for θ = 45 at r =.6386 (modes (3)) r =.797 (mode ()) and r =.7 (modes (3)). This implies that the constructed Moiseev type modal equations can be applicable for the non-dimensional ratios r.6. Figure 8 demonstrates two critical values of r for θ = 6. These are r =.67 (the secondary resonance by the mode ()) and r =.396 (the secondary resonance for the modes (3)). Moreover i 3 is close to for r.5 but i for.55 r. This means that the derived modal equations may need revision accounting for secondary resonances for this semi-apex angle. 7. Stability analysis The hydrodynamic instability of the time-periodic solutions (46) and (48) can be studied by employing the first Lyapunov method. This implies introducing small perturbations of these solutions denoted by α β η γ δ µ and ν i.e. p (t) = p (t) + η(t) p (t) = p (t) + β(t) r (t) = r (t) + α(t) p (t) = p (t) + δ(t) r (t) = r (t) + γ (t) p 3 (t) = p 3 (t) + ν(t) r 3 (t) = r 3 (t) + µ(t) and constructing the following linear differential variational equations with respect to α β η γ δ µ ν: η + σ η + d ( r α + p β + αr + β p ) + d 8 ( αṙ + β ṗ ) + σ g (αr + βp ) = 8

19 α + σ α + d (α(ṗ + ṙ + p p + r r ) + r ( β p + αr + p β + β ṗ + αṙ )) +d (p ( α p βr + α ṗ ) + β(ṗ ṙ + p r r p ) α( p p + ṗ ) + β(ṙ p ṗ r )) + d 3 ( βr α p + βṙ α ṗ + γ ṗ δṙ + γ p δ r ) +d 4 (β r α p + γ p δr ) + d 5 ( α p + α ṗ + ηṙ + η r ) + d 6 (α p + ηr ) +σ (g (αp + ηr ) + g (βr αp + γ p δr ) +g 3 (βp r + α(p + 3r ))) = β + σ β + d (β(ṗ + ṙ + p p + r r ) + p ( β p + β ṗ + αr + αṙ + α r )) +d (r ( βr + βṙ α p ) + α( p r + ṗ ṙ r p ) β( r r + ṙ ) + α(ṗ r ṙ p )) + d 3 ( β p + β ṗ + αr + αṙ + δ ṗ + δ p + γ ṙ + γ r ) +d 4 ( δ p + γ r + α r + β p ) + d 5 ( β p + β ṗ + η ṗ + η p ) + d 6 (β p + η p ) +σ (g (βp + ηp ) + g (αr + βp + γ r + δp ) +g 3 (αp r + β(3p + r ))) = γ + σ γ + d 9( p α + r β + α p + βr ) + d 7 ( α ṗ + βṙ ) + σ g 4(αp + βr ) = δ + σ δ + d 9( r α β p αr p β) + d 7 ( β ṗ αṙ ) + σ g 4(βp αr ) = µ + σ 3 µ + d ( α(p r ) + β p r + α( p p r r ) + β( p r + r p )) +d ( α(p ṗ r ṙ ) + α(ṗ ṙ ) + β(p ṙ + r ṗ ) + β ṗ ṙ ) +d 3 ( α p + βr + δ r + γ p ) + d 4 (α p + β r + γ p + δr ) +d 5 ( α ṗ + βṙ + γ ṗ + δṙ ) + σ 3 (g 5(αp + βr + δr + γ p ) +3g 6 (α(p r ) + βp r )) = ν + σ 3 ν + d ( β(p r ) α p r + β(p p r r ) α(r p + p r )) +d ( β(p ṗ r ṙ ) α(r ṗ + p ṙ ) + β(ṗ ṙ ) α ṗ ṙ ) +d 3 ( β p αr + δ p γ r ) + d 4 (β p α r + δ p γ r ) +d 5 ( β ṗ αṙ + δ ṗ γ ṙ ) + σ 3 (g 5(βp αr + δp γ r ) +3g 6 (β(p r ) αp r )) =. Here d i (i =... 5) and g j ( j =... 6) are the coefficients of derived nonlinear modal system. Equations above constitute a system of linear ordinary differential equations with periodic coefficients. Its fundamental solution can be obtained by employing the Floquet theory suggesting the solution α(t) = e λt ψ (t) β(t) = e λt ψ (t) γ (t) = e λt ψ 3 (t) δ(t) = e λt ψ 4 (t) η(t) = e λt ψ 5 (t) µ(t) = e λt ψ 6 (t) ν(t) = e λt ψ 7 (t) where λ is the characteristic exponent and ψ i are the π/σ periodic functions. The instability of (46) and (48) as follows from the expressions (54) depends on the values λ. At least one of the values should have the positive real part. (54) 9

20 .8.7 A K M N M B N N M K N M. K N M σ/σ N.. M σ/σ Figure 9. The response curves for planar and swirling resonant steady-state sloshing drawn for the semi-apex angle θ = 3 the non-dimensional ratio r =.5 and the non-dimensional excitation amplitude H =.. The amplitude parameters A and B imply longitudinal and transverse wave components. To get the characteristic exponent approximate values we pose the periodic functions ψ (t) and ψ (t) in the Fourier series ψ (t) = a cos σ t + a sin σ t + ψ (t) = b cos σ t + b sin σ t + (55) and substitute them together with (54) into equations in variations. Using the Bubnov Galerkin method leads to the following system of linear homogeneous equations: C a + C a + C 3 b + C 4 b = C a + C a + C 3 b + C 4 b = C 3 a + C 3 a + C 33 b + C 34 b = C 4 a + C 4 a + C 43 b + C 44 b = with respect to a i and b i (i = ) where coefficients C i j (i j =... 