Narakorn Srinil

Giuseppe Rega

The effects of kinematic condensation on internally resonant forced vibrations of shallow horizontal cables

Keywords: Horizontal cable, Kinematic condensation, Modal interaction, Non-linear forced vibration, Primary resonance, Internal resonance

This study aims at comparing non-linear modal interactions in shallow horizontal cables with kinematically non-condensed vs condensed modeling, under simultaneous primary external and internal resonances. Planar 1:1 or 2:1 internal resonance is considered. The governing partial-differential equations of motion of non-condensed model account for spatiotemporal modification of dynamic tension, and explicitly capture non-linear coupling of longitudinal/vertical displacements. On the contrary, in the condensed model, a single integrodifferential equation is obtained by eliminating the longitudinal inertia according to a quasi-static cable stretching assumption, which entails spatially-uniform dynamic tension. This model is largely considered in the literature. Based on a multi modal discretization and a second-order multiple scales solution accounting for higher-order quadratic non-linear effects of infinite modes, coupled/uncoupled dynamic responses and the associated stability are evaluated by means of frequency-and force-response diagrams. Direct numerical integrations confirm the occurrence of amplitude-steady or -modulated responses. Non-linear dynamic configurations and tensions are also examined. Depending on internal resonance condition, system elasto-geometric and control parameters, the condensed model may lead to significant quantitative and/or qualitative discrepancies, against the non-condensed model, in the evaluation of resonant dynamic responses, bifurcations and maximal/minimal stresses. Results of even shallow cables reveal meaningful drawbacks of the kinematic condensation and allow us to detect cases where the more accurate non-condensed model has to be used.

A considerable number of publications have been concerned with geometrically non-linear vibrations of elastic suspended cables. Most of them are devoted to horizontal and shallow cables, whose non-linear modal interactions due to internal resonance phenomena are quite interesting subjects. Understanding of cable structural mechanics and non-linear dynamic behaviors due to internal/external resonance effects has been obtained by various analytical, numerical and experimental treatments [START_REF] Rega | Nonlinear dynamics of suspended cables, Part I. Modeling and analysis, Part II. Deterministic phenomena[END_REF].

As regards mechanical modeling and formulation, some simplifications, worth easing the analytical procedures, are frequently found in the relevant literature. The kinematic condensation technique is a well-established one, which considers zeroing the longitudinal inertia based on the quasi-static stretching assumption of the cable in motion, and links the corresponding longitudinal displacements to the transversal ones. As far as linear vibrations are concerned, the kinematically condensed model proves to be very useful for, e.g., identifying the low-order natural frequencies and modes of low-extensible and shallow cables [START_REF] Irvine | The linear theory of free vibrations of a suspended cable[END_REF]. However, when moving towards the non-linear regime, this model essentially disregards higher-order contributions of longitudinal dynamic deformation to the physics of the problem, by admitting spatiallyindependent non-linear stretching in cable kinematics. In other words, the system longitudinal and transversal dynamics are considered as non-linearly uncoupled, and the system vibrations are governed by integro-differential equations describing the solely transversal motion. In contrast, by accounting for both spatial and temporal variability of cable stretching via exact [START_REF] Srinil | Large amplitude three-dimensional free vibrations of inclined sagged elastic cables[END_REF][START_REF] Srinil | Three-dimensional nonlinear coupling and dynamic tension in the large amplitude free vibrations of arbitrarily sagged cables[END_REF] or approximate [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF][START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF] element kinematics, overall non-linear coupling is captured through the ensuing partial-differential equations (PDEs) of motion, and the space-time varying non-linear dynamic responses and tensions can be thoroughly assessed.

From a modal interaction viewpoint, suspended cables exhibit various activable planar and non-planar (e.g., 1:1, 2:1) internal resonances due to the geometrical combination of quadratic and cubic (cable sag and stretching) nonlinearities and to the vanishing non-linear orthogonality properties of modes [START_REF] Lacarbonara | Resonant nonlinear normal modes, Part I. Analytical treatment for structural one-dimensional systems[END_REF][START_REF] Lacarbonara | Resonant nonlinear normal modes, Part II. Activation/orthogonality conditions for shallow structural systems[END_REF]. For a non-planar interaction, both 2:1 and 1:1 resonances have been investigated in [START_REF] Rao | Internal resonance and non-linear response of a cable under periodic excitation[END_REF][START_REF] Perkins | Modal interactions in the non-linear response of elastic cables under parametric/external excitation[END_REF][START_REF] Lee | Nonlinear oscillations of suspended cables containing a two-to-one internal resonance[END_REF] and [START_REF] Pakdemirli | Analysis of one-to-one autoparametric resonance in cables-discretization vs. direct treatment[END_REF][START_REF] Zhao | Non-linear dynamic analysis of the twodimensional simplified model of an elastic cable[END_REF], respectively, whereas multiple internal resonances involving several planar/non-planar modes have been explored, e.g., in [START_REF] Rega | Multiple resonances in suspended cables: Direct versus reduced-order models[END_REF][START_REF] Nayfeh | Multimode interactions in suspended cables[END_REF]. All of these analytical studies, which describe fundamental and rich forced dynamics of a weakly non-linear system, were based on the condensed horizontal cable model. More recently, based on the noncondensed model [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF], the 2:1 resonance effects on non-linear planar free vibrations have been highlighted for both horizontal/inclined cables [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF]. Within an independent numerical framework, the strong modal interaction features characterizing large-amplitude free or forced vibrations due to internal resonance effects have been investigated in [START_REF] Srinil | Large amplitude three-dimensional free vibrations of inclined sagged elastic cables[END_REF][START_REF] Srinil | Three-dimensional nonlinear coupling and dynamic tension in the large amplitude free vibrations of arbitrarily sagged cables[END_REF] and [START_REF] Zheng | Super-harmonic and internal resonances of a suspended cable with nearly commensurable natural frequencies[END_REF][17] through a finite difference-or finite element-based, non-condensed, cable model, respectively.

In this study, we aim at making a thorough comparison of cable non-linear forced responses with kinematically non-condensed (NC) vs condensed (CC) modeling. The main goal is to highlight the effects of kinematic condensation on non-linear modal interactions due to 1:1/2:1 internal resonances in shallow horizontal cables as a result of different involvement of symmetric/anti-symmetric modes. Attention is focused on planar dynamics, wherein the effects of the two alternative models already fully display, and the analysis is conducted in such a way to highlight higher-order contributions from both resonant/non-resonant modes to the response of various -crossover/non-crossover -cables. The paper is organized as follows. In Sect. 2, approximate non-linear PDEs of motion of NC/CC models are addressed, with a comparison of system quadratic/cubic coefficients obtained through a multi-mode Galerkin expansion. In Sect.3, the asymptotic solution based on second-order multiple scales is summarized for each resonance case. Modal contributions and a comparison of interaction coefficients are discussed in Sect. 4, prior to examining the non-linear response through a continuation technique in Sect. 5. Direct numerical integrations of modulation equations are also performed to verify continuation results.

A comparison of space-time varying non-linear dynamic configurations and tensions is carried out in Sect. 6. The paper ends with the conclusions in Sect. 7.

A c c e p t e d

m a n u s c r i p t 4

THEORETICAL BACKGROUND

A horizontal suspended cable subjected to a uniformly distributed vertical harmonic excitation, F(x, t) = FcosΩt, is schematically shown in Fig.

H H H H x y d u x y d u X X X X = = = = , , , , , , C H H H H H C C w X X v t gH EA gH v t c c F F X X X w H H w H gH α = = = = = Ω=Ω
and cable parameter ( ) ( )

2 3 1 C C w S EA H λ π π = [2]
are introduced, with g being the gravity, E the Young's Modulus, A the cross-sectional area, w C the self-weight per unit unstretched length, S C the equilibrium length, c the viscous damping coefficient, F and Ω the amplitude and frequency of harmonic excitation, respectively. Here, x is the spatial variable, and t denotes time, the prime and dot being the relevant derivatives. The flexible cable is assumed to be homogenous and linear elastic with negligible torsional, bending and shear rigidities. Consider also that the cable has a small initial sag-to-span ratio (less than 1:8) and low static strain, the associated static configuration being described through the parabola 4 (1 ) y dx x = - [2]. In the following, the (∼) notation is dropped and zero boundary conditions, i.e., u(0, t

) = u(1, t) = v(0, t) = v(1, t)= 0 are
imposed. Symmetric (anti-symmetric) modes correspond to symmetric (anti-symmetric) v and anti-symmetric (symmetric) u components.

Kinematically Non-condensed Modeling

With the assumption of moderately large vibration amplitudes leading to a small dynamic strain, approximate element kinematics of total axial strain (e f ) of cables is given by [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF] ( )

2 2 2 1 1 , 1 2 f d
e e e e u yv u v y

⎛ ⎞ ′ ′ ′ ′ ′ = + = + + + + ⎜ ⎟ ′ + ⎝ ⎠ (1)

A c c e p t e d m a n u s c r i p t

5 where e(x) is the static strain, and e d (x, t) is the dynamic strain expressed through its Lagrangian measure. Due to geometrically non-linear stretching, planar forced damped vibrations of horizontal cables about static equilibrium are governed, in non-dimensional form, by [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF]:

( ) ( ) ( )
2 2 2 3 2 3 3 3 1 , 2 2 u c u u u y v u y u v u v u u v α α α ρ ρ ρ ρ ρ ′ ⎧ ⎫ ⎛ ⎞ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ + = + + + + + + + + ⎨ ⎬ ⎜ ⎟ ⎝ ⎠ ⎩ ⎭ (2) ( ) ( ) ( ) 2 2 2 2 2 3 3 3 3 cos , 2 2
y v c v v y u y v uv y v u v u v v F t α α α ρ ρ ρ ρ ρ ρ ′ ′ ⎧ ⎫ ⎛ ⎞ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ + = + + + + + + + + + Ω ⎨ ⎬ ⎜ ⎟ ⎝ ⎠ ⎩ ⎭ (3)
where

2 1 y ρ ′ = +
. This mechanical model, which holds also for inclined cables [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF][START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF], nonlinearly couples the u and v dynamics, and contains both quadratic/cubic non-linear terms even in the absence of initial sag (i.e., the taut string case). By omitting nonlinearities, damping and forcing terms, natural frequencies and (u/v) modal shapes are determined based on an N-terms assumed sine-based series [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF]. It is worth noting that, consistent with the assumed parabolic profile, here we put ρ ≈ 1 in both the linear dynamics analysis and the linear terms of Eqs. (2) and (3), so that the expected differences between NC/CC models mainly result from non-linear

terms. Yet, higher-order effects of ρ 3 will be accounted for in non-linear terms since numerical results in [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF] have highlighted their significant contribution to non-linear response.

