Assumed Mode Method Research Articles

Dynamic modeling and performance analysis of a lower-mobility parallel robot with a rotatable platform.

Recently, applications of high-speed, lightweight parallel robots have been gaining increasing interest. Studies have shown that their elastic that their elastic deformation during operation often affects the robot's dynamic performance. In this paper, we designed and studied a 3 DOF parallel robot with a rotatable working platform. We developed a rigid-flexible coupled dynamics model consisting of a fully flexible rod and a rigid platform by combining the Assumed Mode Method with the Augmented Lagrange Method. The driving moments under three different modes were used as feedforward in the model's numerical simulation and analysis. We conducted a comparative analysis demonstrating that the flexible rod's elastic deformation under a redundant drive is significantly smaller than that of a non-redundant one, leading to a better suppression effect on vibration. The system's dynamic performance under the redundant drive was significantly superior compared to that of the non-redundant one. Additionally, the motion accuracy was higher and the driving mode b was better than that of the driving mode c. Finally, the proposed dynamics model's correctness was verified by modeling it in Adams.

Flutter Analysis of a 3D Box-Wing Aircraft Configuration

The aim of this paper is to present a flutter analysis of a 3D Box-Wing Aircraft (BWA) configuration. The box wing structure is considered as consisting of two wings (front and rear wings) connected with a winglet. Plunge and pitch motions are considered for each wing and the winglet is modeled by a longitudinal spring. In order to exert the effect of the wing-joint interactions (bending and torsion coupling), two ends of the spring are located on the gravity centers of the wings tip sections. Wagner unsteady model is used to simulate the aerodynamic force and moment on the wing. The governing equations are extracted via Hamilton’s variational principle. To transform the resulting partial integro-differential governing equations into a set of ordinary differential equations, the assumed modes method is utilized. In order to confirm the aerodynamic model, the flutter results of a clean wing are compared and validated with the previously published results. Also, for the validation, the 3D box wing aircraft configuration flutter results are compared with MSC NASTRAN software and good agreement is observed. The effects of design parameters such as the winglet tension stiffness, the wing sweep and dihedral angles, and the aircraft altitude on the flutter velocity and frequency are investigated. The results reveal that physical and geometrical properties of the front and rear wings and also the winglet design have a significant influence on BWA aeroelastic stability boundary.

Optimum control of a flexible single link manipulator with Artificial Bee Colony Algorithm

This paper presents a stable control for a single link flexible manipulator. The flexible manipulator is considered as Euler–Bernoulli beam, and its dynamic model is developed based on assumed mode method. Position and vibration control are realised with a single controller as motor torque. The controller has three parameters that must be selected. In this paper, the main motivation is to find the suitable parameter values to generate optimum torque values in every sampling time. In order to achieve this goal, Artificial Bee Colony Algorithm is performed, and the controller parameters are optimised simultaneously in every sampling period. Simulations verify that the manipulator achieves the position control objective, and the vibration is suppressed simultaneously even with different payloads with the proposed optimisation method.

Dynamic Instability of Functionally Graded Carbon Nanotubes-Reinforced Composite Joined Conical-Cylindrical Shell

In this paper, dynamic instability of functionally graded carbon nanotubes (CNTs)-reinforced composite joined conical-cylindrical shell in supersonic flow is analyzed numerically. The higher-order shear deformation theory is applied to describe the stress–strain state of thin-walled structure. The assumed-mode method is used to derive the finite degrees-of-freedom dynamical system, which describes the structure motions. The structure motions are expanded by using the eigenmodes, which are obtained by the Rayleigh–Ritz method. The trial functions, which satisfy the continuity conditions at the cylindrical-cone junction, are used to obtain the eigenmodes. The properties of free vibrations of thin-walled structure are analyzed numerically. The dynamic instability of the joined conical-cylindrical shell in supersonic flow is analyzed using the characteristic exponents. As follows from the numerical study, the dynamic instability is arisen due to the Hopf bifurcation. The dependences of the supersonic flow critical pressure on the Mach number and the type of CNTs distribution are analyzed numerically.