4) are functions of the hydrodynamic coefficients d i and g j of the original nonlinear modal equations system as well as of λ ( λ = λ/σ ) and the amplitude parameters A and B of the generalized coordinates p (t) and r (t) whose expressions are given by formulae (B.3). Requiring a non-trivial solution of (56) with respect to a i and b i (i = ) leads to the zero-determinant condition C C C 3 C 4 C D(λ) = C C 3 C 4 C 3 C 3 C 33 C = (57) 34 C 4 C 4 C 43 C 44 whose roots are the required characteristic exponents λ. (56) 8. The response curves Figures 9 and present the response curves (in terms of the amplitude parameters A and B ) associated with the steady-state wave motions. Accounting for the secondary resonance analysis in section 6 and the related limitations of the Moiseev-type modal equations the focus is on the semi-apex angles θ = 3 and 45 and the ratios r =.5 and.4 respectively.

21 .8.7 A K M N M B N N M K N M. K N M σ/σ N.. M σ/σ Figure. The same as in figure 9 but for θ = 45 r =.4 and H =.. Analyzing the response curves makes it possible to estimate the effective frequency ranges for planar and/or swirling sloshing. The amplitude parameter A measures the longitudinal wave component but B corresponds to the cross-wave component. The solid lines mark stable steady-state wave regimes but dashed lines are used to denote their instability. The figures demonstrate that the response curves are qualitatively similar to those known for non-truncated conical tanks Gavrilyuk et al (5). Firstly the planar sloshing (branches K K and M M ) is always unstable in a neighborhood of the linear resonance (σ/σ = ); the instability is expected for the forcing frequencies laying between the abscissas of K and M. Here K is the turning point but M is the Poincaré bifurcation point from which the branch M M 3 corresponding to unstable swirling emerges. Secondly stable swirling exists at σ/σ =. The swirling branch N N is divided by the Hopf bifurcation point N so that subbranch N N corresponds to stable steady-state wave motions (the abscissa of N is less than ) but the subbranch N N implies unstable steady-state swirling. Thirdly the interval between the abscissas of N and M marks the frequency range where both planar and swirling steady-state wave regimes are unstable. In this frequency range irregular (chaotic) waves are expected. The literature on experimental studies devoted to nonlinear resonant sloshing in a truncated conical tank is almost empty. Being interested in these experiments to validate our theoretical results we paid attention to Casciati et al (3) where appropriate experiments were mentioned in the context of the tuned liquid dampers equipped with a conical tank. Thanks to Professor Fabio Casciati and Dr Emiliano Matta (Politecnico di Torino Italy) we have got a more detailed report on these experiments documented in the PhD Thesis by Matta (). In the Thesis the experimental tank with the semi-apex angle θ = 3 was used for measuring the hydrodynamic force occurring due to a horizontal harmonic tank excitation. The thesis reports a set of the hydrodynamic force recordings as well as trying to classify the liquid motions based on both measurements and observations. Because the experimental series were conducted on a relatively short time scale the classification was only partly successful. In some cases it was possible to conclude on almost steady-state wave regime (planar or swirling) but many of the Matta s experimental series reported strong breaking waves and irregular motions which may be explained as either continuing transients or hydrodynamic instability. However these irregular and almost steady-state liquid motions were found as they follow from our analysis: irregular waves were established for σ/σ < a few stable model series demonstrated swirling for σ/σ > and stable planar waves were detected far from the linear resonance σ/σ =. This qualitatively supports our theory. As for a quantitative validation we think that it is difficult to do because the experimental series were relatively

