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 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.
In the paper, the static sensitivity of complex structures with respect to random design parameters is presented. Using the adjoint system method, based on the mean-point second-order perturbation method, the first two probabilistic moments of time-independent sensitivity are formulated with means and cross-covariances of random design parameters as the input data. It enables one to obtain the second-order accuracy of the solution. The presented formulations are illustrated by a number of numerical examples. The influence of finite element mesh density for the obtained results is discussed using the analysis of the spatial bar dome.
In high-rise buildings earthquake ground motions induce bending deformation of the host structure. Large dynamic displacements at the top of the building can be observed which in turn lead to the excitation of the cables/ropes within lift installations. In this paper, the stochastic dynamics of a cable with a spring-damper and a mass system deployed in a tall cantilever structure under earthquake excitation is considered. The non-linear system is developed to describe lateral displacements of a vertical cable with a concentrated mass attached at its lower end. The system is moving slowly in the vertical direction. The horizontal displacements of the main mass are constrained by a spring-viscous damping element. The earthquake ground motions are modelled as a filtered Gaussian white noise stochastic process. The equivalent linearization technique is then used to replace the original non-linear system with a linear one with the coefficients determined by utilising the minimization of the mean-square error between both systems. Mean values, variances and covariances of particular random state variables have been obtained by using the numerical calculation. The received results were compared with the deterministic response of the system to the harmonic process and were verified against results obtained by Monte Carlo simulation.
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