Kinematically Condensed Modeling

For CC model, it is further assumed, with respect to previous assumptions, that (i) the secondorder term 2 u′ of the longitudinal gradient is negligible with respect to unity, and that, besides being ρ ≈ 1 in statics and linear dynamics, (ii) the higher-order ρ 3 effects in the non-linear dynamics are also negligible. Furthermore, by assuming that (iii) the cable non-linearly stretches in a quasi-static manner [START_REF] Irvine | The linear theory of free vibrations of a suspended cable[END_REF] in the absence of longitudinal external loading, the corresponding inertia and viscous damping effects are eliminated. Accounting for Eq. ( 2) and relevant boundary conditions, the dynamic strain in Eq. ( 1) becomes ( ) Thus, the CC model exhibits a space-independent dynamic tension (EAe d ) averaged over the spatial integral, in contrast with Eq. ( 1) that allows for both spatial and temporal strain variation.

1 2 0 1 ( ) . 2 d e t y v v dx ′ ′ ′ = + ∫ (4)
Similar to [START_REF] Rega | Multiple resonances in suspended cables: Direct versus reduced-order models[END_REF][START_REF] Nayfeh | Multimode interactions in suspended cables[END_REF], planar forced damped dynamics of condensed horizontal cables are governed by a single integro-differential equation associated with v only,

1 1 1 1 2 2 0 0 0 0 cos , 2 2 v cv v y y v dx v y v dx y v dx v v dx F t α α α α ′′ ′′ ′ ′ ′′ ′ ′ ′′ ′ ′′ ′ + = + + + + + Ω ∫ ∫ ∫ ∫ (5)
whereas the u displacement turns out to be v-dependent. Closed-form expressions of natural frequencies (ω) and modes of this CC system can be found in [START_REF] Irvine | The linear theory of free vibrations of a suspended cable[END_REF][START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF].

Multi Modal Discretization and Interaction Coefficients

Eqs. ( 2) and (3) or Eq. ( 5) are cast in state-space (first-order) form [START_REF] Lacarbonara | Resonant nonlinear normal modes, Part I. Analytical treatment for structural one-dimensional systems[END_REF]. Accounting for the orthonormality properties of linear eigenfunctions, the derived equations are projected onto the system full eigenbasis by letting , ,

J J J J m m m m m m U f V p ζ ζ ∞ ∞ = = = = ∑ ∑ in which J=1, 2 (J=2) for NC (CC) model, U 1 =u, U 2 =v, V 1 = u , V 2 = v , 1 m m ζ φ = , 2 m m ζ ϕ =
, f m and p m being the unknown displacement and velocity coordinates associated with both the longitudinal (φ m ) and vertical (ϕ m ) shape functions of the m mode [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF]. The Galerkin approach is then applied, thereby providing the infinite set of non-linearly coupled ODEs:

2 1 1 1 1 1 0, 2 cos , m m m m m m m mij i j mijk i j k m i j i j k f p p p f f f f f f Z t µ ω ∞ ∞ ∞ ∞ ∞ = = = = = - = + + = Λ + Γ + Ω ∑∑ ∑∑∑ (6)
for m=1, 2,…,∞, wherein ( )

1 2 2 0 2 m m m c d x µ φ ϕ = + ∫ and 1 0 m m Z F dx ϕ = ∫
are modal damping and forcing terms. With F being a constant, Z m is never (always) zero when ϕ m is a symmetric (antisymmetric) mode. The quadratic and cubic interaction coefficients of NC model are given, respectively, by

1 3 0 1 3 1 3 , 2 2 2 2 mij m i j i j i j m i j i j i j y y y d x α φ φφ φϕ ϕϕ ϕ φφ φϕ ϕϕ ρ ′ ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ Λ = - + + + + + ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎩ ⎭ ∫ (7) ( ) ( ) { } 1 3 0 1 , 2 mijk m i j k i j k m i j k i j k dx α φ φφ φ φϕ ϕ ϕ φφ ϕ ϕϕ ϕ ρ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ Γ = - + + + ∫ (8)
whereas those of CC model read

1 1 1 1 0 0 0 0 1 , 2 mij m i j m i j dx y dx y dx dx α ϕ ϕ ϕ ϕ ϕϕ ⎧ ⎫ ′′ ′ ′ ′′ ′ ′ Λ = + ⎨ ⎬ ⎩ ⎭ ∫ ∫ ∫ ∫ (9)
A c c e p t e d m a n u s c r i p t

7 1 1 0 0 . 2 mijk m i j k dx dx α ϕ ϕ ϕ ϕ ⎧ ⎫ ′′ ′ ′ Γ = ⎨ ⎬ ⎩ ⎭ ∫ ∫ (10)
Preliminary observation of these physical coefficients provides some hints about possible differences in the associated numerical outcomes [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF]:

• Both quadratic/cubic coefficients depend on the mechanical parameter α and the static solution (y′, y′′), but only the NC model accounts for ρ 3 -term effects.

• The NC (CC) model explicitly accounts for (ignores) non-linear coupling of longitudinal φ and vertical ϕ displacements, which possibly plays a meaningful role in system response depending on also cable sag and extensibility [18].

• Looking at the terms involving the solely ϕ functions in Eq. ( 9) or ( 10) against ( 7) or ( 8), it can be seen how the kinematic condensation entails approximating the exact integrals of products of the shape functions of NC model through products of their integrals.

SECOND-ORDER MULTIPLE SCALES SOLUTION

For a weakly non-linear response, periodic solutions to Eq. ( 6) are determined using the method of multiple scales (MMS) in condition of primary external and 1:1 or 2:1 internal resonance. The second-order asymptotic analysis is considered, capturing combined effects due to higher-order quadratic and cubic nonlinearities [START_REF] Lacarbonara | Resonant nonlinear normal modes, Part I. Analytical treatment for structural one-dimensional systems[END_REF]. With ε denoting a small bookkeeping parameter, we order the damping µ m and excitation Z m amplitude so that they appear in the same ε 3 -order problem, in both resonance cases, namely µ m → ε 2 µ m and Z m → ε 3 Z m . The relationships of external and 1:1 internal resonances are quantified through Ω=ω s + ε 2 σ f , ω s =ω r + ε 2 σ, whereas those of external and 2:1 internal resonances are expressed as Ω=ω s + εσ f , ω s =2ω r +εσ, with σ f and σ being external and internal detuning parameters, respectively. In both cases, the external excitation is assumed to put energy into the system via a symmetric (s) mode because Z s ≠ 0. A companion r mode -which may be either symmetric or anti-symmetric -is driven in the response via relevant internal resonance. Following the MMS analysis of free-undamped

A c c e p t e d m a n u s c r i p t

8 dynamics in [START_REF] Lacarbonara | Resonant nonlinear normal modes, Part I. Analytical treatment for structural one-dimensional systems[END_REF] and accounting for modal interaction of the two coupled (r, s) modes, approximate solutions of the forced-damped dynamics are obtained and herein summarized.

One-to-One Internal Resonance

Because 1:1 internal resonance is associated with cubic nonlinearities, the ensuing secular effects appear at ε 3 order. The real-valued modulation equations describing the slowly-varying evolution of amplitudes (a r , a s ) and relevant phases (β r , β s ) are obtained as [18] ( )

3 2 2 1 2 3 2 3 2 2 3 1 2 3 2 2 3 1 3
K a K a a K a a a a K a a K a K a a K a a K a a a K a a K a a Z K a a a a µ ω ω ω γ σ σ ω ω ω ω ω γ µ ω ω ω ω ∆ ∆ ∆ = - + + + ∆ ∆ ∆ = + + + + + + ∆ ∆ ∆ = - - - - + 3 2 2 2 3 1 3 2 3 cos cos 2 cos cos , 8 8 8 8 8 2 ss s rs s r r s r s s s r s s f s s s s s s s K a K a a K a a K a a Z K a a γ γ σ ω ω ω ω ω ω ∆ ∆ ∆ = + + + + + + (11-14) in which γ r = (σ f +σ)t -β r , γ s = σ f t -β s , ∆ = γ r -γ s ,
and K are second-order coefficients depending on the infinite-dimensional modal series and governing quadratic and cubic contributions. They read:

( )

2 2 2 1 2 1 3 , 4 hh hhm hmh mhh hhhh m m m h K ω ω ω ∞ = ⎡ ⎤ ⎛ ⎞ = Λ +Λ Λ + + Γ ⎢ ⎥ ⎜ ⎟ - ⎝ ⎠ ⎣ ⎦ ∑ h = r, s ( ) ( )( ) ( ) ( ) ( ) 2
K ω ω ω ω ω ω ω ∞ = ⎡ ⎤ ⎛ ⎞ Λ ⎢ ⎥ = Λ + Λ + Λ + Λ Λ + Λ + ⎜ ⎟ ⎜ ⎟ - + - - ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ + Γ +Γ +Γ ∑ ( ) ( )( ) (
)

1 2 2 2 2 1 , 4 ssm sms mrs msr mss srm smr sssr ssrs srss m m s m s r K ω ω ω ω ω ∞ = ⎡ ⎤ Λ +Λ Λ +Λ Λ = Λ +Λ + +Γ +Γ +Γ ⎢ ⎥ - - - ⎢ ⎥ ⎣ ⎦ ∑ ( ) 2 2 2 2 1 2 1 3 , 4 srm smr mrr srrr m m m r K ω ω ω ∞ = ⎡ ⎤ ⎛ ⎞ = Λ +Λ Λ + + Γ ⎢ ⎥ ⎜ ⎟ - ⎝ ⎠ ⎣ ⎦ ∑ ( ) ( )( ) ( ) 3
K ω ω ω ω ω ∞ = ⎡ ⎤ Λ +Λ Λ +Λ Λ = Λ +Λ + +Γ +Γ +Γ ⎢ ⎥ - - - ⎢ ⎥ ⎣ ⎦ ∑ (15-19)
Note that the 1:1 resonant interaction is activated only if at least one of K 1 , K 2 and K 3 is nontrivial [START_REF] Lacarbonara | Resonant nonlinear normal modes, Part II. Activation/orthogonality conditions for shallow structural systems[END_REF]. Based on the assumed full-basis Galerkin expansion, the second-order coupled forced

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2
, cos cos

cos 2 2 cos 2 2 1 , 2 cos 2 cos J J J r r r s s s J J J J s s ss ss r r rr rr J J s r r s r s r s r s U x t a t x a t x a t x x a t x x a a t x x γ ζ γ ζ γ ψ κ γ ψ κ γ γ ψ γ γ κ ≈ Ω - + Ω - + ⎧ ⎫ ⎡ ⎤ ⎡ ⎤ Ω - + + Ω - + + ⎪ ⎣ ⎦ ⎣ ⎦ ⎪ ⎨ ⎬ ⎡ ⎤ Ω -- + - ⎪ ⎪ ⎣ ⎦ ⎭ ⎩ (20)
with J = 1, 2 (J = 2) for NC (CC) model. The pertinent spatial corrections ,

J J ij ij
ψ κ accounting for quadratic effects of all infinite modes can be found in [START_REF] Lacarbonara | Resonant nonlinear normal modes, Part I. Analytical treatment for structural one-dimensional systems[END_REF].

