Time Optimal Transfer Function of a Mechanism in the Presence of Dissipative Forces

2004 ◽  
Author(s):  
A. A. Goun ◽  
O. K. Sliva
Keyword(s):  
Dry Friction ◽  
Optimization Problems ◽  
Mechanical Energy ◽  
Operating Time ◽  
Variational Calculus ◽  
Viscous Friction ◽  
Optimal Position ◽  
Dissipative Forces ◽  
Time Optimal ◽  
Theoretical Mechanics

A variety of optimization problems in theoretical mechanics are related to the problem of finding the quickest operating mechanism among all possible mechanisms. Let us consider a mechanical system with one degree of freedom consisting of leading and lagging links with masses M1 and M2 respectively. Source of mechanical energy is attached to the leading link and its potential energy is known as a function of the position of the leading link. Positions of the leading - x and lagging links - y are related through the position function. Now the optimization problem can be formulated in the following way: find the position function such that the lagging link is transferred from initial to the final position in the shortest time. The analytic solution for this problem for the case when dissipative forces can be neglected was found using variational calculus method. In the case when dissipative forces can be described as viscous friction the problem can be solved using iterative methods. The differential equation that describes the stationary point of these iterations was obtained. The dependence of the optimal position function on the magnitude of friction is analyzed. In the case when dissipative forces are of the dry friction type the approach based on the variational calculus fails. We were able to find the optimal position function problem using Maximum Principle. New qualitative features of the solution arising due to dry friction are discussed. Approaches developed in this paper can be generalized for a variety of mechanisms where the operating time is critical.

Time Optimal Transfer Function of a Mechanism

Volume 2 ◽  
2004 ◽  
Author(s):  
A. A. Goun ◽  
O. K. Sliva ◽  
Vyacheslav Murzin
Keyword(s):  
Transfer Function ◽  
Mechanical System ◽  
Optimization Problems ◽  
Mechanical Energy ◽  
Operating Time ◽  
Variational Calculus ◽  
Initial Position ◽  
Optimal Transfer ◽  
Time Optimal ◽  
Switch Design

A variety of optimization problems in theoretical mechanics are related to the problem of finding the quickest operating mechanism among all possible mechanisms. It can be formulated in the following way. Let us consider a mechanical system with one degree of freedom consisting of leading and lagging links with masses M1 and M2 respectively. Source of mechanical energy is attached to the leading link and its potential energy is known as a function of the position of the leading link. Positions of the leading - x and lagging links - y are related through the position function. Now the optimization problem can be formulated in the following way: find the position function such that the lagging link is transferred from initial position to the final position in the shortest time. We have obtained an analytic solution to this problem using variational calculus methods in the case then dissipative forces acting on the system can be neglected. We have shown that this problem is analogous to the classical brachistochtone problem. The qualitative feature of this result is that the optical mechanical system tends to accelerate the leading link first in order to maximize the power being extracted from the mechanical energy source. Analytical solution is applied for the optimization of the fast acting electric switch design. Leading link load and operating time reductions with the use of the optimal transfer function is demonstrated. This approach can be generalized for a variety of mechanisms where the operating time is critical.

A Numerical Scheme for a Class of Parametric Problem of Fractional Variational Calculus

2012 ◽  
Vol 7 (2) ◽  
Author(s):  
Om P. Agrawal ◽  
M. Mehedi Hasan ◽  
X. W. Tangpong
Keyword(s):  
Analytical Solutions ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimization Problems ◽  
Practical Interest ◽  
Variational Calculus ◽  
Functional Minimization ◽  
Orthogonal Functions ◽  
Order Problem ◽  
Parametric Problem

Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in the future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such properties as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered, and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.

