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This article explores the synchronization, stability and motion characteristics of the generalized dynamical model with two rigid frames (RFs) driven by two groups (even number) of coaxial rotating exciters. In light of this system’s generalized coordinates, we use Lagrange’s equations to derive the generalized differential equation of motion. The responses of absolute and relative motion of the generalized system are obtained using the transfer function method. The synchronization and stability criteria of multiple exciters are derived using the average method and Hamilton’s theory, respectively. Taking a dynamical model driven by two pairs of exciters as an example, the localized studies of generalized results are carried out. The stable synchronous solutions for phase differences and excitation frequency, the stability ability coefficient curves and the response curves are graphically presented considering the effect of two crucial dimensionless parameters on the stable synchronous states. The simulation results of the specific system are obtained using the fourth-order Runge-Kutta algorithms and compared with the numerical qualitative analysis results to reveal the high consistency between them and clarify the used methods’ effectiveness. The strength of this work stems from its use in the field of high power and large scale self-synchronization vibrating machines.

The linear and nonlinear dynamics of a bolted cylindrical shell under a point excitation were experimentally investigated. The tests have been carried out through methods of the hammer test and sweeping frequency test. This paper aims to discover the effect of bolt joints on shell vibration, including the effects of bolt number, tightening torque, and excitation levels. The natural characteristics were analyzed by utilizing natural frequency and mode shapes. The nonlinear dynamic characteristics were analyzed through amplitude-frequency curves. The resonance frequency, resonance amplitude, and damping ratio under different tightening torques and excitation levels were investigated. Results show that: the number of bolts can not only change the value and order of the natural frequencies but also the shapes of the mode; The low tightening torque and larger excitation levels strongly lead to more complex nonlinear dynamics of shell structures.

Abstract Bolt connection is one of the main fixing methods of cylindrical shell structures. A typical bolted connection model is considered as a tuned system. However, in the actual working conditions, due to the manufacturing error, installation error and uneven materials of bolts, there are always random errors between different bolts. To investigate the influence of non-uniform parameters of bolt joint, including the stiffness and the distribution position, on frequency complexity characteristics of cylindrical shell through a statistical method is the main aim of this paper. The bolted joints considered here were simplified as a series of springs with random features. The vibration equation of the bolted joined cylindrical shell was derived based on Sanders’ thin shell theory. The Monte Carlo simulation and statistical theory were applied to the statistical analysis of mode characteristics of the system. First, the frequency and mode shape of the tuned system were investigated and compared with FEM. Then, the effect of the random distribution and the random constraint stiffness of the bolts on the frequency and mode shape were studied. And the statistical analysis on the natural frequencies was evaluated for different mistuned levels. And some special cases were presented to help understand the effect of random mistuning. This research introduces random theory into the modeling of bolted joints and proposes a reference result to interpret the complexity of the modal characteristics of cylindrical shells with non-uniform parameters of bolt joints.

In this paper, a dynamic model with two vibrator-driven pendulums is presented. Each pendulum is motivated by a pair of vibrators mounted on it. According to the deduced synchronization criteria of the pendulums and the vibrators, the synchronous states of pendulums are determined by the phase differences of vibrators. The two pendulums achieve anti-phase synchronization when the phase differences of arbitrary-two vibrators are all stabilized around zero. If the phase differences of the two vibrators mounted on the same pendulum are zero whereas those on different pendulums are π, the pendulums will arrive at the in-phase synchronous state. Besides, the system shows a relatively good isolative effect under the anti-phase synchronous motion of the two pendulums. Based on the theoretical results, some numerical qualitative analyses are presented and simulations are implemented for verification. The dynamic model in this paper can offer guidance to design a kind of vibrating jaw crusher. Here, the two pendulums serve as movable jaws and work by the anti-phase synchronous swing motion trajectory.