Нелінійні коливання сандвіч-пластини зі стільниковим наповнювачем, виго-товленим методом 3d-друку

A three-layer sandwich plate with a FDM-printed honeycomb core made of polycarbonate is considered. The upper and lower faces of the sandwich are made of a carbon fiber-reinforced composite. To study the response of the sandwich plate, the honeycomb core is replaced with a homogeneous layer with appropriate mechanical properties. To verify the honeycomb core model, a finite-element simulation of the representative volume of the core was performed using the ANSYS software package. A modification of the high-order shear theory is used to describe the structure dynamics. The assumed-mode method is used to simulate nonlinear forced oscillations of the plate. The Rayleigh–Ritz method is used to calculate the eigenfrequencies and eigenmodes of the plate, in which the displacement of the plate points during nonlinear oscillations are expanded. This technique allows one to obtain a finite-degree-of-freedom nonlinear dynamic system, which describes the oscillations of the plate. The frequency response of the system is calculated using the continuation approach applied to a two-point boundary value problem for nonlinear ordinary differential equations and the Floquet multiplier method, which allows one to determine the stability and bifurcations of periodic solutions. The resonance behavior of the system is analyzed using its frequency response. The proposed technique is used to analyze the forced oscillations of a square three-layer plate clamped along the contour. The results of the analysis of the free oscillations of the plate are compared with those of ANSYS finite-element simulation, and the convergence of the results with increasing number of basis functions is analyzed. The comparison shows that the results are in close agreement. The analysis of the forced oscillations shows that the plate executes essentially nonlinear oscillations with two saddle-node bifurcations in the frequency response curve, in which the periodic motion stability of the system changes. The nonlinear oscillations of the plate near the first fundamental resonance are mostly monoharmonic. They may be calculated using the describing function method.

Dynamic characteristics of a simply supported elastic beam with three induction motors

In the paper, the dynamic characteristics of a three induction motor driven system on a simply supported elastic beam are studied. Firstly, a dynamic coupling model of the whole system is established according to the Hamilton principle and discretized by applying the assumed mode method. Next, the synchronization solutions containing the average rotation speeds and phase differences of three motors and their stability are calculated and analyzed by using the average perturbation method. As a result, the synchronization and stability conditions of self-synchronization of three motors are established. Finally, the feasibility of the average perturbation method is verified by comparing the theoretical results with the simulation results. The results indicate that the elastic support has a large influence on the motion states of three motors. There is the self-synchronization phenomenon of three motors on the elastic beam. There appear the Sommerfeld effect (rotational speed capture and jump phenomena) near the resonance of the beam and the synchronous states of three motors will rapidly change before and after the resonance of beam. The large changes of the motion states of three motors cause the impacts of the responses of the elastic beam structure. Additionally, the effects of the structure parameters such as unbalance mass, power frequency, beam size and vibration modes on the dynamic characteristics of the system are discussed. This paper can provide some references for the design of multi-motor systems on the elastic foundations.

Speed control method for dual-flexible manipulator with a telescopic arm considering bearing friction based on adaptive PI controller with DOB

A dual-flexible manipulator with a telescopic arm is affected by joint flexibility, load flexibility, and bearing friction, which will cause angular velocity fluctuation. Besides, the motion of the telescopic arm leads to dynamic parameters of the flexible load in the servo system time-varying. The variation of dynamic parameters increases fluctuations of angular velocity. Fluctuations in angular velocity will aggravate the vibration of the manipulator and affect the control accuracy of end-effectors. In this paper, the adaptive PI control strategy with the disturbance observer (DOB) suppresses the vibration of the manipulator by weakening the angular velocity fluctuations. Firstly, dynamics equations of the dual-flexible manipulator servo system considering the bearing friction are established by using the assumed mode method. Next, the low-pass filter in DOB is designed according to the robust stability theorem, so that the servo system can satisfy the stability under dynamic parameters time-varying and friction torque simultaneously. Then, the adaptive law and the over-load compensation control law are designed by Lyapunov’s stability theorem. Finally, numerical simulation analysis and control experiments of the manipulator are carried out. The simulation and experimental results show that: 1) The bearing friction torque can cause angular velocity fluctuations, but the DOB can effectively reduce the influence of the disturbance torque. 2) The adaptive control strategy can effectively reduce angular velocity errors under conditions that the telescopic arm is in different lengths. Therefore, the adaptive PI control strategy with the DOB can effectively suppress angular velocity fluctuations of the manipulator.