22 short in the time. Moreover the experiments were done with relatively small (almost shallow) liquid depths causing the ratio r =.85. Section 6 shows that unfortunately this ratio is too close to r =.896 (the secondary resonance by the mode ()) and r =.835 (the secondary resonance by the mode (3)). This means that the Moiseev type modal equations are most likely inapplicable to this shallow water sloshing to get a quantitative agreement due to these secondary resonances. The secondary resonances are implicitly confirmed in observations of Matta (). Breaking waves and overturning are almost always detected for the forcing frequencies in a neighborhood of σ/σ = where our theory predicts irregular waves or swirling. As was discussed in chapters 8 and 9 by Faltinsen and Timokha (9) these phenomena are a typical attribute of multiple secondary resonances especially for the shallow water case. 9. Conclusions Employing the non-conformal mapping technique by Lukovsky (99) and the Moiseevtype asymptotics we derived approximate weakly nonlinear modal equations which describe resonant liquid sloshing in the V -shaped truncated conical tank. The modal system couples seven generalized coordinates of the considered infinite-dimensional mechanical system. The generalized coordinates are associated with perturbations of the seven lowest natural sloshing modes. The considered weakly nonlinear resonant sloshing is assumed to be due to a smallmagnitude excitation of the lowest natural sloshing frequency and there are no secondary resonances amplifying higher generalized coordinates. Arguments in choosing the seven generalized coordinates (see section 4.3) are based on physical circumstances and referring to earlier successful nonlinear modal systems for an upright circular cylindrical tank. Along with ideas and derivation details the present paper presents the numerical nondimensional hydrodynamic coefficients which can be useful for practically oriented readers. We studied the limit cases to ensure that there are no algebraic and arithmetic errors. Using the nonlinear Moiseev-type modal equations we studied the resonant steady-state sloshing occurring due to harmonic tank excitations with the forcing frequency close to the lowest natural sloshing frequency. Combining the Bubnov Galerkin and asymptotic schemes we constructed a time-periodic solution of the nonlinear modal equations and using the first Lyapunov method studied stability of this solution. Physically the time-periodic solution yields two types of steady-state wave motions planar and swirling. The planar sloshing implies liquid motions in the excitation plane but swirling means a rotary wave. The response curves were drawn to show a similarity between the steady-state sloshing in truncated conical tanks and upright circular cylindrical tanks. Qualitative agreement was found with experimental observations on steady-state wave motions by Matta (). Acknowledgment The authors acknowledge the financial support of the German Research Council (DFG). Appendix A. Technical details of derivations The employed natural sloshing modes are ϕ = ψ ϕ = sin x 3 ψ ϕ 3 = cos x 3 ψ ϕ 4 = sin x 3 ψ ϕ 5 = cos x 3 ψ ϕ 6 = sin 3x 3 ψ ϕ 7 = cos 3x 3 ψ that implies f (x x 3 ) = β + f (x )p + f (x )p cos x 3 + f (x )r sin x 3 + f (x )p cos x 3 + f (x )r sin x 3 + f 3 (x )p 3

