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The contact state and stiffness of linear guide are affected by the complex external loads, and the vibrations in different directions are coupled with each other. However, existing dynamic models are inaccurate because they ignore displacement coupling and do not consider the position deviation of guide rail. This paper presents a five degree-of-freedom dynamic model for table system that incorporates the effects of the displacement coupling and position deviation of guide rail. The ball-to-groove contact model and position deviation model are established by using the Hertz contact theory and homogeneous coordinate transformation, respectively. Dynamic equations are solved by combining the Monte Carlo and Newmark methods to determine the displacements of table system, and the accuracy of proposed dynamic model is verified through experimental method. The results show that the vibrations of table system are coupled with each other, and the position deviation of guide rail only affects the vibration amplitudes in the region other than the resonance regions. Proposed model is thus suitable for improving the machining accuracy and stability of machine tools.

Based on the finite element theory and the loaded tooth contact analysis, an analytical-finite-element model considering the complex foundation types and the crack propagation paths is proposed to calculate the mesh stiffness of spur gears. The complex gear foundation types (including web foundation and slots foundation) and the crack propagation paths (including tooth fracture and rim fracture) are considered in the proposed method, and the effects of various foundation types, crack propagation paths and crack lengths on the mesh stiffness are analyzed. In order to verify the proposed method, the simulated crack paths are confirmed by those derived from the experimental and the extended finite element methods, and the mesh stiffness obtained from the proposed method is validated by that of the potential energy method and the three-dimensional finite element method. Compared with the potential energy method, the proposed model can be used to calculate the mesh stiffness of spur gears with different gear foundation types and real crack propagation paths. Meanwhile, the proposed method has higher computational efficiency compared with the finite element method (e.g., the finite element method takes about 3.5 h for a mesh cycle and the proposed method costs 25 s). The results indicate that the web thickness has greater influence on the mesh stiffness of spur gears than the rim thickness. The results show that two crack types, namely the tooth fracture and rim fracture, have more obvious influence on the mesh stiffness of single-tooth contact zone relative to that of the double-tooth contact zone.

Combining the loaded tooth contact analysis (LTCA) method and the gear dynamic model, a wear prediction model for spur gears with tip relief is established. The simulated wear depth is verified by the experimental results in a published reference. Based on the wear profile, the mesh stiffness and the tooth root stress obtained from the proposed model are compared with those obtained from the finite element model. Relative to the finite element method, the proposed method can greatly reduce the computation time. The effects of the tip relief and surface wear on the meshing and the dynamic characteristics are discussed. The tip relief can greatly decelerate the surface wear. Appropriate tip relief and slight wear can decrease the tooth interference and reduce the system vibration. Furthermore, the effects of the rotational speed on the wear depth are also studied and it is found that the quasi-static model is applicable to wear prediction when the rotational speed is away from the harmonic resonance peak.

This paper investigates the effect of a gear tooth spalling in a helical gear. An analytical model was developed to study the effect of the damage to the time-varying mesh stiffness, contact stress and tooth root fillet stress. In order to validate the efficiency and the accuracy of the proposed method, a finite element method is presented for simulation, such to compare the mesh stiffness and stresses obtained from the analytical method under different spalling variables, including the spalling lengths, widths, axial positions (in the direction of face width) and tooth orientation positions (from dedendum to addendum). The results show that the proposed analytical method has higher computation efficiency than the finite element method, and the mesh stiffness obtained from two methods shows a good agreement. However, some errors exist between the time-varying mesh stiffness and stresses obtained by the two methods, but change laws of the tooth root stress and contact stress obtained from the two methods are similar. This is because the effects of flank-tip contact and spalling edge contact are considered in the finite element method, which makes the results more accurate than the results obtained by the proposed analytical method.

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.

We will build a theoretical system of innovative design with Chinese characteristics, a digital, networked, green and intelligent design technology system. We will build a knowledge base platform for innovative design, focusing on industry-based platforms such as design knowledge database, materials science database, commercial big data and patent big data. We will push forward the all-digital simulation pilot of product design, strengthen the infrastructure of the information network, big data, cloud computing and other co-creation platforms, build a world-class broadband network and application system covering the whole country, and form an open source “customer creation” network sharing platform. We should strengthen the support of virtual reality/augmented reality/hybrid reality (VR/AR/MR) and artificial intelligence (AI) to the design technology environment and platform.

Based on the thin slice assumption, a revised time-varying mesh stiffness (TVMS) model of spur gear pairs with tooth modifications is developed. The spur gear is divided into many individual slices along tooth width, and considering the revised fillet-foundation stiffness, the nonlinear contact stiffness, the extended tooth contact and the tooth profile errors, the stiffness of each slice gear pair is figured out. According to the relationship between the deformation and the total stiffness in mesh period, the TVMS of spur gear pairs can be worked out. Meanwhile, relative to the finite element (FE) method, the errors of the proposed method under different modification quantities are discussed. The proposed method is more accurate than those previous methods but there are still some errors. Taking the FE model as a benchmark, the TVMS is further revised based on a simple model updating technique. Based on the revised model, the effects of the tooth width and torque on mesh stiffness are also studied. The result shows that based on the proposed method, the TVMS under any given modification quantities in a suitable range can be calculated accurately.

