Non-stationary oscillations of an instantly loaded oscillator under conditions of nonlinear resistance

The motion of an oscillator instantaneously loaded with a constant force under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, are considered. Using the first integral of the equation of motion and the Lambert function, compact formulas for calculating the ranges of oscillations are derived. In order to simplify the search for the values of the Lambert function, asymptotic formulas are given that, with an error of less than one percent, express this special function in terms of elementary functions. It is shown that as a result of the action of the resistance force, including dry friction, the oscillation process has a finite number of cycles and is limited in time, since the oscillator enters the stagnation region, which is located in the vicinity of the static deviation of the oscillator caused by the applied external force. The system dynamic factor is less than two. Examples of calculations that illustrate the possibilities of the stated theory are considered. In addition to analytical research, numerical computer integration of the differential equation of motion was carried out. The complete convergence of the results obtained using the derived formulas and numerical integration is established, which confirms that using analytical solutions it is possible to determine the extreme displacements of the oscillator without numerical integration of the nonlinear differential equation. To simplify the calculations, the literature is also recommended, where tables of the Lambert function are printed, allowing you to find its value for interpolating tabular data. Under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, the process of oscillations of an instantly loaded oscillator has a limited number of cycles. The dependences obtained in this work using the Lambert function make it possible to determine the range of oscillations without numerical integration of the nonlinear differential equation of motion both for an oscillator with quadratic viscous resistance and dry friction, and for an oscillator with quadratic resistance and positional and dry friction. Keywords: nonlinear oscillator, instantaneous loading, quadratic viscous resistance, Lambert function, oscillation amplitude.

Determining the energy efficiency of a resonance single-mass vibratory machine whose operation is based on the Sommerfeld effect

This paper reports determining the energy efficiency of a vibratory machine consisting of a viscoelastically fixed platform that can move vertically, and a vibration exciter whose operation is based on the Sommerfeld effect. The body of the vibration exciter rotates at a steady angular speed while there are the same loads in the form of a ball, a roller, or a pendulum inside it. The load, being moved relative to the body, is exposed to the forces of viscous resistance, which are internal within the system. It was established that under the steady oscillatory modes of a vibratory machine's movement, the loads are tightly pressed to each other, thereby forming a combined load. Energy is productively spent on platform oscillations and unproductively dissipated due to the movement of the combined load relative to the body. With an increase in the speed of the body rotation, the increasing internal forces of viscous resistance bring the speed of rotation of the combined load closer to the resonance speed, and the amplitude of platform oscillations increases. However, the combined load, in this case, increasingly lags behind the body, which increases unproductive energy loss and decreases the efficiency of the vibratory machine. A purely resonant motion mode of the vibratory machine produces the maximum amplitude of platform oscillations, the dynamic factor, the total power of viscous resistance forces. In this case, the efficiency reaches its minimum value. To obtain vigorous oscillations of the platform with a simultaneous increase in the efficiency of the vibratory machine, it is necessary to reduce the forces of viscous resistance in supports with a simultaneous increase in the internal forces of viscous resistance. An algorithm for calculating the basic dynamic characteristics of the vibratory machine's oscillatory motion has been built, based on solving the problem parametrically. The accepted parameter is the angular speed at which a combined load gets stuck. The effectiveness of the algorithm has been illustrated using a specific example

Mechanosensitive recruitment of stator units promotes binding of the response regulator CheY-P to the flagellar motor

Reversible switching of the bacterial flagellar motor between clockwise (CW) and counterclockwise (CCW) rotation is necessary for chemotaxis, which enables cells to swim towards favorable chemical habitats. Increase in the viscous resistance to the rotation of the motor (mechanical load) inhibits switching. However, cells must maintain homeostasis in switching to navigate within environments of different viscosities. The mechanism by which the cell maintains optimal chemotactic function under varying loads is not understood. Here, we show that the flagellar motor allosterically controls the binding affinity of the chemotaxis response regulator, CheY-P, to the flagellar switch complex by modulating the mechanical forces acting on the rotor. Mechanosensitive CheY-P binding compensates for the load-induced loss of switching by precisely adapting the switch response to a mechanical stimulus. The interplay between mechanical forces and CheY-P binding tunes the chemotactic function to match the load. This adaptive response of the chemotaxis output to mechanical stimuli resembles the proprioceptive feedback in the neuromuscular systems of insects and vertebrates.

Mechanosensitive recruitment of stator units promotes binding of the response regulator CheY-P to the flagellar motor

Reversible switching of the bacterial flagellar motor between clockwise (CW) and counterclockwise (CCW) rotation is necessary for chemotaxis, which enables cells to swim towards favorable chemical habitats. Increase in the viscous resistance to the rotation of the motor (mechanical load) inhibits switching. However, cells must maintain homeostasis in switching to navigate within environments of different viscosities. The mechanism by which the cell maintains optimal chemotactic function under varying loads is not understood. Here, we show that the flagellar motor allosterically controls the binding affinity of the chemotaxis response regulator, CheY-P, to the flagellar switch complex by modulating the mechanical forces acting on the rotor. Mechanosensitive CheY-P binding compensates for the load-induced loss of switching by precisely adapting the switch response to a mechanical stimulus. The interplay between mechanical forces and CheY-P binding tunes the chemotactic function to match the load. This adaptive response of the chemotaxis output to mechanical stimuli resembles the proprioceptive feedback in the neuromuscular systems of insects and vertebrates.