Two-to-One Internal Resonance

The secular effects due to a 2:1 internal resonance appear at ε 2 order which is associated with quadratic nonlinearities only. However, one has to account for also the higher ε 3 -order effects associated with both quadratic and cubic nonlinearities so as to refine the asymptotic-based solution [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF]. We consider primary resonance of a high-frequency symmetric s mode, instead of a low-frequency r one which may be either symmetric or anti-symmetric. The relevant real-valued modulation equations read [18] sin , 4

r s r r r r r a a a a γ µ ω ℜ = - + (21) ( ) ( ) 2
K a K a a Z a a a γ γ γ σ ω ω ω ω ℜ = + + + + ( 24
)
where γ r = σt -2β r + β s , γ s = σ f tβ s , and the first-order interaction coefficient

2 srr ℜ = Λ [5, 7].
The second-order coefficient K ss is identical to Eq. ( 15) of 1:1 resonance case, whereas

( ) ( ) 2
K ω ωω ω ∞ = ≠ ⎡ ⎤ ⎛ ⎞ = Λ Λ +Λ + Λ +Λ Λ + + Γ ⎢ ⎥ ⎜ ⎟ - ⎝ ⎠ ⎣ ⎦ ∑ A c
K ω ω ω ω ω ω ω ω ω ∞ = ≠ ⎡ ⎤ ⎛ ⎞ Λ ⎢ ⎥ = Λ +Λ + Λ +Λ Λ +Λ + ⎜ ⎟ ⎜ ⎟ - + - - ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ Λ +Λ + + Λ Λ + Γ + Γ + Γ ∑
The 2:1 resonant interaction is activated only if ℜ is non-trivial [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF][START_REF] Lacarbonara | Resonant nonlinear normal modes, Part II. Activation/orthogonality conditions for shallow structural systems[END_REF], a circumstance which herein certainly occurs because the high-frequency mode is symmetric. In turn, the second-order coupled forced dynamic configurations of a 2:1 resonant suspended cable read

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2
γ γ ζ γ ζ γ ψ κ γ γ ψ κ γ γ ψ γ γ κ Ω ⎛ ⎞ ≈ -- + Ω- + ⎜ ⎟ ⎝ ⎠ ⎧ ⎫ ⎡ ⎤ ⎡ ⎤ Ω - + + Ω -- + + ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎪ ⎨ ⎬ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ Ω - - + Ω - + ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎪ ⎭ ⎩ (27)
where the spatial shape functions ( ,

J J ij ij
ψ κ ) can be found in [5, 7].

Mention must be made that, for the sake of generality, we account for also higher-order effects of σ both in second-order coefficients, Eqs. ( 16), ( 17), ( 19) and (26), of modulation equations, and in spatial displacement corrections of Eqs. ( 20) and ( 27), whereas these effects were disregarded in [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF][START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF]. On accounting for higher-order quadratic effects from every mode in the coupled amplitudes and displacements, the major drawback of discretization solution -in which the spatial variation in non-linear problem is assumed to be the same as that in linear problem -has been overcome. Indeed, it is seen from Eqs. ( 20) and ( 27) how, in NC (CC) model, both (only) u and v (v) are (is) spatially influenced by contributions from all of the modal eigenfunctions. This occurs irrespective of the fact that the amplitudes obtained with the two models might be already different due to their different coefficients (see Sects. 2.3, 4 and 5).

Thus, it is conjectured that overall errors in the associated non-linear dynamics may be meaningful. In the following, a prerequisite analysis of interaction coefficients is presented. The non-crossover (λ/π ≈ 2.95) cable involves symmetric (s,r) modes [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF], whereas the crossover (λ/π ≈ 2.02, 4.03) cables involve, though being different depending on the activated internal resonance, anti-symmetric low-frequency r modes, see also v shapes in Fig. 2. The convergent number N of the assumed sine series in linear dynamics increases with d and α, and the modal integral ϒ s affecting the forcing magnitude (Z s = Fϒ s ) changes with the solely v shape function.

Modal Interaction Perspectives

Actual features of internal resonance activation and possible solutions are drawn as follows.

I. Nearly (ω m ≈2ω r ≈2ω s ), instead of perfectly, tuned cables are considered in Γ Γ Γ and srrr Γ are zero in both K 1 and K 2 , too. In contrast, K 3 is always non-trivial, and the 1:1 resonance seems to be always activated at crossovers [START_REF] Lacarbonara | Resonant nonlinear normal modes, Part II. Activation/orthogonality conditions for shallow structural systems[END_REF].

III. As regards the 2:1 resonance, both NC and CC models never reveal zero values of ℜ when the symmetric high-frequency mode is involved in the internal resonance. Thus, such 2:1 resonance is always activated in horizontal cables [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF][START_REF] Lacarbonara | Resonant nonlinear normal modes, Part II. Activation/orthogonality conditions for shallow structural systems[END_REF].

IV. Analysis of Eqs. ( 11)-( 14) or ( 21)-(24) reveals two kinds of possible steady-state response ( 0 a γ = = ), namely the uncoupled-mode (a r =0, a s ≠ 0) solution of the directly excited symmetric mode, and the coupled-mode (a r ≠ 0, a s ≠ 0) solution. However, while in the 1:1 resonance the uncoupled solution occurs regardless of the low-or high-frequency mode being excited (i.e., σ may be positive or negative) only due to the vanishing K 1 and K 2 , it always (never) exists in the 2:1 resonance involving the high-frequency (low-frequency) excited mode.

Second-Order Quadratic Modal Contributions

Because both NC/CC models exhibit similar modal participation with the same number of retained modes [18], we use the more general NC model to discuss the modal contributions to second-order quadratic (a superscript q) coefficients due to a 1:1 ( 3 , q q ii K K ), in comparison with a 2:1 ( q ii K ), resonance. Accounting for the first 10 modes of 1:1 resonant first-crossover (λ/π ≈ 2.02) and 2:1 resonant non-crossover cables (λ/π ≈ 2.95), the pertinent percent contributions to 2, which distinguishes between resonant (underlined) and non-resonant, as well as symmetric (S) and anti-symmetric (A), modes. For the 1:1 resonant cable, both symmetric and anti-symmetric (low-order) modes contribute to 3 , q q rs K K , whereas only symmetric modes contribute to , q q rr ss K K . In accordance with Sect.4.1 (II), these differences occur because of the non-linear orthogonality: the two resonant (s, r) mixed (S, A) modal-based functions (e.g., mrs Λ ) substantially affect q rs K and 3 q K in Eqs.( 16) and ( 19), respectively, whereas only purely based -either s or r (S or A) -modal functions (e.g., hhm Λ , h = r or s) appear in Eq. ( 15) of q rr K and q ss K . In turn, such orthogonality allows for solely nonvanishing symmetric modal contributions in the 2:1 resonance case (λ/π ≈ 2.95), see also [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF] for λ/π ≈1.28 and 5.48. To sustain these aspects, the larger-sagged cable at second crossover (λ/π ≈ 4.03) -which may exhibit either 1:1 or 2:1 resonance -is considered in Table 3, by accounting for the first 15 modes. Because of involving the same high-frequency mode (s = 4) and no antisymmetric contribution, the symmetric modal contributions to q ss K are practically identical in both resonance cases. Besides the role played by a higher (order) number of symmetric nonresonant modes due to the increased sag effect, it can now be seen that anti-symmetric modes come into play when the cable exhibits not only 1:1 ( 3 , q q rs K K ) but also 2:1 (i.e., q rs K ) resonance, because of involving mixed resonant S/A modes.

In view of Table 2 or 3, anti-symmetric modal contributions of resonant (1 st or 3 rd ) mode are much higher than the negligible ones of non-resonant modes. It is also noticed (Table 3) that, when the 1:1 (2:1) resonance involving the anti-symmetric 3 rd (1 st ) mode is activated, even the contributions from the anti-symmetric low-frequency (high-frequency) 1 st (3 rd ) mode -which might be involved in the coexisting 2:1 (1:1) resonance -are negligible. Thus, one may discard a priori anti-symmetric non-resonant modes from the relevant discretization. This makes possible to avoid the expected singularity of the MMS solution for a perfectly tuned cable, see Sect. for smaller-sagged cables (Table 2), but up to the 9 th mode for larger-sagged cables (Table 3).

Overall, the second-order quadratic modal contributions due to 1:1 or 2:1 resonances at crossovers are in agreement with those due to 2:1 resonances away from crossovers [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF], thus confirming the importance of accounting for both resonant and non-resonant (higher-order, mostly symmetric) modes in the asymptotic solution of even shallow cables. This is always valid unless a very shallow cable (e.g., λ/π ≈ 1.28) is considered, wherein non-resonant modes are negligible [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF]. To get solution convergence, the non-resonant modes become as more important as the cable sag is increased towards higher λ/π values (e.g., λ/π ≈ 5.48), which require more retained modes. By way of example, non-resonant modal contributions to , , q q q rr ss rs K K K are about 7.03, 7.55 and -10.76 % for the first-crossover cable (Table 2), whereas they become 26.39, 28.86 and -30.98 % for the second-crossover cable (Table 3), respectively. Such meaningful effects have been overlooked in several papers that investigated modal interactions near crossovers by accounting for only resonant modes in the second-order analysis or by constraining the analysis to just the first order [START_REF] Rega | Nonlinear dynamics of suspended cables, Part I. Modeling and analysis, Part II. Deterministic phenomena[END_REF]. Based on the MMS, the lowest-dimensional discretization yields quantitatively-inaccurate and/or qualitatively-crude results with respect to the coinciding infinite-dimensional discretization [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF] or direct [START_REF] Pakdemirli | Analysis of one-to-one autoparametric resonance in cables-discretization vs. direct treatment[END_REF][START_REF] Rega | Multiple resonances in suspended cables: Direct versus reduced-order models[END_REF] perturbation.