23 cos 3x 3 + f 3 (x )r 3 sin 3x 3 in (6) where the time-dependent function β (t) follows from the volume conservation condition π x ( x x f + x f + ) 3 f 3 dx dx 3 = and takes (neglecting the higher-order terms) the form β = k p + k (p + r ) + k 3 p (r + p ) + k 4(p r r p (r + p )) + k 5(p + r ) + k 6(p 3 + r 3 ) + k 7(p r r 3 + p p p 3 r r p 3 + r p r 3 ) +. The coefficients k i are computed by the formulae k = e h t x k = e x h k 3 = e t h t x k 4 = e h t x k 5 = e h t x k 6 = e 33 h t x k 7 = e 3 h t x k 8 = π ( h 4 4 t h 4 ) b x k 9 = π h t e k = πh t e k = π h t e k = π h t e 33 k 3 = πh t e k 4 = πh t e k 5 = πh t e 3 k 6 = 3π ( ) e 8 e x k 7 = π 4 e 3 where h t = h = r cot θ h b = h = r cot θ and e i jk = x x f i (x ) f j (x ) f k (x )dx. The vector (l l l 3 ) by (4) reads as π x f +ht π x f +ht l = ρ x 3 x dx dx dx 3 l = ρ x 3 x cos x 3 dx dx dx 3 h b h b l 3 = ρ π x f +ht and takes the form h b x 3 x sin x 3 dx dx dx 3 l = λ + λ ( p + r ) + (A.) l = λ p + λ p ( p + r ) + λ3 p p + λ 4 (p p + r r ) + (A.) l 3 = λ 3 r + λ 3 r ( p + r ) + λ33 p r + λ 34 (p r p r ) + (A.3) where λ = k 8 λ = k 9 λ = λ = λ 3 = λ i λ = λ 3 = λ i λ 3 = λ 33 = λ i3 and λ 4 = λ 34 = λ i4 are computed by the formulae λ i = πρh 3 t s λ i = ( x s3 4 ) s s λi3 = 3πρh t s λ i4 = 3 πρh t s s = e and 3πρh t 4x quadratures si jkl (i j k l = 3) are defined by x si jkl = x ( f (x )) i ( f (x )) j ( f (x )) k ( f 3 (x )) l dx where the coefficients i j k l mean the power of functions f m (x ). Coefficients λ i jk are given by the formulae λ = λ 3 = λ i λ 3 = λ 33 = 4 (3λ i (4 o () + o () )λ i3 (4 o () + o () )λ i4) λ 3 = λ 33 = 4 (λ i (4 o () o () )λ i3 + (4 o () 3 o () )λ i4) λ 33 = λ 333 = 9 4 (λ i o () λ i3 o () λ i4). Explicit representation of (A.) appearing in (3) takes the form l = k 8 + k 9 (p + r ) + k p + k (p + r )+ k (p3 + r 3 )+ k 4(p r r + p (p r ))+ k 3 p (r + p )+ k 6(p r + p 4 + r 4) + k 5(r (p r 3 p 3 r ) + p (r r 3 + p p 3 )) + k 7 (p p 3 (p 3r ) + r r 3 (3p r )). 3

24 Within the introduced seven generalized coordinates the asymptotic expansion of (4) leads to A = b + b p + b 3 (p + r ) + b 4 p + b 5(r + p ) + b 6 p (p + r ) +b 7 (p p + p r r p r ) A = b 8 r + b 9 r (p + r ) + b p r + b (p r r p ) + b (p r 3 r p 3 ) +b 3 (r 3 (p r ) p r p 3 ) A 3 = b 8 p + b 9 p (p + r ) + b p p + b (p p + r r ) + b (p p 3 + r r 3 ) +b 3 (p 3 (p r ) + p r r 3 ) A 4 = b 4 r + b 5 p r + b 6 p r + b 9 p p r + b 7 r (p + r ) + b 8(p r 3 p 3 r ) A 5 = b 4 p + b 5 (p r ) + b 6 p p + b 7 p (p + r ) + b 8(p p 3 + r r 3 ) + b 9 p (p r ) A 6 = b r 3 + b r (3p r ) + b 4 p r 3 + b (r p + p r ) + b 3 r 3 (p + r ) A 7 = b p 3 + b p (p 3r ) + b 4 p p 3 + b (p p r r ) + b 3 p 3 (p + r ) A = b 5 + b 6 p + b 7 (p + r ) A = b 8 r + b 9 p r + b 3 (p r r p ) + b 3 r (p + r ) A 3 = b 8 p + b 9 p p + b 3 (p p + r r ) + b 3 p (p + r ) A 4 = b 3 r + b 33 p r + b 34 (p r 3 r p 3 ) A 5 = b 3 p + b 33 (p r ) + b 34(p p 3 + r r 3 ) A 6 = b 35 r 3 + b 36 (p r + r p ) A 7 = b 35 p 3 + b 36 (p p r r ) A = b 37 + b 38 p + b 39 p + b 4 p + b 4r + b 4(p p 3 + r r 3 ) A 3 = b 39 r + b 43 p r + b 4 (r p 3 p r 3 ) A 4 = b 44 p + b 45 p 3 + b 46 p p + b 47 p p + b 48 r r + b 49 p (p + r ) A 5 = b 44 r b 45 r 3 b 46 p r b 47 p r + b 48 r p b 49 r (p + r ) A 6 = b 5 p + b 5 (p r ) + b 5 p p 3 + b 53 r r 3 A 7 = b 5 r b 5 p r b 5 p r 3 + b 53 r p 3 A 33 = b 37 + b 38 p b 39 p + b 4 r + b 4 p b 4(p p 3 + r r 3 ) A 34 = b 44 r b 45 r 3 + b 46 p r b 47 r p + b 48 p r + b 49 r (p + r ) A 35 = b 44 p b 45 p 3 + b 46 p p + b 47 r r + b 48 p p + b 49 p (p + r ) A 36 = b 5 r + b 5 p r b 5 r p 3 + b 53 p r 3 A 37 = b 5 p + b 5 (p r ) + b 5r r 3 + b 53 p p 3 A 44 = b 54 + b 55 p + b 56 (p + r ) + b 57(p p 3 r r 3 ) A 45 = b 57 (r p 3 + p r 3 ) A 67 = A 46 = b 58 p + b 59 p p + b 6 (p p + r r ) + b 6 p (p + r ) A 47 = b 58 r b 59 p r + b 6 (r p p r ) + b 6 r (r p ) A 55 = b 54 + b 55 p + b 56 (p + r ) + b 57(r r 3 p p 3 ) 4