Time-Optimal Coordination Control for the Gear-Shifting Process in Electric-Driven Mechanical Transmission (Dog Clutch) without Impacts

2020 ◽  
Vol 9 (2) ◽  
pp. 155-168
Author(s):  
Ziwang Lu ◽  
◽  
Guangyu Tian ◽  
Keyword(s):  
Control Strategy ◽  
Position Control ◽  
Control Method ◽  
Synchronization Control ◽  
Mechanical Transmission ◽  
Coordination Control ◽  
Optimal Position ◽  
Drive Motor ◽  
Time Optimal ◽  
Optimal Coordination

Torque interruption and shift jerk are the two main issues that occur during the gear-shifting process of electric-driven mechanical transmission. Herein, a time-optimal coordination control strategy between the the drive motor and the shift motor is proposed to eliminate the impacts between the sleeve and the gear ring. To determine the optimal control law, first, a gear-shifting dynamic model is constructed to capture the drive motor and shift motor dynamics. Next, the time-optimal dual synchronization control for the drive motor and the time-optimal position control for the shift motor are designed. Moreover, a switched control for the shift motor between a bang-off-bang control and a receding horizon control (RHC) law is derived to match the time-optimal dual synchronization control strategy of the drive motor. Finally, two case studies are conducted to validate the bang-off-bang control and RHC. In addition, the method to obtain the appropriate parameters of the drive motor and shift motor is analyzed according to the coordination control method.

Modeling and Analysis of a Ball and Beam System Including Impacts and Dry Friction

2014 ◽  
Author(s):  
Diego Colón ◽  
Átila Madureira Bueno ◽  
Ivando S. Diniz ◽  
Jose M. Balthazar
Keyword(s):  
Differential Inclusion ◽  
Dry Friction ◽  
Viscous Friction ◽  
Mathematical Object ◽  
Contact Forces ◽  
Modeling And Analysis ◽  
Typical Situation ◽  
Beam System ◽  
The Impact ◽  
Ball And Beam System

The Ball and Beam system is a common didactical plant that presents a complex nonlinear dynamics. This comes from the fact that the ball rolls over the beam, which rotates around its barycenter. In order to deduce the system’s equations, composition of movement must be applied, using a non-inertial reference frame attached to the beam. In the Literature, a common hypothesis is to suppose that the ball rolls without slipping. If a viscous friction is supposed to be present, a simpler situation is obtained, where Lagrangean mechanics can be applied, and no contact force is known. Even then, the dynamics is very nonlinear. However, this model does not include all the relevant phenomena, such as ball’s slipping at higher beam’s inclination angles, dry friction between the ball and the beam, and impacts between: 1) the ball and the ends of the beam, and 2) the beam and the base (ground). These additions to the model impose the necessity to calculate, in a simulation setting, the contact forces, and the Newton’s approach to determine the system’s equations becomes more convenient. Also, discontinuities in the model are introduced, and the simpler mathematical object for model such systems are the differential inclusion systems. In this work, we deduce the Ball and Beam differential inclusion system, including dry friction and the impact between the ball and beam. We also present simulation results for the corresponding differential inclusion system in a typical situation.

scholarly journals Investigation of Turbulent Transition in Plane Couette Flows Using Energy Gradient Method

2011 ◽  
Vol 3 (2) ◽  
pp. 165-180 ◽  
Author(s):  
Hua-Shu Dou ◽  
Boo Cheong Khoo
Keyword(s):  
Energy Loss ◽  
Gradient Method ◽  
Poiseuille Flow ◽  
Mechanical Energy ◽  
Viscous Friction ◽  
Critical Value ◽  
Plane Couette Flow ◽  
Turbulent Transition ◽  
Energy Gradient ◽  
Total Mechanical Energy

