Elastic Dynamic Model Research Articles

Chiral Standing Spin Waves and Unidirectional Waves of Odd Elastic Cylindrical Shells

Abstract Rotating waves can be observed in structures with periodic conditions, such as cylinders and spheres. Compared with traveling waves and standing waves, rotating waves have received less attention. In this paper, an odd elastic dynamic model of the cylindrical shells is established, and the dispersion relation, traveling waves, and standing waves are investigated. The non-Hermitian rotating waves and single-handedness chiral standing spin waves are reported, which are novel dynamic phenomenon caused by odd elastic effects. Waves generally cannot propagate in passive materials with vanishingly small elastic modulus. However, a unidirectional wave with the highest cut off frequency can occur in an odd elastic cylindrical shell with vanishingly small elastic modulus. For incompletely restrained end displacements, the odd elastic cylindrical shell can also generate a hybrid mode combining standing spin waves with unidirectional waves.

Effects of inter-cell connections on the multi-stable dynamics of dual-cell stacked Miura-origami structures

Origami has become an important source for the development of mechanical metamaterials because folding can often lead to enhanced or unconventional mechanical properties. Metamaterials are always made of repeating cells by stacking and tessellating. The inter-cell connection thus plays a key role in determining the overall mechanics and dynamics of the metamaterial, but current studies mainly focus on the effects of inter-cell connection on statics. To overcome the shortcomings and advance the state of the art, this research employs the dual-cell stacked Miura-origami (SMO) structure as a platform to comprehensively dissect the effects of inter-cell connection forms on the dynamics. The two constituent SMO cells, by design, can be bi-stable such that the dual-cell structure possesses four stable states that are fundamentally different in configuration. Two inter-cell connection forms, a rod connection and a crease connection, are examined both theoretically and experimentally in this research. Assuming rigid-foldability of the constituent SMO structures and by equivalently quantifying the inter-cell connection constraint via additional elastic potential energy, a refined and processible dynamic model of the multi-stable dual-cell SMO structure is developed. Comprehensive numerical calculations reveal, on the one hand, the rich and complex dynamics of the dual-cell structure and, on the other hand, uncover the significant differences caused by the inter-cell connection forms. With relatively weak inter-cell constraint (e.g., the rod connection), dynamic configuration switches would occur at low excitation amplitudes, and the dynamic response types of the two cells can be fundamentally different; however, with relatively strong inter-cell constraint (e.g., the crease connection), more energy input is required to switch the configuration, and the dynamic response types of the two cells remain consistent. Such findings are qualitatively verified via dynamic experiments on the dual-cell SMO structures. We also extend our research to global dynamics by analyzing the basins of attraction and basin stability, which demonstrates from a probabilistic viewpoint that stronger inter-cell constraint would converge the dynamics to synchronous response types. The results of this research would offer a solid foundation for regulating the multi-stable dynamics of multi-cell origami structures/metamaterials and advancing their dynamic applications.

Verification of a Body Freedom Flutter Numerical Simulation Method Based on Main Influence Parameters

The body freedom flutter characteristics of an airfoil and a fly wing aircraft model were calculated based on a CFD method for the Navier–Stokes equations. Firstly, a rigid elastic coupling dynamic model of a two-dimensional airfoil with a free–free boundary condition was derived in an inertial frame and decoupled by rigid body mode and elastic mode. In the fluid–solid coupling method, the rigid body was trimmed by subtracting the generalized steady aerodynamic force from the structural dynamic equation. The flutter characteristics were predicted by the variable stiffness method at a fixed Mach number and flight altitude. Finally, validation of the predicted body freedom flutter characteristics was performed through a comparison of theoretical solutions based on a Theodorsen unsteady aerodynamic model for airfoil and experimental results for a fly wing aircraft model. The mechanism of the influence of the bending mode stiffness and the position of the center of gravity on the body freedom flutter characteristics were briefly analyzed.

Prediction and validation of dynamic characteristics of a valve train system with flexible components and gyroscopic effect

Flexibility and inner stress of transmission components are main considerations in the development of high-precision and lightweight engines. Gyroscopic effect of rotary shaft and rotor should also be considered in high-speed engines. Therefore, an elastic dynamic model of a valve train system is proposed. The model includes multiaxial flexibilities of camshaft and linkages, gyroscopic effect of the camshaft and cam rotor, centrifugal force of cams and friction force and moment of the cam–tappet pair. Then, the natural frequency, mode shape, pushrod stress and housing vibration are predicted using the proposed model. Results show that the gyroscopic effect of the rotors and camshaft apparently affects the natural frequency of the system by changing the camshaft stiffness. The proposed model can effectively predict the dynamic stress, as proven by a stress experiment. In addition, idling and maximum working speeds are recommended using the predicted model and verified by a vibration experiment. Thus, the proposed model can be an important reference in engine adjustment, improvement of component durability and determination of idling and maximum speeds.