In the traditional rubbing model, the overall displacement (rigid body displacement) of the casing is always taken into account, while the local flexible deformation is either ignored or simply assumed as a spring element. Due to the small thickness of the casing (about 2–4 mm), there may exist a large error for predicting the normal rubbing force using the traditional model, which can lead to the error when the local deformation and wear extent are calculated. Aiming at this phenomenon, this paper deduces an improved rotating blade-casing rubbing model based on elastic compatibility conditions. During the derivation, the centrifugal stiffening, spin softening and rubbing softening are considered and the quasi-static bending displacement of blade is calculated based on the theory of mechanics of materials. The casing is regarded as a flexible ring with spring support in the radial direction, and the displacements are composed of rigid-body and flexible displacements. Ignoring the effects of the kinetic energy, according to Hamilton principle, the structural stiffness of the casing under the quasi-static condition is obtained. The proposed model is verified by comparing with the experimental data obtained from both rigid casings and flexible casings in published references. Based on a rotating-blade-flexible-casing system, the rubbing-induced vibration behaviors from both finite element (FE) model in ANSYS software and the proposed model are figured out and compared with each other for testing the effectiveness of the proposed model again. A frequency sweep is performed in order to better understand the resonances of the blade and casing. Vibration responses obtained from the traditional rubbing model and the proposed model are also compared with each other so that the advantages of the proposed model is emphasized. The comparison results show that the normal rubbing forces obtained from the proposed model are closer to the experimental results because the proposed rubbing model can accurately describe the flexible deformations of the casing under both quasi-static and dynamic conditions. The rubbing-induced blade vibration responses obtained from the traditional and proposed rubbing models agree well, however, the dynamic rubbing forces obtained from the proposed model are smaller than those from the traditional model. In addition, different combined nodal-diameter modes of the casing can be excited under different rotating speeds.

Modeling and free vibration analysis of bolted joined cylindrical-cylindrical shell were presented in this paper. First, a new analytical model was established for bolted joints. Second, based on Sanders’ shell theory, the dynamic equation of substructure of the assembly structure was formed. Then, by using the connection condition, into which the coupling stiffness between the segments was considered, the vibration differential equation of the assembly structure was established. Finally, the vibration characteristics of bolted joined cylindrical-cylindrical shell were obtained. To test the accuracy of the present method, the frequencies and mode shapes under some special connecting condition were examined with the results obtained by ANSYS and a good agreement was shown. Further, the number of bolts, connecting stiffness and length ratio of the two segments have a significant influence on natural frequency parameters and mode shapes of symmetrical and unsymmetrical structures. The results reveal the vibration characteristics of bolted joined cylindrical-cylindrical shell and are presented to provide useful information for future research. Moreover, some new results would be of signification for assembly structure design and vibration control.

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.

This paper addresses the problem of composite synchronization of three eccentric rotors (ERs) driven by induction motors in a vibrating system. The composite synchronous motion of three ERs is composed of the controlled synchronous motion of two ERs and the self-synchronous motion of the third ER. Combining an adaptive sliding mode control (ASMC) algorithm with a modified master-slave control structure, the controllers are designed to implement controlled synchronous motion of two ERs with zero phase difference. Based on Lyapunov stability theorem and Barbalat's lemma, the stability of the designed controllers is verified. On basis of controlled synchronization of two ERs, self-synchronization of the third ER is introduced to implement composite synchronous motion of three ERs. The feasibility of the proposed composite synchronization method is analyzed by numerical method. The effects of motor and structure parameters on composite synchronous motion are discussed. Experiments on a vibrating test bench driven by three ERs are operated to validate the effectiveness of the proposed composite synchronization method, including a comparison with self-synchronization method.

A new dynamic model of shrouded blades with elastic support and variable cross-section is derived based on the Timoshenko beam theory, in which factors to be considered include centrifugal stiffening, spin softening, Coriolis force, blade stagger angle, and twist angle. The proposed model (PM) is verified by comparing the dynamic frequency results of the finite element model (FEM) and the PM. An impact-friction model of shrouded blades is established which can take into account the inertia effects of two passive blades. By comparing the vibration response of the two models (PM and FEM), the established impact-friction model is verified. Finally, the influences of normal contact stiffness, blade root support stiffness, and shroud position on the vibration response of the system are analyzed. Meanwhile, bifurcation diagrams, time-domain waveforms, phase diagrams, Poincare maps, and frequency spectrum are utilized to analyze the nonlinear vibration characteristics of shrouded blades. The results show that the impact and friction between adjacent shrouds will cause the complex nonlinear vibration of the blades. Under small initial gaps, hard nonlinear phenomena can be observed in the vibration response of the shrouded blade. While under small initial preloads, it shows an obvious soft nonlinear phenomenon. Furthermore, with the increase of normal contact stiffness and blade root support stiffness, the resonance peak of the blade-tip flexural displacement decreases. When the shroud is closer to the blade-tip of the blade, the vibration reduction effect of the shrouded blade is better. The active blade experiences the chaos, period-one motion due to the nonlinear behavior of the impact-friction coupling at the adjacent shroud contact interfaces.

In view of the design problem of the working amplitude and working frequency of vibrating icebreaking machine, the stiffness and damping value of the ice-snow was identified with the mechanical vibration theory and finite element method. The two-degree-of-freedom mathematical model was established in this paper. And numerical simulation was performed. Then the related icebreaking experiment was carried out. A lot of test and experiment results had proved that experimental data was consistent with the theoretical analysis of mathematical model of vibrating icebreaking system. It was confirmed that the correctness of vibrating icebreaking system’s mathematical model and the rationality of its design could be proved.

The effects of three different gear crack types such as, for example, the crack along tooth width uniformly and the crack propagating in the depth direction (crack type 1, CT1), the crack along tooth width non-uniformly and the crack propagating in both the depth and the tooth width directions (crack type 2, CT2), and the spatial crack propagating in the depth, the tooth width and the tooth profile directions (crack type 3, CT3) on the time-varying mesh stiffness (TVMS) of spur gear pairs are investigated in this study. Firstly, an analytical model for studying these three types of cracks is established based on potential energy method. A finite element (FE) model of the cracked spur gear pair is also built in the ANSYS software as well. In order to verify the analytical method, the TVMS obtained from analytical method is compared with that obtained from FE method under different crack types. Moreover, the effects of the depth, the length and the height of crack are discussed. The equivalent stress, contact pressure and displacement of tooth are also analyzed under different crack types by using the FE method. The results show that the effect of crack depth on TVMS is the largest, while that of the crack height is the smallest, and the non-penetrating crack for CT2 and CT3 will generate the non-uniform load distribution along tooth width.