A Comparison of Effective Non-linear Coefficients

The effective first-(ℜ) and second-order (K) coefficients of NC/CC cables in Tables 2 and3 are now compared in Tables 4a and4b, respectively. Apart from overall significant quantitative discrepancies, it can be seen that, for 1:1 resonant first (Table 4a) and second (Table 4b) crossover cables, there is no sign difference in all of the K rr , K ss , K rs , K 3 values; in particular, the CC model predicts smaller (larger) absolute values of K rr (K ss , K rs , K 3 ), thus entailing weaker hardening effects [START_REF] Lacarbonara | Resonant nonlinear normal modes, Part I. Analytical treatment for structural one-dimensional systems[END_REF]. As regards 2:1 resonant cables, the CC model predicts larger absolute values of ℜ. Moreover, while only -though substantial -quantitative difference is amenable to

A c c e p t e d m a n u s c r i p t

15 K rr and K ss , a remarkable sign difference occurs in K rs for both non-crossover (Table 4a) and second-crossover (Table 4b) cables. As discussed in [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part I. Theoretical formulation and model validation[END_REF]18], such circumstance occurs because in K rs the difference of the embedded cubic coefficients between NC/CC models is substantially larger than that of the embedded quadratic coefficients. The underlying physical meaning is that, based on second-order analysis, the CC model reduces (strengthens) the degree of hardening (softening) nonlinearities, since meaningful higher-order effects of longitudinal dynamic deformation due to cable stretching are neglected through the kinematic condensation. Such remarkable differences in overall values and/or sign of some coefficients are capable of influencing the ensuing non-linear dynamics and bifurcations of the two cable models.

A COMPARISON OF NONLINEAR DYNAMIC RESPONSES

A NC/CC comparison of non-linear dynamic responses is made by means of frequency-and force-response diagrams. To evaluate the equilibrium (fixed points) and dynamic solutions, the Cartesian form of Eqs. ( 11)-( 14) or ( 21)-( 24) is considered. Following [19], the fixed points representing steady-state motion are evaluated by the Newton-Raphson procedure, whereas the dynamic solutions representing limit cycles are evaluated based on the shooting application, with overall paths being traced out, upon varying a control parameter, via the continuation approach.

Stability of fixed points is analyzed based on eigenvalues of the relevant 4x4 Jacobian matrix, whereas stability of limit cycles is evaluated based on Floquet multipliers. As mentioned in Sect.

First-Crossover Resonant Cable

The first-crossover cable, commonly considered in the literature [START_REF] Rega | Nonlinear dynamics of suspended cables, Part I. Modeling and analysis, Part II. Deterministic phenomena[END_REF], is first analyzed in the 1:1 resonant interaction case. The first comparison is made through the frequency-response curves shown in Fig. 3 with a small forcing amplitude F = .002 (Z s ≈ .0016). The qualitative resemblance of overall pictures and bifurcation types with NC (Fig. 3a) and CC (Fig. 3b) models is remarkable: both exhibit the stable/unstable coupled solution (a s -a r ) curves on both sides of the zero-σ f axis, which ensue from the uncoupled a s branches via PF 1 and PF 2 bifurcations. In particular, both diagrams exhibit softening-type nonlinearities, and the stable coupled solutions persist over a wide σ f range. The directly (internally-resonant) excited a s (a r ) amplitude prevails left (right) of perfect external tuning (σ f = 0). These dynamic characteristics are in good qualitative agreement with the responses in [START_REF] Rega | Multiple resonances in suspended cables: Direct versus reduced-order models[END_REF], where, however, also multi-modal interactions of CC cables due to multiple internal resonances are considered, thus highlighting a more complex dynamics and bifurcation scenario in a certain σ f range.

Small quantitative differences occur between Figs. 3a and3b, being appreciable mostly as regards the large-amplitude coupled branches left to the zero-σ f axis. The CC model underestimates a s and slightly overestimates a r , by affecting the former greater than the latter.

The overall distinctive outcomes are due to the fact that, with the two models, the difference in the uncoupled branch is solely controlled by the different coefficient K ss , whereas the difference in the coupled branch is thoroughly affected by all of the different coefficients K rr , K ss , K rs and K 3 (see Eqs. 11-14). As a result, the CC model exhibits more softening non-linear response than the NC one, as already discussed in Table 4a. In addition, all of the associated bifurcations (SN 1 , PF 1 , PF 2 ) slightly shift to the left.

It is of interest to compare and discuss the force-response diagrams, so as to realize how much the discrepancies evolve when varying the forcing amplitude F. The ensuing NC (CC) results are depicted in Figs.4a (4b) and 4c (4d) for the given detunings σ f = -.12 and .12, respectively.

Depending on the initial conditions, multiple steady-state solutions coexist in both NC/CC (F ≈ .00296, .01720) and 4b (F ≈ .002645, .00825), whereas they remain stable and isolated from the coupled solutions over the whole considered F range in Figs. 4c and4d, respectively.

Overall, quantitative discrepancies in force-response curves and bifurcation locations between NC/CC models are observable, especially in the predominantly stable coupled solutions, and the differences become more remarkable when increasing F. While Figs. 4a and 4b look substantially similar, there are significant qualitative differences between Figs. 4c and4d as regards bifurcation features. Indeed, the NC model (Fig. 4c) shows the occurrence of a large region of dynamic solutions in between the two Hopf bifurcations at F ≈ .0161 (HF 1 ) and 0.0482 (HF 2 ) in the stable coupled branches, whereas the CC model (Fig. 4d) reveals none of them.

Similar to Fig. 3, the CC model underestimates (overestimates) a s (a r ) values with respect to the NC one. As a result, in the considered F range, the stable coupled a r amplitudes in Fig. 4d are seen to be always greater than the corresponding a s amplitudes, in contrast with Fig. 4c, where a s dominates the coupled response in the larger F range.

Second-Crossover Resonant Cable

A comparison of frequency-response diagrams of 1:1 resonant second-crossover cable is presented in Fig. 5. In this region, using the CC model, Pakdemirli et al. [START_REF] Pakdemirli | Analysis of one-to-one autoparametric resonance in cables-discretization vs. direct treatment[END_REF] also investigated a 1:1 resonance, however involving in-plane/out-of-plane modes, whereas a NC/CC comparison of responses due to planar 2:1 resonance is reported in [18]. With the same assigned parameters and σ f range as of the first-crossover cable, the softening dynamic responses of NC (Fig. 5a) and CC (Fig. 5b) models are still in qualitative agreement, and they are similar to those in Figs. 3a and3b because the sign of K rr , K ss , K rs and K 3 does not change for the higher crossover cable (Table 4b A c c e p t e d m a n u s c r i p t 18 vs 4a). However, overall response in Fig. 5 is more softening, as expected due to the larger sag.

Again, the CC model predicts more softening behavior than the NC one.

As regards stability and bifurcations, some remarkable changes occur in both models with respect to Fig. 3. First, there are two Hopf bifurcations in the coupled branches before they experience the jump via SN 1 to the uncoupled branches, when σ f is swept up. The HF 1 and HF 2 points occurring around σ f =0 are responsible for coupled-mode periodic solution. Second, the PF 2 bifurcation evidently shifts to lower σ f values. It is also worth remarking that the range of coupled solution (between SN 2 -PF 1 in Fig. 5), where there is no coexisting stable uncoupled solution, is greater than that of the first crossover cable. Moreover, some major extent of energy transfer from the directly excited a s mode to the driven a r mode is seen to occur. A comparison of dynamic solution branches emerged from HF 1 and HF 2 with NC/CC models is depicted in Figs. 5c and5d. Apart from being the σ f ranges of occurrence slightly shifted with respect to each other, both models predicts a sub-critical (super-critical) bifurcation HF 1 (HF 2 ) because the associated born limit cycle is unstable (stable), and a jump phenomenon of the stable branch onto the steady response via cyclic-fold bifurcation when sweeping σ f down.

To verify the existence of amplitude-steady or -modulated solutions and to better characterize the modal interactions, as distinguished by the stability, the continuation NC/CC results are complemented by numerical integrations of modulation equations. The fourth-order Runge-Kutta scheme is considered, with initial conditions of the NC/CC fixed points. Associated with Figs. 5a and 5b in which F = .002 and σ = 0, the time histories showing the coupled a r (solid lines) and a s (dashed lines) amplitudes of NC and CC cables are illustrated in Figs. 6a and6b for σ f = -0.1, and in Figs. 6c and6d for σ f = 0, respectively. Considering still σ f = 0 and σ = 0, but a larger F It can be seen that, after experiencing initial short-interval transient dynamics, both NC (Fig. 6a) and CC (Fig. 6b) time responses settle down onto steady-state coupled responses, as predicted by the continuation for σ f = -0.1. Analogously, for σ f = 0, the predicted coupled-mode dynamic solutions after Hopf bifurcations (Figs. 5a and5b) are confirmed by the periodically amplitude-modulated time histories established in Figs. 6c and6d upon imposing the unstable fixed-point NC and CC solutions as initial conditions. Thus, the NC and CC temporal laws are in agreement -though showing some quantitative differences -as regards stationary and nonstationary amplitude response features. However, for the same second-crossover cable, when increasing F while keeping other parameters and initial conditions unchanged, more marked amplitude-dependent limit cycles and stronger interaction features are seen in Figs. 6e and6f with respect to 6c and 6d, respectively. Essentially, qualitative differences -in terms of both the extent and duration of interaction -occur between NC (Fig. 6e) and CC (Fig. 6f) models according to the fact that the associated dynamic amplitudes are substantially different at σ f = 0 (Figs. 5c vs 5d). Overall, the numerical results validate the continuation outcomes, yet providing a further comparison in terms of non-linear temporal behaviors.