Tribo-dynamics model of a spur gear pair with gyroscopic effect and flexible shaft

This study proposes a tribo-dynamic model of a spur gear by coupling an elastic dynamic model and an isothermal elastohydrodynamic lubrication (EHL) model. In this model, the gear shafts are modelled as flexible bodies by the resultant elements of Timoshenko beam and bar elements, and the gyroscopic effect of gear rotors and shafts is also included. The transient and non-Newtonian effects are considered in the EHL model to predict tooth friction forces and moments. Then, an example is provided to analyze the interaction between gear tribology and dynamic behavior. Results indicate that the fluctuations of mesh force and surface tangential velocity caused by gear vibration considerably affect the pressure and thickness of the lubricant film. The gyroscopic effect exerts a significant influence on the dynamic and tribology properties of the gear drive at high speed. The proposed model has important guidance for the design of lubrication systems and dynamic properties in gear drives.

Study on the effect of piston skirt profile on the vibration behavior of non-road high pressure common rail diesel engine

The piston secondary motion is the most predominant excitation source for the mechanical vibration of diesel engine, which is directly affected by the piston profile. The effects of piston skirt profile on the dynamic behavior of piston and the vibration characteristics of engine block is studied synthetically by the method of a combination of experimental test and numerical simulation. The temperature distribution of the piston top surface under rated working condition is tested by a hardness plug, which is used as the boundary condition to carry out the thermodynamic simulation of diesel engine working process, to obtain the hot profile of the piston. Then, the dynamics model of piston assemblies are formally established, and the skirt profiles of different bump positions of the piston skirt are optimally designed. Based on the above analysis, a comprehensive study on the influence of bump position on the piston dynamic behavior is carried out. Then, applying the modal testing method for the engine block and crankshaft to establish the elastic multi-body dynamics model of diesel engine and load the piston slapping force at different bump positions, the vibration response of the surface structure under the action of different piston slapping forces is also studied. The results show that the mechanical vibration of engine block will become more severe with the hiking up of bump position at the piston skirt. Also, the vibration behavior of the engine block in the middle and high frequency section is very sensitive to the bump position of the piston skirt, especially in the frequency range of 1600 Hz–3150 Hz. In addition, vibration noise performance can be significantly improved by adjusting the bump position of piston skirt downwards. Particularly, the vibration noise performance can be best achieved when the bump position of piston skirt is 5 mm.

A dynamic parameter identification method of industrial robots considering joint elasticity

Considering the joint elasticity, a novel dynamic parameter identification method is proposed for general industrial robots only with motor encoders. Firstly, the unknown parameters of the elastic joint dynamic model are analyzed and divided into two types. The first type is the motion-independent parameter only including the joint stiffness, which can be identified by the static force/torque-deformation experiments without the dynamic model. The second type is the motion-dependent parameter composed of the rest of the parameters, which needs the dynamic excitation experiments. Therefore, these two types of parameters can be identified separately. Meanwhile, it is found that the rotor inertia parameters can be obtained from the manufacturer, which reduces the identification difficulty of other parameters. After obtaining the rotor inertia and joint stiffness, an approximate processing algorithm is proposed considering the motor friction to establish the linear identification model of other parameters. Hence, the least squares can be employed to identify the parameters, and the independence of the inertia and joint viscous friction parameters are not affected. Meanwhile, the exciting trajectories can be optimized throughout the robot workspace, which reduces the effect of measurement noise on identification accuracy. With the proposed separated identification strategy and approximate processing algorithm, the dynamic parameters can be obtained precisely without double encoders on each joint. Finally, a series of simulations are conducted to evaluate the good performance of the proposed method.

Elastic dynamics modelling and analysis for a valve train including oil film stiffness and dry contact stiffness

An elastic dynamics model for a valve train including combined stiffness of oil film and dry contact between a cam and a tappet, is proposed. Multi-directional elastic deformations of the valve train are solved by discretizing slender components into Rayleigh beam and bar elements. The oil film stiffness is determined by non-Newtonian elastohydrodynamic lubrication in the line contact, whereas the dry contact stiffness is achieved by using finite element contact analysis. A numerical solution for the dynamic model is developed and verified by conducting a dynamic stress experiment. The influences of rotation speed on the oil film stiffness and contact force between the cam and the tappet are discussed, and the combined effects of rotation speed and oil film stiffness on a dynamic transmission error of the valve train are analyzed. Results show that the oil film stiffness decreases when the cam rotation speed increases within a certain range, whereas the sharp increased contact force may enlarge the oil film stiffness when the speed exceeds a critical value. The oil film stiffness variation slight influences the contact force, but such variation significantly affects the dynamic transmission error.