Mediation of lubricated air films using spatially periodic dielectrophoretic effect

A stone thrown in a lake captures air as it collides with water and sinks; likewise a rain drop falling on a flat surface traps air bubbles underneath and creates a spectacular splash. These natural occurrences, from bubble entrapment to liquid ejection, happen as air fails to escape from the closing gap between liquid and solid surfaces. Trapping of air is devastating for casting, coating, painting, and printing industries, or those intolerant of water entry noise. Attempts to eliminate the interfering air rely on either reducing the ambient pressure or modifying the solid surfaces. The former approach is inflexible in its implementation, while the latter one is inherently limited by the wetting speed of liquid or the draining capacity of air passages created on the solid. Here, we present a “divide and conquer” approach to split the thin air gap into tunnels and subsequently squeeze air out from the tunnels against its viscous resistance using spatially periodic dielectrophoretic force. We confirm the working principles by demonstrating suppression of both bubble entrapment and splash upon impacts of droplets on solid surfaces.

CESSATION OF FREE VIBRATIONS OF A NONLINEAR ELASTIC SYSTEM DUE TO VISCOUS RESISTANCE

The paper deals with free vibrations of a system with power-law nonlinear elasticity subjected to power-law viscous resistance. The relation between the nonlinearity indices is determined when the impact of the viscous resistance force causes the vibrations to die away. In this case the vibrations are limited in time i.e. consist of a finite number of cycles analogous to a system with Coulomb dry friction. The research exploits the energy balance method. The periodic Ateb-functions are used to obtain an approximate formula for the work of dissipative force over a semi-cycle of vibrations. A recursive power-law equation for the vibration swings is derived from the condition of equality of the work to the potential energy change. By analyzing the change of the coefficient in the equation, which is related to the change of the semi-cycle number as well as the vibration swings, the condition for the equation to have no positive root is determined, which means that the vibrations die away. The condition is formulated in the form of an inequality. It is shown to generalize the results previously known. The theoretical inferences are verified by numerical integration of the nonlinear differential equation of motion. It is shown that under the conditions proposed in the paper the free vibrations consist of a finite number of cycles even if dry friction is absent from the system. Special cases are highlighted, when the approximate energy balance method results into exact computational formulae. The length of the cycles increases during the motion since it depends on the swing of damped vibrations in the essentially nonlinear system with rigid force characteristics considered.

Optimal parameters of tuned mass dampers for an inverted pendulum with two degrees of freedom

In reality, an inverted pendulum can be used to model many real structures as the fluid tower, super-tall buildings, or articulated tower in the ocean, etc. However, for the inverted pendulum with two degrees of freedom, to the best knowledge of the author, there is no study to determine optimal parameters of two tuned mass dampers (TMD) by using the maximization of equivalent viscous resistance method. Therefore, the current study presents the analytical solutions to the optimization of two orthogonal TMDs, which is used to eliminate vibration of the inverted pendulum with two degrees of freedom. The parameters considered in optimizing are the natural frequency ratios and damping ratios of the two TMDs. The new results of this paper can be summarized as follows: Firstly, the equivalent resistance forces of the two TMDs acting on the inverted pendulum with two degrees of freedom are established. Secondly, the quadratic torque matrices of the vibration response of the inverted pendulum attached with two TMDs is revealed. Thirdly, the optimal expressions are derived using the maximization of equivalent viscous resistance method. The obtained formulae provide exact solutions for the proposed problem. Finally, to confirm the effectiveness of the obtained formulae, parametric studies on vibration are performed for sample articulated tower in the ocean with and without optimal TMDs. Numerical results show that vibrations of the articulated tower attached with optimal TMDs are effectively eliminated. This confirms that the optimal parameters of the two TMDs are determined in this paper are reliable and accurate.

FREE OSCILLATOR OSCILLATIONS IN THE PRESENCE OF QUADRATIC VISCOUS RESISTANCE AND DRY FRICTION

The article is devoted to the derivation of formulas for calculating the ranges of free damped oscillations of a double nonlinear oscillator. Using the Lambert function and the first integral of the nonlinear differential equation of motion, formulas are derived for calculating the ranges of free damped oscillations of a linearly elastic oscillator under the combined action of the forces of quadratic viscous resistance and Coulomb dry friction. The calculations involve a table of the specified special function of the negative argument. It is shown that the presence of viscous resistance reduces the duration of free oscillations to a complete stop of the oscillator. The set dynamics problem is also approximately solved by the energy balance method, and a numerical integration of the nonlinear differential equation of motion on a computer is carried out. The satisfactory convergence of the numerical results obtained in various ways confirmed the suitability of the derived closed formulas for engineering calculations. In addition to calculating the magnitude of the oscillations, the energy balance method is also used for an approximate solution of the inverse problem of dynamics, by identifying the values of the coefficient of quadratic resistance and dry friction force in the presence of an experimental vibrogram of free damped oscillations. An example of identification is given. This information on friction is needed to calculate forced oscillations, especially under resonance conditions. It is noted that from the obtained results, in some cases, well-known formulas follow, where the quadratic viscous resistance is not associated with dry friction.