Generalized multiple scale approach to the problem of a taut string traveled by a single force

The strongly nonlinear dynamics of taut strings, traveled by a force moving with uniform velocity, is analyzed. A change of variable is performed, which recasts the equations of motion in terms of a linearized dynamic displacement, measured from the nonlinear quasi-static response. Under the hypothesis the load velocity is far enough from the celerity of the string, the system appears in the form of linear PDEs whose coefficients are slowly variable in time. Since the classic perturbation methods are not applicable to such kind of equations, the Generalized Method of Multiple Scales is developed, by directly attacking the PDEs, to derive asymptotic solutions. The validity of the analytical predictions is assessed by comparisons with numerical simulations, aimed to prove the accuracy of (1) linearization, and (2) the asymptotic approach.

by vehicles, machine tools, vehicle disc, guideways in robotics. There are many hundreds of published papers on moving-load dynamics; a brief overview is given in [1]. Pioneering studies on the subject date back to Frỳba [2], who investigated the effects of moving loads on elastic/inelastic solids and structures. Moving loads are typically schematized as: (i) moving force model, if the mass involved in the motion is not significant compared to that of supporting systems (e.g., [3,4]); (ii) moving mass model, when possible effects of loadstructure interaction are negligible (e.g., [5,6]); (iii) moving oscillator model, if stiffness and damping couplings between load and support are important (e.g., [7,8]). Among these, the moving force model is the simplest and widely used, since it is often sufficient to deal with problems of practical interest. Here, the focus is on moving loads which travel taut strings, for which the literature is full of applications (e.g., [9][10][11][12][13], most on traveling mass points). These systems are representative of ropeways (cableways), which are the subject of great interest (e.g., [14][15][16]) as they are used in the winter sports, in the material transportation (material ropeways) and also in urban transport. Particularly, urban ropeways appeared in some large world metropolises (such as Berlin, London, Lisbon and even New York), where they are seen as a good solution to speed up travel and to lighten the traffic on the road network; the urban cable cars move with low velocities, that are far enough from the critical traveling value (i.e. U U c , with U c the transverse wave speed of the string). As regards the taut string, it is an idealized model of cable with evanescent sag; it is quite accurate in the case of the prestressed cables of small sag, where the Irvine parameter is λ 2 < 4π 2 ( [17]). Indeed, such a model is used in literature to study several phenomena relating to prestressed cables, as e.g., the vibration control ( [18]) and the dynamic of inclined cables of cablestayed bridges ( [19]). Inclined cables of small sag, indeed, can be studied as horizontal taut strings, simply projecting the external forces onto the normal to the chord, to within a small error which is due to the tangential forces neglected. This approach, which is commonly followed in technical environment, has been proven to be asymptotically consistent, as a first step of a perturbation approach [20]. Whereas the linear taut string has been extensively analyzed (e.g., [21]), the situation is different for the nonlinear taut string, whose geometric nonlinearity is rarely addressed in engineering applications. Recently, Ferretti et al. [22,23] studied the combined effects induced by moving forces and nonlinearity on Kirchhoff taut strings. Particularly, in [22] a weakly nonlinear dynamics is investigated, where perturbation solutions are derived via the multiple scale method (MSM); the cyclical increment of tension is found as the main effect. Such a phenomenon deserves to be better investigated in a practical context, possibly relating to the fatigue in taut cables. Then, in [23] the studies have been extended to the strongly nonlinear dynamics, for which the classic perturbation methods are no longer applicable. A semi-analytical approach is proposed there, based on the decomposition of the total displacement into a nonlinear quasi-static part and a linear dynamic contribution; as a consequence, a new set of linearized governing (incremental) equations of motion is derived. Whereas a closed-form expression is provided for the quasi-static response, the dynamic increment is numerically found via the Galerkin Method (GM).
In this paper, the model [23] is revisited with the aim of providing closed-form solutions also for the incremental dynamic problem. After performing the same change of variable as in [23], and assuming loading velocities far enough from the celerity of the string, the equations of motion appear in the form of PDEs whose coefficients slowly vary over time. It is known that, for such kind of systems, the classic perturbation methods do not work ( [24,25]). As an alternative method, a generalized approach of Multiple Scales is developed here, by directly attacking the PDEs. It represents the major contribution of the work. Indeed, several exam-ples of the method are present in the literature, but most of them refer to discrete systems (e.g., [24,25]); an exception seems to be represented only by [13,26] with reference to infinite taut strings on the Winkler foundation subjected to a mass-spring system. The resulting asymptotic model turns out to be very accurate, when the analytical predictions are compared with the numerical results.
The layout of the paper is as follows. In Sect. 2, the mathematical model of a planar nonlinear string traveled by a moving force is briefly recalled. The change of variable is performed, leading to new equations of motion in the form of linear PDEs with slowly varying coefficients. In Sect. 3, the failure of the Multiple Scales Method for this kind of equations is shown for a simple example. In Sect. 4, the generalized multiple scale method (GMSM) is applied, and asymptotic solutions are derived. In Sect. 5, numerical results are shown for several sample systems, by proving the accuracy of the model. Finally, in Sect. 6, the main findings of the work are summarized. Three appendixes close the paper, two of which (Appendix A and B) supply some details, while the third one (Appendix C) includes a numerical application dealing with a case study.

