Abstract A new formulation is developed for dynamic analysis of a rotating planar Timoshenko beam. The configuration of Timoshenko beam is described using its slope angle and axial and shear strains; hence, the shear locking problem can be naturally avoided. While six boundary conditions are needed for choices of trial functions of three dependent variables, there are only four boundary conditions that can be determined and two boundary conditions are undetermined. An accurate global spatial discretization method is used, where dependent variables are divided into internal and boundary-induced terms. Internal terms only need to satisfy homogeneous boundary conditions, which can be easily chosen as trigonometric functions. Boundary-induced terms are interpolated using dependent variables at boundaries that are taken as generalized coordinates. When the hub rotates at a constant angular velocity, nonlinear governing equations can be linearized for vibration analysis. Frequency veering and mode shift phenomena occur. Nonlinear dynamic responses of the system are then calculated and compared with those from the commercial software ADAMS, and they are in good agreement. Axial and shear strains of the beam and their spatial derivatives are also calculated. Since trial functions in the assumed modes method cannot satisfy undetermined boundary conditions, inaccurate results of strains and their spatial derivatives are obtained using the assumed modes method. Hence, use of the accurate global spatial discretization method in the current formulation is essential.

Abstract The track is mainly composed of track shoes, track pins and rubber bushing elements. In order to suppress the transversal vibration of the upper track during the smooth running process of the tracked vehicle, it is necessary to study the important factors affecting the frequency characteristics of the kinematic chain and their interaction. Unlike the conventional chain drive system, the track in the natural state has a bending rigidity due to the action of the rubber bushing. Based on the dynamic theory of axially moving beam, the differential equation of transversal vibration of a beam element is established. The entire upper track is assumed to be a continuous multi-span axially moving Euler-Bernoulli beam with an axial tension. Based on the Transfer Matrix Method of Multibody System, the transfer equation is obtained. According to the boundary conditions, the natural frequency of the system is solved. The correctness of the beam model hypothesis is verified by experiments. The results show that the first-order natural frequency of the upper track increases with the increase of the tension and the decrease of the vehicle speed. Through frequency analysis, the main excitation source for the transversal vibration of the track is the polygon effect produced by the meshing of the track and the sprocket. This study provides a theoretical basis for the vibration analysis and stability control of the upper track on the tracked vehicle.

Crop phenotyping is frequently used by breeders and crop scientists to monitor growth of plants and to relate them to plants genotypes. Seemingly, this contributes to better crop growth and results in higher yield. Instead of traditional crop monitoring, which is labor intensive, high-throughput phenotyping (HTP) platforms using ground-based vehicle have several advantages (in speed, efficiency, and cost) over manual methods. A wheeled mobile platform for HTP was developed, and automated data collection were performed for different traits of canola and wheat. These data were compared with manual measured data. In this paper, vibration analysis of a relatively long cantilever boom attached to a vehicle is reported. The paper investigates how different factors influence the boom attached to a regular farming machine, its vibration, and effects on phenotyping sensors attached to the boom.

Natural vibration characteristics and dynamics response of multiple launch rocket system (MLRS) are of fundamental importance from the viewpoint of vibration levels, firing dispersion, and stability. In this study, a new launch vehicle-supports-rockets coupling dynamic model for a practical MLRS is established. Rui method, namely the transfer matrix method for multibody systems (MSTMM) is a new and efficient method for multibody system dynamics (MSD) and is used to obtain the vibration characteristics and dynamics response. The dynamics model, the topology figure of dynamics model, the transfer equations of elements, the overall transfer equation, the eigenfrequency equation, the body dynamics equations, the generalized coordinate equations, and the dynamics simulation system for the MLRS are established. Based on the advantages of MSTMM in studying MSD, the vibration characteristics and dynamics response of complex MLRS are computed rapidly. Finally, the new model is validated in three ways: (1) modal experiment of MLRS, (2) launch dynamics experiment of non-full loading rockets, and (3) launch dynamics experiment of full loading rockets. The results show that the proposed model can not only simulate the natural vibration characteristics of the MLRS but also effectively perform dynamic simulations of the MLRS during launching process.