Abstract In previous studies about the synchronization of vibrators, the restoring forces of springs are mainly treated as linear directly, whereas the nonlinear features are rarely considered in vibrating systems. To make up this drawback, a dynamical model of a nonlinear vibrating mechanical system with double rigid frames (RFs), driven by two vibrators, is proposed to explore the synchronization and stability of the system. In this paper, the nonlinearity is reflected in nonlinear restoring forces of springs characterized by asymmetrical piecewise linear, where the nonlinear stiffness of springs is linearized equivalently using the asymptotic method. Based on the average method and Hamilton’s principle, the theory conditions to achieve synchronization and stability of two vibrators are deduced. After the theory analyses, some numerical qualitative analyses are given to reveal the coupling dynamical characteristics of the system and the relative motion properties between two RFs. Besides, some experiments are carried out to examine the validity of the theoretical results and the correctness of the numerical analyses results. Based on the comparisons of the theory, numeric and experiment, the ideal working regions of the system are suggested. Based on the present work, some new types of vibrating equipment, such as vibrating discharging centrifugal dehydrators/conveyers/screens, can be designed.

Abstract The length of flexible manipulators with a telescopic arm alters during movement. The dynamic parameters of telescopic flexible manipulators exhibit significant time-varying characteristics owing to variations in length. With an increase in the manipulators’ length, the nonlinear terms caused by flexibility in the manipulators’ dynamic equations cannot be ignored. The time-varying characteristics and nonlinear terms of telescopic flexible manipulators cause fluctuations in rotation angles, which affect the operation accuracy of end-effectors. In this study, a control strategy based on a combination of fuzzy adjustment and an RBF neural network is utilized to improve the control accuracy of flexible telescopic manipulators. First, the dynamic equation of the manipulators is established using the assumed mode method and Lagrange’s principle, and the influence of nonlinear terms is analyzed. Subsequently, a combined control strategy is proposed to suppress the fluctuation of the rotation angle in telescopic flexible manipulators. The variation ranges of the feedforward PD controller parameters are determined by the pole placement strategy and length of the manipulators. Fuzzy rules are utilized to adjust the controller parameters in real-time. The RBF neural network is utilized to identify and compensate the uncertain part of the dynamic model of the flexible manipulators. The uncertain part comprises time-varying parameters and nonlinear terms. Finally, numerical simulations and prototype experiments prove the effectiveness of the combined control strategy. The results prove that the proposed control strategy has a smaller standard deviation of errors. Therefore, the combined control strategy is more suitable for telescopic flexible manipulators, which can effectively improve the control accuracy of rotation angles.

In this paper, a novel adaptive attitude tracking control method is investigated to enhance tracking performance and robustness for a quadrotor unmanned aerial vehicle (UAV) against the modeling uncertainties and external disturbances. First, the attitude dynamics of a quadrotor UAV based on the modified Rodrigues parameter (MRP) is derived. Then, an adaptive dynamic surface technique with robust integral of the sign of the error (RISE) control approach are designed to improve the tracking performance and robustness of the quadrotor attitude control system suffering from the uncertainties and disturbances. A prescribed performance function is employed to improve the tracking performance of the quadrotor UAV by restraining the tracking errors range. The stability analysis proves the presented control scheme can guarantee all the close-loop control system signals are bounded and control system is uniformly ultimately bounded. Simulation results are carried out to demonstrate the superior and effective performance of the presented control method.

Friction has a vital role in studying materials’ and systems’ behavior. The friction between two objects and the inner friction of materials under the condition of vibration usually can present different characteristics. These characteristics are different from the conventional conditions. It is shown in practice that vibration can reduce the friction coefficient and friction force between two objects. Vibration can lighten abrasion of objects and reduce energy consumption. All of these can give great efficiency, but, until now, the vibration friction-reducing mechanism has not been fully revealed. In this manuscript, the friction-reducing mechanism of materials under arbitrary vibration forces is investigated. The results show that the effective friction coefficient of materials under arbitrary vibration forces is always the minimum. The relationship between the effective friction coefficient and the negative gradient is investigated in this research. When the vibration force direction projects are in the first and the third quadrants, the negative gradient of the effective friction coefficient gets larger slowly, and then it becomes stable. When the vibration force direction projects are in the second and the fourth quadrants, the negative gradient of the effective friction coefficient decays to zero at the initial stage and then increases rapidly.

