Equations Of Motion For System Research Articles

Time-varying dynamic analysis for a helical gear pair system with three-dimensional motion due to bearing deformation

The dynamic response of a helical gear pair system is investigated. A new dynamic model for a helical gear pair system, considering three-dimensional motion due to bearing deformation, is proposed. The proposed model considers the helix angle, gear pair center distance, transverse pressure angle, and the contact ratio as time-dependent variables, which are considered as constants in other models. In fact, three-dimensional motion due to bearing deformation will lead to the changes in a series of dynamic responses. The system equations of motion were obtained by applying Lagrange’s equation and the dynamic responses are computed by the fourth-order Runge–Kutta method. The time-varying dynamic displacements, helix angle, gear pair center distance, transverse pressure angle, and the contact ratio are investigated with bearing deformation, different radial bear stiffness, different axial bear stiffness, and gear eccentricity. The results show that, due to the time-varying effect, this new helical gear pair model provides more accurate dynamic responses than those previous models which are considered as constant. In the future, this study can provide some useful information for the time-varying dynamic design of a helical gear pair system.

Dynamic analysis of tapping mode atomic force microscope (AFM) for critical dimension measurement

Transient dynamics of tapping mode atomic force microscope (AFM) for critical dimension measurement are analyzed. A simplified nonlinear model of AFM is presented to describe the forced vibration of the micro cantilever-tip system with consideration of both contact and non-contact transient behavior for critical dimension measurement. The governing motion equations of the AFM cantilever system are derived from the developed model. Based on the established dynamic model, motion state of the AFM cantilever system is calculated utilizing the method of averaging with the form of slow flow equations. Further analytical solutions are obtained to reveal the effects of critical parameters on the system dynamic performance. In addition, features of dynamic response of tapping mode AFM in critical dimension measurement are studied, where the effects of equivalent contact stiffness, quality factor and resonance frequency of cantilever on the system dynamic behavior are investigated. Contact behavior between the tip and sample is also analyzed and the frequency drift in contact phase is further explored. Influence of the interaction between the tip and sample on the subsequent non-contact phase is studied with regard to different parameters. The dependence of the minimum amplitude of tip displacement and maximum phase difference on the equivalent contact stiffness, quality factor and resonance frequency are investigated. This study brings further insights into the dynamic characteristics of tapping mode AFM for critical dimension measurement, and thus provides guidelines for the high fidelity tapping mode AFM scanning.

Maggi’s Equations Used in the Finite Element Analysis of the Multibody Systems with Elastic Elements

The main method used to determine the equations of motion of a multibody system (MBS) with elastic elements is the method of Lagrange’s multipliers. The assembly of equations for the whole system represents an important step in the elastodynamic analysis of such a system. This paper presents a new method of approaching this stage, by applying Maggi’s equations. In this way, the links that exist between the finite elements and the connections that exist between different bodies of the MBS system are conveniently taken into account, each body having a distinct velocity and acceleration field. Although Maggi’s equations have been used, sporadically, in some applications so far, we are not aware that they have been used in the study of elastic systems using the finite element method. Finally, an algorithm is presented that uses the Maggi formalism to obtain the equations of motion for an MBS system.

Non-intrusive hierarchical coupling strategies for multi-scale simulations in gravitational dynamics

Hierarchical code coupling strategies make it possible to combine the results of individual numerical solvers into a self-consistent symplectic solution. We explore the possibility of allowing such a coupling strategy to be non-intrusive. In that case, the underlying numerical implementation is not affected by the coupling itself, but its functionality is carried over in the interface. This method is efficient for solving the equations of motion for a self-gravitating system over a wide range of scales. We adopt a dedicated integrator for solving each particular part of the problem and combine the results to a self-consistent solution. In particular, we explore the possibilities of combining the evolution of one or more microscopic systems that are embedded in a macroscopic system. The here presented generalizations of Bridge include higher-order coupling strategies (from the classic 2nd order up to 10th-order), but we also demonstrate how multiple bridges can be nested and how additional processes can be introduced at the bridge time-step to enrich the physics, for example by incorporating dissipative processesor. Such augmentation allows for including additional processes in a classic Newtonian N-body integrator without alterations to the underlying code. These additional processes include for example the Yarkovsky effect, dynamical friction or relativistic dynamics. Some of these processes operate on all particles whereas others apply only to a subset.The presented method is non-intrusive in the sense that the underlying methods remain operational without changes to the code (apart from adding the get- and set-functions to enable the bridge operator). As a result, the fundamental integrators continue to operate with their internal time step and preserve their local optimizations and parallelism. Multiple bridges can be nested and coupled hierarchically, allowing for the construction of a complex environment of multiple nested augmented bridges. While the coupling topology may become rather complicated, we introduce the hierarchical coupling language (HCL), a meta language in which complex bridge topologies can be described. The meta language is meant for stimulating the discussion on even more complex hierarchies in which the bridge operators are introduced as patternsWe present example applications for several of these cases and discuss the conditions under which these integrators can be applied. Typical applications range over 10 orders of magnitude in temporal and spatial scales when we apply the method to simulating planetary systems (au spatial and year-temporal scale) in a star cluster that orbits in the Galaxy (100 kpc-spatial and 10 Gyr-temporal scale).