In this study, a finite element model of an elastic-support cantilever beam with crack and offset boundary is established by using mixed elements in ANSYS software. In the proposed model, different contact elements are adopted to describe the breathing effect of crack and offset boundary, and spring elements are used to simulate the elastic support, and the model is also validated by comparing the natural frequencies with those in published literatures. Based on the developed model, the combined effects of the crack and offset boundary on the system dynamic characteristics are studied. The results indicate that the amplitude of double frequency component (2fe) firstly decreases and then increases with the offset values when the crack position is on the opposite side of offset boundary. 2fe may disappear when the crack and the offset boundary locate at a certain position. In addition, the more distant the offset boundary is, the more intense the system nonlinearity becomes. The amplitude of 2fe increases with the offset values when the crack position is on the same side of offset boundary under a constant crack depth and location. Moreover, it also shows some complicated frequency components due to the gradually strengthened nonlinearity of the system with the increasing offset values, and the obvious distortion phenomenon in the phase plane portraits can be observed near the super-harmonic resonance region. This study can provide some basis for the diagnosis of beam-like structures with crack.

On the basis of our previous work (Ma et al., 2016, Journal of Sound and Vibration, 378, 92–108) [36] , an improved analytical model (IAM) of a rotating twisted shrouded blade with stagger angle simulated by flexible beam with a tip-mass is established based on Timoshenko beam theory, whose effectiveness is verified using finite element (FE) method. The effects of different parameters such as shroud gaps, contact stiffness, stagger angles and twist angels on the vibration responses of the shrouded blades are analyzed using two different impact models where the adjacent two shrouded blades are simulated by massless springs in impact model 1 (IM1) and those are simulated by Timoshenko beam in impact model 2 (IM2). The results indicate that two impact models agree well under some cases such as big shroud gaps and small contact stiffness due to the small vibration effects of adjacent blades, but not vice versa under the condition of small shroud gaps and big contact stiffness. As for IM2, the resonance appears because the limitation of the adjacent blades is weakened due to their inertia effects, however, the resonance does not appear because of the strong limitation of the springs used to simulate adjacent blades for IM1. With the increase of stagger angles and twist angles, the first-order resonance rotational speed increases due to the increase of the dynamic stiffness under no-impact condition, and the rotational speeds of starting impact and ending impact rise under the impact condition.

Time-varying mesh stiffness (TVMS) and the dynamic coupling between the helical gears have a great influence on the vibration characteristics of a helical gear rotor system. Considering the effects of TVMS and adopting two coupling models (lateral-torsional coupling model and lateral-torsional-axial-swing coupling model), the dynamic behavior of a helical gear system was studied. First, an analytical model was used to analyze TVMS of a helical gear pair where the helical tooth is simulated by many spur tooth slices along the direction of the tooth width and the mesh stiffness of each slice is calculated using the energy method. Then, considering the effects of the TVMS excitation, the finite element model of a helical gear rotor system was established. Gear mesh was simulated by the above-mentioned two coupling models to investigate the effects of coupling forms on the system vibration characteristics. The strain energy was used to distinguish the dominant mode and dominant shaft of a gear system in natural characteristics analysis. The results show that the full coupling model can analyze accurately the vibration characteristics of the system and the axial and swing motions cannot be ignored in vibration analysis. Finally, the effects of helix angle on TVMS and vibration responses of a helical gear system were also studied.

The influence of blade vibration on the nonlinear characteristics of rotor–bearing system is non-ignorable in estimating system performance. The extensive studies simplify the rotor system as lumped mass points. The influence of shaft’s bending and shear and the flexibility are usually ignored. The present paper is aim to analyze the nonlinear dynamic behavior of a continuum model. The continuum model of flexible blade–rotor–bearing coupling system is established, simplifying the shaft as Timoshenko beam. The Lagrange method is utilized to derive the differential equation of motion of system. Then, the nonlinear equations of coupling system are numerically solved using the Newmark- $$\upbeta $$ method. The results obtained through the proposed model are compared with the rotor–bearing system without the blades. The effect of several parameters such as rotational speed, the damping coefficient and the length of blade on the nonlinear dynamics of rotor system have been investigated. Inclusive of the analysis methods of bifurcation diagram, three-dimensional spectral plots, time-base analysis, Poincare maps and spectral plots are used to analyze the behavior of the coupling system under different operating conditions, which exhibits rich dynamic behavior of the system.

In order to simulate the complicated dynamic phenomena of cantilever beam structures with different crack types and levels in the engineering machinery, three types of cracks are assumed, i.e., non-penetrating parabolic crack (NPPC), penetrating trapezoid crack (PTC) and uniform-penetrating crack (UPC). Based on ANSYS software, mixed elements combining beam elements and solid elements are adopted to establish the finite element (FE) models of cracked cantilever beams where the crack levels are evaluated by introducing the area damage factor, i.e., the ratio of the damage area to the cross-sectional area. Then vibration responses and crack level identification of the system under three cases of crack severity are discussed by the spectrum cascades, acceleration-velocity and velocity-displacement phase portraits, and contact pressure nephograms. The results show that the magnitudes of constant components in the spectra increase with the increase of crack severity. Lateral velocity-displacement phase portraits, perpendicular to the excitation direction, are more sensitive to the appearance of the small crack than acceleration-velocity phase portraits in the excitation direction, and the combination of the two phase portraits can be used to identify the crack severity. The crack breathing effects can be evaluated using contact pressures distributions, and local contact phenomenon can be observed during the positive half circle of the excitation period under the larger crack conditions.

This paper presents a study of a novel nonlinear damper on vibration control of rotor system with variable damping by using a smart material- shear thickening fluid (STF), whose viscosity and damping will dramatically increase with shear rate. A STF damper was self-developed and dynamic model of the STF damper-rotor system was established by using the Bouc-Wen model as mechanical model of the smart material damper. The incremental harmonic balance (IHB) method combined with the arc-length method was presented to investigate the dynamic response of rotor system with hysteresis nonlinear damping force theoretically. Experimental study of STF damper on suppressing vibration of rotor system was carried out by using a Bently rotor test rig with two STF dampers. The damping effect under different damper installation direction, vibration frequency and amplitude were investigated theoretically and experimentally. Study shows that the nonlinear damper exhibits larger damping force and fuller hysteresis characteristics, so it has better effect on resonance energy absorption compared with the traditional linear damper; and its output damping force can increase dramatically with the change of frequency and amplitude. It is verified that the unbalance vibration, especially the resonance amplitude of rotor system can be effectively suppressed by nonlinear damping of STF damper. The study shows the feasibility and advantages of using the STF damper for potential applications of adaptive or semi-active vibration control of rotating machinery.

