Non-linear dynamic response of a cable system with a tuned mass damper to stochastic base excitation via equivalent linearization technique

Meccanica https://doi.org/10.1007/s11012-020-01169-3 (0123456789().,-volV) ( 01234567 89().,-volV) RECENT ADVANCES IN NONLINEAR DYNAMICS AND VIBRATIONS Non-linear dynamic response of a cable system with a tuned mass damper to stochastic base excitation via equivalent linearization technique Hanna Weber . Stefan Kaczmarczyk . Radosław Iwankiewicz Received: 4 November 2019 / Accepted: 15 April 2020 Ó The Author(s) 2020 Abstract Non-linear dynamic model of a cable– mass system with a transverse tuned mass damper is considered. The system is moving in a vertical host structure therefore the cable length varies slowly over time. Under the time-dependent external loads the sway of host structure with low frequencies and high amplitudes can be observed. That yields the base excitation which in turn results in the excitation of a cable system. The original model is governed by a system of non-linear partial differential equations with corresponding boundary conditions defined in a slowly time-variant space domain. To discretise the continuous model the Galerkin method is used. The assumption of the analysis is that the lateral displacements of the cable are coupled with its longitudinal elastic stretching. This brings the quadratic couplings between the longitudinal and transverse modes and H. Weber (&)  R. Iwankiewicz West Pomeranian University of Technology in Szczecin, Szczecin, Poland e-mail: R. Iwankiewicz e-mail: S. Kaczmarczyk Faculty of Arts, Science and Technology, University of Northampton, Northampton, UK e-mail: R. Iwankiewicz Institute of Mechanics and Ocean Engineering, Hamburg University of Technology, Hamburg, Germany cubic nonlinear terms due to the couplings between the transverse modes. To mitigate the dynamic response of the cable in the resonance region the tuned mass damper is applied. The stochastic base excitation, assumed as a narrow-band process mean-square equivalent to the harmonic process, is idealized with the aid of two linear filters: one second-order and one first-order. To determine the stochastic response the equivalent linearization technique is used. Mean values and variances of particular random state variable have been calculated numerically under various operational conditions. The stochastic results have been compared with the deterministic response to a harmonic process base excitation. Keywords Cable–mass system  Tuned mass damper  Stochastic dynamics  Equivalent linearization technique 1 Introduction Moving cable systems carrying inertia elements such as rigid-body masses are applied in many engineering systems. In some applications the length of ropes and cables vary during the motion, which results in nonstationary behaviour of the system. For example in elevator and mine lifting installations, the length of the cable varies during the moving with some speed. As a result the variation of natural frequencies of the system 123 Meccanica can be observed [1]. The host structures are often subjected to external dynamic loads such as wind or earthquakes [2]. This causes the excitation of the structural system and corresponding response of the cable that can be described by using the deterministic models. On the other hand, because of nondeterministic nature of wind load or earthquakes, these systems should be considered with stochastic methods [3–5]. The wind action can be assumed as a wide-band random process, however, due to the damping effect inside the system the response of the structure can be regarded as narrow-band random process. In this paper two models of a cable–mass system are presented: the nonlinear deterministic model under harmonic excitation and corresponding stochastic model with the excitation represented as a narrowband process mean-square equivalent to the harmonic process. The horizontal displacements of main mass are constrained by applying an auxiliary springdamper-mass combination to act as a tuned mass damper (TMD) [6]. TMD is used to reduce the negative effects of the resonance phenomenon [7]. It is very difficult to consider the behaviour of this type of structure by applying the analytical methods due to the non-stationarity and non-linearity of the process [8]. Therefore the numerical techniques should by used. In paper [6] an approximated linear model was expanded by neglecting the non-linear terms in original set of equations of motion. This paper presents different approach, where the equivalent linearization technique is used to replace the nonlinear system by an equivalent linear one, whose coefficients are obtained from the conditions of mean-square minimization of the error between both systems and are given by the terms of expectations of nonlinear functions of the response process. The statistical linearization technique has been applied to consideration of many various nonlinear stochastic problems since using by Caughey [9]. Other examples of this method can be found e.g. in [10–13]. This technique was also applied to problems of nonGaussian excitations such as non-linear system under a Poisson impulse excitation e.g. [14–16] or in combination of various advanced method of stochastic dynamics e.g. [17–21]. In this paper the mean values and variances are calculated for different values of auxiliary damping filter ratio. The expected values of particular random 123 state variables are compared with the deterministic results obtained for original nonlinear system subjected to harmonic excitation. 2 Non-linear model In the model presented in Fig. 1 the main mass M moves downwards at the transport speed denoted as V, and is suspended by a metallic elastic cable of length L ¼ LðtÞ which is varying over time. The total height of the host structure is Z0 . The characteristic values of the cable such as cross-sectional area, modulus of elasticity, mass per unit length and mean quasi-static tension are denoted as A, E, m and T i , respectively. The tension magnitude can be obtained by using the following expression T i ¼ ½M þ md þ mðL  xÞðg  aÞ; ð1Þ where md is a small auxiliary mass attached to the main mass by a spring-dashpot system with the coefficients of stiffness and viscous damping assumed as kd and cd , respectively. The horizontal displacement of auxiliary mass is denoted as zd . The main mass M is constrained in the lateral direction by a spring of the coefficient of stiffness k. The horizontal and vertical displacements of main mass are assumed as uM ðtÞ and (a) (b) Fig. 1 Schematic model of cable–mass system: a undeformed setting, b deformed setting Meccanica vM ðtÞ. The lateral time-dependent displacements of the cable v(x, t) are coupled with the longitudinal vibrations u(x, t). The Hamilton’s principle expressed in terms of the kinetic energy and the potential energy of the system and external work of nonconservative forces yields the partial differential equations of motion in the following form D2 u  EAex ¼ 0 Dt2 2 D v m 2  Tvxx þ mðg  aÞðxvxx þ vx Þ  (...truncated)