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SUMMARYA new, passive, vibroprotective device of the rolling-pendulum tuned mass damper type is presented that, relying on a proper three-dimensional guiding surface, can simultaneously control the response of the supporting structure in two mutually orthogonal horizontal directions. Unlike existing examples of ball vibration absorbers, mounted on spherical recesses and effective for axial-symmetrical structures, the new device is bidirectionally tuneable, by virtue of the optimum shape of the rolling cavity, to both fundamental structural modes, even when the corresponding natural frequencies are different, in such a case recurring to an innovative non-axial-symmetrical rolling guide.Based on Appell's non-holonomic mechanics, a non-linear dynamic model is first derived for the bidirectional absorber mounted on a 1-storey 3-degrees-of-freedom linear structure translating under the effect of both imparted base motion and applied dynamic forces. A laboratory-scaled prototype of the device is then tested to experimentally demonstrate the bidirectional tuning capability and to validate the mathematical model. The design procedure and the seismic performance of the absorber are finally exemplified through numerical simulation.
SUMMARYA new, passive, vibroprotective device of the rolling-pendulum tuned mass damper type is presented that, relying on a proper three-dimensional guiding surface, can simultaneously control the response of the supporting structure in two mutually orthogonal horizontal directions. Unlike existing examples of ball vibration absorbers, mounted on spherical recesses and effective for axial-symmetrical structures, the new device is bidirectionally tuneable, by virtue of the optimum shape of the rolling cavity, to both fundamental structural modes, even when the corresponding natural frequencies are different, in such a case recurring to an innovative non-axial-symmetrical rolling guide.Based on Appell's non-holonomic mechanics, a non-linear dynamic model is first derived for the bidirectional absorber mounted on a 1-storey 3-degrees-of-freedom linear structure translating under the effect of both imparted base motion and applied dynamic forces. A laboratory-scaled prototype of the device is then tested to experimentally demonstrate the bidirectional tuning capability and to validate the mathematical model. The design procedure and the seismic performance of the absorber are finally exemplified through numerical simulation.
The paper proposes computer algebra system (CAS) algorithms for computer-assisted derivation of the equations of motion for systems of rigid bodies with holonomic and nonholonomic constraints that are linear with respect to the generalized velocities. The main advantages of using the D'Alembert-Lagrange principle for the CSA-based derivation of the equations of motion for nonholonomic systems of rigid bodies are demonstrated. Among them are universality, algorithmizability, computational efficiency, and simplicity of deriving equations for holonomic and nonholonomic systems in terms of generalized coordinates or pseudo-velocities Keywords: computer algebra system, equations of motion for systems of rigid bodies, holonomic and nonholonomic systems, D'Alembert-Lagrange principleIntroduction. Problems of nonholonomic mechanics are complicated regarding the formalization of description. They play a special role in studying mechanical systems and are important for engineering applications [2,[5][6][7][8][13][14][15]. There are different methods for describing, deriving, and integrating the equations of motion for nonholonomic systems [3,4,9,10]. These methods employ different representations of kinetic energy (or the Lagrangian function): as a function of generalized coordinates and velocities (equation with Lagrange multipliers); as a function of generalized coordinates and independent velocities (Chaplygin's and Voronets' equations); and as a function of generalized coordinates and pseudovelocities (Lagrange-Euler equations derived by Hamel). The acceleration energy function introduced by Appell [3] and the modified Hamiltonian function [10] are used with the same purpose. The listed methods are not universal-resulting expressions are rather awkward and do not guarantee correct results; and if the mechanical system changes its structure, then the equations should be derived anew. Therefore, the methods are of little use in the development of universal problem-oriented systems (POSs) for PCs with the purpose of research automation and solution of nonholonomic mechanics problems. The paper [12] proposed a theory and analytic algorithms for computer-algebra systems (CASs) used to develop a POS for PCs with the purpose of research automation and solution of basic problems in the mechanics of holonomic rigid-body systems with elastic and dissipative constraints based on the D'Alembert-Lagrange principle.In what follows, we will first calculate the number of operations to demonstrate the numerical efficiency of the equations of motion for holonomic systems and the fundamental possibility of deriving the equations of motion for nonholonomic systems using the method proposed in [12]. Then we will discuss algorithms (based on the D'Alembert-Lagrange principle) for computer-assisted derivation of the equations of motion for systems with nonholonomic constraints linear with respect to generalized velocities. We will also outline modifications of the algorithms from [12] that make it possible to use one CAS to find analytic and...
The motion of a heavy homogeneous cylinder is considered as rolling without slipping along an unknown curve. A functional in the form of the total time of rolling is found and minimized by solving a variational problem. The algebraic equation of the quickest-descent directrix is derived in parametric form Keywords: heavy homogeneous cylinder, rolling without slipping, variational problem, functional, quickest-descent directrixIntroduction. The main performance criterion for many vibration-protection devices, shock-absorbers, dampers, and stabilizers is minimum displacements of some points of the bearing object or minimum forces (moments) in the most critical sections during forced vibrations [6,9,13,15,19,20]. Stability of buildings, mobile robots [1,5,8], wheeled vehicles, and carrying and carried bodies is analyzed in [9,12,14,16,18] taking into account the possible bifurcations of their dynamic equilibrium. However, there are a number of vibration-protection devices for which the main performance criterion is the minimum time it takes to reduce the amplitudes of forced vibrations of load-bearing structures to an admissible level [4,7,10,14].Here we use such a criterion for roller-type vibration-protection devices [2,3,9,11,17]. In this connection, it is necessary to identify the directrix of a cylindrical surface along which a heavy cylinder descends the quickest. It is assumed that the motion of the cylinder, which is considered homogeneous, along the directrix is rolling without slipping.Thus, we deal with a variational problem for a heavy homogeneous cylinder that rolls without slipping in a cylindrical hollow with a quickest-descent directrix. This study is a generalization of Bernoulli's study where he derived the equation of a brachistochrone along which a material point descends the quickest. This curve is known to be a cycloid [1,5,8].1. Problem Formulation and Objective Functional. Consider a homogeneous cylinder of mass m and radius r rolling without slipping from a point A without initial velocity over a cylindrical valley with a directrix AKL (Fig. 1). The curve AKL lies in a vertical plane. Let us determine the time it takes the cylinder to move from the point A to the point K. We choose the origin of coordinates at the point O and direct the OZ-axis vertically downwards and the OX-axis horizontally to the right (the point A is on the OZ-axis).The inward unit normal vector to the unknown curve at its arbitrary point K is expressed as
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