Non-Crossover Resonant Cable

Moving to the non-crossover cable (λ/π ≈ 2.95) exhibiting 2:1 resonance, major quantitative and qualitative discrepancies are visible between the frequency-response curves of NC (Fig. 7a)

and CC (Fig. 7b) models. The uncoupled a s solution now exhibits a hardening non-linearity and the indirectly-excited a r amplitudes play a dominant -or comparable -role in the coupled responses with respect to the directly-excited a s amplitudes. The differences between the two models are significant mostly as regards the coupled branches emanated from the uncoupled branches via PF 1 and PF 2 bifurcations. This occurs even though the overall forcing magnitude is reduced, with respect to the crossover cable cases, to Z s ≈ .0005, while fixing F = .002, due to a decreased ϒ s (Table 1). Remarkably, the CC model (Fig. 7b) predicts overestimated a r values, with the associated coupled responses being confined in a marginal σ f range with respect to NC model (Fig. 7a). Moreover, the former reveals the existence of dynamic solutions with two HF 1 /HF 2 bifurcations, which do not occur in the latter.

Apart from the very large quantitative errors, the qualitative differences of NC and CC models are definitely apparent in the force-response diagrams of Fig. 8, with σ f = 0 corresponding to Fig. 7. The NC coupled and uncoupled branches in Fig. 8a are seen to be isolated from each other in the considered F range, with the latter being stable over the entire interval. On the other hand, the CC model reveals in Fig. 8b how the uncoupled branch loses stability via PF 1 at a very low F and regains stability at PF 2 . The stable coupled a s branch is almost coincident with the unstable uncoupled one. This shows that the stable coupled solution takes part in only a certain F range, unlike Fig. 8a, where the stable coupled/uncoupled branches nearly always coexist with their amplitudes monotonously increasing when increasing F. These outstanding qualitative differences may be attributed -besides to the overall quantitative differences in the first-and second-order coefficients governing the amplitudes at different orders -also to the sign difference in the resulting K rs of the two models discussed in Sect. 4.3.

A COMPARISON OF NONLINEAR DYNAMIC DISPLACEMENTS AND TENSIONS

A comparison of non-linear dynamic displacements furnished by NC/CC models is now presented, in conjunction with the significance of space-time non-linear dynamic tensions [START_REF] Srinil | Large amplitude three-dimensional free vibrations of inclined sagged elastic cables[END_REF][START_REF] Srinil | Three-dimensional nonlinear coupling and dynamic tension in the large amplitude free vibrations of arbitrarily sagged cables[END_REF][START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF], which has been overlooked in many studies. Based on the obtained stable amplitudes, the 1:1 or 2:1 resonant coupled dynamic configurations accounting for second-order spatial corrections are constructed through Eq. ( 20) or (27), respectively. The NC dynamic tensions (T d ) accounting

for both u and v displacements are evaluated through the dynamic strain e d in Eq. ( 1), whereas the approximate -spatially constant -CC dynamic tensions are evaluated based on Eq. ( 4). In all cases, 51 cable nodes from the left support are considered. [-.0021,.0021] and [-.001,.002], respectively. Overall, it is evident that, due to mixed modal interactions at first (Fig. 9a) and second (Fig. 9b) crossovers, the 1:1 resonant displacements are actually asymmetric with respect to middle span, whereas the 2:1 resonant ones, due to purely symmetric modal interaction, still exhibit the symmetric spatial character (Fig. 9c). These results are in qualitative agreement with numerical and analytical results in [START_REF] Srinil | Three-dimensional nonlinear coupling and dynamic tension in the large amplitude free vibrations of arbitrarily sagged cables[END_REF] and [START_REF] Srinil | Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, Part II. Internal resonance activation, reduced-order models and nonlinear normal modes[END_REF], which highlighted 1:1 and 2:1 resonant modes in non-linear free vibrations, respectively.

Since the resonant (a r -a s ) amplitudes of NC/CC models are quantitatively different (see Figs.3, 5 and 7), there are quantitative discrepancies in the relevant displacement distributions of all cables. For 1:1 resonant cables, the qualitative agreement in Figs.9a and 9b is satisfactory, whereas there are some qualitative differences in 2:1 resonant profiles. Namely, as mostly visible at t=.4τ or .5τ in Fig. 9c, the NC model shows the symmetric second mode-like configuration (see Fig. 2) because the driven a r amplitude (≈ .000784) dominates the coupled response and contributions from the corresponding a s amplitude (≈ .000168) are relatively small (Fig. 7a), whereas the CC model exhibits a clear combination of 2 nd and 5 th modal shapes because the amplitudes (a r ≈ .000547, a s ≈ .000415) of the two participating modes are nearly comparable to each other (Fig. 7b). Thus, the largest difference at t = .4τ or .5τ is seen in Fig. 9c at cable midspan, where the maximum static curvature and quadratic drift effects take place.

With reference to Figs. and10d. For both σ f , dynamic tensions of the associated stable higher- or lower-amplitude uncoupled a s responses (dotted lines) are also evaluated, to better determine actual maximum stresses. In all cases, the T d values -which can be either positive (tensile) or negative (compressive) -are normalized with respect to the associated maximum static tension T H , and evaluated over a half period τ, whose time steps sequence follows the arrow in Fig. 10.

Overall, the spatially asymmetric (symmetric) character of the coupled (uncoupled) 1:1 resonant configurations of NC model, (see, e.g., Fig. 9a for the coupled configuration), reflects the asymmetric (symmetric) feature of dynamic tension distributions in Figs. 10a and10c. With σ f = 0, some NC tension distributions of coupled and uncoupled responses (Fig. 10a) are nearly spatially uniform and trivial, e.g. at t = 0.2τ, because the cable vibrates in a neighborhood of its static equilibrium, giving rise to small displacement gradients (see, e.g., Fig. 9a). In both Figs. 10a and10b, some of the estimated dynamic tensions of the uncoupled responses seem to be more tensile (t = 0-.2τ) or compressive (t = 0.4τ -.5τ) than those of the coexisting coupled responses. Depending on internal resonance condition and system elasto-geometric/control parameters, the parametric investigation highlights that, even for shallow cables, the condensed model may lead to significant quantitative and/or qualitative discrepancies in non-linear dynamic responses, bifurcation properties, as well as non-linear tensile/compressive stresses, with respect to the noncondensed model. Actual errors are seen to be significant in coupled -particularly higheramplitude -responses, with respect to the uncoupled ones, because the dynamics of the former are thoroughly controlled by the differences in all of the interaction coefficients, whereas those of the latter are solely controlled by the difference in one of them.

For this reason, to obtain more accurate approximate solutions and overcome some inherent drawbacks of the condensed model, it is recommended, based on our explorations, to consider the more general non-condensed model in the analytical-numerical treatment of cable non-linear resonant dynamics. In this respect, the drawbacks are expected to be somehow enhanced

A c c e p t e d m a n u s c r i p t 3

 3

A c c e p t e d m a n u s c r i p t 6

 6

A c c e p t e d m a n u s c r i p t 9 dynamic

 9 configurations associated with the u (J=1) or v (J=2) displacement component of a 1:1 resonant suspended cable are expressed as

A c c e p t e d m a n u s c r i p t 11 4.

 11 ANALYSIS OF 1:1/2:1 RESONANT INTERACTION COEFFICIENTS Planar 1:1 resonances involving mixed (symmetric/anti-symmetric) modes may occur near crossovers (λ/π ≈ 2n, n = 1, 2,…) in the frequency spectrum of Fig.

A c c e p t e d m a n u s c r i p t 12 II

 12 . The NC model involving a 1:1 resonance exhibits -though being not explicit -vanishing K 1 and K 2 through the embedded quadratic (Eq.7) and cubic (Eq.8) coefficients, in agreement with CC model through Eqs. (9) and (10)[START_REF] Lacarbonara | Resonant nonlinear normal modes, Part II. Activation/orthogonality conditions for shallow structural systems[END_REF] 18]. This is because the non-linear orthogonality properties hold between the eigenfunctions of mixed modal types at crossovers. With m being the anti-symmetric (symmetric) resonant or non-resonant mode, the ensuing identically zero in K 1 and K 2 , respectively, and all of the associated cubic coefficients , , sssr ssrs srss

A c c e p t e d m a n u s c r i p t 13 each

 13 of the coefficients are reported in Table

4 . 1 A c c e p t e d m a n u s c r i p t 14 (

 4114 I). For a proper reduced-order model selection, it is suggested accounting for up to the 5 th mode

4. 1 (

 1 IV), both coupled (a s -a r ) and uncoupled (a s ) solutions exist in both 1:1/2:1 resonances. For the sake of comparison, we choose small damping c such that µ r =.005 and µ s =.006, and consider perfect tuning σ =0. These parameters are kept constant in all cases. In the following, solid lines indicate stable fixed points, whereas dashed and dotted lines indicate unstable fixed points whose stability is exchanged through saddle-node (SN) or pitchfork (PF) and Hopf (HF) bifurcations, respectively. Filled (open) circles indicate stable (unstable) limit cycles.

A c c e p t e d m a n u s c r i p t 16

 16

A c c e p t e d m a n u s c r i p t 17 diagrams

 17 , which exhibit, at most, five (one stable and two unstable uncoupled, one stable and one unstable coupled) or three (one stable uncoupled, one stable and one unstable, or two unstable, coupled) fixed points in Figs.4a, b or 4c, drespectively. The corresponding uncoupled solutions lose and gain their stability via PF 1 and PF 2 bifurcations at low and higher F in Figs.4a

(

  .005), the associated time histories are shown in Figs. 6e (NC) and 6f (CC). A convergent time step equal to .0001 is considered in all cases.