A dynamic model of rotating shrouded blades with impacts among adjacent shrouded blades is established considering the effects of the centrifugal stiffening, spin softening and Coriolis force, and the model is validated using finite element method. In the proposed model, the shrouded blade is simplified as a cantilever Euler–Bernoulli beam with a mass point at the free end, and the flexural dynamic stiffness of shrouded blade is selected as contact stiffness during collision. Based on the developed model, the effects of symmetric and asymmetric shroud gaps, rotational speeds, and aerodynamic force amplitudes on the dynamic characteristics of shrouded blades are analyzed through Newmark-β numerical method. The results indicate that (1) the vibro-impact responses of shrouded blades under some asymmetric gaps are more complicated than that under symmetric gap. (2) With the increase of rotational speed from 6000 to 10,000 rev/min, the system vibration experiences from period-three motion, through chaotic motion, finally to period-one motion during collision process because the increasing rotational speed changes the flexural dynamic stiffness of rotating blade. (3) The vibration displacements of shrouded blades increase linearly, and impact force increases linearly with the increase of aerodynamic force amplitude.

A cyclic sector corresponding to blade-disk structure with dovetail connection (1/38 blade-disk) is studied and the finite element (FE) model of this structure is established based on ANSYS software. A revised normal rubbing force model is developed and a pulse force model is established to simulate the local rubbing phenomenon between the blade and elastic casing based on the revised model. The effects of the rubbing under different rotating speeds and penetration depths on the blade vibration response and contact behaviors of dovetail interface are analyzed. The results show that the rubbing will cause amplitude amplification phenomenon when the multiple frequency components are close to the first bending and first torsion natural frequencies. The arch bending of the blade caused by blade-tip rubbing can be identified by evaluating the displacement and stress of the blade in the radial direction ( y-direction). The dynamic stress in the process of rubbing gradually changes from alternation between tension and compression stress to the tension stress with the increasing rotating speed. Maximum contact sliding distance may change dramatically when the rubbing force is greater than the centrifugal force. With the increase of rotating speed, the contact pressure increases under the centrifugal force and its fluctuation under rubbing is smaller at higher rotating speeds.

Considering the effects of extended tooth contact (ETC), revised fillet-foundation stiffness under double-tooth engagement region, nonlinear contact stiffness and tooth spalling defect, an analytical model for time-varying mesh stiffness (TVMS) calculation of spur gears is established. In addition, the analytical model is also verified by comparing the TVMS under different spalling widths, lengths and locations with that obtained from finite element method. The results show that gear mesh stiffness decreases sharply with the increase of spalling width, especially during the single-tooth engagement; the spalling length only has an effect on the beginning and ending of gear mesh stiffness reduction; the spalling location can affect the range of gear mesh stiffness reduction, and the range will reduce when the spalling location is close to the addendum. This study can provide a theoretical basis for spalling defect diagnosis.

This paper presents a new active disturbance rejection controller to solve the altitude and attitude control problem for a quadrotor unmanned aerial vehicle. The proposed method requires only the output information of the system. Using the pitch subsystem as an example, the proposed controller is designed by using dynamic surface control strategy incorporated with tracking differentiator, and extended state observer, which is used to estimate the uncertain disturbance. The estimate states of extended state observer are used to design the dynamic surface control law for altitude and attitude tracking problem of the quadrotor unmanned aerial vehicle. The stability analysis proves that a sufficient condition of the asymptotic stability of the extended state observer is achieved, the asymptotic stability of the closed-loop system can be guaranteed, and the tracking feedback error can made arbitrarily small by adjusting the controller parameters. Several simulation results are presented to corroborate that the proposed control method has better effectiveness and robustness.

This research has experimentally investigated on the influence on modal parameters of thin cylindrical shell coated with constrained layer damping (CLD). Firstly, based on vibration behavior of thin walled shell structure, an experiment system is established to measure modal parameters of the shell with high accuracy and efficiency, and the corresponding test methods and identification techniques which are suitable for the shell coated with CLD are proposed. Then, the shell is coated with different numbers and sizes of CLD pieces or in different coating positions of CLD rings, and modal parameters are obtained by the proposed test method. Finally, based on the accurate measured data, the influence on natural frequencies, mode shapes and damping ratios of TCS coated with CLD are analyzed and discussed in details. This research can provide technical support for CLD application on thin walled shell structure.

In this paper, phase and speed synchronization control of four eccentric rotors (ERs) driven by induction motors in a linear vibratory feeder with unknown time-varying load torques is studied. Firstly, the electromechanical coupling model of the linear vibratory feeder is established by associating induction motor׳s model with the dynamic model of the system, which is a typical under actuated model. According to the characteristics of the linear vibratory feeder, the complex control problem of the under actuated electromechanical coupling model converts to phase and speed synchronization control of four ERs. In order to keep the four ERs operating synchronously with zero phase differences, phase and speed synchronization controllers are designed by employing adaptive sliding mode control (ASMC) algorithm via a modified master-slave structure. The stability of the controllers is proved by Lyapunov stability theorem. The proposed controllers are verified by simulation via Matlab/Simulink program and compared with the conventional sliding mode control (SMC) algorithm. The results show the proposed controllers can reject the time-varying load torques effectively and four ERs can operate synchronously with zero phase differences. Moreover, the control performance is better than the conventional SMC algorithm and the chattering phenomenon is attenuated. Furthermore, the effects of reference speed and parametric perturbations are discussed to show the strong robustness of the proposed controllers. Finally, experiments on a simple vibratory test bench are operated by using the proposed controllers and without control, respectively, to validate the effectiveness of the proposed controllers further.