Continuous model
A planar string of length , mass per-unit-length m, fixed at the two end points and subjected to the prestress T 0 , is considered. The curvature due to the self-weight is assumed to be negligible. The string is traveled by a transverse force of intensity P, which moves to the right with a constant velocity U , that is assumed to be much less than the critical traveling value (i.e., U U c , with U c = √ T 0 /m the transverse wave speed of the string). At the initial instant t = 0 the force is at the left end of the string (s = 0); it occupies the instantaneous position C, of abscissa s C = U t, at the current time t (see Fig. 1).

Equations of motion
The in-plane equations of motion of the model are briefly recalled; they read (equivalent notation in [23]): is the transverse displacement of the taut string, with s ∈ (0, ) the abscissa along the string axis and t the time; E A is the axial stiffness; a dash denotes partial space differentiation, a dot partial time differentiation and the double bracket denotes a jump (e.g., ; moreover, the string is assumed at rest at the initial time. By introducing the following nondimensional quantities: Equations (3) are now discussed. Equation (3a) is the classic field equation of the Kirchhoff model of a nonlinear taut string [28]. There, the coefficient of v is the nondimensional dynamic tension: in which the (positive) increment with respect to the static prestress 1 appears; since this is proportional to (the usually large) α, it constitutes the main source of nonlinearity. The jump conditions (Eqs. (3b,c)) require that, at the singular point (s = U t), the string displacement is continuous, whereas its slope is discontinuous.
In particular, Eq. (3b) was derived in [23,29] from a moving boundary problem, whose balance condition involves a small finite portion of string including the singular point C. The following result was found: which is known as the Hugoniot-Rankine condition for the taut string traveled by a moving force. Equation (5) points out that the jump of tension at the singular point is not only related to the external force (as it happens in statics), but it also depends on the jump of velocity (entailing an instantaneous change of the kinetic energy of the infinitesimal element of string passing from the right to the left of the force). The jump, moreover, has to satisfy the kinematic relation v C = −U v C , known as the Hadamard fundamental condition of compatibility ( [30]), which assures continuity of the total time derivative d dt = U ∂ ∂s + ∂ ∂t of the displacement v(U t; t) at the singular point. When this condition is substituted in Eq. (5), Eq. (3b) is finally obtained. It is incidentally observed that the moving load reduces the linear geometric stiffness of the string, from 1 to 1−U 2 ; therefore, at the critical value U c := 1 (i.e., when the load velocity equates the transverse wave speed) the taut string has zero linear stiffness.
It is worth noticing that the classic engineering formulation makes use of the Dirac generalized function ( [2]), according to which the concentrated load P at s C is treated as a distributed load on the domain. Here, an alternative formulation has been preferred, because the Dirac delta formulation is difficult to manage when the problem strategy involves change of variables and derivatives, as will be seen later.