A c c e p t e d m a n u s c r i p t 19

 19

A c c e p t e d m a n u s c r i p t 20

 20

A c c e p t e d m a n u s c r i p t 21 Coupled v displacements of 1 :

 211 1 resonant first-crossover (σ =σ f = 0, F = .002) and secondcrossover (σ = 0, σ f = -0.1, F = .002), and 2:1 resonant non-crossover (σ =σ f = 0, F = .002) cables are comparatively visualized in Figs.9a, 9b and 9c, respectively. The solid (dashed) lines denote time-varying spatial distributions over a half period (t = 0-.5τ) of forced vibrations with NC (CC) model. The hidden vertical axes in Figs.9a, 9b and 9c have a range of [-.005,.005],

A c c e p t e d m a n u s c r i p t 22 (

 22 3a and 3b of the 1:1 resonant first-crossover cable, the spatio-temporal dynamic tensions of NC model (solid lines) are shown in Figs.10a and 10c for σ f = 0 and σ f ≈ dashed lines) in Figs.10b

A c c e p t e d m a n u s c r i p t 24 deformation

 24 forced vibrations of shallow horizontal cables due to simultaneous primary external and planar 1:1 or 2:1 internal resonances are analytically investigated with the kinematically non-condensed vs condensed modeling. The former accounts for spatio-temporal dynamic tension modification and explicitly captures non-linear coupling of longitudinal/vertical displacements, whereas, in the latter, meaningful higher-order effects of longitudinal dynamic are eliminated, thereby entailing spatially-constant dynamic tension. Second-order quadratic non-linear modal contributions and orthogonality properties of symmetric/antisymmetric modes are discussed, clarifying the modal participating capacity between 1:1/2:1 (mixed/symmetric) resonant cables (modal interactions), and between crossover/non-crossover cables. Analyses put into evidence the significance of accounting for both resonant/non-resonant modes in all resonance cases, and meaningful differences in the values and/or sign of effective non-linear coefficients with the two distinctive models.Coupled/uncoupled, steady-state/periodic responses and their stability (bifurcations) are evaluated by means of frequency-and force-response diagrams. Direct numerical integrations are utilized to confirm the existence of steady-state or modulated amplitude responses. Moreover, coupled displacement profiles accounting for spatial corrections from all of the retained modes due to quadratic nonlinearities are constructed, and space-time varying non-linear dynamic tensions are analyzed. Results shed light on the actual spatially asymmetric/symmetric character of both dynamic configurations and tension distributions furnished by the non-condensed model.

A c c e p t e d m a n u s c r i p t 25 wheneverA c c e p t e d m a n u s c r i p t

 25 considering a multiple internal resonance involving coupled planar/non-planar modes, as well as highly extensible and/or non-shallow (i.e., non-parabolic) suspended cables. V., Martinelli L., Perotti F., Vestroni F., Nonlinear oscillations of cables under harmonic loading using analytical and finite element models, Comput. Methods Appl. Mech. Eng. 193, 2004, 69-85. 18. Srinil N., Rega G., Resonant non-linear dynamic responses of horizontal cables via kinematically non-condensed/condensed modeling, The 3 rd European Conference on Computational Mech. Solids, Struct. Coupled Problems in Eng., Lisbon, 2006, 1-20. 19. Seydel R., Practical Bifurcation and Stability Analysis, Springer, New York, 1994. 20. Gaul, L., The influence of damping on waves and vibrations, Mechanical Systems and Signal Processing 13, 1999, 1-30. 21. Kounadis A.N., Hamiltonian weakly damped autonomous systems exhibiting periodic attractors, ZAMP 57, 2006, 324-350.

Table 1

 1

	ω	2 m	-	(	s ω ω +	r	)	2	or 2 m ω	4 ω	h	2

, to avoid singularities due to small divisor terms in Eqs. (

15

)-(

19

) or (

25

)-(26), i.e.,

-

, h = r or s.

  Depending on cable kinematic modeling, it is worth remarking that not all of the obtained stable large-amplitude results always guarantee positive cable total tension during vibration. We have just drawn overall pictures and made a thorough comparison of non-linear responses with different modeling. In spite of this, analyses based on NC model have highlighted how dynamic tensions induced in even shallow cables significantly change in both space and time. This accounted variability, on the one hand, allows us to accurately predict the stress demand for cables; on the other hand, it allows us to cope with the associated finite-amplitude vibrations, whether unavoidable, by properly improving other cable properties such as damping[20, 21]. In contrast, analyses based on CC model, besides being definitely constrained to spatially-constant tensions, give rise to considerably underestimated dynamic stresses. This may result in unreliable design, particularly when compressive forces might actually occur.

	occurs at node 38 (26) with T d /T H ≈.1569 (-.1546). With CC model, the maximum tensile
	(compressive) stresses of coupled/uncoupled responses occur at t=0 (t=.5τ) with T d /T H ≈ .4004
	(-.3025) and .1564 (-.1526), respectively. Consequently, there are greater quantitative differences
	(> 40%) in dynamic stresses of coupled responses provided by the two models, with the NC one
	With σ f ≈ -.12, because of low-amplitude responses, it is obvious that all of the dynamic A c c e p t e d m a n u s c r i p t furnishing a larger time excursion range.
	tension distributions of uncoupled NC solution (Fig.10c) are nearly spatially uniform, like those
	of CC model (Fig.10d); consequently both models have comparable values at each time step and
	one may use the CC model for such situation. On the other hand, there are greater differences in
	the larger-amplitude coupled responses, where the associated spatial variation of NC dynamic
	strain is actually important putting into evidence a strong asymmetric feature. The coupled

Thus, the former produces larger non-linear tensile or compressive stresses than the latter. With NC (CC) model, the maximum tensile stress of uncoupled responses occurs at t = 0, at node 37, with T d /T H ≈ .3709 (.3202), whereas that of coupled responses occurs at t = 0, at node 28, with T d /T H ≈ .3492 (.2208). Apart from overlooking the essential asymmetric spatial features, the CC model predicts lower dynamic stresses in both coupled/uncoupled responses. Yet, the greater difference (36.77 %) with respect to NC model actually occurs in the coupled response. 23 responses also produce larger dynamic stresses than the uncoupled ones, with respect to the case σ f = 0. With NC model, the maximum tensile (compressive) stress at t = 0 (t = -.5τ) of coupled response occurs at node 12 (45) with T d /T H ≈.6791 (-.5269), whereas that of uncoupled response

Acknowledgement: Dr. Narakorn Srinil is extremely grateful to the support from the University of Rome 'La Sapienza', Italy, through a Postdoctoral Fellowship.

A c c e p t e d m a n u s c r i p t

List of Figures

The schematic model of a shallow horizontal cable.

2 Planar frequency spectrum and crossover phenomena of horizontal cables, with vertical shapes of r and s modes involved in internal resonances.

3

A comparison of frequency-response curves and bifurcations of 1:1 resonant first-crossover cable (λ/π ≈ 2.02) with σ = 0, F = .002: (a) NC and (b) CC model.

8

A comparison of force-response diagrams and bifurcations of 2:1 resonant non-crossover cable (λ/π ≈ 2.95) with σ = 0,σ f = 0: (a) NC and (b) CC model.

9

A comparison of non-linear coupled dynamic configurations over a half period of forced vibration with NC (solid lines) and CC (dashed lines) models: Modal contributions (%) 1:1 Modal contributions (%) 1:1

List of Tables


Dimitrios E Panayotounakos

Anastasia B Sotiropoulou

Nikos B Sotiropoulos

Manos Manios

Exact analytic solutions of the porous media and the gas pressure diffusion ODEs in nonlinear mechanics

Keywords: Exact analytic solutions, Nonlinear ODEs, Kidder's equation, Gas pressure diffusion equation

Two kinds of second-order nonlinear ordinary differential equations (ODEs) appearing in mathematical physics and nonlinear mechanics are analyzed in this paper. The one concerns the Kidder equation in porous media and the second the gas pressure diffusion equation. Both these equations are strongly nonlinear including quadratic first order derivatives (damping terms). By a series of admissible functional transformations we reduce the prescribed equations to Abel's equations of the second kind of the normal form that they do not admit exact analytic solutions in terms of known (tabulated) functions. According to a mathematical methodology recently developed concerning the construction of exact analytic solutions of the above class of Abel's equations, we succeed in performing the exact analytic solutions of both Kidder's and gas pressure diffusion equations. The boundary and initial data being used in the above constructions are in according with each specific problem under considerations.

Introduction

The Kidder nonlinear ordinary differential equation (ODE) governs the problem of the unsteady flow of a gas through a semi-infinite porous medium [START_REF] Kidder | Unsteady flow of gas through a semi-infinite porous medium[END_REF][START_REF] Davis | Introduction to Nonlinear Differential and Integral Equations[END_REF]. The origin of this equation is attractive, since it appears in the one dimensional problem obtained from the gas flow nonlinear partial differential equations. Until now, because of its strong nonlinearity (including quadratic first-order derivative damping term), only approximate or numerical solutions have been constructed under suitable boundary conditions. Furthermore, the equation itself consists a guideline for several relative problems appearing in the gas flow theory. In this paper using series of admissible functional transformations we reduce the Kidder nonlinear ODE to an equivalent Abel equation of the second kind of the normal form. It is well known [START_REF] Kamke | Differentialgleichungen[END_REF][START_REF] Polyanin | Handbook of Exact Solutions for Ordinary Differential Equations[END_REF] that this class of equations do not admit exact analytic solutions in terms of known (tabulated) functions. Only very special types can be analytically solved in parametric form depending on the kind of its second free member [START_REF] Polyanin | Handbook of Exact Solutions for Ordinary Differential Equations[END_REF].

A relative equation appearing in the theory of gas pressure problems is the onedimensional gas pressure diffusion equation [START_REF] Vardoulakis | Bifurcation Analysis in Geomechanics[END_REF]. We prove that this strongly nonlinear second-order ODE can be reduced to an equivalent generalized Emden-Fowler, or normal Emden-Fowler, or finally, to an Abel equation of the second kind of the normal form. Thus, according to the prescribed mathematical methodology, the construction of the exact analytic solution of this equation is obtained too.

The mathematical technique introduced is general and can be applied to a very large class of unsolvable (in terms of known tabulated functions) ODEs in nonlinear A c c e p t e d m a n u s c r i p t 2 mechanics and, generally, in mathematical physics (Blasius' eq.; Emden's eq.; Duffing's eq.; Van der Pol's eq.; Langmuir's eq., etc.)

Preliminaries -Notation

The equation due to R. F. Kidder [START_REF] Kidder | Unsteady flow of gas through a semi-infinite porous medium[END_REF] is the following

x y = = 0, , ( ) 0 x y → ∞ ∞ →
A semi analytical solution of the equation leading to expressions including the probability integral was constructed by means of the power series technique (see Ref.