In this paper, the dynamics of a mechanical exciter and three cylindrical rollers (CRs) with the non-identical friction coefficients interacting through a rigid platform is considered. Sufficient conditions for the existence and stability of synchronous solutions in the coupled system are derived by using the average method of modified small parameters and Routh-Hurwitz principle. The obtained theoretical results are illustrated and analysed based on numerical calculations. In the analysis, the numerical results are presented for simple one-parameter variation, as well as for a group of varied parameters, when the influence of the coupling structure’s parameters on synchronization and stability is studied. An appropriate selection of the key parameters will eventually lead to desired synchronization performance. Finally, the theoretical and numerical results are supported by computer simulations. The stable synchronized states can be observed in the simulations even when there are unavoidably small differences in the three friction coefficients. If we mismatch the friction coefficients of the CRs, they are seen to synchronize with a constant phase difference. The key feature of the proposed coupled system is the dynamic coupling torque, which serves as the vehicle for transferring energy from an induction motor to three CRs without the direct driving sources and the synchronization controller for maintaining the originally synchronous and stable states against the disturbance in the simulations.

In this paper, the dynamic model of a pre-twisted cracked rotating blade is established using self-programed beam element (CBE) and its nonlinear characteristics are analyzed. Firstly, based on the Castigliano’s theorem, the CBE with an open crack model is established. Secondly, a new breathing model of CBE is proposed in terms of the change of the crack contact region. Subsequently, the finite element (FE) model of a pre-twisted cracked rotating blade is established, and then verified by the ANSYS model. Finally, the effects of the crack depth, the crack position, the aerodynamic amplitude and the rotating speed on blade vibration characteristics and crack breathing status are investigated. The results show that the proposed model relative to the ANSYS model is of high efficiency and accuracy, besides, the larger crack depth and higher rotating speed result in a non-breathing crack, while the larger crack position and aerodynamic amplitude result in a breathing crack.

In this paper, a high accuracy attitude tracking control is presented for a quadrotor unmanned aerial vehicle (UAV) under the model uncertainties and external unpredictable disturbances by using a robust feedback control method. First, the cascade attitude dynamics of a quadrotor UAV is derived with the lumped disturbances. Then, a backstepping adaptive modified robust integral of the sign of the error (RISE) control is designed to improve the robustness of the control system against the disturbances. For arriving high accuracy tracking under the disturbances, a prescribed performance control algorithm is employed in combination to hold the transient state performance where the convergence rate is faster than the prescribed value, and the maximum overshoot is less than the prescribed value. The stability analysis is carried out to prove that the presented controller could guarantee that the closed-loop system is asymptotically stable, and the tracking feedback errors can converge to an arbitrarily small value near zero. The superior performance of the proposed control method is illustrated in simulations against the nonlinear lumped disturbances, and the benefits of the proposed robust controller is corroborated by comparing with the existed controller.

In this work, a continuum model is proposed to simulate and interpret the coupling vibration characteristics of a rotating disk-beam system with the dovetail interfaces. The dovetail interface feature is represented by a macroslip dry friction model. The present study also derives a new mode function to simulate the vibration of a rotating beam with loosely assembled dovetail attachment. The new proposed mode function is validated by comparing the natural characteristics and vibration response with those obtained from a finite element (FE) model. At last, based on the nonlinear response results obtained by the Newmark-β method, the effects of different parameters on the nonlinear dynamics of the coupling system are discussed. The following interesting phenomena have been revealed: the flexible disk can impose different influences on the response characteristics of the rotating beam. The effects of dry friction on the beam’s energy dissipation are significant, especially at a low rotational speed. Further analysis yields that the excitation level and friction coefficient also exhibit a significant impact on the damping effect of dry friction. Consequently, it should be noted that the optimal values of the contact surface’s parameters allow achieving a better damping effect in the engineering practice.

From the perspective of theoretical derivation, numerical simulation, and engineering application, the vibratory synchronization characteristics of a dual-mass vibrating system driven by two exciters, were studied. The differential motion equations of the total system were calculated using Lagrange’s equations, and the responses of the vibrating system in the steady state were derived by Laplace transform. The synchronization criterion between two exciters was deduced by using the averaging method. Based on the Hamilton principle, the stability criterion of the vibrating system in synchronous states is given. According to the theoretical results, the coupling characteristics between two exciters such as synchronization and stability were analyzed numerically. Some analyses of the numerical simulation of the system were carried out, which fully support the theoretical results. The rotational speed of two exciters, their phase difference, responses, and difference of responses of two rigid bodies were studied quantitatively in the subresonant state and super-resonant state of the system. This paper presents a practical example of vibratory synchronization of a dual-mass system driven by two exciters in engineering.