Transfer matrix method is a practical technology for vibration analysis of engineering mechanics. In this paper, Differential Quadrature Discrete Time–Transfer Matrix Method (DQ-DT-TMM) is presented for solving vibration mechanics. Firstly ordinary differential equations of the sub-structure or the element of the mechanical system are determined by classical mechanics rule and transformed as a set of algebraic equations at some discrete time points by the application of differential quadrature method. Then by extending the state vector of transfer matrix method, new transfer equations and transfer matrices of the sub-structures of the mechanical system are developed. The Riccati transform can be used to improve the computational convergence of the method. Several numerical examples show the proposed method can be regarded as an efficient tool for transient response analysis of vibration system.

The paper presents the dynamic response of an Euler-Bernoulli beam supported by an elastic foundation and subjected to a moving step load. The Riccati transfer matrix method for linear multibody systems (Riccati MSTMM) is employed to find eigenfrequencies and mode shapes of the supported beam. A comparison of results obtained with the finite element method (FEM) indicates that the Riccati MSTMM is more accurate when using the same number segments. Based on these results, the dynamic response of the beam with moving step load is investigated for different propagation velocities by mode superposition, and the effect of loads is discussed.

Vibration performances of roller bearings are greatly affected by various localized defects. Thus, it is very important to analyze vibration characteristics of the roller bearings with the localized defects. In this paper, a nonlinear dynamic model is developed to formulate the effect of a localized defect on the vibration characteristics of a cylindrical roller bearing (CRB). A bump defect is formulated in this model. The defect profile is defined as a spherical one. The time-varying displacement excitation caused by the defect is modelled. Effects of the defect sizes on the vibration characteristics of the roller bearing are discussed. The simulation results show that the developed method can provide some guidance for understanding the vibration characteristics of the CRB with a bump defect.

This paper presents an analytical model for the vibration analysis of V-shaped beam electrothermal microactuators. With the consideration of the unique geometrical feature, i.e., the V-shaped beam with the two inclined half-spans interconnecting at the center of the beam, both lateral and longitudinal flexible deformations are included the vibration model established in this work. Furthermore, the vibrations in the lateral and longitudinal directions are coupled, therefore both the boundary and continuity conditions are defined to obtain the frequency equation. The natural frequencies can be calculated by solving the frequency equation numerically. To verify the established vibration model, the comparison is conducted between analytical results and finite-element (FE) simulation results using commercial software ANSYS. The comparison of the modal shapes and frequencies demonstrate the results based on the established analytical model well agree with FE simulation results. This work provides the first insight study on the vibration analysis of the V-shaped beam electrothermal microactuators with aims to lay the foundation of establishing thermo-mechanical model that could be used for the control of V-shaped actuators, and the improvement design in achieving optimal dynamic performance.

This paper focuses on nonlinear forced vibration analysis of a free-form conveying fluid nanotube. Non-Uniform Rational B-Splines (NURBS) is used to model the free-form curvature of the nanotube, analytically. In order to develop the ordinary differential equations of motion, the Euler-Bernoulli beam theory and Galerkin method are implemented and the frequency response and the primary resonance of the nanotube under a harmonic excitation are studied. The effects of the free-form curvature of the nanotube and its physical characteristic on the nonlinear vibration behavior of the system are discussed as a parametric study.

Aeolian vibrations are the major cause for the failure of conductor cables. Using a Stockbridge damper reduces these vibrations and increases the life span of the conductor cable. Designing an efficient Stockbridge damper that suits the conductor cable requires a robust mathematical model with minimum assumptions. However it is not easy to analytically model the complex geometry of the messenger. Since, the messenger is a bunch of thin helical wires with nonlinear contact conditions. Therefore an equivalent stiffness must be determined so that it can be used in the analytical model. This paper examines the bending stiffness of the cable and discusses the effect of this stiffness on the natural frequencies. The obtained equivalent stiffness compensates for the assumption of modeling the messenger as a rod. The results from the free vibration analysis of the analytical model with the equivalent stiffness is validated using the full scale finite element model of the Stockbridge damper.

