Alexander V. Pesterev

On asymptotics of the solution of the moving oscillator problem

Asymptotic behavior of the solution of the moving oscillator problem is examined for large and small values of the spring stiffness for the general case of non-zero beam initial conditions. In the limiting case of infinite spring stiffness, it is shown that the moving oscillator problem for a simply supported beam is not equivalent, in a strict sense, to the moving mass problem. The two problems are shown to be equivalent in terms of the beam displacements but are not equivalent in terms of stresses (the higher order derivatives of the two solutions differ). In the general case, the force acting on the beam from the oscillator is shown to contain a high-frequency component , which does not vanish and can even grow when the spring coefficient tends to infinity. The magnitude of this force and its dependence on the oscillator parameters can be estimated by considering the asymptotics of the solution for the initial stage of the oscillator motion. It is shown that, for the case of a simply supported beam, the magnitude of the high-frequency force depends linearly on the oscillator eigenfrequency and velocity. The deficiency of the moving mass model is principally that it fails to predict stresses in the supporting structure. For small values of the spring stiffness, the moving oscillator problem is shown to be equivalent to the moving force problem. The discussion is amply illustrated by results of numerical experiments.

Revisiting the moving force problem

Abstract The problem of the vibration of a beam subject to a travelling force is considered. The purpose of the study is to develop simple tools for finding the maximum deflection of a beam for any given velocity of the travelling force. It is shown that, for given boundary conditions, there exists a unique response–velocity dependence function. A technique to determine this function is suggested, which is based on the assumption that the maximum beam response can be adequately approximated by means of the first beam mode. To illustrate this, the maximum response function is calculated analytically for a simply supported (SS) beam and constructed numerically for a clamped–clamped beam. The effect of the higher modes on the maximum response is investigated, and the relative error of the one-mode approximation for a SS beam is constructed. The estimates obtained substantiate the assumption about adequacy of the one-mode approximation in a wide range of velocities; in particular, the relative error in the neighborhood of the velocity that results in the largest response is less than one percent.

A new method for calculating bending moment and shear force in moving load problems

In this paper, a new series expansion for calculating the bending moment and the shear force in a proportionally damped, one-dimensional distributed parameter system due to moving loads is suggested. The number of moving forces, which may be functions of time and spatial coordinate, and their velocities are arbitrary. The derivation of the series expansion is not limited to moving forces that are a priori known, making this method also applicable to problems in which the moving forces depend on the interactions between the continuous system and the subsystems it carries, e.g., the moving oscillator problem. A main advantage of the proposed method is in the accurate and efficient evaluation of the bending moment and shear force, and in particular, the shear jumps at the locations where the moving forces are applied. Numerical results are presented to demonstrate the rapid convergence of the new series representation.

Response of a Nonconservative Continuous System to a Moving Concentrated Load

The problem of calculating the response of a general class of nonconservative linear distributed parameter systems excited by a moving concentrated load is investigated. A method of solution based on the series expansion of the response in terms of complex eigenfunctions of the continuous system is proposed. A set of ordinary differential equations in the time-dependent coefficients of the expansion is established in terms of the unknown force of interaction on the continuum, which allows one to investigate different models of concentrated loads. For the case of a conservative oscillator moving with arbitrarily varying speed, the coefficients of the equations are obtained in explicit terms. Some results of numerical experiments involving a proportionally damped beam are presented.

A novel approach to the calculation of pothole-induced contact forces in MDOF vehicle models

Abstract A technique is developed to predict the dynamic contact forces arising after passing road surface irregularities by a vehicle modelled as an undamped multiple-degrees-of-freedom (MDOF) system. An MDOF system moving along an uneven profile is decomposed into an aggregate of independent oscillators in the modal space, such that the response of each oscillator can be calculated independently. An equation relating the contact forces in the physical space to the modal forces is established. The technique developed is applied to the calculation of the coefficients of the harmonic components of the contact forces arising after the passage of a “cosine” pothole. The application of the technique to various problems, such as evaluation of the effect of parameter modifications on the vehicle dynamics and reduction of vehicle models in bridge-related problems, as well as its extension to the damped case, are also discussed. One interesting phenomenon reported in the DIVINE project [1], regarding the replacement of a steel suspension by an air suspension, resulting in an increase of the maximum response of short-span bridges, is explained by applying the technique suggested. The discussion is amply illustrated by examples of the application of the technique to the calculation of the tire forces due to a pothole for two simple—quarter-car and half-car—vehicle models.