SIMULATION OF ANTHROPOMORPHIC ROBOTS WITH ELASTIC DRIVES BY INTRODUCING VIRTUAL LINKS

Anthropomorphic walking robots are among the most promising robot types, due to the possibility to introduce them into the urbane environment through the use of the existing infrastructure. Control systems developed for such robots require access to the exact mathematical models of these robots, taking into account the properties of actuators, gears and sensors. In this paper, we consider approaches to describing the model of a bipedal walking robot with elastic drives. The robot is a three-link mechanism that moves in the sagittal plane and performs verticalization (sit-to-stand transfer). Two variants of describing the dynamics of the robot are shown. In the first variant, the number of equations describing the movement of the robot is doubled due to the introduction of elastic drives, in comparison with the case when there are no elastic elements present. In the second variant, there is robot model and the elastic element dynamics model, and bothare described separately. The advantages of this method include the fact that it allows us to preserve the structure and properties of the equations of motion of the mechanism used in constructing control methods in cases when the elastic properties of the gears are not taken into account, and it also allows to conserve the structure of the generalized inertia matrix. The simulation results are presented in two described previously variants, their comparison is made. It is established that both mathematical models behave almost identically, with the most significant differences manifested in the formation of control actions generated by the regulator.

UNSTEADY OPERATING MODES OF THE MAIN DRIVE UNIT OF A MACHINE-TOOLS FOR SOLID RAILWAY WHEELS PROCESSING

Purpose. The paper deals with determining the conditions of the dynamic overload formation for a drive unit in the wheel-processing machine in various unsteady operating modes and suggests ways to reduce it. Methodology. The problem was solved by creating a dynamic computational multi-mass chain model followed by modeling of its loading by external force options, the values of which were obtained by analytical and experimental methods. The solution to equations of motion with determining the load of the drive unit in the transient acceleration mode and in the tie-in one was carried out by means of MathCad. Findings. The design scheme for the drive unit of the rotational movement of the machine faceplate is represented by a linear three-mass row system with two elastic connections and with three absolute degrees of freedom. The analytical determination of the cutting force moments takes into account changes in the kinematic parameters of the cutting regime when machining the different sections of the profile with cup-tip tool. The worst case of load application is accepted with simultaneous insertion of tools. The equivalent torque of the cutting forces is defined as the sum of the moments from each of the tools at multi-cut machining. The moments of elasticity of bonds in the steady state of the machine drive unit are determined. Dynamic loads with coefficients of dynamism in the bonds Kd12 = 1.13 and Kd23 = 1.04 are insignificant in the tie-in mode and rapidly decrease their value. Absolute values of the twisting elastic moment in the C23 bond in the start-up mode are insignificant and much less than the values of the steady load. Using the method of calculation and experimental research, a significant dynamic overload of the machine drive unit in the mode of rheostat start-up by a directcurrent motor was established. The reactive elastic moment with the periodic change is 2.7 times higher than the nominal moment of steady processing regime. Originality. Authors created the elastic dynamic model of the machine drive unit for the processing of railway wheels. In accordance with the model, the equations of mass motion and bond loading are compiled and solved. The theoretical conditions for reducing the drive unit loading are determined. Practical value. Based on the simulation of the transient drive start-up process, significant overloads of its elements have been determined. They are several times higher than the response of elastic bonds during cutting. To prevent this, it is recommended to change the mode of multistage starting of the engine with a decrease of the starting current by at least two times up to a value of not more than 1 200 A.

Bifurcation and chaotic behaviors of 4-UPS-RPS high-speed parallel mechanism

In order to grasp the nonlinear characteristics of high-speed spatial parallel mechanism, the bifurcation and chaotic behaviors of 4-UPS-RPS mechanism are analyzed. Firstly, the nonlinear elastic dynamic model of the mechanism is established by using the Lagrange equation and the finite element method. Then the effects of parameters including driving angular velocity, the radius of motion trajectory, the material of driving limbs, the diameter of driving limbs, and the mass of moving platform, on the bifurcation and chaotic behaviors of high-speed spatial parallel mechanism are studied. The results show that the above parameters all have a certain influence on nonlinear characteristics of the 4-UPS-RPS high-speed spatial parallel mechanism. The research can provide important theoretical basis for the further research on the non-linear dynamics of spatial parallel mechanism.