Quasi-static solution
For reasons that will be clear later, the associated quasistatic problem is analyzed. It is derived from Eqs. (3) by neglecting all the dynamic terms, i.e.: wherev is the unknown displacement of the point C. Equation (7) satisfies the field equations and the boundary conditions, except for the mechanical jump condition. Then, by substituting Eq. (7) in the latter, one obtains: from which a closed-form expression is derived forv. Accordingly, the quasi-static tension of the taut string reads:

Dynamic increment of the response
The following change of variable is performed: according to which the transverse displacement of the string v(s, t) is decomposed into a (large, but known) quasi-static part v(s, t) and a (supposed small and unknown) incremental dynamic contribution w(s, t). By substituting Eq. (10) into Eq. (3) and linearizing in the dynamic increment w, new equations of motion are obtained: where the position c 2 (U t) :=T (U t) holds. It is observed that the initial conditions are non-trivial, since, while the static deflection vanishes at t = 0, i.e. v (s, 0) = 0, the quasi-static field velocity is nonzero, . It is should be noted that the termv in Eq. (11a) is small, since it is the acceleration of the quasi-static component of the displacement; this is also evident from the next Equation (25a). Moreover, based on the hypothesis U 1, Eq. (11) is susceptible to the following mechanical interpretation; it represents a linear taut string: (i) subjected to a distributed transverse load of intensity −v; (ii) constrained by a weakly moving spring of weakly variable stiffness 2α P 2 /c 4 (U t); (iii) traveled by a weakly variable force PU 2 /c 2 (U t), weakly moving together with the spring.
In the new variable, the total tension T reads: where use has been made, in the order, of Eq. (4), (10) and, moreover, of Eq. (7), which states thatv is stepwise constant. This entails that: in which Eq. (11d) has been exploited; finally Eq (6c) has been used. It is worth noting that Eq. (12) is nonlinear, and therefore not consistent with the linearization process; however, one can experience that including the nonlinear term w 2 improves the numerical results. In a regime of low load velocity (U 1), the equations of motion Eq. (11) appear as PDEs with coefficients slowly varying over time, via c = c(U t) and s C = U t. It is known that, in tackling such problems, the Multiple Scales Method fails; a simple example showing the drawback is reported below, with reference to a single degree of freedom oscillator.

Failure of MSM for slowly variable coefficients
Let us consider the following single degree of freedom system: in which the frequency ω > 0 is slowly variable with the time t, since 0 < η 1. According to the classic version of the MSM, the solution q(t) of Eq. (14) is assumed as a periodic signal, which is slowly modulated on slower independent and linear time-scales: Accordingly, q (t) = q (t 0 , t 1 , · · · ) and: Moreover, an integer series expansion for the dependent variable is assumed: By substituting the series of Eqs. (16), (17) in Eq. (14), and separately equating to zero terms with the same power of η, the following perturbation equations are derived: Equation (18a) admits the solution: where A is a complex amplitude depending on slower time scales and c.c. stands for complex conjugate terms. Once Eq. (19) is substituted in the η order perturbation Eq. (18b), it reads: By imposing the compatibility condition on this latter, i.e., by removing secular terms, the following equation is obtained: or, equivalently: Since the left side of Eq. (22) is a function of the slow scale t 1 , and the right side equal to t 0 , the classic MSM leads to a contradiction. An uniform approximation to q (t) cannot be obtained by using the linear time scales in Eq. (15).