Component-Based Modeling and Simulation of Nonlinear Drill-String Dynamics

Abstract In this article, we present a dynamic model for a generic drill-string. The model is developed with the intention for component-based simulation with coupling to external subsystems. The performance of the drill-string is vital in terms of efficient wellbore excavation for increased hydrocarbon extraction. Drill-string vibrations limit the performance of rotary drilling; the phenomenon is well known and still a subject of interest in academia and in industry. In this study, we have developed a nonlinear flexible drill-string model based on Lagrangian dynamics to simulate the performance during vibrations. The model incorporates dynamics governed by lateral bending, longitudinal motion, and torsional deformation. The elastic property of the string is modeled by the assumed mode method, representing the elastic deformation, with a finite set of modal coordinates. By developing a bond graph model from the equations of motion, we can ensure correct causality of the model toward interacting subsystems. The model is analyzed through extensive simulations in case studies, comparing the qualitative behavior of the model with state-of-the art models. The flexible drill-string model presented in this article can aid in developing system simulation case studies and parameter identification for offshore drilling operations.

Quenching Vibration on a Harmonically Excited Symmetric Laminated Composite Plate

Abstract In this paper, a simple and efficient method is developed to quench the steady-state vibration of a harmonically excited, damped and symmetric laminated composite rectangular plate. This is achieved by enforcing points of zero displacement, or nodes, at some specified locations on the laminated composite plate using properly tuned damped oscillators. Using the assumed-modes method, the governing equations of the laminated composite plate carrying the damped oscillators are first formulated. A set of constraint equations is established by enforcing nodes at user-specified locations on the plate. Two attachment scenarios are considered: when the attachment and node locations coincide and when they are distinct. Numerical experiments show that for both cases, the damped oscillator parameters can be readily determined and the desired node locations can be successfully imposed. More importantly, enforcing nodes can suppress vibration in the vicinity of the node locations, thereby keeping that region of the laminated composite plate nearly stationary.

Transient response of functionally graded carbon nanotubes reinforced composite conical shell with ring-stiffener under the action of impact loads

The mathematical model of transient response of functionally graded carbon nanotubes reinforced conical shell with ring-stiffener is derived. The deformation state of the conical shell is described by the Reddy higher-order shear deformation theory. The deformation of the isotropic ring-stiffener is described by the Euler-Bernoulli hypotheses. The Rayleigh-Ritz method is used to analyze the eigenmodes of the nanocomposite structure. Displacements projections and rotations of middle surface normal are the main unknowns of the problem.The mathematical model of the structure transient response is obtained as the linear dynamical system with respect to generalized coordinates. The assumed mode method is used to obtain this dynamical system.The influence of the type of carbon nanotube composite reinforcement on the amplitudes of the transient responses is analyzed. As follows from the numerical simulations, the top ring-stiffener significantly decreases the amplitudes of the transient response of the nanocomposite structure.

Nonlinear Flutter Analysis of Porous Functionally Graded Plate in Yawed Hypersonic Flow

An aerothermoelastic analysis of functionally graded plate containing porosities in ‎yawed hypersonic flows is investigated. Due to some incorrect manufacturing processes, two ‎different types of porosity, namely, even and uneven distributions are taken into account. The ‎third-order piston theory is utilized to estimate the unsteady aerodynamic pressure induced by the ‎hypersonic airflow. The material properties of a plate are assumed to vary across the thickness ‎direction according to a simple power law. Based on classical plate theory, the motion equations are ‎developed with geometric nonlinearity taking into consideration of von Karman strains. The ‎formulations are established based on Hamilton’s principle while the generalized differential ‎quadrature method is employed to solve the nonlinear aerothermoelastic equations. Due to lower ‎computational efforts‎‏, ‏the method of generalized differential quadrature (GDQM) is used to obtain ‎accurate results. Moreover, the assumed mode method along with the Runge-Kutta integration ‎algorithm is used as a solution method. The reliability and precision of the obtained results are ‎validated by published literature. Then, the influence of porosity distribution, porosity coefficient, ‎and yawed flow angle are discussed in detail. In general, this paper shows that even porosity ‎distribution would have a more destabilizing effect compared with the ‎uneven porous model.‎ And also, for both porosity distributions, the chaotic behavior appears in higher top surface ‎temperature but even porosity distribution has a profound effect on chaotic motion.‎

Dynamics of a rotating hollow FGM beam in the temperature field

Abstract Dynamic responses and vibration characteristics of a rotating functionally graded material (FGM) beam with a hollow circular cross-section in the temperature field are investigated in this paper. The material properties of the FGM beam are assumed to be temperature-dependent and vary along the thickness direction of the beam. By considering the rigid-flexible coupling effect, the geometrically nonlinear dynamic equations of a hub–FGM beam system are derived by employing the assumed modes method and Lagrange’s equations. With the high-order coupling dynamic model, the effect of temperature variations under two different laws of motion is discussed, and the free vibration of the system is studied based on the first-order approximate coupling model. This research can provide ideas for the design of space thermal protection mechanisms.