AbstractThe energy gradient method has been proposed with the aim of better understanding the mechanism of flow transition from laminar flow to turbulent flow. In this method, it is demonstrated that the transition to turbulence depends on the relative magnitudes of the transverse gradient of the total mechanical energy which amplifies the disturbance and the energy loss from viscous friction which damps the disturbance, for given imposed disturbance. For a given flow geometry and fluid properties, when the maximum of the function K (a function standing for the ratio of the gradient of total mechanical energy in the transverse direction to the rate of energy loss due to viscous friction in the streamwise direction) in the flow field is larger than a certain critical value, it is expected that instability would occur for some initial disturbances. In this paper, using the energy gradient analysis, the equation for calculating the energy gradient function K for plane Couette flow is derived. The result indicates that K reaches the maximum at the moving walls. Thus, the fluid layer near the moving wall is the most dangerous position to generate initial oscillation at sufficient high Re for given same level of normalized perturbation in the domain. The critical value of K at turbulent transition, which is observed from experiments, is about 370 for plane Couette flow when two walls move in opposite directions (anti-symmetry). This value is about the same as that for plane Poiseuille flow and pipe Poiseuille flow (385-389). Therefore, it is concluded that the critical value of K at turbulent transition is about 370-389 for wall-bounded parallel shear flows which include both pressure (symmetrical case) and shear driven flows (anti-symmetrical case).

scholarly journals DISSIPATIVE OSCILLATORS’ POWER CHARACTERISTIC NON-SYMMETRY DYNAMIC EFFECT

2021 ◽  
pp. 65-75
Author(s):  
Vasiliy Olshanskiy ◽  
Stanislav Olshanskiy
Keyword(s):  
Differential Equation ◽  
Dry Friction ◽  
Elastic Characteristic ◽  
Approximation Error ◽  
Dynamic Effect ◽  
Viscous Friction ◽  
Equation Of Motion ◽  
Lambert Function ◽  
Coulomb Dry Friction ◽  
Non Linear Oscillator

The features of motion of a non-linear oscillator under the instantaneous force pulse loading are studied. The elastic characteristic of the oscillator is given by a polygonal chain consisting of two linear segments. The focus of the paper is on the influence of the dissipative forces on the possibility of occurrence of the elastic characteristic non-symmetry dynamic effect, studied previously without taking into account the influence of these forces. Four types of drag forces are considered, namely linear viscous friction, Coulomb dry friction, position friction, and quadratic viscous resistance. For the cases of linear viscous friction and Coulomb dry friction the analytical solutions of the differential equation of oscillations are found by the fitting method and the formulae for computing the swings are derived. The conditions on the parameters of the problem are determined for which the elastic characteristic non-symmetry dynamic effect occurs in the system. The conditions for the effect to occur in the system with the position friction are derived from the energy relations without solving the differential equation of motion. In the case of quadratic viscous friction the first integral of the differential equation of motion is given by the Lambert function of either positive or negative argument depending on the value of the initial velocity. The elastic characteristic non-symmetry dynamic effect is shown to occur for small initial velocities, whereas it is absent from the system when the initial velocities are sufficiently large. The values of the Lambert function are proposed to be computed by either linear interpolation of the known data or approximation of the Lambert function by elementary functions using asymptotic formulae which approximation error is less than 1%. The theoretical study presented in the paper is followed up by computational examples. The results of the computations by the formulae proposed in the paper are shown to be in perfect agreement with the results of numerical integration of the differential equation of motion of the oscillator using a computer.

Nonconvex Time-Optimal Trajectory Planning for Robot Manipulators

2019 ◽  
Vol 141 (11) ◽  
Author(s):  
Ákos Nagy ◽  
István Vajk
Keyword(s):  
Motion Planning ◽  
Degrees Of Freedom ◽  
Optimization Problem ◽  
Robot Manipulator ◽  
Viscous Friction ◽  
Global Optimum ◽  
Convex Optimization Problem ◽  
Optimal Motion Planning ◽  
Time Optimal ◽  
Active Research

Time-optimal motion-planning has been a topic of active research in the literature for a while. This paper presents a new approach for velocity profile generation, which is a subproblem in motion-planning. In the case of simplified constraints, profile generation can be translated to a convex optimization problem. However, some practical constraints (e.g., velocity-dependent torque, viscous friction) destroy the convexity. The proposed method can obtain the global optimum of the nonconvex optimization problem. The experimental results with a three degrees-of-freedom (DOF) robot manipulator are also presented in this paper.