Generalized Multiple Scale Method
Differently from the (classic) MSM, the generalized multiple scale method turns out to be a valid tool to obtain uniformly valid asymptotic solutions also for this class of problems, as discussed by [25] for discrete systems; however, here, the method is adapted to directly attack the partial differential equations of motion.
The small load velocity U is taken as perturbation parameter, according to which the following timescales are introduced: where θ is a slow time-scale and τ k are infinitely many fast-time scales, with f k (θ ) arbitrary functions to be determined by the procedure itself. Accordingly, the chain rule supplies: Assuming w = w (s, θ, τ 1 , τ 2 , . . .) and beingv = v (s, θ), the equations of motion become: By taking the following expansion for the unknown displacement: the perturbation equations are finally derived at the U 0 , U 1 orders:

Generating solution
The U 0 -order equation admits the generating solution: Here: A k (θ ) are unknown complex modulating amplitudes, which vary on the slow time-scale; (ω k (θ ) , φ k (s, θ)) are eigenpairs of a spatial boundary value problem, defined as follows: It turns out that φ k (s, θ) is an eigenfunction associated to the unknown wave-number β k (θ ), all depending on the slow time-scale. This solution satisfies the boundary conditions at the ends; to satisfy the geometrical and mechanical conditions at s = θ , the following equations must hold: where the dependence of β k , a 1k , a 2k on θ has been omitted. Moreover, the normalization condition 1 0 φ k (s, θ) 2 ds = 1 must be enforced.
The k-th wave number β k (θ ) and the amplitudes a ik (θ ) are determined asymptotically, by perturbing the "smooth" eigenfunctions of the string, β k0 := kπ , φ k0 = √ 2 sin (kπ ) (relevant to the case in which c = const and the "moving spring and force" are removed). By following the procedure illustrated in Appendix A, the following results are found: where a 1k is determined by the normalization condition.
Alternatively, the eigenvalue problem in Eq. (30) can be numerically solved by finding the roots of the characteristic equation, namely: which calls for building up the functions β k (θ ), using a sufficiently small steps Δθ, in which the θ ∈ (0, 1) interval is divided. Substituting the roots β k in Eq. (30b) and imposing the normalization condition, the amplitudes a 1k , a 2k are also derived. A comparison between asymptotic and numerical results is performed in the Appendix A.

Solving arbitrariness of the f k 's functions
A key step in the procedure consists in solving the arbitrariness of the functions f k (θ ), appearing in the definition of the fast time scales τ k . Since the reason of the failure of the classic MSM relies on the time dependence of the frequencies, the infinitely many functions f k (θ ) are chosen in such a way to make the infinitely many frequencies ω k (θ ) independent of θ , e.g., by requiring ω k (θ ) = 1 (k = 1, 2, · · · ). Consequently, from Eq. (29a), it follows, that d f k dθ = β k (θ ) c (θ ), entailing: With this result, the generating solution, Eq. (28), reads: Substituting it in U 1 -order equation, one obtains:

Solvability condition
The terms of the type g k (s, θ) e i τ k in Eq. (35a) would produce solutions divergent when τ k → ∞. To eliminate these secular terms, it need to enforce infinitely many solvability conditions, which require zeroing the scalar product of g k (s, θ) and φ k (s, θ) in the interval [0, θ) ∪ (θ, 1], that is: where the dependence of f k , A k on θ has been omitted. By recasting Eq. (36) as a sum of integrals and moving the terms that do not depend on the space s out of the integrals, one gets: Here, 1 0 φ 2 k (s, θ) ds = 1 according to the normalization condition, from which the following equation also holds: ∂ ∂θ Using the "Leibniz integral rule", it reads:

Amplitude Modulation Equations and first-order solution
Finally the solvability condition (Eq. (37)) provides the following Amplitude Modulation Equations (AME): These nonlinear equations can be integrated, to supply: with C k ∈ C arbitrary constants. Finally, once Eq. (41) and Eq. (33) are substituted in the generating solution, Eq. (28), reconstituting and coming back to the original time variable, one obtains: The arbitrary constants C k must be determined by enforcing the initial conditions, w (s, 0) = 0 anḋ w (s, 0) = −PU (s − 1); after that, the following, leading order, asymptotic solution is finally drawn: When this latter is introduced in Eqs. (10) and (12), closed-form expressions for the total transverse displacement v (s, t) and the dynamic tension T (t) of the string are derived, respectively.