Theoretical analysis of a parametrically excited rotor system with electromechanically coupled boundary condition

In this paper, the dynamic behavior of a rotor system with electromechanically coupled boundary condition under periodic axial load is studied, where the boundary condition is derived from a ring-shaped piezoelectric damper which is developed for vibration control of rotor system. This damper is based on the piezoelectric shunt damping technique, which can produce much or little damping performance depends on the selection of shunt circuit parameters. By using assumed mode method and Lagrange equation, the equations of motion are derived. Actually these equations can also be derived from a general forced parametrically excited gyroscopic system. Thus, to analyze such system, a method of multiple scales is developed firstly, where a general procedure is proposed to establish solvability conditions. Subsequently, this procedure is applied to obtain the analytical instability boundaries and forced vibration responses. The analytical results show that the additional combination instability regions are created due to the introduction of shunt circuit. When the parametrically excited eccentric rotor is rotating, the parametric vibrations are superimposed on the unbalanced responses. As the resistance value of shunt circuit is increasing, both parametric vibrations and unbalance vibrations can be significantly suppressed. For the unbalanced responses, both the resonance and anti-resonance phenomena can be observed; whereas for the parametric vibrations, only the resonance phenomena exist. These phenomena indicate that we may achieve great vibration control performance for the rotor parametric vibrations by introducing the piezoelectric shunt damping, as long as the circuit parameters are well adjusted. To validate the obtained analytical expressions, the numerical methods are applied. Specifically, the discrete state transition matrix method (DSTM) is applied to validate the analytical instability boundaries and the Runge-Kutta method is conducted to verify the analytical frequency response functions.

Robust fuzzy sliding mode control and vibration suppression of free-floating flexible-link and flexible-joints space manipulator with external interference and uncertain parameter

AbstractThe flexibility of the free-floating flexible space manipulator system’s link and joint may affect the control accuracy and cause vibrations. We studied the dynamics modeling, joint trajectory tracking control, and vibration suppressing problem of free-floating flexible-link and flexible-joints space manipulator system with external interference and uncertain parameter. The system’s dynamic equations are established using linear momentum conservation, angular momentum conservation, assumed mode method, and Lagrange equation. Then, the system’s singular perturbation model is established, and a hybrid control is presented. For the slow subsystem, a robust fuzzy sliding mode control is proposed to realize the joint desired trajectory tracking. For the fast subsystem, a speed difference feedback control and a linear-quadratic optimal control are designed to suppress the vibration caused by the flexible joints and the flexible link separately. The simulation comparison experiments under different conditions are taken. The simulate results demonstrate the proposed hybrid control’s validity.

PZT Actuators’ Effect on Vibration Control of the PRRRP 2-DOF Flexible Parallel Manipulator

Thanks to their advantages over rigid ones, interest for lightweight parallel manipulator was increased. Besides, structural flexibility effects at high operational speeds are more significant. Thus, developing an appropriate model for the assessment of the dynamic properties of flexible mechanisms and linkages to gain effective vibration control will raise high demand. Therefore, this paper represents the dynamic and kinematic modeling using the assumed mode method and first-type Lagrange equations of the 2-DOF planar parallel manipulator with two flexible links. To truly predict vibrations of the manipulator without any major simplifying assumptions, nonlinear dynamic modeling, which thoroughly attempts to represent the flexible behavior of the links, is considered. As a result, an active damping approach is being studied with PZT actuators. The results show that this approach is effective in damping the vibrations of the links that give accurate trajectory control.

Dynamic Instability of Shallow Shells Interacting with a Three-Dimensional Potential Gas Flow

To study the interaction of a vibrating shallow shell with a three-dimensional subsonic gas flow, we deduce a system of hypersingular integral equations for the aerodynamic derivatives of the pressure drop. This system of equations is convenient for the solution of the problems of aeroelasticity. We solve the system of hypersingular integral equations by using a numerical approach based on the discrete vortex method. To model vibrations of shallow shells, we deduce a system of ordinary differential equations with the help of the assumed-mode method. We also perform the numerical investigation of dynamic instability of the equilibrium state of a shallow shell in the subsonic gas flow.