Two types of blade-tip rubbing due to the static misalignment of the bladed-disk center and casing center and casing deformation are simulated. By applying aerodynamic load in the blade lateral/flexural direction, vibration responses due to blade-casing rubbing are analyzed under the run-up process with constant angular acceleration and the steady-state process at 10000 rev/min. The effects of some parameters, such as the static misalignment e c, casing stiffness k c and casing deformation n p, on the system vibration responses are also illustrated by spectrum cascades, time-domain waveforms of displacement, normal rubbing forces, amplitude spectra and the impulse P in a single blade-casing rubbing period. The results show that blade-tip rubbing will cause amplitude amplification and harmonic resonance phenomena when the multiple frequencies (nf r) of rotational frequency (f r) coincide with the first three flexural dynamic frequencies of the blade (f n1, f n2 and f n3). For example, the displacement amplitudes at 3f r, 14f r and 38f r are large and the vibration is dominant near f n1. In addition, the casing deformation mainly excites the dominant Blade passing frequency (BPF), which is related to the casing deformation coefficient n p. By comparing these impulse values, for the selected parameters in this paper, the casing stiffness has a greater effect on impulse than the static misalignment and casing deformation coefficient. The impulse shows a linear increase trend with the increasing static misalignment, and it decreases under the large n p because the contact time decreases with the increase of n p.

Due to the effects of gear flexibility, the extended tooth contact (ETC) can appear, which is the phenomenon that the incoming tooth pair gets into contact ahead of the theoretical start of contact and the outgoing tooth pair is out of contact later than the theoretical end of contact. A large calculation error for the time-varying mesh stiffness (TVMS) calculation can be caused if the effects of ETC are ignored, especially under the larger torques. In this paper, an improved analytical method (IAM) suitable for gear pairs with tip relief is established to determine time-varying mesh stiffness (TVMS), where the effects of ETC, nonlinear contact stiffness, revised fillet-foundation stiffness, and tooth profile modification are considered. Based on the improved analytical model, TVMS under different torques, lengths, and amounts of profile modification is compared with that obtained from analytical finite element approach [29] and from FE method. The results show that TVMS obtained from the IAM agrees well with that from FE method and from analytical FE approach [29], and the computational efficiency of the IAM is also much higher than that of FE method.

This paper aims at the blade-casing rubbing in a shaft–disk–blade (SDB) system including shaft, disk, blade and bearing, and focuses the effects of stagger angles of blades, rotational speeds and casing stiffness on the rubbing-induced vibration responses of the SDB system and casing. Firstly, a finite element (FE) model of an SDB system is developed, and the rubbing between the blade-tip and casing is simulated using contact dynamics theory. In the proposed model, Timoshenko beam elements are adopted to simulate the shaft and the blade, and shell elements to simulate the disk, and spring-damping elements to simulate the ball bearings. A point–point contact element is adopted to simulate the blade-casing rubbing. Moreover, the augmented Lagrangian method is utilized to deal with contact constraint conditions, and the Coulomb friction model is used to simulate the friction between the blade and casing. The proposed model is also validated by comparing the natural frequencies with those obtained from the published literature. The results indicate that (1) amplitude amplification phenomena can be observed when the multiple frequency components coincide with the torsional natural frequency of the SDB system and the bending natural frequencies of the blades under rotational state; (2) the torsional vibration features of the SDB system with blade-tip rubbing are more significant than the lateral vibration features of the shaft; (3) the torsional vibration of the SDB system increases, and the blade bending vibration reduces with the increase of the stagger angle of the blade; (4) period-2 motion may appear under the large casing stiffness and high rotational speeds, and the torsional vibration of the SDB system and blade bending vibration tend to increase with the increasing casing stiffness.

In this paper, a mesh stiffness model is developed for profile shifted gears with addendum modifications and tooth profile modifications (TPMs). The time-varying mesh stiffness (TVMS), load sharing factor (LSF), loaded static transmission error (LSTE) and non-loaded static transmission error (NLSTE) of a profile shifted spur gear pair with TPMs are obtained by the analytical model. The optimum profile modification curve under different amounts of TPM is determined by analyzing the LSTE first. Then, considering the effect of NLSTE, finite element (FE) model of a geared rotor system is established. The system vibration responses under different TPM curves are analyzed and the optimum modification curve is further evaluated by amplitude frequency responses. The results show that the optimum modification curve is related to the amount of TPM and modification coefficients. The comparison of the optimum profile modification curves is evaluated by LSTE and vibration responses, which shows that the optimum modification curve should be determined by evaluating the vibration response of the geared rotor system in the low mesh frequency range.

An analysis of nonlinear behaviors of pressure thin-wall pipe segment with supported clearance at both ends was presented in this paper. The model of pressure thin-wall pipe segment with supported clearance was established by assuming the restraint condition as the work of springs in the deformation directions. Based on Sanders shell theory, Galerkin method was utilized to discretize the energy equations, external excitation, and nonlinear restraint forces. And the nonlinear governing equations of motion were derived by using Lagrange equation. The displacements in three directions were represented by the characteristic orthogonal polynomial series and trigonometric functions. The effects of supporting stiffness and supported clearance on dynamic behavior of pipe wall were discussed. The results show that the existence of supported clearance may lead to the changing of stiffness of the pipe vibration system and the dynamic behaviors of the pipe system show nonlinearity and become more complex; for example, the amplitude-frequency curve of the foundation frequency showed hard nonlinear phenomenon. The chaos and bifurcation may emerge at some region of the values of stiffness and clearance, which means that the responses of the pressure thin-wall pipe segment would be more complex, including periodic motion, times periodic motion, and quasiperiodic or chaotic motions.

The load distribution of balls is the key to affect the gradual wear and the life of the ball screw. Because the structure of ball screw mechanism is complex and closed, it is difficult to experimentally determine the contact loads of balls between the nut and the screw. This paper presents a novel method to analyze the static load distribution of ball screws, which takes into account the nut position variation. In order to investigate the load distribution conveniently, an appropriate transformation coordinate system is established. The axial, torsional and lateral elastic deformations are comprised into the geometric relationship of the groove centers of the screw and the nut by representing the ball screw as beam elements. The nonlinear equations, which take into account the Hertzian contact deformation, are solved iteratively by using the Newton-Raphson numerical method to determine the contact load of each ball. The detailed analysis for the distribution of contact load, contact angle and torsion angle when the nut is located at different positions of the screw. Furthermore, the effects of axial load and original contact angle on load distribution are analyzed.

