Variable Mass Systems Research Articles

Poisson brackets formulation for the dynamics of a position-dependent mass particle

Following the history of mechanics (Dugas in A history of mechanics, Dover, New York, 1988), we read that Poisson’s theorem figured in the 1811 edition of the celebrated Mecanique Analytique. Indeed, as it can be observed in the classical textbooks, Poisson brackets formulation is one of the cardinal chapters of analytical mechanics. Given the natural motivation to accommodate variable-mass systems at the level of classical analytical mechanics, we will herein direct our attention toward Poisson brackets formulation. To wit, considering the case of a position-dependent mass particle, in which we will assume that the absolute velocity of mass ejection or aggregation is a linear function of the generalized velocity, we will endeavor to provide such position-dependent mass problems with an appropriate Poisson brackets formulation. To our very best knowledge, this means an original contribution to the research field of the analytical mechanics of variable-mass systems. We will start establishing the Poisson brackets definition for the dynamics of a position-dependent mass particle, which will be posited in harmony with the classical mathematical portrait of analytical mechanics. Therefrom, we will demonstrate consequent results which give rise to the required formulation, namely, Jacobi’s identity, canonical equations expressed by means of such Poisson brackets and Poisson’s theorem. Last, we will apply Poisson’s theorem to evaluate the relationship between the conservation laws which are at our disposal in the domain of position-dependent mass problems.

Nonlinear dynamics and control of a laterally mass varying fighter aircraft

This paper presents dynamic analysis and nonlinear control of laterally mass varying combat aircraft based on a six degree-of-freedom model derived for a variable mass system. The objective is to investigate the effects of asymmetric lateral mass variation, due to mass ejection, from two different perspectives – the open loop dynamic behavior through bifurcation analysis; and the closed loop control performance while carrying out some demanding maneuvers. Bifurcation analysis reveals considerable coupling between the longitudinal and lateral-directional channels even for a modest lateral shift in the center-of-mass due to store ejection, which in turn is observed to give rise to a spiral-dive like divergent mode even at low angles-of-attack. Thereafter, the well-known high angle-of-attack cobra maneuver is implemented for the symmetric center-of-mass case using sliding mode control technique and using a new single loop control formulation in contrast with the conventional inner-outer loop formulation. The proposed single loop control formulation is further extended to handle the lateral movement of center-of-mass due to asymmetric store ejection using the asymmetric dynamics model proposed and derived in this paper.

Stochastic stability of variable-mass Duffing oscillator with mass disturbance modeled as Gaussian white noise

The stochastic asymptotic stability with probability one of the stochastic variable- mass Duffing oscillator under weakly parametric excitation is investigated by using the largest Lyapunov exponent, in which the small disturbance of mass is modeled as Gaussian white noise. First, the original system is replaced by an approximate system without mass disturbance based on an equivalent transformation. Then the averaged Ito equation of the approximate system is derived by applying the stochastic averaging method based on the generalized harmonic function. Finally, the Lyapunov exponent of the linearized averaged Ito equation is derived and a necessary and sufficient condition for the stability of the trivial solution is obtained approximately by letting this Lyapunov exponent be negative. The effects of mass disturbance on system stability are explored adequately. It turns out that mass disturbance deteriorates system stability remarkably. It is remarkable that the influences of linear stiffness on the stability of a variable-mass system are significantly different from that of an associated invariable-mass system.

Stochastic stationary response of a variable-mass system with mass disturbance described by Poisson white noise

Variable-mass systems have received widespread attention and show prominent significance with the explosive development of micro- and nanotechnologies, so there is a growing need to study the influences of mass disturbances on systems. This paper is devoted to investigating the stochastic response of a variable-mass system subject to weakly random excitation, in which the mass disturbance is modeled as a Poisson white noise. Firstly, the original system is approximately replaced by the associated conservative system with small disturbance based on the Taylor expansion technique. Then the stationary response of the approximate system is obtained by applying the stochastic averaging method. At last, a representative variable-mass oscillator is worked out to illustrate the effectiveness of the analytical solution by comparing with Monte Carlo simulation. The relative change of mean-square displacement is used to measure the influences of mass disturbance on system responses. Results reveal that the stochastic responses are more sensitive to mass disturbance for some system parameters. It is also found that the influences of Poisson white noise as the mass disturbance on system responses are significantly different from that of Gaussian white noise of the same intensity.

A differential equation for the flow rate during silo discharge: Beyond the Beverloo rule

We present a differential equation for the flow rate of granular materials during the discharge of a silo. This is based in the energy balance of the variable mass system in contrast with the traditional derivations based on heuristic postulates such as the free fall arch. We show that this new equation is consistent with the well known Beverloo rule, providing an independent estimate for the universal Beverloo prefactor. We also find an analytic expression for the pressure under discharging conditions.