This article presents the dynamic model of three eccentric rotors driven by induction motors and respectively derives the response equations of the three directions with different frequency in a vibrating system. On account of the difficulty to realize multifrequency self-synchronization, the controlling method is used to realize the synchronization of the speed and phase. In this article, the fuzzy PID method is introduced to control the vibrating system which is based on a master–slave controlling strategy. And a tracking method based on the phase ratio is proposed. The effectiveness and stability are shown in the numerical simulation and the experiment validation is given to certify the accuracy of the theory. In the meanwhile, the arbitrary of the multifrequency is also demonstrated. Finally, some conclusions are summarized to present the significant in the engineering.

This paper investigates stability of the synchronous states for three homodromy exciters in the vibrating system, some theoretical analyses and simulation results on which are given. Based on Lagrange equations, the differential equations of motion of the system are obtained. Using the average method yields the dimensionless coupling torque balanced equations of three exciters and the simplified analytical expressions for synchronization criterion of the system are deduced, the stability criterion of the synchronous states complies with Routh–Hurwitz principle. The dynamic characteristics of the system for different frequency ratios are discussed numerically. In order to verify the validity of theoretical methods, the simulation results by a Runge–Kutta routine are carried out; it indicates that the motion state of the vibrating system can be classified into two types: sub-resonant state and super-resonant state. The motion type of the rigid frame is strong positive superposition vibration in a sub-resonant state; while in a super-resonant state, the exciting forces of three exciters are mutually cancelled and the rigid frame is motionless. During the design process of the vibrating machines, the ideal working points are only selected in a sub-resonant state, which makes the vibrating machines work efficiently.

The dynamic vibration absorbers have been applied to attenuate the critical or unbalanced vibration but may create the fluid-induced vibration instability in the rotor/seal system. The major purpose of this study is devoted to the effects of the dynamic vibration absorber on the nonlinear dynamic behavior and stability of the fluid-induced vibration in the rotor/seal system. The dynamic vibration absorber is attached on the shaft in the perpendicular directions. The model of the rotor/seal-dynamic vibration absorber system is established as the modified Jeffcott rotor system, and Muszynska nonlinear seal force is applied. The numerical method is used for the dynamic behavior analysis. The effects of the natural frequency and damping ratio of the dynamic vibration absorber on the dynamic behavior are discussed. The stability of the rotor/seal-dynamic vibration absorber system is judged by the eigenvalue theory. The variations of the instability threshold with the parameters of the dynamic vibration absorber are obtained. The results show that the instability threshold and instability vibration frequency are changed by the dynamic vibration absorber. The parameters of the dynamic vibration absorber must be selected carefully to avoid reducing the instability threshold and causing the instability vibration to occur in advance when the dynamic vibration absorber is applied to attenuate the critical or unbalanced vibration of the rotor/seal system.

A dynamic model of the rotating shrouded blade is established, considering the shroud mass, the Coriolis force, and the centrifugal stiffening effect. And a macroslip model of dry friction with variable normal load is established to simulate the separation-contact-stick-slip state of the shroud. The Lagrangian equation is utilized to solve the differential motion equation, and the Galerkin method is used for discretization. The influence of shroud structure’s parameters such as rotational speed, contact angle, friction coefficient, clearance, and shroud position on the damping effect of the shroud is reviewed by means of amplitude-frequency response and energy through the Newmark-β numerical method. The results demonstrate that the damping effect of the shroud by contact is more obvious than by friction and the amplitude-frequency curve of the shrouded blade shows a strong hard nonlinear phenomenon.