Research on Elastic Dynamic Model of Trolley System for Anti-sway Control System of Bridge Crane

For bridge cranes, the movement of the trolley will cause the vibration of the cart and the vertical deformation of the cranes, including the elastic deformation of the main beam, the elastic deformation of the cart running mechanism and the supporting legs. This paper analyzes the influence of the crane vertical deformation on anti-sway control, presents a novel three-mass three-degree-of-freedom elastic dynamic model of the trolley system when the cart is stationary and only the trolley is running. The validity of the model is verified by Matlab.

Dynamic response analysis and chaos identification of 4-UPS-UPU flexible spatial parallel mechanism

In order to realize the dynamic response analysis and the chaos identification of flexible spatial parallel mechanism, the nonlinear elastic dynamics model of 4-UPS-UPU flexible parallel mechanism is established under the ideal situation, and the dynamic response, phase diagrams, Poincare map and largest Lyapunov exponent of the spatial parallel mechanism are investigated. Based on the finite element method, the driving limbs of spatial parallel mechanism are divided into elements. The kinetic energy equation and potential energy equation of units are built. Then the nonlinear elastic dynamics model of 4-UPS-UPU parallel mechanism is acquired by Lagrange equation. The dynamic response of kinematic error for 4-UPS-UPU flexible parallel mechanism is analyzed. In addition, the chaos phenomenon contained in the mechanism is identified by phase diagrams, Poincare map and largest Lyapunov exponent, respectively. Subsequently, the relationship between the basic parameters of parallel mechanism and largest Lyapunov exponent is discussed. The results indicate that there exists chaotic phenomena in the 4-UPS-UPU flexible parallel mechanism, and the basic parameters, including the material of driving limbs, diameter of driving limbs, mass of moving platform and the distribution radius of hinges of moving platform all have great effect on chaotic motion of 4-UPS-UPU flexible parallel mechanism. The researches can provide important theoretical for the further nonlinear dynamics behaviors research and optimal design of 4-UPS-UPU flexible spatial parallel mechanism.

Elastic dynamic research of high-speed multi-link precision press considering structural stiffness of rotation joints

An elastic dynamic model of high-speed multi-link precision press considering structural stiffness of rotation joints was established by the finite element method. In the finite element model, rotation joint was established by four bar elements with equivalent stiffness, and connected link was established by beam element. Then, the elastic dynamics equation of the system was established, and modal superposition method was used to solve the dynamic response. Compared with the traditional elastic dynamic model with perfect constraint of the rotation joints, the elastic dynamic response value of the improved model is larger. To validate the presented new method of elastic dynamics analysis with stiffness of rotation joints, a related test of slider Bottom dead center (BDC) position in different speed was designed. The test shows that the model with stiffness of rotation joints is more reasonable. So it provides a reasonable theory and method for dynamic characteristics research of sucha multi-link machine.

Dynamic precision analysis and experimental verification of high-speed precision punch press

In order to investigate the impact factors and their affection on high-speed precision multilink punch press (MPP), the dynamic model with different joint clearance was established, and the influence of different clearance and speed on the dynamic positional repeatability of bottom dead center (BDC) was analyzed. The elastic dynamic model of high-speed MPP was established, the affection of the elastic deformation and elastic wave on the positional repeatability of the BDC were presented by using modal superposition method to solve dynamic equation. Meanwhile, experiments on the dynamic repeatability of the BDC of the punch during working were completed. At last, the comparison of the experimental results with the analyzed results was given, and based on which, it can be concluded that the clearance, elastic wave and deformation are the key factors of the dynamic repeatability precision of the BDC.

Comprehensive analysis of the position error and vibration characteristics of Delta robot

Taking Delta robot as research object, its characteristics of position error and vibration are analyzed synthetically. Using the position characteristic of driven arms, based on the method of geometry space vector, establish the mechanism error model; Taking use the principle of mathematical statistics and space vector as the basic, deriving the joint clearance error model; Based on finite element theory, on the basic of elastic dynamic model, establishing the flexible error model. Then, the kinematics and dynamics of branched chain with joint clearance generalized collision force are studied. Based on the definition of rod virtual length, combining the generalized collision force with the elastic dynamics model which includes mechanism error, joint clearance error and flexible error, establishing the comprehensive elastic dynamic model. Finally, by use of simulation software, numerical calculation and experiment, verified the correctness of comprehensive elastic dynamic model, and the characteristics of position error and vibration are analyzed.

A corneal elastic dynamic model derived from Scheimpflug imaging technology.