An improved rotor-blade dynamic model is developed based on our previous works (Ma et al. in J Sound Vib, 337:301–320, 2015; J Sound Vib 357:168–194, 2015). In the proposed model, the shaft is discretized using a finite element method and the effects of the swing of the rigid disk and stagger angles of the blades are considered. Furthermore, the mode shapes of rotor-blade systems can be obtained based on the proposed model. The proposed model is more accurate than our previous model, and it is also verified by comparing the natural frequencies obtained from the proposed model with those from the finite element model and published literature. By simplifying the casing as a two degrees of freedom model, the single- and four-blade rubbings are studied using numerical simulation and experiment. Results show that for both the single- and four-blade rubbings, amplitude amplification phenomena can be observed when the multiple frequencies of the rotational frequency $$(f_\mathrm{r})$$ coincide with the conical and torsional natural frequencies of the rotor-blade system, natural frequencies of the casing and the bending natural frequencies of the blades. In addition, for the four-blade rubbing, the blade passing frequency (BPF, $$4f_\mathrm{r})$$ and its multiple frequency components also have larger amplitudes, especially, when they coincide with the natural frequencies of the rotor-blade system or casing; the four-blade rubbing levels are related to the rotor whirl, and the most severe rubbing happens on the blade located at the right end of the whirl orbit.

A new dynamic model of rotor–blade systems is developed in this paper considering the lateral and torsional deformations of the shaft, gyroscopic effects of the rotor which consists of shaft and disk, and the centrifugal stiffening, spin softening and Coriolis force of the blades. In this model, the rotating flexible blades are represented by Timoshenko beams. The shaft and rigid disk are described by multiple lumped mass points (LMPs), and these points are connected by massless springs which have both lateral and torsional stiffness. LMPs are represented by the corresponding masses and mass moments of inertia in lateral and rotational directions, where each point has five degrees of freedom (dofs) excluding axial dof. Equations of motion of the rotor–blade system are derived using Hamilton׳s principle in conjunction with the assumed modes method to describe blade deformation. The proposed model is compared with both finite element (FE) model and real experiments. The proposed model is first validated by comparing the model natural frequencies and vibration responses with those obtained from an FE model. A further verification of the model is then performed by comparing the model natural frequencies at zero rotational speed with those obtained from experimental studies. The results shown a good agreement between the model predicted system characteristics and those obtained from the FE model and experimental tests. Moreover, the following interesting phenomena have been revealed from the new model based analysis: The torsional natural frequency of the system decreases with the increase of rotational speed, and the frequency veering phenomenon has been observed at high rotational speed; The complicated coupling modes, such as the blade–blade coupling mode (BB), the coupling mode between the rotor lateral vibration and blade bending (RBL), and the coupling mode between the rotor torsional vibration and blade bending (RBT), have also been observed when the number of blades increases.

This paper investigates the dynamic behaviors of a perforated gear system considering effects of the gear crack propagation paths and this study focuses on the effects of a crack propagating through the rim on the time-varying mesh stiffness (TVMS) and vibration responses. Considering the effects of the extended tooth contact, a finite element (FE) model of a gear pair is established based on ANSYS software. TVMS of the perforated gear with crack propagating through tooth and rim are calculated by using the FE model. Furthermore, a lumped mass model is adopted to investigate the vibration responses of the perforated gear system. The results show that there exist three periods related to slots of the gear body in a rotating period of the perforated gear. Gear cracks propagating through tooth and rim both reduce the gear body stiffness and lead to reduction of TVMS besides the crack tooth contact moment, and the TVMS weakening for the former is less than that for the latter. Moreover, the results also show that the gear crack propagating through the rim (CPR) has a greater effect on vibration responses than the gear crack propagating through the tooth (CPT) under the same crack level. Vibration level increases with the increasing crack depth, especially for the gear with CPR.

Based on our previous work (Ma et al., 2014, Engineering Failure Analysis , 44, 179–194), this paper presents an improved analytical model (IAM) for the time-varying mesh stiffness (TVMS) calculation of cracked spur gears. In the improved analytical model, the calculation error of TVMS under double-tooth engagement due to repeatedly considering the stiffness of the fillet-foundation is revised, and the effects of reduction of fillet-foundation stiffness of cracked gears and extended tooth contact (ETC) are also considered, which have a great influence on TVMS, especially under the condition of large torques and crack levels. Moreover, the comparisons among the IAM, traditional analytical model (TAM) and finite element (FE) model are also carried out under different torques and crack depths. IAM is also verified by comparing TVMS and vibration responses obtained by FE model, which can be considered as a gauge to evaluate the calculation error. The results show that the maximum error of IAM is about 12.04%, however, that of TAM can be up to 32.73%.

Dynamic characteristics of cracked gear systems, also known as cracked-gear rotor systems, have received increasing interests among industry and academy in the past two decades. This paper reviews published papers on the dynamics of cracked gear systems. These studies mainly focused on three topics: crack propagation prediction, time-varying mesh stiffness (TVMS) calculation and vibration response calculation; Study objects involve the spur gear, helical gear and planetary gear; Different modeling methods including analytical method, finite element (FE) method, combined analytical-FE approach were adopted. More specifically, this review is composed of three related parts according to the above three topics. The first part involves the prediction of the crack propagation path based on two-dimensional (2D) or three-dimensional (3D) gear models, which provides a basis for the hypothesis of crack path in the process of TVMS calculation of cracked gear pairs. The second part summarizes the TVMS calculation methods including analytical methods, FE methods, combined analytical-FE approaches and experimental methods. The final part reviews the dynamic models for vibration analysis of cracked gear systems including lumped mass models and FE models, where the crack effects are characterized by introducing TVMS of cracked gear pairs into the system dynamic models. The well known open problems about cracked gear dynamics are finally stated, and some new research interests are also pointed out. The review will provide valuable references for future studies on dynamics of cracked gears.