Numerical results
The numerical results refer to a sample system already analyzed in [23], on the basis of which a sensitivity analysis is carried out, varying the parameters that govern the dynamics of the nonlinear taut string, namely the string elasticity parameter α, the magnitude of the (dimensionless) load P and its (dimensionless) velocity U . In addition, another numerical application closely related to a practical context is added in Appendix C. The accuracy of the asymptotic solutions is detected by comparisons with "exact" results, derived via the Galerkin Method. The latter is applied: (i) to the original problem Eq. (3) (in order to check the error related to the linearization of the dynamic part of the displacement), and (ii) to the linearized equations of motion Eq. (11) (in order to check the accuracy of the asymptotic solution), as detailed in Appendix B. The two benchmark solutions, exact and linearized, are denoted as GME and GML, respectively. In the following, all the  Figure 2 shows the string response in terms of displacements (-a,b) and total tension (-c). The asymptotic solution (solid gray line) is compared to both the GME solution (solid black line) and GML solution (dashed black line); an excellent agreement is detected. Here, the nonlinearity of the problem is evident from the comparison with the linear solution (dashed red line). Furthermore, with the aim of pointing out the dynamic contribution, the quasistatic response (solid blue line) is also reported. As main global effect, a variation of the total tension T (static plus dynamic) occurs; it is increased by up to 10% compared to the initial prestress. Particularly, it is seen that the dynamics involves a modification of the response frequency, causing small tension increments ( T d := T −T ) with respect to the quasistatic response, related to a certain number of cycles n. The latter are unknown and mainly depend on the load velocity; namely, n increases when U decreases, as a result of successive reflections of the waves at the right end and at load instantaneous location. For the examined case, n = 5 and T d max = 0.02. In general, such a phenomenon could cause fatigue mechanisms and, therefore, deserves to be investigated. Moreover, with reference to Fig. 2a,b, it is seen that the dynamic increments constitute a small part of the total response, consistently with the problem hypotheses; accordingly, the total response appears as a perturbation of the quasistatic one.
Similar results are shown in Fig. 3 for an other set of parameters (α = 500, P = 0.05, U = 0.1), which differs from the previous one for a greater intensity of the load. Here, the nonlinearity of the problem is higher, as evidenced by the deviation of the response from the linear solution. However, an excellent agreement is still found between the GME solution, the GML solution and the asymptotic solution. Now, the total tension (static plus dynamic) is increased by up to 25% compared to the initial prestress. As regards the only dynamic effects, peaks of tension are detected, with a maximum value of T d max = 0.03 and an unchanged number of cycles with respect to the previous case.
Then, a case of highly nonlinear dynamics is studied, with reference to the following set of parameters (α = 5000, P = 0.03, U = 0.1), where a very high elasticity of the string α is taken. The results are shown in Fig. 4, where the response deviates a lot from the linear solution. However, the GME solution is well approximated by the GML one, which, moreover, is found to be in excellent agreement with the asymptotic solution. A greater increment of the total tension is detected, which is increased by up to 60% compared to the initial prestress; the maximum dynamical increment is T d max = 0.1.
Finally, an "extreme" set of parameters (α = 500, P = 0.03, U = 0.5) is taken, which is characterized by a very high load velocity, not consistent with the perturbation hypotheses. Despite this, the results shown in Fig. 5 depict an asymptotic solution that fits perfectly with the GML one, highlighting the efficiency of the perturbation method. However, a not-negligible error with respect to the GME solution is detected. It can be proved that a better approximation of the "exact" solution is obtained when the dynamic term U 2 is included, although inconsistently, in the quasistatic problem Eq. (6). That was, indeed, done in Fig. 7 of [23], although not specified. Moreover, it is seen that the velocity of the load strongly influences the dynamic response of the system; especially, in this case, n = 1 and T d max = 0.18.

