Dynamic response of objects with shock-absorbers to seismic loads

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A dynamic model for a multimass system with a seismic damper as a nonlinear inertial component is reduced via balancing. The seismic damper is subject to a nonideal constraint. As an example, the order of the system of differential equations is reduced by one for a dynamic three-mass system. A comparison of the processes in the original and reduced systems confirms that it is possible to use balancing to simplify such models Introduction. Most kinematic pairs are subjected, to a greater or lesser extent, to friction, which in certain cases has a strong effect on dynamic processes in various engineering devices, machines, and mechanisms [2,8,10,14,18]. The friction in seismic dampers and structural elements of buildings and other structures contributes to the damping of seismic vibrations and to the decrease of the extreme dynamic loads [9,11,12,15]. Theoretical studies of dynamic processes play an important role in designing seismic isolation systems. This is why there is an objective need for mathematical models that would adequately describe their major properties and meet the requirements of numerical integration (see, e.g., [17]) such as that the system of differential equations must not be stiff [5,6,16,19], which frequently leads to the need to reduce the system.Due to the increasing interest to the seismoprotection of buildings and other structures [9, 20], we will outline the model of an in-line system that includes a passive seismic damper as a nonlinear inertial via balancing [16,19] for not only linear and nonlinear systems [5,13], the latter including inertialess and inertial components, but also systems that have a nonlinear inertial component with a nonideal constraint. The balancing method is detailed in [5,6,16,19].It is natural that the approach outlined here can be applied to not only buildings subject to seismic effects, but also mining, metallurgy, and farm machinery, machining tools, etc. subjected to their typical loads. Mathematical Model of a System with a Seismic Damper that Includes a Nonideal Constraint.First, we will briefly outline a model for a system with seismic dampers that include ideal constraints. With such an assumption, the reactions R i at the contact points between the bodies I and O are normal to the surface, i.e., the angles of friction r i between the normal and total reactions R are zero (Fig. 1).The body O undergoes plane translation and has only one degree of freedom because two of the three possible degrees of freedom of a free body in plane motion are excluded by the two normal constraints to the surfaces at the contact points between the bodies I and O. Each additional damper of the same shape and size (third, fourth, …, nth) would introduce a repeated (passive) constraint [4] not affecting the law of motion of the system, but making it (singly, doubly, (n -2)-tuply) statically indeterminate. The reactions in each damper cannot be found through kinetostatics. For the translating system in Fig. 1, it is convenient to use two equations of motion of the body O and an ...

Dynamic-model reduction for a multimass system with nonideal constraint in a seismic damper

Antonyuk

Timokhin

2008

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Determining the constraint reactions of awheeled robotic vehicle with one steerable wheel

Larin

2008

The reactions of the nonholonomic constraints of a wheeled robotic vehicle with one steerable wheel are determined. Simplified (asymptotic) relations are derived in addition to the exact ones. They are used to estimate the reactions. The efficiency of the approximate formulas is demonstrated by an example Keywords: nonholonomic constraints, wheeled transport robot, one steerable wheel, reactions of constraints Introduction. One of the tasks in the development of manipulators is to choose the velocity to move a workpiece along a set trajectory with allowance for the constraints imposed on the dynamic parameters of the actuating mechanisms (see, e.g., [12, 18, 19]). A similar task arises in relation to wheeled robotic vehicles [10][11][12][13][14][15][16][17]. However, constraints here are associated with the admissible reaction of nonholonomic constraints rather than with the capabilities of the actuating mechanisms [2,4,5,8,9]. This reaction should not generally exceed the robot's weight multiplied by the dynamic coefficient of friction. Note that it may appear of primary necessity to determine constraint forces (for example, holonomic) in various applied problems (see, e.g., [7]). Generally, analytic expressions for the reaction forces of nonholonomic constraints are rather complex and, therefore, there is a need to derive simpler expressions that would permit relatively easy determination of the constraint forces.The present paper uses a model of a wheeled robotic vehicle with one steerable wheel (Lineikin model [4], kinematic car [17], see also [10,11,13]) to analyze the problem of determining constraint forces (obtain exact equations) and derive approximate (asymptotic) relations to estimate the constraint forces. The efficiency of the approximate formulas is demonstrated by way of examples [11][12][13]17].1. General Equations [9]. The motion of a mechanical system with nonholonomic constraints is described by the equations

Quickest-descent curve in the problem of rolling of a homogeneous cylinder

Legeza¹

2008

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