Time-Varying Dynamic Analysis of a Helical-Geared Rotor-Bearing System with Three-Dimensional Motion Due to Shaft Deformation

The rotordynamics of a helical-geared rotor-bearing system were investigated. A new dynamic model for a helical-geared rotor-bearing system, which takes into account three-dimensional (3-D) motion due to rotating shaft deformation, was proposed. The proposed model considers the time-varying effect, which in other models, is considered constant. The system equations of motion were obtained by applying Lagrange’s equation, and the dynamic responses were computed by the fourth-order Runge–Kutta method. The time-varying dynamic responses of the helix angle, transverse pressure angle, gear pair center distance, and total contact ratio were investigated. The numerical results show that the time-varying effect is an important factor in gear vibration analysis and cannot be neglected when the helical geared rotor-bearing system has a lower stiffness.

Influence of Blade Flexibility on the Dynamic Response Simulation of a Turbomolecular Pump on Magnetic Bearings

Abstract This paper proposes the simulation of a complete mechanical model of a turbomolecular pump rotor, including rotor and blades flexibility, suspended by controlled active magnetic bearings. The mechanical model is composed of an eight stage blisk, attached to a shaft. Magnetic forces are linearized by the first-order Taylor expansion around a given point. Including blades and rotor flexibility makes the mechanical system asymmetric, so the equations of motion for the coupled system have periodic terms. A modal controller was designed to control rigid body modes, since the number of sensors is limited and no state observer is implemented. PID controllers are used for low frequency modes combined with the second order filters to damp high frequency modes. First of all, stability analysis was carried out for the axisymmetric case. Second, blades flexibility was included. Forced response of the whole system to an impulsive force was studied. Divergent responses for the system in rotation were obtained as a second order filter pole possibly interacting with blades modes. Taking the second order filters off the control loop allowed the system to be stable. These results show that the analysis method developed here is efficient to evaluate the performance of a controller in closed loop with the complete flexible system. This method may be used in industrial design processes as computation times for the complete system are very short.

Dynamic analysis of a bevel gear system equipped with finite length squeeze film dampers for passive vibration control

Though finite length squeeze film dampers (FLSFD) are practically and effectively applied to vibration attenuation, they have not been popularly used in bevel gear systems (BGS) due to the limitations of the existing models of FLSFD. The present work proposes a general approach for modeling and calculating the nonlinear oil-film force of FLSFD, which is introduced into system motion equations to model the BGS supported on FLSFD mathematically. Comparisons with conventional methods show the superiorities of the proposed model in terms of computational accuracy and versatility. The damping characteristics of FLSFD are examined by comparing the dynamic behaviors of the BGS with and without FLSFD. Besides, the response sensitivity of the system to different parameters, including the length and radial clearance of FLSFD, is also discussed. The results show that FLSFD can significantly reduce the vibration amplitude passing through the critical speeds and suppress the nonlinear characteristics of the system, such as bistable state responses and jump phenomena. The results also reveal a dependence of the damping performance on the choice of the FLSFD design parameters.

Dynamic response and sensitivity analysis for mechanical systems with clearance joints and parameter uncertainties using Chebyshev polynomials method

A mechanical system with clearance joints exhibits non-linear characteristics. Even a small variation of one parameter may lead to a drastic change of the overall system response. The coupling of clearance and uncertainty would therefore significantly influence the system performance. In this paper, an analysis method for dynamic response and parameter sensitivity of mechanical systems is proposed with the consideration of both clearance joints and uncertainties. In this method, elements of a revolute clearance joint are modeled as colliding bodies, where continuous contact force model and modified friction force model are employed to describe the impact-contact behavior. The clearance joint model is then coupled with the system motion equations to obtain the system dynamic response. Furthermore, by introducing the multi-dimensional Chebyshev polynomials approach, dependence of the system dynamic response on its parameters can be established. The bounds of the dynamic response could be obtained by using the interval operations, and the parameter sensitivity is interpreted as the rate of the change of the system dynamic response with respect to the parameter variation, revealing the contribution of each parameter on the dynamic performance. Finally, dynamics of a crank-slider mechanism with clearance and uncertainties is investigated. Results show that a larger clearance would cause severe oscillation in the response bounds and larger expansion of the interval length, and clearance effect restraint zones can be obtained through the sensitivity analysis, where the expansion of the response interval length could be reduced.