[2], pp. 410-411).

A similar to the above problem was recently provided in Ref. [START_REF] Vardoulakis | Bifurcation Analysis in Geomechanics[END_REF] and concerns the nonlinear gas-pressure diffusion equation appearing in the formulation of gas flow problems, where the fluid compressibility must be taken into account. This relative nonlinear ODE is of the type with the following boundary conditions:

(2.4) 0 0 for , 1 , and for 0 , 0 1.

x y y x x y ∞ → ∞ = = = = < <
In what follows we will prove that by a series of admissible functional transformations both Kidder's equation (2.1) and the gas pressure diffusion equation (2.3) can be reduced to several well-known nonlinear ODEs of the generalized Emden-Fowler's, or of the Emden-Fowler's, or finally, of Abel's types of equations. The prescribed equations do not admit exact analytic solutions in terms of known (tabulated) functions, since only very special cases of them can be analytically solved in parametric form (see Ref. [START_REF] Polyanin | Handbook of Exact Solutions for Ordinary Differential Equations[END_REF], pp. 277-301; 241-250, and 29-55).

The reduction procedure a) Kidder's equation

Consider the Kidder equation Eq. (3.1) results in the following generalized Emden-Fowler equation { }

1 2 0 0 1 2 ; , , 1 , , 2 for 0 , 1 , and for , 1
.

xx x z xz z n m l x x z z a x z z - ∞ 1 , ⎧ ⎫ ′′ ′ = - = - ⎨ ⎬ ⎩ ⎭ = = = = - → ∞ = = (3.3)
Here, we utilize the triad notation { }

, , n m l to denote the specific exponents of the reduced equation.

The -transformation

F 1 1 : x z dz z dx dx x dz ′ = = = ′ F (3.4) reduces (3.
3) into the alternative generalized Emden-Fowler equation

( ) { } 2 1 2 0 1 2 ; , , ,1 , 2 for 1
, 0 , and for 1 , .

zz z x z x x n m l z a x x z z x - ∞ 2 , ⎧ ⎫ ′′ ′ = =-⎨ ⎬ ⎩ ⎭ = - = = = = →∞ (3.5)
Since in this case we have 1 2 1 , 1 0 and 2 1 n m l = -≠ -= ≠ = ≠ , the coordinate transformations (see Ref. [START_REF] Polyanin | Handbook of Exact Solutions for Ordinary Differential Equations[END_REF], p. 300) ( ) ( )

1 1 12 : , l n z z w z z t x x - + ′ ′ = = = = S 1 , - (3.6)
transform Eq. (3.5) to the generalized Emden-Fowler equation for with the parameters changed:

( ) w w t = ( ) { } { } 3 1 0 0 8 ; , , 1 ,1 ,3 , for 1 , 1
, and for 1 , 1.

tt t w t w w n m l z z a w w a z z w - ∞ ′′ ′ = - = - = = - = = - = = → (3.7)
But, since

) ( ) t t w w w t = ⇒ = 2 1 2 1 2 , z z z w ww w w ′ ′ = ⇒ = ⇒ =
while Eq. (3.5) can be rewritten as ( )

1 2 z z x x z ′ ′ = -
. Also, the second of (3.6) permits us to write the last equation as 2 2

z wz t x z t w t w x t 4 w ′ ′ ′ ′ ′ = - = = ⇒ = - .
Thus, the solution of Eq. (3.5) in parametric form results in ( )

1 2 1 , 4 t z w x w -. ′ = =- (3.9)
As we can see from the classification tables, concerning classes of second order nonlinear ODEs of Ref. In what follows it will be shown that both Emden-Fowler equations of the normal form (3.8) and generalized Emden-Fowler equation (3.5) can be reduced to Abel equations of the second kind of the normal form respectively. This can be achieved using convenient admissible functional transformations. We distinguish the following two cases.

Case a:

The Emden-Fowler equation of the normal form (3.8).

Since the pair of indices { }

, n m of this equation satisfies, and

1 1 m = -≠ 1 2 3 19 m n = ≠ --= -5,
there exists an admissible functional transformation that can reduce it to an Abel equation of the second kind of the normal form (see Ref. [START_REF] Polyanin | Handbook of Exact Solutions for Ordinary Differential Equations[END_REF], p. 250). To show this we introduce the new variables s and u such that ( )

x yy y f x ′ -= 2 2 1 1 2 3 2 : , 1 1 n n m m w n m n s w t u w w t m m + + - - + + + ⎛ ′ = = ⎜ - - ⎝ ⎠ H , t ⎞ + ⎟ (3.10) that is to say such that 3 2 3 2 3 : 2 , 2 w s w t u w wt t - - ⎛ ′ = - = - ⎜ ⎝ ⎠ H , ⎞ ⎟ (3.11)
obtaining the following Abel equation

2 2 1 2 2 s s u s s u s u ⎛ ⎞ ′ -+ = - ⎜ ⎟ ⎝ ⎠ , , (3.14)
which, by the substitution where n and ξ are to be determined, Eq. (3.20) becomes ( )

(3.15) * 1 1 u u s - = -+ performs the simplified Abel equation of the second kind * * * 2 2 1 3 1 1 5 2 2 2 2 s u u s u s s s s ⎛ ⎞ ⎛ ′ = - + + - ⎜ ⎟ ⎜ ⎝ ⎠ ⎝ . 2 ⎞ ⎟ ⎠ ( 3
ξξ ξ ξ ξ ξ ξ ξ - - ∞ ⎛ ⎞ ′′ ′ = = = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ → ∞ = = = ⎜ ⎟ ⎝ ⎠ (3.24)
An alternative form of (3.24) can be obtained by the already introduced Ftransformation. In fact, setting 1 :

n n ξ , ξ ′ = ′ F (3.25)
we reduced (3.24) into the alternative generalized Emden-Fowler equation: In the next Section, according to the mathematical methodology recently developed in Refs. [START_REF] Panayotounakos | Exact analytic solutions of unsolvable classes of first and second order nonlinear ODEs (Part I: Abel's equations)[END_REF][START_REF] Panayotounakos | Exact analytic solutions of nonlinear boundary value problems in fluid mechanics (Blasius equation)[END_REF], we will provide the exact analytic solution of the Kidder equation (2.1). Exactly the same mathematical technique follows the gas pressure diffusion equation (2.14).

ξ ξ ξ ξ ξ ξ - - ∞ ⎛ ⎞ ′′ ′ = = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ = = →∞ = = < < ⎜ ⎟ ⎝ ⎠ (

Exact analytic solution of the Kidder equation

Consider the general Abel equation of the second kind of the normal form ( ) ,

s uu u f s ′ -= (4.1)
where ( ) f s is an arbitrary smooth function. In the present case (Eq. (3.12)), ( ) f s is specified by the expression It was recently proved [START_REF] Vardoulakis | Bifurcation Analysis in Geomechanics[END_REF][START_REF] Panayotounakos | Exact analytic solutions of unsolvable classes of first and second order nonlinear ODEs (Part I: Abel's equations)[END_REF] that the Abel equation (4.1) admits an exact analytic solution in terms of known (tabulated) functions. Summarizing the results of the above reference, we write the solution of the Abel equation (3.12) as follows:

( ) ( ) ( ) ( ) ( ) 1 1 1 * * 2 1 3 * 1 ( ) 2 3 ( ) , 2 2 2 1 1 ( ) , 4 2 3 2 ( ) 3 sin cos cos 4 cos 1 4 2 4 s N s s C u s G As Bs u N s s C N s A A G As Bs A ; ξ ξ ξ ξ ξ ξ ξ - - ⎡ ⎤ + + ⎢ ⎥ ⎣ ⎦ = ⎡ ⎤ + + ⎡ ⎤ ⎣ ⎦ ′ = + + ⎢ ⎥ ⎡ ⎤ ⎣ ⎦ + + ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎛ ⎞ + + + ⎜ ⎟ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ⎡ ⎤ + + =- ⎣ ⎦ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠
(2)

(3) 3 ( ) 2 cos , 3 3
( ) 2 cos , ( )
2 cos ; 3 3 3 3 cos , 0 ; 2 3 p a N s p a p a N s N s q a a p π π π = - - + = -- = -- = - < < ⎛ ⎞ -⎜ ⎟ ⎝ ⎠ (4.4) Case b: Q > 0 3 3 ( ) 2 2 q q N s Q Q = -+ + -- (4.5) Case c: Q = 0 (1) (2) (3) 3 ( ) 2 , ( )
( ) 2 2 q N s N s N s = - = = -- 3 q (4.6)
where:

2 3 2 3 1 1 a 2 1 ( ) ( ) ( ) , ( ) , ( ) a a , 27 4 3 27 3 Q s p s q s p s b q s b c = + =-+ = - + (4.7) and ( ) ( ) ( ) ( ) ( )

A c c e p t e d m a n u s c r i p t

The subsidiary function is determined by the second of Eqs. ( 4.

3). ( ) G s

The already constructed analytic solutions defined by equations (4.3) to (4.8) constitute the intermediate integral (the solution in the phase plane) of the original Kidder equation (2.1). To show this one follows the inverse course through the already introduced admissible functional transformations. Thus, by the transformation (3.11), that reduces the Emden-Fowler equation (3.8) to the Abel equation of the normal form (3.12), we have the system of equations (3.11). This system, by means of Eqs. (4.3) to (4.8), can be equivalent written in the form:

1 2 1 3 2 2 1 2 ( ) , ( ) ; 3 1 ( ) , ( ) , 4 2
w w t f s w t f s f s u s f s s - - ′ = = = + = - (4.9)
where the expression is given by Eq. ( 4.3). Dividing the two first of (4.9) between them and using the second of them, we obtain the equivalent functional relations: The second of these expressions furnishes

( ) u u s = 1 2 1 3 2 2 ( ) , ( ) .
f w w f f ′ ′ ′ ′ ′ ′ ′ ′ = = = ⇒ = = + ⇒ ′ ′ ′ = - (4.12)
Integrating we extract

2 2 1 2 2 exp , 2 3 x f w C ds f f ′ ⎛ ⎞ = - ⎜ - ⎝ ⎠ ∫ ⎟ (4.13)
or, according to (4.9) we extract also 2 exp , 2 3

ds w C u s ⎛ = - ⎜ + ⎝ ⎠ ∫ ⎞ ⎟ (4.14)
where is the solution of the Abel equation (3.12) given by one of the expressions (4.4) to (4.6), and is a second constant of integration.

