The fluctuation-dissipation theorem and the autocorrelation function of thermal radiation

A new relatively simple derivation of the fluctuation-dissipation theorem (FDT) is presented. The generalized coordinate of the system is changed by an external force and is expressed by means of causal susceptibility, its Fourier transform – the transfer function, generalized impedance and active resistance. These characteristics describe heat dissipation on the resistor and the result is generalized to the dissipative system which is under the action of macroscopic force. The fluctuation voltage on the resistor is obtained by decomposing the thermal chaotic motion of free charges along the conductor into a Fourier series. The number of standing waves and the average energy of the quantum oscillation state at a fixed temperature give the thermal power of charge transfer. By comparing with the Joule-Lenz law and by generalizing the result to an arbitrary isothermal system, the mean square of the fluctuating force and dispersion of the generalized coordinate caused by the thermal motion are obtained. The autocorrelation functions of the generalized coordinate and the random force, and their spectral densities are expressed through the considered characteristics. The content of FDT is that the power of heat release, the spectral densities of the fluctuating force and the autocorrelation are proportional to the imaginary part of the transfer function of the system. The result is used for thermal radiation in a cavity the walls of which contain electric dipoles excited by thermal motion. The transfer function, the fluctuating force acting on the charge, the dispersion of the electric field strength, time autocorrelation of the electric field strength and its spectral density are obtained. Real and imaginary components, the modulus and phase are found for complex relative autocorrelation of the electric field strength and the coherence time is determined.

Mathematical Model of the Tensioning in the Collet Clamping Mechanism with the Rotary Movable Input Link on Spindle Units

Increasing machining productivity causes the cutting forces acting on tools or workpieces to grow and requires extra clamping forces for their fixation reliably. In the research, a mathematical model of the operation of the clamping mechanism for fixating cylindrical objects on the spindle of machine tools at the stage of tension is presented. The presented design of the mechanism contains screw gear and provides self-braking. Based on the calculation model, mathematical dependencies are developed to describe the relationship among the movements of the parts of the mechanism when clamping forces are growing. The presented analytical dependencies allow considering the stage of growing clamping forces separately when the conservative type of forces are prevailing in the mechanism’s operation. That stage of work when both types of forces of dissipative and potential characters exist is considered. The developed dependencies describe the position of parts of the clamping mechanism depending on the generalized coordinate. The angle of rotation of the input rotating link is used as the generalized coordinate. This fact allows calculating the position of the elements of the clamping mechanism of this type depending on time. Results of the research enhance understanding the pattern of the change in the interaction of the elements and forces that act in the mechanism during the final stage of clamping. The obtained mathematical dependencies are a precondition for the development of design methodology for mechanisms of this type.

Trapezium-Type Inequalities for an Extension of Riemann–Liouville Fractional Integrals Using Raina’s Special Function and Generalized Coordinate Convex Functions

In this paper, the authors analyse and study some recent publications about integral inequalities related to generalized convex functions of several variables and the use of extended fractional integrals. In particular, they establish a new Hermite–Hadamard inequality for generalized coordinate ϕ-convex functions via an extension of the Riemann–Liouville fractional integral. Furthermore, an interesting identity for functions with two variables is obtained, and with the use of it, some new extensions of trapezium-type inequalities using Raina’s special function via generalized coordinate ϕ-convex functions are developed. Various special cases have been studied. At the end, a brief conclusion is given as well.

Kinematic analysis of the method increasing the accuracy of treatment of conic surfaces

A method for processing high-precision conical surfaces under conditions of free grinding on a lever machine is proposed, which allows adjusting the process parameters taking into account the technological heredity of the workpiece from the point of view of the distribution pattern of the allowance to be removed from its surface. The scheme of implementation of the method in which the main links are a tool in the form of a faceplate and a separator for maintaining the required accuracy of its flatness. In order to develop a method of targeted control of the actuation of the working surface of the tool to maintain the required accuracy of its flatness, mathematical modeling of the motion patterns of the links of the actuator of the base machine, on which the proposed device for processing conical parts is installed, was performed. In the process of mathematical simulation structural analysis of the executive mechanism of the lever machine is carried out, connection between the generalized coordinate and provisions of links of the executive mechanism of the lever machine is established, position of the lever and the upper link in the vertical plane is probed, geometrical parameters of reflexive rotational motion of the upper link are considered, the ratio for calculation of length of a crank of the executive mechanism of the machine depending on amplitude of figurative movement of its output link is obtained. An analytical relationship has been established between the input and output links of the machine actuator, which makes it possible to calculate the length of the first of the marked links depending on the required amplitude of the oscillatory motion of the second of them and creates the prerequisites for the development of engineering methods for controlling the process of forming high-precision conical surfaces under conditions of free grinding.

Dynamics of a Linkage Mechanism Using Sample Entropy

The dynamics of a slider-crank mechanism is developed using Kane's equations of motion. The motor torque is a function of the derivative of the generalized coordinate. The nonlinear equations of motion are solved using MATLAB numerical techniques. The sample entropy is calculated for different angular velocities of the crank.

Modeling and Simulation for Multi-Rotor Fixed-Wing UAV Based on Multibody Dynamics

Accurate dynamic modeling lays foundation for design and control of UAV. The dynamic model for the multi-rotor fixed-wing UAV was looked into and it was divided into fuselage, air-body, multi-rotors, vertical fin, vertical tail and control surfaces, based on the multibody dynamics. The force and moment model for each body was established and derived into the Lagrange equation of the second king by virtual work. By electing quaternion as generalized coordinate and introducing Lagrangian multiplier, the dynamic modeling was deduced and established. Finally, the comparison between the simulation results and the experimental can be found that the present dynamic model accurately describes the process of dynamic change of this UAV and lay foundation for the control of UAV.

Self-adaptive grasp analysis of a novel under-actuated cable-truss robotic finger

This paper presents the self-adaptive grasp of a novel cable-driven finger, which is an underactuated finger comprising cable-truss units. The underactuated cable-truss finger uses tendon-pulley transmission and parallel four-linkage mechanism to realize grasps. The working principle of the underactuated cable-truss finger and the self-adaptation at the grasp-closing stage are introduced. A self-adaptive grasping model is constructed to analyze the self-adaptation, and a new analysis method that considers the position and posture of grasping point in distal phalange is proposed. A new generalized coordinate, which directly shows the contacting position and the relative angular displacement in the distal phalange, is established. The expression of general static grasping force is established by using the virtual work principle, which reveals the relationship among the driving force, the equivalent torques on joints and the grasping forces. The workspace of the underactuated cable-truss finger and distributions of grasping force in new generalized coordinate are assessed through numerical analysis. The balance conditions of the self-adaptive grasp and the corresponding statuses are theoretically illustrated. Valid and adequate self-adaptive grasping experiments are conducted to verify the accuracy of self-adaptive grasping analysis.