An inclined FGM beam under a moving mass considering Coriolis and centrifugal accelerations

In this paper, the dynamic behaviour of an inclined functionally graded material (FGM) beam with different boundary conditions under a moving mass is investigated based on the first-order shear deformation theory (FSDT). The material properties vary continuously along the beam thickness based on the power-law distribution. The system of motion equations is derived by using Hamilton\'s principle. The finite element method (FEM) is adopted to develop a general solution procedure. The moving mass is considered on the top surface of the beam instead of supposing it on the mid-plane. In order to consider the Coriolis, centrifugal accelerations and the friction force, the contact force method is used. Moreover, the effects of boundary conditions, the moving mass velocity and various material distributions are studied. For verification of the present results, a comparative fundamental frequency analysis of an FGM beam is conducted and the dynamic transverse displacements of the homogeneous and FGM beams traversed by a moving mass are compared with those in the existing literature. There is a good accord in all compared cases. In this study for the first time in dynamic analysis of the inclined FGM beams, the Coriolis and centrifugal accelerations of the moving mass are taken into account, and it is observed that these accelerations can be ignored for the low-speeds of the moving mass. The new provided results for dynamics of the inclined FGM beams traversed by a moving mass can be significant for the scientific and engineering community in the area of FGM structures.

Dynamic Response of a Casting Crane Rigid‐Flexible Coupling System to High Temperature

To determine the influence of temperature on the mechanical properties of crane metal structures, three Q355 alloy steel samples were processed and their elastic moduli were tested at different temperatures using a metal tension test bed. The constitutive equation for the elastic modulus of Q355 alloy steel at different temperatures was predicted using test data and a neural network algorithm. Based on crane structural characteristics and the principle of system dynamics, a coupling vibration model was established that included the crane flexible girder, cabin, trolley, crane, and temperature. System motion equations were established according to the Lagrange equation, and the approximate solution of nonlinear system vibration was solved by the direct integration method (the Newmark method). The dynamic characteristics of the main beam and cabin were analyzed at different temperatures, as well as safety during service. The results show that, with increasing temperature, the maximum midspan displacement of the main beam increases gradually, by 14.3%, 21.4%, and 57.1% at temperatures of 300°C, 400°C, and 600°C, respectively. The cabin vibration displacement increases with temperature, by up to 32.5% at 600°C, but the influence of temperature on cabin vibration acceleration is not obvious. It was concluded that the influence of temperature on the dynamic characteristics of the main beam must be considered during the design stage of cranes. The proposed model and analysis method provide a theoretical basis for the design of casting cranes according to temperature.

Optimal control of propellant consumption during insertion of rocket into a circle orbit of the Earth

The problem of launching a rocket into the Earth's orbit has already been solved using the regularization method in previous studies. But the regularization method remains relevant for application to solving integral equations of the first kind, which determine the components of speed and acceleration. The problem of optimal control of propellant consumption during the insertion of a rocket into a circle orbit of the Earth is solved using regularized solutions of integral equations of the first kind which are solutions of corresponding Euler equations on discrete-time net. The influence of the regularization parameter and some additional parameters on precision of discredited problem is investigated. Calculations are carried out for existing chemical rocket engine and promising plasmic one. Considered algorithm is summed up easily to problem of suborbital flights by setting desired coordinate system and modifying motion equations. Conclusions were drawn about the required speed for the lowest fuel consumption, as well as about the problem for a single-stage rocket. Thus, the development of a plasma rocket engine with an exhaust velocity is more than ten times higher than that of a chemical one.

The traffic performance of the wheeled carrying liquid materials study

A feature of the wheeled vehicles capable of carrying liquid materials is the presence of the external disturbances in the form of interaction with the roadbed. This applies to peculiar transport vehicles, carrying liquid materials considered as non-Newtonian fluid. The internal disturbances in the form of the liquid material free surface interaction is the presence of the external disturbances in the form of interaction with the roadbed. Internal disturbances in the form of the wheeled concrete transport machine working vessel fluid free surface interaction should also be taken into account. The article describes the first part of the task - the interaction of the system with the roadbed. The simplest form of the system motion equations with rolling is in the accompanying coordinate system when written in the form of equations in quasi-coordinates. Based on the apparatus of nonholonomic mechanics, M.V. Keldysh built the theory of the rolling wheels with an elastic tire. Under certain assumptions, the above-mentioned theory in the article is extended to the case of curvilinear motion. These equations are represented as a system of first-order equations convenient for numerical solution.