To simultaneously extract the corneal Young's modulus and the damping ratio from Scheimpflug imaging data. A spherical diaphragm model can better represent the geometry and physics of an eyeball than the popular mass-spring-damper model. This research derived the dynamic model of a water-filled spherical diaphragm based on the hydrodynamics and wave propagation theories. By applying modal analysis on the model, one can decouple the cornea vibration into individual modes and reconstruct the air puff vibration from the decoupled responses. By matching this response with the Scheimpflug imaging data from the Corvis(®) ST, it was then possible to extract multiple physiological properties as desired. The dynamic modal analysis was employed to extract the corneal physiological properties of 25 Taiwanese normal subjects. Specifically, the corneal Young's moduli and damping ratios were estimated. In fact the model is dependent on the physiological parameters such as cornea thickness, densities, and intraocular pressure. It is thus also possible to extract these parameters through multi-goal minimisation processes. The spherical diaphragm model was able to better describe the dynamic response of the eyeball. The model analysis also provides additional corneal physiological properties that were not available through other means.

Study on Elastic Dynamic Model for the Clamping Mechanism of High-Speed Precision Injection Molding Machine

This work centered on the double-toggle clamping mechanism with diagonal-five points for the high-speed precise plastic injection machine. Based on Lagrange equations, the differential equations of motion for the beam elements are established, in a rotating coordinate system and an absolute coordinate system, respectively. 43 generalized coordinates and a model matrix for the mechanism are created and some coordinate matrices are derived. By coupling the coordinate transformation and matrix manipulation, a high nonlinear and strong time-variant elastic dynamic model is obtained. Based on the dynamic model, a Kineto-Elasto Dynamics (KED) analysis and a Kineto-Elasto Static (KES) analysis are carried out, respectively. By comparing and analyzing the simulation results of KED and KES, the regularity of elastic vibration of the clamping mechanism in high-speed clamping process has been revealed.

Theoretical and Experimental Aspects Concerning Elastic Behavior in the Grinding Technological System

In this paper will be treated the problem of strains and stresses within the connections and elements from the grinding technological system. These system deformations can lead to positive or negative effects. Main negative effects will be occurrence of processing errors on the part profile, but also the excessive tension from system elements. Cause of processing errors and the low safety due of the system rigidity must be sought in its joints between parts, stiffness of moving pieces, low number and reduced rigidity of parts used to assembling (bolts, cotter joints, pins). Elastic properties of the assembly are modeled by the inserted connections with specific characteristics. The elastic deformation of the work piece must be taken into account, besides the elastic behavior of the device which appears in manufacturing process. A very important aspect in the study of elastic behavior of the whole system is how the abrasive tool interacts with the work piece, namely the displacement of contact zone. Contact stresses lead to a deformation of the grinding wheel and the work piece, and as a result the work piece loses from precision of tooth profile. The rigidity of system can be characterized by two aspects, one static and one dynamic and we will describe both aspects using CAE simulations and classical mathematical models. It will be presented some results of numerical simulation of the stiffness of gear grinding system using finite element method (FEM). The elastic dynamic model was design using the forces of inertia and gravity that occurs during grinding process. Also by permanent pursuing of the technological forces variation during the grinding process will be sought the dynamic deformation of the system. The study its concentrated around the low stiffness of components inside the system, in order to find errors that may affect the precision on the horizontal, vertical and rotational direction of technological system elements. Since through this analysis it tried to find how the forces influence on the ETS stiffness, it will present numerical values of the system displacements and stress distribution. Knowing the direction, sense and numerical values of these errors can be made interpretation of the results, namely the removal of these consequences.

Vibration Control of a High-Speed Parallel Robot Using Discrete Variable Structure Control Methodology

Abstract A discrete variable structure control methodology is presented to suppress vibration of high-speed flexible parallel robot. The elastic dynamic model of the flexible robot is built by using the finite element method and experimental modal testing. Considering uncertain external disturbances and measurement noise, and uncertain parameters of the system, we can utilize invariance of the sliding mode of variable structure control in order to eliminate these adverse influences. The discrete variable structure control strategy is employed to construct the vibration robust controller which enables the flexible robot to be zero state so that the system states are away from the equilibrium one due to external disturbances and uncertain parameters. The discrete Kalman filter is constructed to as state estimator because the state variables cannot be directly measured. The first four natural frequencies and damping ratio are obtained by using experimental modal testing, and the correctness of theoretical model is verified. The experimental vibration control system is constructed and experimental validation of the controller is carried out using dSPACE real-time control system and MATLAB/Simulink tool. The experimental results showed that the proposed controller can effectively alleviate the vibration response of the high-speed flexible robot.