Experimental Modeling of Wear Behavior of Filled Elastomer SBR under Dry Friction-Influence of Roughness

2015 ◽  
Vol 640 ◽  
pp. 13-20
Author(s):  
Rachid Djeridi ◽  
Mohand Ould Ouali
Keyword(s):  
Weight Loss ◽  
Contact Pressure ◽  
Dry Friction ◽  
Sliding Friction ◽  
Operating Time ◽  
Wear Behaviour ◽  
Styrene Butadiene Rubber ◽  
Butadiene Rubber ◽  
Filled Elastomer ◽  
Tribological Parameters

The wear behaviour of a filled styrene butadiene rubber (SBR) is investigated in this paper. The material contact used is plan /plan with a hard steel XC38. The influence of tribological parameters such as type contact (contact plan/plan under dry friction), relative motion between the contact surfaces (rotational disc/fixed elastomer sample), topography of the surface contact (roughness), loading (normal load or contact pressure), sliding friction and operating time or number of cycles is investigated. The highlighting of these parameters influence and analysis results permits us to formulate a wear model for the filled elastomer SBR. The model is based on the Archard law developed for metallic materials. The modification concerns the introduction of material parameters to take accounts the hyperelastic behaviour of elastomer due to the presence of amorphous phase. Particular interest is given to the influence of the surface state of the indenter given by the counterface arithmetic roughnessRaon the weight loss of the elastomer due to the wear phenomena. For a lower value (little to 6.3μm) of the arithmetic roughness, the weight loss is insignificant for different contact pressure and various sliding speeds. This effect is more noticeable at higher values of roughness and dependent on other tribological parameters. This results comfort other conclusion on the literature that express the influence of roughness by the geometric parameters of the micro-waves in the surface. The effects of the roughness can be explained by the ratio between the amplitude and wavelength of the corrugation. Indeed, we relate the roughness influence at the strain energy restored by material hyperelastic which also is, necessarily, a function of the velocity sliding and pressure contact.

scholarly journals SOME PROBLEMS OF OPTIMIZATION AND CONTROL OF THE NATURAL FREQUENCIES OF AN ELASTICALLY SUPPORTED RIGID BODY

2021 ◽  
Vol 3 (2) ◽  
pp. 88-102
Author(s):  
S. Bekshaev ◽  
Keyword(s):  
Rigid Body ◽  
Fundamental Frequency ◽  
Natural Frequencies ◽  
Optimization Problems ◽  
Free Vibrations ◽  
The Body ◽  
Natural Vibrations ◽  
Natural Oscillations ◽  
Optimal Position ◽  
Elastic Supports

The article analytically investigates the behavior of the frequencies and modes of natural vibrations of a rigid body, based on point elastic supports, when the position of the supports changes. It is assumed that the body is in plane motion and has two degrees of freedom. A linear description of body vibrations is accepted. The problems of determining such optimal positions of elastic supports at which the fundamental frequency of the structure reaches its maximum value are considered. Two groups of problems were studied. The first group concerns a body supported by only two supports. It was found that in the absence of restrictions on the position of the supports to maximize the fundamental natural frequency, these supports should be positioned so that the basic natural vibrations of the body are translational. Simple analytical conditions are formulated that must be satisfied by the corresponding positions of the supports. In real practical situations, these positions may be unreachable due to the presence of various kinds of restrictions due to design requirements. In this paper, optimization problems are considered taking into account a number of restrictions on the position of supports, typical for practice, expressed analytically by equations and inequalities. For each of the considered types of constraints, results are obtained that determine the optimal positions of the supports and the corresponding maximum values of the main natural frequencies. The approach applied allows us to consider other types of restrictions, which are not considered in the article. In the second group of problems for a body resting on an arbitrary number of supports, the optimal position of an additional elastic support introduced in order to maximize the fundamental frequency in fixed positions and the stiffness coefficients of the remaining supports was sought. It was found that this position depends on the value of the stiffness coefficient of the introduced support. Results are obtained that qualitatively and quantitatively characterize this position and the corresponding frequencies and modes of natural oscillations, including taking into account practically established limitations. The research method uses a qualitative approach, systematically based on the well-known Rayleigh theorem on the effect of imposing constraints on the free vibrations of an elastic structure.

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