Dynamics of an Inhomogeneous Ball on a Vibrating Base with Two-Component Viscous Friction

The problem of forced oscillations of a heavy inhomogeneous ball placed on a horizontal base is considered. It is assumed that the base moves in a horizontal direction according to harmonic law and at the point of contact between the surface of body and the base, the force and moment of viscous friction act. The equations of motion for a mechanical system are derived. Their solution is obtained by the averaging method. The dependences of the oscillation amplitudes for the excitation frequencies close to the eigenfrequency of the oscillations of the inhomogeneous ball on an absolutely smooth base are constructed. Both plane and spatial modes of oscillations have been found.

Singularity-Free Trajectory Planning of Free-Floating Multiarm Space Robots for Keeping the Base Inertially Stabilized

In a multiarm space robotic system, one or more manipulators can be used to stabilize the base through counteracting the disturbance caused by other manipulators performing on-orbital tasks. However, singularities are inevitably present in the traditional methods based on differential kinematics solutions. In this paper, we propose a singularity-free trajectory planning method to simultaneously keep the attitude and centroid position of the base stabilized in inertial space; the balance arms are also designed. First, we derive the coupling motion equations of a free-floating multiarm space robotic system. Then, the singularity problems are theoretically analyzed, and the theoretical basis for singularity-free trajectory planning is established. Second, we decompose the six degrees of freedom pose (attitude and position) stabilization problem into two 3DOF subproblems related to attitude and position balancing. We then design two robotic arms: 1) a position balance arm and 2) an attitude balance arm, to maintain the base centroid position and attitude, respectively. Third, we plan the coordinated trajectories of the two balance arms according to holonomic and nonholonomic constraints. As long as the desired motion is not beyond its balance ability, the reasonable joint variables can always be determined without encountering a singularity problem. Finally, the proposed methods are verified using simulations of typical on-orbital missions, including joint trajectory tracking and target capturing.

Well posed formulations of holonomic mechanical system dynamics and design sensitivity analysis

Manifold theoretic ordinary differential equations of motion for holonomic mechanical systems that depend on problem data, or design variables, are shown to be well posed; i.e., they have a unique solution that depends continuously on problem data. It is proved that these differential equations are equivalent to the d’Alembert variational formulation and the index 3 Lagrange multiplier formulation of differential-algebraic equations of motion, which are also shown to be well posed. These results provide a foundation for dynamic system design sensitivity analysis, which requires differentiability of solutions of the equations of motion with respect to design variables.

Investigation of Turbulence Effects on the Nonlinear Vibration of a Rigid Rotor Supported by Finite Length 2-Lobe and Circular Bearings

AbstractThis study investigated the effect of turbulence on the nonlinear vibration of a symmetrical rigid rotor supported by two identical journal bearings. The bearings consisted of various length to diameter (L/D) ratio circular and 2-lobe bearings with differing pad preloads. Two turbulent (Ng–Pan–Elrod and Constantinescu model) and one laminar Reynolds equations were selected for comparison, and they were solved using a finite difference method to obtain nonlinear bearing forces. The nonlinear equations of motion for the rotor-bearing system were solved using a shooting method and arclength continuation to obtain limit cycles for each bearing configuration. Floquet multiplier analysis was then utilized to identify the stability of the obtained limit cycles. For the cases of the circular and 2-lobe bearing without pad preload, the turbulent Reynolds equations yielded a lower onset speed of instability and L/D ratio at which the bifurcation type changed from supercritical to subcritical than the laminar Reynolds equation. However, at higher pad preloads (preloads of 0.25 or 0.5), the turbulence effects increased the onset speed of instability, especially for L/D ratios > 0.7, and only supercritical bifurcation was observed. For all bearing configurations, the ratio of the limit cycle whirl frequency to shaft rotational speed for both turbulence bearing models was higher than that of the laminar bearing model, and the Ng–Pan–Elrod turbulence model always generated lower onset speed of instability than the Constantinescu model.