Conclusions
The strongly nonlinear dynamics of taut strings traveled by a moving force has been studied. A mathematical model has been formulated, in which a change of variable has been introduced, which leads to equations of motion expressed in terms of a (supposed) small dynamic incremental displacement superimposed to the nonlinear quasi-static response. Under the hypothesis of low velocity of the moving force, the incremental equations of motion appear in the form of linear PDEs with slowly time-varying coefficients. The latter have been solved via of a generalized multiple scale method, here developed ad-hoc and directly applied to the PDEs. As a main result, closed-form solutions have been provided for the string response, both in terms of displacement and tension. In relation to this aspect, the authors believe that a closed-form solution, even if approximate, is preferable to a purely numerical solution for the following reasons: (a) it allows low-cost sensitivity analyses, (b) it often reveals the true physical essence of the phenomenon.
The accuracy of the asymptotic model has been investigated by analyzing several sample systems, in which different sets of the parameters have been chosen (namely, string elasticity, magnitude and velocity of the moving force), spanning from moderate to large nonlinear systems. Particularly, comparisons with benchmark solutions, obtained through the Galerkin method, have proved that, unlike the MSM, the GMSM provides very precise analytical solutions, even when the nonlinearities are high. As regards the physic of the problem, the main global effect due to the high nonlinearity is in the variation of the total tension (static plus dynamic) of the string, which, in some cases, can increase up to 60% with respect to the initial prestress. Another relevant aspect is closely related to the dynamics, which modifies the response frequency, by involving cyclic increments in tension with respect to the quasi-static component. Even if these increments are generally low, they occur according to a number of cycles which depends on the load velocity. Particularly, it has been proved that lower velocities produce a higher number of cycles. Such a behavior could deserve to be investigated in relation to fatigue phenomena.
In closing, this paper involves some future perspectives, among which the extension of the proposed method to different moving loads as (i) the moving oscillator model and (ii) the train of moving forces.
Funding Open access funding provided by Università degli Studi dell'Aquila within the CRUI-CARE Agreement. The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Data Availability
The datasets generated during the current study are available from the corresponding author on reasonable request.

Declarations
Competing Interests The authors have no relevant financial or non-financial interests to disclose.
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A.1 Analytic expressions for the eigenpairs of Eq.
The problem Eq. (30) is solved via a perturbation method with the aim of obtaining closed-form expres-sions for β k (θ ) , a ik (θ ). The string elasticity coefficient is rescaled as α → αη, with η a dimensionless perturbation parameter; then, the following formal series are assumed: Substituting the latter series in Eq. (30) and requiring this is satisfied ∀η, the perturbation equations are obtained: The characteristic equation of the generating problem reads: from which, β k0 = kπ and the associated vector of the amplitudes is a 1k0 , a 1k0 (−1) k , with a 1k0 an arbitrary constant. By substituting all in the η 1 -order problem, since the operator is singular, a compatibility (or solvability) condition has to be enforced on the known term in order to the problem does admit solution; namely, it requires that the known term is orthogonal to the kernel of the adjoint operator. Thus, one obtains: By the same steps carried out at the η 2 -order problem, one gets: β k2 = 4α 2 P 4 sin 3 (π θk)(π(2θ − 1)k cos(π θk) − sin(π θk)) π 3 k 3 c(θ) 12 , a 2k2 =α 2 P 4 sin 2 (π θk)(−2 sin(2πθk)+π(4θ−1)k cos(2πθk)+π k) The final solutions read as in Eq. (31).