Harnessing bulging or sloshing modes to design locally resonant liquid-solid metamaterials

This paper proposes harnessing bulging or sloshing modes to design locally resonant liquid-solid metamaterials that can be applied to vibration isolation engineering. The longitudinal dynamic effective mass of the proposed bulging or sloshing cell is analyzed through the assumed mode method and the finite element simulation. The meta-structure consisting of a sequence of unit cells is studied as well. In a certain frequency regime, the proposed bulging metamaterial can block the propagation of elastic longitudinal waves and convert the energy of the host structure to the bending energy of localized covering plates. This longitudinal-to-transverse conversion is triggered by the coupled vibration of the internal liquid and the covering plate. This feature can be applied to energy harvesting. In terms of the sloshing metamaterial, it can attenuate elastic longitudinal waves in an extremely low-frequency range. The local resonance frequency and the bandwidth of the sloshing metamaterial can be tuned by altering the liquid height. Additionally, we exhibit equivalent pendulum-like models to understand the vibration behavior of the bulging and sloshing cells. The proposed liquid-solid metamaterials can provide new approaches for vibration isolation and mechanical energy harvesting. The concept presented here may also give inspirations to designing other multi-physics metamaterials.

Electrostatically actuated double walled piezoelectric nanoshell subjected to nonlinear van der Waals effect: nonclassical vibrations and stability analysis

Abstract In this paper, nonlinear vibration and frequency response analysis of double walled piezoelectric nanoshell (DWPENS) is investigated using nonclassical approach of the Gurtin–Murdoch surface/interface (GMSIT) theory. The piezoelectric nanoshell is simultaneously subjected to visco-Pasternak medium, the nonlinear van der Waals and electrostatic forces. Hamilton’s principles, the assumed mode method combined with Lagrange–Euler’s are used for the governing equations and boundary conditions. Complex averaging method combined with Arc-length continuation is used to achieve the nonlinear frequency response and stability analysis of the DWPENS. It is found that the electrostatic and piezoelectric voltages, the length to radius ratio, the nanoshell gap width, van der Waals (vdW) coefficients and other parameters can effectively change the flexural rigidity of the system which in turn affects the nonlinear frequency response. And also, increasing or decreasing of some parameters lead to increasing or decreasing the resonance amplitude, resonant frequency, the system’s instability, nonlinear behavior, and bandwidth.

A lattice sandwich structure with the active variable stiffness device under aerodynamical condition

Herein, a novel pyramid lattice sandwich structure with the active variable stiffness (AVS) is proposed, which consists of a proportion integration (PI) active controller, a piezoelectric actuator, and a variable stiffness device. The principle of AVS is to process the real-time vibration feedback signal of the lattice sandwich beam by using the PI controller and the piezoelectric actuator to change structural stiffness continuously. The structure has excellent self-adjusting and active vibration reduction effect. Based on the supersonic piston theory, Hamilton's principle, and the assumed modes method, the dynamics equations of the whole structure are established. The vibration reduction and energy dissipation of lattice sandwich beams with AVS and without AVS are analyzed by using the Runge-Kutta algorithm. Moreover, the relevant parameters of PI controller, positive stiffness, negative stiffness, incoming flow velocity, damping coefficient, and installation position are compared and analyzed, respectively. The results indicate that the novel lattice sandwich structure with AVS can achieve ideal vibration reduction effect, and the efficiency can be improved by adjusting the parameters appropriately.

Dynamic Analysis of a Rotating FGM Beam with the Point Interpolation Method

The dynamic characteristics of a hub-functionally graded material beam undergoing large overall motions are studied. The deformation field of the flexible beam is described by using the assumed mode method (AMM), the finite element method (FEM) and the point interpolation method (PIM). Assuming that the physical parameters of functionally graded materials follow certain kind of power law gradient distribution and vary along the thickness direction. The longitudinal deformation and transversal deformation of the beam are both considered, and the nonlinear coupling term which is known as the longitudinal shortening caused by transversal deformation is also taken into account. The rigid-flexible coupling dynamics equations of the system described by three different discrete methods which have a uniform form are derived via employing Lagrange’s equations of the second kind. The validity of the point interpolation method established in this paper is verified by comparing with the numerical simulation results of the assumed mode method and the finite element method. On this basis, the influence of the functional gradient distribution rules on the dynamic characteristics of flexible beams undergoing large overall motions is discussed. The results show that the assumed mode method cannot deal with large deformation problem. Remaining other physical parameters of functionally graded materials beam unchanged, the maximum displacement of the beam increases with the increase of functionally graded materials index. The natural frequency of transverse bending of beam increases with the increase of rotational speed, when rotational speed is constant, the natural frequency will decrease with the increase of functional gradient index.