In this work the derivation of a vibration model for an H-rotor/Giromill blade is investigated. The blade is treated as a uniform straight elastic Euler-Bernoulli beam under transverse bending and twisting deformation. The derivation of the energy equations for the bending and twisting blade and a simplified aerodynamic model is issued. Lagrange’s equations are applied to assumed modal coordinates to obtain nonlinear equations of motion for bend and twist. A single quasi-steady airfoil theory is applied to obtain the aeroelastic loads. The behavior of the linearized equation for bend only is examined.

Vibration analysis is currently the most efficient, non-invasive way to monitor the condition of the gears. Faults in gears can be of two distinct types, distributed or local. Many fault detection methods are effective for one type of fault or the other but not both. Also, many methods have the inconvenience that they are not simple and/or require initial information about the state of gear. In this paper, several methods are proposed with the objective of finding a filter-free, simple and efficient method for the detection of both types of faults. The calculus enhanced energy operator (CEEO), previously designed for fault detection in bearings, is proposed here for the first time on gears. Two new methods, the EO123 and EO23, based on the original energy operator are also proposed and evaluated. All the proposed methods are filter free, computationally simple and can handle a certain level of noise and interference. With the exception of low rotational frequencies of the gears, it is demonstrated via simulated and experimentally obtained signals that the CEEO method can handle noise better than the other proposed methods and that the EO23 method can handle interference better than the others. Different conditions determine the effectiveness of the methods.

The aim of this study is to model and investigate the nonlinear transversal vibration of a carbon nanotube carrying an intermediate mass along the structure considering the nonlocal and non-classical theories. Due to the application of the proposed system in sensors, actuators, mass detection units among others, the analysis of forced vibration of such systems is of an important task being considered here. The governing equation of motion is developed by combining the Euler-Bernoulli beam theory and the Eringen non-local theory. The Galerkin approach is employed to obtain the governing differential equation of the system and the transient beam response for the clamped-hinged boundary condition. A strong perturbation method is utilized to solve the equation obtained and the system responses subjected to a harmonic excitation is examined. The steady-state motion is studied and the frequency response in an analytical form is obtained. Finally, results are evaluated for some numerical parameter values and their effect on the frequency responses are presented and fully discussed.

This paper presents vibration analysis and structural optimization of a self-assembled structure for swimming. The third mode shape of the structure in the longitudinal direction resembles the body waveform of a swimming eel fish. At the final destination, the box self-assembles using shape memory alloys. MFCs (Piezoelectric Micro Fiber Composites) are actuated at the fundamental natural frequency of the structure. This excites the primary mode of resonance. We optimize the thickness of the panels and the stiffness of the joints to most efficiently generate the swimming waveforms that resembles the body waveform of eel. Traveling wave is generated using two piezoelectric batches actuators bonded on the first and fourth segments of the beams in the longitudinal direction. Excitation of the piezoelectrics results in coupled system dynamics equations that can be translated into generation of waves. Theoretical analysis based on the distributed parameter model was conducted in this paper. A scalar measure of the traveling to standing wave ratio was created using 2-dimensional Fourier Transform of the wave form. The results then were compared to common method in the literature for assessment of standing to traveling wave ratio.

Nonlinear harmonic vibration analysis of a clamped-clamped micro-beam is studied in this paper. Nonlinear forced vibration of a special kind of micro-actuators is examined for the first time. Galerkin method is employed to derive the equation of motion of the micro-beam with two symmetric potential wells. An electric force composed of DC and AC components is applied to the structure. Multiple Scales method (MSM) is used to solve the nonlinear equation of motion. Primary and secondary resonances are taken into account and steady-state response of the microbeam is obtained. A parametric study is then carried out to investigate the effects of different parameters on the amplitude-frequency curves. Finally, phase plot and Poincare map have been taken into consideration to investigate the influence of the amplitude of the harmonic excitation on stability of the microelectromechanical system.