Motion control for a wheeled robot following a curvilinear path

A control synthesis problem for planar motion of a wheeled robot with regard to the steering gear dynamics is considered. The control goal is to bring the robot to a given curvilinear path and to stabilize its motion along the path. The trajectory is assumed to be an arbitrary parameterized smooth curve satisfying some additional curvature constraints. A change of variables is found by means of which the system of differential equations governing controlled motion of the robot reduces to the form that admits feedback linearization. A control law is synthesized for an arbitrary target path with regard to phase and control constraints. The form of the boundary manifold and the phase portrait of the system for the case of the straight target trajectory are studied. Results of numerical experiments are presented.

Construction of invariant ellipsoids in the stabilization problem for a wheeled robot following a curvilinear path

The synthesis control problem for the plane motion of a wheeled robot is studied. The goal of the control is to bring the robot to an assigned curvilinear trajectory and to stabilize its motion along it in the presence of phase and control constraints. For a synthesized control law, invariant ellipsoids—quadratic approximations of the attraction domains of the target trajectory—are constructed, which allow one to check in the course of the robot motion whether the control law can stabilize motion along the current trajectory segment. To take into account constraints on the control, methods of absolute stability theory are applied. The construction of the invariant ellipsoids reduces to solving a system of linear matrix inequalities.

Smoothing curvature of trajectories constructed by noisy measurements in path planning problems for wheeled robots

A path planning problem for a wheeled robot is considered. The problem consists in constructing a trajectory that approximates a given ordered sequence of points on the plane and satisfies certain smoothness requirements and curvature constraints. Such a problem arises, for example, when it is required to follow in an automated mode a path stored as a discrete set of points measured in the course of the first passage of this path in a manual mode. Due to errors inherent in the data points, the shape of the curve approximating the desired path may turn out inappropriate or even unacceptable from the control standpoint. The shape of the curve can be improved by applying the so-called fairing, which consists in moving the original data points with the aim to minimize some functional. Adequate small variations of the data points (within the measurement error) preserve the proximity of the resulting path to the original data points and, at the same time, may considerably improve its shape. In the paper, a new global fairing method for improving shape of curves consisting of elementary B-splines is proposed. The improvement is achieved through minimization of jumps of the spline third derivative. The problem of finding desired variations reduces to solving a quadratic programming problem with simple constraints. The discussion is illustrated by numerical examples.

Stabilization Problem for a Wheeled Robot Following a Curvilinear Path on Uneven Terrain

A control synthesis problem for a wheeled robot moving on uneven terrain is studied. The terrain is assumed to be described by a sufficiently smooth function that does not vary too much at distances of the order of the platform size, which makes it possible to employ a planar robot model. The terrain model is not a priori known, and the information on the local terrain configuration is made available for the robot through measuring its pitch and roll angles. The control goal is to bring the robot to a given curvilinear path and to stabilize robot’s motion along it. A change of variables is found by means of which the system of differential equations governing controlled motion of the robot reduces to the form that admits feedback linearization. A numerical example presented demonstrates advantages of the synthesized control compared to that derived without regard to the terrain unevenness. It is shown that the latter is generally not capable of stabilizing robot’s motion with a desired accuracy.

Canonical representation of the path following problem for wheeled robots

For the problem of stabilizing motion of an n-dimensional nonholonomic wheeled system along a prescribed path, the concept of a canonical representation of the equations of motion is introduced. The latter is defined to be a representation that can easily be reduced to a linear system in stabilizable variables by means of an appropriate nonlinear feedback. In the canonical representation, the path following problem is formulated as that of stabilizing the zero solution of an (n−1)-dimensional subsystem of the canonical system. It is shown that, by changing the independent variable, the construction of the canonical representation reduces to finding the normal form of a stationary affine system. The canonical representation is shown to be not unique and is determined by the choice of the independent variable. Three changes of variables known from the literature, which were earlier used for synthesis of stabilizing controls for wheeled robot models described by the third- and fourth-order systems of equations, are shown to be canonical ones and can be generalized to the n-dimensional case. Advantages and disadvantages of the linearizing control laws obtained by means of these changes of variables are discussed.