A.2 Numerical results
The reliability of the previous closed-form solutions Eq. (31) is proved for a given sample system, having α = 2000, P = 0.03, U = 0.1. Figure 6 shows the wave number β k and the amplitudes a ik for k = 1, 2, 3, as the load position θ varies; the asymptotic results (gray lines) are compared with the numerical ones (black dots). A very good agreement is detected. The shape of the corresponding eigenfunctions, as defined in Eq. (29b) for k = 1, 2, 3, is represented in Fig. 7, for two assigned load abscissas: θ = 1 4 and θ = 1 2 . It is seen as the load position affects the form of the modes, which present singularity points (more or less evident) at the coordinate s = θ .
Although the small corrections of the eigenfunctions appearing in Figs. 7 could induce to believe that the 'smooth' modes of the free string can successful be employed, it has been checked, that such a choice would entail remarkable errors. Indeed, if one repeats the analysis of the system in Fig. 3, by using Eqs. (31) truncated at the first term, one finds the results displayed in Fig. 8.

B Galerkin method
The classic Galerkin Method (GM) is used to obtain benchmark solutions for: (i) the original dynamic problem Eq. (3); (ii) the linearized incremental dynamic problem Eq. (11). The method is applied to partial differential equations to derive a space-discrete model, as detailed in [22] and briefly resumed here.

B.1 GM applied to the original dynamic problem
The transverse displacement of the string is assumed as: Here: ψ k (s) = √ 2 ω k sin (ω k s) are normalized eigenfunctions of the linear taut string, where ω k = kπ (with k a positive integer number), q k (t) are the corresponding time-varying amplitudes and N the number of the considered eigenfunctions. Accordingly, the generic jth Galerkin equation reads: where the frequency Ω j := ω j U = jπU and the term in brackets represents the total tension T in the Galerkin approximation. Equation (51) is a system of ordinary differential equations, that can be numerically solved via standard techniques (like Runge-Kutta method). Once the generic amplitude has been determined, the solution of the original problem is reconstructed by means of Galerkin expansion (50). Moreover, when α = 0, Eq. (51) describes the corresponding linear (discrete) problem, which admits the closed-form solution: q j = Γ j e iΩ j t + Λ j e iω j t + c.c. where: where y k (t) is the k-th time-varying amplitude associated with the eigenfunction ψ k (s) of the linear taut string (above defined). The j-th Galerkin equation thus reads: Once the unknown amplitudes are determined by means of standard numerical techniques, the solution of the incremental linearized problem is reconstructed through the Galerkin expansion Eq. (54).

C Numerical results for a case study
In order to show the utility of this study in a practical context, the case study of a ropeways is analyzed. An aerial cableway (cable car system) of 1.65 km, divided into four parts of length about = 410 m, is considered. The combined cabin and suspension arrangement has a mass of 1700 kg with a full load of passengers, i.e. P = 16660 kN. The maximum operating speed is U = 15 m/s. The track rope has nominal mass per unit length m = 10.25 kg/m, effective section area A = 1963.5 mm 2 and nominal modulus of elasticity E = 160 kN/mm 2 . Track rope tension is T 0 = 500 kN.
According to Eq. (2), the following dimensionless parameters, α = 353.43, P = 0.033 and U = 0.068 govern the dynamics of the nonlinear taut string. As done in Sect. 5, the string response is shown in terms of displacements and total tension; see Fig. 9. Here, the asymptotic solution (solid gray line) is compared to both the GME solution (solid black line) and GML solution (dashed black line); the quasi-static solution (solid blue line) and the linear solution (dashed red line) are also represented. The main global effect is in the variation of the total tension (static plus dynamic), which is increased by up to 9% compared to the initial prestress. Particularly, the dynamics contribution manifests with cyclic tension increments with respect to the quasi-static component, with a maximum value equal to T d max = 0.01; here, the number of cycles is n = 7.