Motion Equations for the Ball and Beam and the Ball and Arc Systems

Abstract Presently, most derivations of the equations of motion for the ball and beam and ball and arc systems model the ball as a point mass. Other derivations apply assumptions related to the ball that lack physical justification. To understand fully the impact the ball has on the equations of motion and controller design, equations of motion are needed that stem from a derivation invoking few assumptions. At that point, simplifying assumptions applied in a consistent manner are possible. This paper derives the equations of motion for each mechanical system using fewer assumptions than what appears in the literature. The development then applies the assumptions needed to convert the derived dynamics to the well-used equations of motion seen in the literature. The presentation shows that some ball and beam models appearing in the literature do not stem from consistent assumptions or correct kinematics. Incorrect kinematics are also present in some of the ball and arc dynamic models. This paper also introduces dimensionless quantities to the dynamic equations of both systems rendering them in dimensionless form. Such formulations allow for comparisons between different systems through the size of the dimensionless quantities. The dimensionless systems are the means used to demonstrate that the equations of motion of each system are the same as the radius of the arc grows without bound. Finally, the paper shows that comparisons between different ball and beam models using these dimensionless numbers are now possible.

Multidimensional Vibration Suppression Method with Piezoelectric Control for Wind Tunnel Models.

In wind tunnel tests, the low-frequency and large-amplitude vibration of the cantilever sting result in poor test data in pitch plane and yaw plane, more seriously, even threatens the safety of wind tunnel tests. To ensure the test data quality, a multidimensional vibration suppression method is proposed to withstand the vibration from any direction, which is based on a system with stackable piezoelectric actuators and velocity feedback employing accelerometers. Firstly, the motion equation of the cantilever sting system is obtained by Hamilton’s principle with the assumed mode method. Then, the multidimensional active control mechanism is qualitatively analyzed and a negative velocity feedback control algorithm combined with a root mean square (RMS) evaluation method is introduced to realize active mass and active damping effect, meanwhile, a weight modification method is performed to determine the sequence number of the stacked piezoelectric actuators and the weight of control voltages in real time. Finally, a multidimensional vibration suppression system was established and verification experiments were carried out in lab and a transonic wind tunnel. The results of lab experiments indicate that the damping ratio of the system is improved more than 4.3 times and the spectrum analyses show reductions of more than 23 dB. In addition, wind tunnel test results have shown that for the working conditions (α = −4~10° with γ = 0° or α = −4~10° with γ = 45°) respectively at 0.6 Ma and 0.7 Ma, the remainder vibration is less than 1.53 g, which proves that the multidimensional vibration suppression method has the ability to resist vibration from any direction to ensure the smooth process of wind tunnel tests.

Research on the Vibration Control Effect of High-Rise isolation system combined with damping device under Strong ground motions

Considering large deformation of the upper structure for base isolation structure and easily overturned whole structure under strong ground motion, the characteristics of earthquake control and seismic response of a high-rise base isolation structure with viscous dampers are studied in this paper. At first, the motion equation of high-rise base isolation structure and high-rise combined control system is established based on the calculation model of multi-degree of freedom system with lumped masses of isolation structure; Then, the state space method and variable order numerical differential algorithm are used to calculate the seismic response of the structure; Finally, the ordinary ground motion records and near-source pulse-like ground motion records are selected as input, and the seismic response of high-rise base isolation structure and high-rise combined control system are calculated, furthermore, its characteristics of seismic response and control effect are compared. The results show that the high-rise base isolation structure with viscous dampers can reduce the structural displacement response significantly, and the displacement demand of isolation bearing can be reduced dramatically too. it is an effective way to develop the new type bearing with low yield stiffness, large deformation and a high tensile strength to solve these questions of base isolation technology in high-rise structure applications.

Dynamic modeling of a gear transmission system containing damping particles using coupled multi-body dynamics and discrete element method

The reduction in vibration in gear transmission systems is an engineering task. Particle damping technology attenuates vibration by means of friction and inelastic collisions between damping particles. This study proposes a dynamic model for a spur gear transmission system that contains damping particles inside the holes on gear bodies, using two-way coupling with multi-body dynamics and discrete element method. The equations of motion for the multi-body system are derived using Euler–Lagrange formalism. The discrete element method with a soft contact approach is used to model the dynamic behavior of damping particles. Hertzian contact theory and Coulomb friction theory are applied to modeling contacts. The effects of particle radius, coefficient of friction and restitution coefficient on the dynamic characteristics are explored. Numerical results show that vibration in the transmission is appreciably attenuated by the particle damping mechanism and that the contact friction, and not contact damping, dominates the energy dissipation of the multi-body system in such a centrifugal scenario.