As the traditional Fourier transformation method cannot process the non-linear and non-stationary vibration signal of the cracking rotor system, the 3-D waterfall spectrum in combination with reassigned wavelet scalogram method is presented to analyze the temporal frequency characteristics of the crack fault. Three different crack depths of rotor crack faults were simulated on a cracking rotor test-bed. The experiment results show that the present method is effective in determining the frequency structure of the signal, especially the twofold frequency component, which can be measured at half of the critical speed. The present method is an effective tool to diagnose cracks in rotor system and the frequency characteristic can be used as references in the diagnosis.

The dynamic behaviors and stability of the linear guide considering contact actions are studied by multi-term incremental harmonic balance method (IHBM). Based on fully considering the parameters of the linear guide, a static model is developed and the contact stiffness is calculated according to Hertz contact theory. A generalized time-varying and piecewise-nonlinear dynamic model of the linear guide is formulated to perform an accurate investigation on its dynamic behaviors and stability. The numerical simulation is used to confirm the feasibility of the approach. The effects of excitation force and mean load on the system are analyzed in low-order nonlinearity. Multi-term IHBM and numerical simulation are employed to the effect of high-order nonlinearity and show the transition to chaos. Additionally, the effects of preload, initial contact angle, the number and diameter of balls are discussed.

Owing to the effect of gear flexibility, the extended tooth contact occurs, which is a phenomenon that the incoming tooth pair enters contact earlier than the theoretical start of contact, and the outgoing tooth pair leaves contact later than the theoretical end of contact. Considering the effects of the extended tooth contact and tooth root crack on the time-varying mesh stiffness (TVMS), a finite element (FE) model of a spur gear pair in mesh is established by ANSYS software. TVMS under different crack depths at constant rated torque (60 Nm) are calculated based on the FE model. Then, a FE model of a geared rotor system is developed by MATLAB software. Frequency-domain features, statistical features (Kurtosis and RMS) and instantaneous energies based on empirical mode decomposition (EMD) under different crack depths are calculated at 1000 rev/min and the corresponding measured results are also performed by model experiment. The results show that considering the effect of extended tooth contact, TVMS of crack gear pair becomes smooth from double-tooth engagement to single-tooth engagement; the gear crack has reduced the gear body rigidity, which leads to the reduction of stiffness of the healthy teeth and has little effect on the vibration response. The instantaneous energy can be chosen as a distinguishing indicator to qualitatively diagnose the gear cracks with different levels.

The influence of shaft bending on the coupling vibration of rotor-blades system is nonignorable. Therefore, this paper analyzed the influence of shaft bending on the coupling vibration of rotor-blades system. The vibration mode function of shaft under elastic supporting condition was also derived to ensure accuracy of the model as well. The influence of the number of blades, the position of disk, and the support stiffness of shaft on critical speed of system was analyzed. The numerical results show that there were two categories of coupling mode shapes which belong to a set where the blade’s first two modes predominate in the system: shaft-blade (SB) mode and interblade (BB) mode due to the coupling between blade and shaft. The BB mode was of repeated frequencies of (Nb-2) multiplicity for number blades, and the SB mode was of repeated frequencies of (2) multiplicity for number blades. What is more, with the increase of the number of blades, natural frequency of rotor was decreasing linearly, that of BB mode was constant, and that of SB mode was increasing linearly. Natural frequency of BB mode was not affected while that of rotor and SB mode was affected (changed symmetrically with the center of shaft) by the position of disk. In the end, vibration characteristics of coupling mode shapes were analyzed.

The identification of the loss factor of fiber-reinforced composite based on complex modulus method is presented. Firstly, the damping model of fiber-reinforced composite plate is established, and the relation between each loss factor and modal damping ratio is deduced based on the complex modulus method. Then, the least square relative error function is formed by using the modal damping ratio obtained in the experimental test, and the appropriate step-size is selected in the range of 0~10% to calculate the loss factor. Next, the identification procedure of loss factor of such composite material is summarized, and the corresponding identification procedure is realized based on self-designed MATLAB program. Finally, TC300 carbon/epoxy composite plate is taken as an example to carry out a case study, and its loss factors along the longitudinal, transverse, and shear direction are identified by the complex modulus method. By comparing the measured damping results obtained in this paper and the calculated damping results based on the Adams-Bacon model with the same loss factor, it is found that the corresponding maximum deviation between them is less than 15%, so the correctness of such identification method has been verified indirectly, which can be used to identify loss factor of fiber-reinforced composite with high precision and efficiency.

On the basis of Ref. Ma et al. (2013, Mechanical System and Signal Processing, 38, 137–153), the paper adopts a finite element (FE) method to investigate the complicated dynamic characteristics of a rotor system with two types of limiters under multilateral contact and friction conditions when the rubbing occurs between the rotor and limiters. The current study focuses on the effects of different stator/limiter forms on the vibration responses of the rotor–stator coupling system. First, FE models of the rotor with two types of limiters (four pin shaped stators and three pin shaped stators) are established. Then four and three point–point contact elements are used to simulate the rubbing between rotor and the two types of limiters. These contact elements describe the coupling of the rotor and the stators by the augmented Lagrangian method. Complicated vibration responses of the rotor system with two types of limiters under different rotating speeds are analyzed by spectrum cascades, rotor orbits, normal rubbing forces, amplitude spectrum, time-domain waveform and stator acceleration. The results show that the vibration responses of the rotor under the first type of limiter are more stable than those under the second type of limiter by observing the intensity and times of the rubbing, the magnitude of normal rubbing force and the regularity of frequency components caused by rubbing.

Considering the misalignment of gear root circle and base circle and accurate transition curve, an improved mesh stiffness model for a healthy gear pair is proposed and validated by the finite element method (FEM). Based on the improved method, three mesh stiffness calculation methods (method 1: straight lines for crack path and limiting line; method 2: straight line for crack path and parabolic curve for limiting line proposed in Ref. [1] ; method 3: parabolic curves for crack path and limiting line) for cracked gear pair are presented and compared with FEM. The results show that there is a significant difference between method 1 and FEM under large crack condition and the results of methods 2 and 3 are quite close to FEM result, which also shows that the parabolic curve as a limiting line is appropriate. Mesh stiffness of method 2 is very close to that of method 3, which also shows that it is acceptable to assume the crack path to be a straight line.