On Hamilton’s principle for discrete systems of variable mass and the corresponding Lagrange’s equations

Departing from d’Alembert’s principle, the classical deductions of Hamilton’s principle and, from it, of Lagrange's equations, are extended to the case of discrete systems of variable mass. These deductions recover the expressions of the extended Lagrange’s equations previously derived by Pesce [1] and of Hamilton’s principle proposed by Pesce and Casetta [19], for systems with mass explicitly dependent on position, velocity, and time. Alternative forms of Hamilton’s principle and Lagrange’s equations are obtained and presented. Cayleys’s falling chain problem is addressed as an application example.

Influence of tuning of passive TLD on the seismic vibration control of elevated water tanks under various tank-full conditions

This paper focuses on the seismic vibration control of elevated water tanks (EWTs) by the passive tuned liquid damper (TLD). Because of the fluctuating water content, the EWT is a variable-mass system resulting in a continual modification of its natural frequencies. The problem of identifying the optimal tuning of the damper for robust performance over the range of fill depths of interest in the EWT is addressed here. The nonlinear model based on shallow water wave theory simulating sloshing motion of the liquid (here water) in the TLD is utilized. The sloshing of water in the EWT itself is considered by modeling the EWT as a two-degree-of-freedom system, one representing the lateral vibrational mode of the supporting tank structure, tank container and impulsive mass of the water and the other denoting the fundamental sloshing mode of the water. A time domain study on a realistic reinforced concrete shaft supported EWT with TLD subjected to harmonic and recorded accelerogram input indicates that the TLD achieves significant response reduction and that it should be tuned to the structural frequency corresponding to the half-full to three-fourth full tank condition for robust performance. Copyright © 2016 John Wiley & Sons, Ltd.

Theorem on a new conservation law for the dynamics of a position-dependent mass particle

In this Note, we aim at proving a theorem on a new conservation law for the dynamics of a position-dependent mass particle. This new conservation law has the significant particularity of being concisely written in terms of the total energy of the problem. Here, we will consider the special case in which the absolute velocity of mass ejection or aggregation is a linear function of the generalized velocity. Our result is an original contribution in the traditional research field of variable-mass systems.

The dynamic responses of protective construction formworks to the continuous imposition of impacts

The dynamic responses of protective construction formworks are analysed in receiving debris continuously falling from a height within a time period. A Kelvin–Voigt type model is established for the dynamic system. The effects of mass variation and the cushion action of buffer materials are taken into account, leading to the application of the Meshchersky equation to a variable mass system of two degrees of freedom. The dynamic load factors are evaluated for designing the formworks to withstand the impact of the continuously falling debris. Following examples demonstrating the use of the approach, the influences of the basic variables on the dynamic factors are investigated through a parameter study, the results of which may be generally referenced for load determination in the design of formworks.

Numerical and experimental investigations of the dynamics of a variable mass pendulum

In this paper, the numerical and experimental results for the dynamics of pendulum with variable mass were described. Mathematical model was developed taking into account the loss of mass, reactive force, air resistance, and friction. A corresponding test rig was designed and built in order to validate the numerical results. The purpose of the paper is to show that in case of the variable mass systems the second Newton’s law cannot be directly applied in the traditional form. The simple experiment was designed to support the thesis that the modification of Newton’s second law is necessary.

Video analysis of sliding chains: A dynamic model based on variable-mass systems

This paper presents an experimental study of the dynamics of a chain sliding off of a table, using video analysis to test a theoretical model. The model consists of two variable-mass subsystems, with friction between the chain and the table and assumes that all links move at the same speed. In order to check the model, the chain position x(t) is obtained using video analysis. The smoothed function x(t) and its derivatives v(t) and a(t) are numerically computed using a local regression algorithm. In this way, the differential equation governing the motion can be directly tested, instead of comparing the position with the solution of the differential equation. Our procedure is very sensitive to deviations between the model and reality, so we can detect the point at which the chain ceases to be in tension and the model is no longer valid. This experiment shows students the limitations of simplified models and offers an opportunity to assess a model's range of validity.

Active linear quadratic Gaussian control of the vibration of a flexible beam with a time-varying mass

A linear quadratic Gaussian method for active vibration control of a flexible beam with a time-varying mass is studied in this article. As the time-varying mass can change the resonance frequencies of system and induce instability of system, the vibration control of time-varying mass systems using piezoelectric actuators has been the subject of recent interest. The equation of motion of a beam with a time-varying concentrated mass is obtained by using admissible functions and the principle of separation of variables. Then, active control using piezoelectric actuators is simulated for the time-varying mass system. The linear quadratic Gaussian control strategy is employed for the suppression of transverse vibration of the beam with time-varying mass under different conditions. The numerical results show that the proposed scheme can attenuate the vibration of the variable mass system effectively in the interested frequency range. The control method can be used for the beam with constant mass or time-varying mass.