⎛ ⎞ = - ⎜ ⎟ + ⎝ ⎠ ⎛ ⎞ = =- - ⎜ ⎟ + ⎝ ⎠ ⎛ ⎞ ⎛ ′ = = = + - ⎜ ⎟ ⎜ ′ + ⎝ ⎠ ⎝ ∫ ∫ ∫ 3 s ⎞ ⎟ ⎠ (4.15) (ii) Transformation (Eqs. (3.7)): F 1 2 1 exp ; 3 2 3 4 t w C d w t u u s - ⎛ ′ = = ⎜ ′ + ⎛ ⎞ ⎝ ⎠ + ⎜ ⎟ ⎝ ⎠ ∫ s s ⎞ ⎟ (4.16)
(iii) Transformation (Eqs. (3.6) and (3.9)):

S 2 2 2 1 1 1 2 1 2 exp 2 , 2 3
1 2 exp , 2 3
1 3 exp 4; 4 4 2 3 z t ds z w C u s ds x s C t u s ds x w u s C u s - - - - ⎛ ⎞ = = - ⎜ ⎟ + ⎝ ⎠ ⎛ ⎞ ′ = = - - ⎜ ⎟ + ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ′ = - = -+ ⎜ ⎟ ⎜ ⎟ + ⎝ ⎠ ⎝ ⎠ ∫ ∫ ∫ (4.17) (iv) Transformation (3.2) -Final solution: 2 2 1 1 2 1 2 1 1 1 exp 2 , 2 3
3 exp 4 ; 4 2 3
( ) solution of the Abel equation (3.12) including one constant of integration ;

second indegration costant ; = z d s y C a a u s ds x u s C u s u u s C C s - - - ⎡ ⎤ ⎛ ⎞ = = - - ⎜ ⎟ ⎢ ⎥ + ⎝ ⎠ ⎣ ⎦ ⎛ ⎞ ⎛ ⎞ = -+ ⎜ ⎟ ⎜ ⎟ + ⎝ ⎠ ⎝ ⎠ = = = ∫ ∫ parameter . (4.18)
The final results for the gas pressure diffusion equation (2.3), that correspond to the final solution (4.18) concerning the Kidder equation (2.1), are the following:

⎡ ⎤ ⎛ ⎞ ⎛ ⎞ = - ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ = - - ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎣ ⎦ ∫ ∫ ∫ ∫ + (4.19)
where is the solution of the Abel equation ( )

u u s = 1 s uu u As Bs - ′ -= + (4.20) with 2 7 6 1 7 , 49 2 2 A B - ⎛ ⎞ = - = -⎜ ⎟ ⎝ ⎠ . (4.21)
We underline that because of the various types that the function ( ) N s admits (formulae (4.4) to (4.6)), it is possible that the solutions of both Kidder's and gas pressure diffusion equations are not unique inside a main interval ; in other words, they can be divided into several branches of solutions valid separately inside convenient consecutive subintervals. In this case matching of the corresponding solutions must be performed in all values of the main interval that solutions change. In what follows, using the results of previous sections, we provide a solution technique concerning the boundary value problem of the Kidder equation.

Boundary value problem -Solution technique

The boundary conditions and the solution technique that follows concern the Kidder equation (2.1). Similar procedure one also follows for the gas pressure diffusion equation (2.3).

The boundary conditions of the problem under consideration have been already prescribed in Section 2, that is: (1)

= = - = = - ′ = - = ∞ = = ′ = = = - = z z (5.3) A A G As Bs s C A A s C ξ ξ ξ ξ ξ ξ ξ ξ ξ - ⎡ ⎤ ⎛ + + + ⎜ ⎟ ⎢ ⎥ + + ⎣ ⎦ ⎝ = + ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = = = + ⎞ ⎠ (5.10) (iv) Validity of the Cardan equation 1 1 1 1 3(1) (1) (1) (1) 0 ,
, , ( , ] s s

The prescribed analysis demands successive solutions of nonlinear (transcendental) systems which are must be satisfied for each of the above mentioned consecutive subintervals 1 ( , ] i i s s + , where the solution of the problem under consideration (in the phase plane) probably changes according to the formulae (4.4) to (4.6). From now on, the solution in the physical plane is obtained through the formulae (4.18).

Discussion and conclusions

By a series of admissible functional transformations we reduce the nonlinear porous media and gas pressure diffusion ODEs to Abel's equations of the second kind of the normal form. These equations do not admit exact analytic solutions in terms of known (tabulated) functions. This unsolvability is due to the fact that only very special forms of this kind of equation can be solved in parametric form [START_REF] Polyanin | Handbook of Exact Solutions for Ordinary Differential Equations[END_REF]. Our goal is the development of the construction of exact analytic solutions of the above equations based on a mathematical technique leading to the derivation of exact analytic solutions of the Abel equation of the second kind of the normal form (see Refs. [START_REF] Panayotounakos | Exact analytic solutions of unsolvable classes of first and second order nonlinear ODEs (Part I: Abel's equations)[END_REF], [START_REF] Panayotounakos | Exact analytic solutions of nonlinear boundary value problems in fluid mechanics (Blasius equation)[END_REF]).

The reduction procedure introduced in the paper and the constructed solutions are very general, and can be applied to a large number of nonlinear ODEs in mathematical physics and nonlinear mechanics including the Van der Pol nonlinear oscillator, the Blasius equation in fluids [START_REF] Panayotounakos | Exact analytic solutions of nonlinear boundary value problems in fluid mechanics (Blasius equation)[END_REF], the Langmuir equation in current flow, etc.

For verification of the above contentions, we shall examine briefly one of the prescribed equations and compare its solution methodology with the already developed solutions procedure.

The Van der Pol free nonlinear oscillator [START_REF] Davis | Introduction to Nonlinear Differential and Integral Equations[END_REF] is governed by the following nonlinear ODE (6.1) ( )

A c c e p t e d m a n u s c r i p t 4 When

 4 the solution of the transformed equation (3.8) is obtained in the form of , the solution to the original equation (3.5) can be written in the following parametric form. By the first transformation of (3.6) we have

(

  [START_REF] Polyanin | Handbook of Exact Solutions for Ordinary Differential Equations[END_REF], pp. 278-281; 242, none of the above generalized Emden-Fowler, or of the normal form Emden-Fowler equations (3.3), (3.5), (3.7) and (3.8) admits an exact analytic solution.

  does not admit an exact analytic solution in parametric form (see Ref.[START_REF] Polyanin | Handbook of Exact Solutions for Ordinary Differential Equations[END_REF], pp. 29-45).Case b:The generalized Emden-Fowler equation(3.5).The coordinate transformations reduce the first of Eqs. (3.5) to the following Abel equation of the second kind ( )

  .16) Finally, the well-known substitution ([START_REF] Polyanin | Handbook of Exact Solutions for Ordinary Differential Equations[END_REF], p. 46) 16), and thus (3.5), to the following Abel equation of the second kind of the normal form 3.18) does not admit an exact solution in terms of known (tabulated) functions ([START_REF] Polyanin | Handbook of Exact Solutions for Ordinary Differential Equations[END_REF], pp. 29-45). Both equations (3.12) and (3.18) are equivalent to the original Kidder equation (3.1) and their possible solutions constitute its intermediate integrals in the phase plane.b) The gas pressure diffusion equationTo a similar to (3.5) generalized Emden-Fowler equation results also the nonlinear gas pressure diffusion equation (2.14). Indeed, the F -transformation[START_REF] Kamke | Differentialgleichungen[END_REF]

F

  3.26) Furthermore in equation(3.24) the exponents n, m and l of the variables ξ, n and n ξ ′ , s c r i p t which by way of the -transformation, can be written to the Emden-Fowler equation of the normal form and initial data.When the solution of the transformed equation (3.29) is obtained in the form , the solution of the original equation (3.24) can be written in the parametric

  given as in the following three cases:Case a: Q<0 (p<0)

( ) u s 2 C

 2 Summarizing, we list the following results:

7 )

 7 are calculated through Eqs. (4.4) to (4.7). The nonlinear (transcendental) system of Eqs. (5.8) to (5.11) enables us to estimate terms of the initial parameters as well as of the known is known. This, enables us to define the type of the solution valid inside (Eqs. (4.4) to (4.6)).1 2

1 ) 5 )

 15 is reduced to the following Abel equation of the second kind: are of the Abel normal form and they are similar to the Kidder and the gas pressure diffusion equations with different right-hand sides. With a set of similar or boundary initial conditions as in the before mentioned cases, the mathematical methodology for the exact analytic solutions of the above equations follows step by step that prescribed in Sections 3 and 4.

A c c e p t e d m a n u s c r i p t

  The origin of the equation is attractive, since it appears in the one dimensional problem obtained from the gas flow nonlinear partial differential equation. The boundary conditions required by the physical problem are the following.

		1	-	ay y	xx ′′	+	2	xy	x ′	=	0, 0	a < <1	(2.1)
	and appears in the problem of the unsteady flow of a gas through a semi-infinite
	porous medium. Here the notation	, y dy dx y x ′ =	x x ′ ′	=	2 d y dx	2	...	is used for total
	derivatives. for	0 ,		(0) 1	and for		.	(2.2)

A c c e p t e d m a n u s c r i p t 12

The first set of Eqs. (5.3) will be used for the evaluation of the intermediate integral in the phase plane, while the second set for the calculation of the final solution in the physical plane. Thus, combining the first set of the boundary data (5.3) together with the coordinate transformations (3.11), we extract ( ) ( ) ( )

4 u s= (5.4) Consequently, the Abel equation (4.2) and the solution (4.3) result in the following four equations:

where 0 N is the value of ( ) N s at 0 s s = , and is the value of the subsidiary function included in Eqs. (4.7), which is to be determined. 0

G ( ) G s

The set of the four equations (5.5) include five unknowns 0 0 0 0 , , , and

1 . Thus, one more equation is needed for the estimation of the above five unknowns. This equation can be derived by means of the coordinate transformations (3.11). Indeed, by way of (3.11) and (3.8) we evaluate ( ) ( )

Thus, since ( ) ( ) ( , ] s s

( 1 ) ( 1 ) ( 0 )

(5.9) (iii) Validity of the third of Eqs. (5.5) ( ) ( ) ( )