Using a lumped mass model of a single rub-impact rotor system considering the gyroscopic effect, the stability and steady-state response of the rotor system are investigated in this paper. The contact between the rotor and the stator is described by the simple Coulomb friction and piecewise linear spring models. An algorithm combining harmonic balance method with pseudo arc-length continuation is adopted to calculate the steady-state vibration response of a nonlinear system. Meanwhile, Hill’s method is used to analyze the stability of the system. The nonlinear dynamic characteristics of the system are investigated when the gap size, stator stiffness and unbalance are regarded as the control parameters. The results show that the gap size determines the location of the rub-impact; besides, the smaller gap can improve the stability of the system. The unsteady motion can be found as the stator stiffness increases. Moreover, the unbalance directly affects vibration amplitude, which becomes greater with the increasing imbalance.

Extensive studies on rotor systems with single or coupled multiple faults have been carried out. However these studies are limited to single-span rotor systems. A finite element model for a complex rotor-bearing system with coupled faults is presented. The dynamic responses of the rotor-bearing system are obtained by using the rotor dynamics theory and the modern nonlinear dynamics theory in connection with the continuation-shooting algorithm(commonly used for obtaining a periodic solution for a nonlinear system) for a range of rub-impact clearances and crack depths. The stability and Hopf instability of the periodic motion of the rotor-bearing system with coupled faults are analyzed by using the procedure described. The results indicate that the finite element method is an effective way for determining the dynamic responses of such complex rotor-bearing systems. Further for a rotor system with rub-impact and crack faults, the influences of the clearances are significantly different for different rub-impact stiffness. On the contrary, the influence of crack depths is rather small. The instability speeds of the rotor-bearing system increase due to the presence of the crack fault. The results obtained using the new finite element model, presented for computation and analysis of dynamic responses of the rotor-bearing systems with coupled faults, are in accordance with measurements in experiment. The formulations given can be used for diagnosis of faults, vibration control, and safe and stable operations of real rotor-bearing systems.

The finite element method is adopted to establish the dynamical models of blade, bladed disk and blades-diskshaft assemblies. Based on the analysis of mistuned structure and the dynamic characteristics of model in different levels, it gives the vibration mode distribution of different models. The research shows that the characteristics of the bladed disk and shaft have a huge difference, where some modes are caused by the strongly split and coupled vibration. The mistuned effects are likely to cause different coupled vibrations of blades between the blades-disk model and the blades-disk-shaft model. Meanwhile, it shows the frequency separation and concentration, and misses the system mode and the local vibration, which bring some difficulties for designing the blade-rotor system. In this paper, the results may provide a certain reference for blade-rotor system design and diagnosis.

Considering the misalignment of gear root circle and base circle and accurate transition curve, an improved mesh stiffness model for healthy gear is proposed, and it is validated by comparison with the finite element method. On the basis of the improved method, a mesh stiffness model for a cracked gear pair is built. Then a finite element model of a cracked gear coupled rotor system in a one-stage reduction gear box is established. The effects of crack depth, width, initial position, and crack propagation direction on gear mesh stiffness, fault features in time domain and frequency domain, and statistical indicators are investigated. Moreover, fault features are also validated by experiment. The results show that the improved mesh stiffness model is more accurate than the traditional mesh stiffness model. When the tooth root crack appears, distinct impulses are found in time domain vibration responses, and sidebands appear in frequency domain. Amplitudes of all the statistical indicators ascend gradually with the growth of crack depth and width, decrease with the increasing crack initial position angle, and firstly increase and then decrease with the growth of propagation direction angle.

In order to simulate the rubbing dynamics of the multi-blades/flexible casing, the finite element model (FEM) of the blisk-oval casing system with elastic supports is established using the self-programmed beam-shell-spring hybrid elements in combination with two types of self-programmed interfacial coupling elements (ICEs). The rotating effects such as centrifugal stiffening, spin softening and Coriolis effects for the blade and gyroscopic effect for the disk are all included in the system. For improving the computational efficiency, a two-step hybrid Craig-Bampton method is utilized to build the reduced blisk model and one step reduction for the casing is sufficient. Corresponding accuracy is verified via the frequency convergence analysis. Then the central difference method in combination with the Lagrange multiplier method is applied to solve the rubbing dynamic responses of the reduced system under different rotating speeds and supporting stiffness. The results show that (1) the Craig-Bampton method is appropriate for reducing the number of dimensions and improving the computational efficiency under the premise of ensuring model accuracy; (2) the fundamental frequency in the vibration responses of the casing is the product of the rotating frequency of the blisk and the number of contact zone on the casing, and period-1 motion/chaotic motion indicates the equal/unequal and radially symmetric contact zones on the casing, respectively; (3) the effects of the casing supporting stiffness on the rubbing characteristics of the system are more significant than those of the blisk supporting stiffness, and the vibration responses of the casing relative to the blisk are more sensitive to the system state variations like the occurrence of the resonance.

Aiming at the oil film instability of the sliding bearing at high speeds, a rotor test rig is built to study the non-linear dynamic behaviours caused by the first- and second-mode instability. A lumped mass model (LMM) of the rotor system considering the gyroscopic effect is established, in which the graphite self-lubricating bearing and the sliding bearing are simulated by a spring–damping model and a nonlinear oil film force model based on the assumption of short bearings, respectively. Moreover, a finite element model is also established to verify the validity of the LMM. The researches focus on the effects of two loading conditions (the first- and second-mode imbalance excitation) on the onset of instability and nonlinear responses of the rotor-bearing system by using the amplitude–frequency response, spectrum cascade, vibration waveform, orbit, and Poincare map. Finally, experiments are carried out on the test rig. Simulation and experiment all show that oil film instability can excite complicated combination frequency components about the rotating frequency and the first-/second-mode whirl/whip frequency.