Dynamic behaviors of flexible fluid containers with fuel consumption

This study focuses on the dynamic characteristics of the containers with fuel consumption. With fuel consumption, the fuel container system vibrates with decreasing mass, which is a typical variable mass system. In order to investigate the influences of the time-varying liquid on vibration of the containers, this fluid–structure coupling system with time-varying mass is modeled combining finite element method, used to model the structure, and virtual mass method representing the fluid. The dynamic responses and time–frequency analysis of the time-varying mass system are presented based on the numerical simulations. From the results, it is shown that decrease of the mass results in increase of the vibration frequencies of the system and induces an additional negative damping. The additional negative damping is proportional to the rate of mass change and causes the vibration to decay slowly. The results also suggest that the time–frequency analysis possess appealing advantages in processing nonstationary signals and thus are able to provide reliable time–frequency domain information for time-varying mass systems.

Self-adaptive Newmark method of variable-mass beam dynamic system

Self-adaptive Newmark method of variable-mass beam dynamic system

English

This paper evaluates the impact that helical motion of fluid products of combustion within the combustion chamber of a rocket can have on the attitude dynamics of rocket systems. By developing the study presented by Sookgaew (2004), we determined the configuration of the Coriolis moment components, which catch the impact of the combustion product's whirling motion, for the radial and centripetal propellant burn pattern specific to S-5M and S-5K solid rocket motors. We continue the investigation of the effects of internal whirling motion of fluid products of combustion on the attitude behavior of variable mass systems of the rocket type by examining the spin motion and transverse attitude motion of such systems. The results obtained show that internal fluid whirling motion can cause appreciable deviations in spin rate predictions, and also affects the frequencies of the transverse angular velocity components.

New kinetic energy operator for variable mass systems

We have derived a new kinetic energy operator for studying variable mass systems. Our operator is dependent on the space deformation profile \( \mu(x)\) subject to the order parameter \( \varsigma\) . At zeroth order, we recover the standard one-dimensional kinetic energy operator for a constant mass system while, for \( \varsigma\geq 1\) , the operator is interpreted as describing a system endowed with a position-dependent effective mass.

Hamilton's law of variable mass system and time finite element formulations for time‐varying structures based on the law

SUMMARYTraditional principles of mechanics are primarily conceived for constant mass systems, which are only valid if mass is gained or lost at null velocity with respect to an inertial reference frame for variable mass systems, thus the numerical algorithms for time‐varying structures based on these principles are only suitable for this special case. In this paper, Hamilton's law of variable mass system is derived based on Meshchersky's fundamental equation, and two classes of novel time finite element formulations for linear systems with arbitrary continuous time‐varying parameters are developed based on the previous law. The formulations are verified extensively through numerical examples in which the convergence and effectiveness of algorithms are evaluated. Numerical examples demonstrate that compared with the algorithms for time‐varying structures that developed based on traditional principles of mechanics, the proposed algorithms provide extended capabilities in both time‐varying mass problems that mass is gained or lost at any velocity (such as rocket problem) and moving‐mass problems (such as vehicle‐bridge interaction problem) besides the time‐varying stiffness and damping problems, the proposed algorithms have a wider range of application. In particular, Hamilton's law of variable mass system provides a solid theoretical foundation for further research on the algorithm design for time‐varying structures. Copyright © 2014 John Wiley & Sons, Ltd.

О прохождении резонанса в механической системе переменной массы

О прохождении резонанса в механической системе переменной массы

Application of the Dynamics of Variable Mass Systems to the Pelton Turbine

This paper deals with an alternative approach to calculate the performance of a Pelton turbine. The discussion begins with an overview on the statement of Newton's second law for variable mass systems. Then the integral form of the Navier-Stokes equation is presented as a special case, which is useful to evaluate the force exerted by the water jet on the turbine buckets. Although the traditional approach based on the angular momentum theorem is more straightforward, the proposed method allows evaluation of the different contributions to the motive force by a suitable choice of control volumes. This in turn makes for a better understanding of factors that penalize overall performance when the ideal assumptions, under which the theory is usually developed, are no longer verified.

船用闪蒸罐闪蒸性能仿真研究

以船用闪蒸罐为原型，在合理假设基础上，建立了闪蒸罐变质量系统热力学模型。采用Simulink搭建闪蒸仿真程序，进行了不同运行工况下船用闪蒸罐闪蒸性能仿真研究，仿真结果表明随着闪蒸罐初始水温下降，过热度增加，闪蒸过程中流体温度变化率增大。在不同背压和初始温度下得到的压力和温度变化规律与试验值吻合较好，最大相对偏差为5%。变质量系统热力学模型在闪蒸罐闪蒸性能仿真计算中的成功应用，可为船舶蒸汽动力系统设计提供一定的理论依据。 Taken marine flash tank as the prototype, the variable mass system thermodynamics model of flash tank was built based on the reasonable assumptions. The simulation program was compiled based on Simulink code. Study on flash evaporation performance for marine flash tank was conducted at different operating conditions and rules of variation of the key thermal-hydraulic parameters were obtained. Simulation results revealed that the change rate of fluid temperature increased due to lower initial temperature and higher superheat. Meanwhile, the results, which were obtained in different backpressure and initial temperature, agreed well with experimental data. The maximum relative deviation arrived at 5%. The successful application of variable mass system thermodynamics model in the simulation of the flash tank performance can supply some theoretical basis for ship steam power system design.