A theoretical research on eliminating the instability vibration and improving the stability of the rotor/seal system using the inerter-based dynamic vibration absorber (IDVA) is presented in this paper. The modified Jeffcott rotor and Muszynska nonlinear seal force models are employed. The proposed IDVA is the damping element of the traditional dynamic vibration absorber (DVA) replaced by one of the six configurations of the inerter. The instability threshold speed of the system is obtained by applying the Routh–Hurwitz stability criterion. The quantum particle swarm optimization (QPSO) method is utilized to optimize the parameters of the IDVA. The numerical method is applied to investigate the nonlinear dynamic responses and stability. The results show that the IDVA can effectively improve the stability and reduce the instability vibration of the rotor/seal system. Furthermore, the performance of the IDVA is more effective than that of the traditional DVA.

The laser linear scanning method is proposed to measure the modal shape of fiber-reinforced composite thin plate with high efficiency and precision. First, by establishing the laser scanning frame model of the composite plate, the corresponding extraction principle of modal shape data and laser scanning rate selection criterion are explained in detail to clarify the theoretical principle of laser linear scanning method. The corresponding test procedure of modal shape, drawing method from the shape scanning data, and control method of the constant laser scanning rate are also proposed based on the developed laser linear scanning system. Then, a TC300 carbon fiber/resin composite thin plate is taken as a research object to verify the effectiveness and reliability of such a method, through comparing the results obtained by the traditional experimental modal method and finite element method. Moreover, the influences of constraint boundary condition, excitation level, laser scanning rate, scanning spacing, scanning path mode, the fiber angles, and fiber material damage on modal shape results are also discussed. It has been found that laser linear scanning method can improve test efficiency of modal shape of the composite plate with high preciseness. Except for scanning path mode, the other parameters have a major impact on each shape morphology, and their effects can be quantitatively analyzed by identifying the positions and clarity of nodal lines of each modal shape. Especially, the high-density modal shape results and their equal amplitude lines in different projection views can be used to determine whether or not the composite plate is damaged as well as its damaged degrees.

A dynamic model is proposed in this paper for analyzing the nonlinear characteristics of a flexible blade. The dynamical equation of motion for a rotational flexible blade in a centrifugal force field is established based on the finite element method. A macro-stick-slip mechanical model of dry friction is established to simulate the constraint condition of the flexible blade. The combined motion of the external excitation and friction produces a piecewise linear vibration which is actually nonlinear. The numerical integration method is employed to calculate the vibration reduction characteristics of the nonlinear constrained rotating blade. The results show that the nonlinear dry friction force produced by the dovetail interface plays an important role in vibration reduction. And the effect of dry friction vibration reduction is significant when the rotating speed is slow or the friction coefficient is small. Besides, the magnitude of external excitation also has a great impact on the state of the friction. Therefore, some relevant experimental researches should be done in the future.

The dynamic vibration absorbers have been applied to attenuate the rotor unbalance and torsional vibrations. The major purpose of this paper is to research the elimination of the fluid-induced vibration in the rotor/seal system using the absorber. The simplified rotor model with the absorber is established, and the Muszynska fluid force model is employed for the nonlinear seal force. The numerical method is used for the solutions of the nonlinear differential equations. The nonlinear responses of the rotor/seal system without and with the absorber are obtained, and then the rotating speed ranges by which the fluid-induced instability can be eliminated completely and partially are presented, respectively. The absorber parameters ranges by which the instability vibration can be eliminated completely and partially are obtained. The results show that the natural frequency vibration due to the fluid-induced instability in the rotor/seal system can be eliminated efficiently using the absorber. The appropriate natural frequency and damping ratio of the absorber can extend the complete elimination region of the instability vibration and postpone the occurrence of the instability vibration.

Multifrequency controlled synchronization of two homodromy eccentric rotors is investigated. The vibrating screen with two rotors is a typical underactuated system. So, the model of the vibrating screen is converted to a synchronization motion problem. Firstly, the mechanical-electromagnetic coupling dynamical model of the vibrating system is established. And then the fuzzy PID method is used for the two motors which are based on the master-slave control strategy. The slave motor uses the method of the phase ratio to trace the master motor and achieve the synchronous motion. Through the simulation, the two rotors cannot reach the state of multifrequency self-synchronization. And, then, it is realized by multifrequency controlled synchronization. The article presents the motion trails of the vibrating system between one and two times and realizes the zero phase difference after each period. Finally, some experiments are used to verify the accuracy and effectiveness of the conclusion in the simulation and analysis of the feature of the movement tracks.