The well-known concept of normal mode for linear systems has been extended to the framework of nonlinear dynamics over the course of the 20th century, initially by Lyapunov, and later by Rosenberg and a growing community of researchers in modal and vibration analysis. This effort has mainly targeted nonlinear smooth systems — the velocity is continuous and differentiable in time — even though systems presenting nonsmooth occurrences have been increasingly studied in the last decades to face the growing industrial need of unilateral contact and friction simulations. Yet, these systems have nearly never been explored from the standpoint of modal analysis. This contribution addresses the notion of modal analysis of nonsmooth systems. Developments are illustrated on a seemingly simple 2-dof autonomous system, subject to unilateral constraints reflected by a perfectly elastic impact law. Even though friction is ignored, its dynamics appears to be extremely rich. Periodic solutions are sought for given numbers of impacts per period and nonsmooth modes are illustrated for one and two impacts per period in the form of two-dimensional manifolds in the phase space. Also, an unexpected bridge between these modes in the frequency-energy plots is observed.

Nonlinear forced vibration of fluid-conveying nanotubes based on Euler-Bernoulli beam theory under electromagnetic actuation is studied. The nanotube is modeled as cantilever type beam and the effects of fluid motion and external harmonic excitation are considered in the governing equation of the structure vibration. The Galerkin procedure is applied in order to discretize the governing equation of vibration of the system. The well-known multiple scales method is utilized to investigate the primary resonance in the forced vibration of nanotubes. The effects of various parameters, namely, fluid velocity, position of applied force, aspect ratio and electromagnetic excitation on the primary resonance of the system are fully investigated. It is revealed that the electromagnetic excitation is highly influential on the frequency response of the considered system.

In this paper, nonlinear vibration of a triangular shape plate, with several stiffeners, is studied. The governing equation of transversal deflection of the plate, with considering the effects of orthotropic characteristics and external excitation, is analyzed. The ordinary differential equation for the time response of the system, through employing the Galerkin method, is obtained; and the frequency response of the plate-shape structure — using the multiple scale method — is determined. A robust genetic-based multi-objective optimization technique is employed to optimize the system’s response by finding the optimum values of the geometry and locations of the plate’s stiffeners. The influence of various parameters on the optimization results is investigated. According to the results, the optimum design of the stiffeners leads to a better performance of the vibration response.

Several hybrid electric driveline topologies have been introduced for vehicles in order to reduce emissions and improve energy efficiency in the transportation sector. This study introduces a novel configuration for a hybrid electric bus driveline. The designed driveline is capable of operating in pure electric, series and parallel hybrid modes. The torsional vibration analysis of the proposed drivetrain is performed for different driving mode configurations. The finite element method is applied to solve the torsional natural frequencies and mode shapes of the driveline. The model contains mass-elastic data of drive shafts, a diesel engine, a permanent magnet generator and a traction motor, as well as clutches and couplings. Different modelling strategies are discussed, and a sensitivity analysis is performed in order to determine the effect of different modelling parameters on the system’s torsional frequencies and mode shapes. Different excitation frequencies arising from the diesel engine and electric drives are identified and compared to the natural frequencies.

In this paper, vibration response of a bridge under vehicular loads has been analyzed and consequently been controlled by two actuators using active Neural Network (NN) control strategy. The bridge and the vehicles are modeled as a simply-supported Euler Bernoulli beam possessing a simple support precisely located at the middle and one-DOF vehicles respectively. With that goal, equations of the coupled system of the bridge and the vehicles are derived using Hamilton’s principle, and then randomness of vibrations transferred to the bridge due to the alternative velocities of the vehicles added, latterly controlled by the designed NN control algorithm. The randomness includes 10 percent of vehicle initial speed. Deflection of the first quarter-span is considered as the output of the system and tried to be controlled by two symmetric forces whose desirable positions are determined by recent studies. Furthermore the sensitivity of the system response has been investigated by applying the different conditions in which the number of the moving vehicles varies. It is shown that the increment in the number of moving vehicles has led to unsolicited deflection specifically when two or more cars moves concurrently, whereas the NN algorithm could either perfectly or effectively get over the issue.