Modeling and Analysis of a Ball and Beam System Including Impacts and Dry Friction

The Ball and Beam system is a common didactical plant that presents a complex nonlinear dynamics. This comes from the fact that the ball rolls over the beam, which rotates around its barycenter. In order to deduce the system’s equations, composition of movement must be applied, using a non-inertial reference frame attached to the beam. In the Literature, a common hypothesis is to suppose that the ball rolls without slipping. If a viscous friction is supposed to be present, a simpler situation is obtained, where Lagrangean mechanics can be applied, and no contact force is known. Even then, the dynamics is very nonlinear. However, this model does not include all the relevant phenomena, such as ball’s slipping at higher beam’s inclination angles, dry friction between the ball and the beam, and impacts between: 1) the ball and the ends of the beam, and 2) the beam and the base (ground). These additions to the model impose the necessity to calculate, in a simulation setting, the contact forces, and the Newton’s approach to determine the system’s equations becomes more convenient. Also, discontinuities in the model are introduced, and the simpler mathematical object for model such systems are the differential inclusion systems. In this work, we deduce the Ball and Beam differential inclusion system, including dry friction and the impact between the ball and beam. We also present simulation results for the corresponding differential inclusion system in a typical situation.

Nonlinear Dynamics of a Beam-Rigid Body Microgyroscope

The nonlinear dynamics of a cantilever-beam-rigid-body MEMS gyroscope near primary resonance are studied by using a shooting method and long time integration. The microsensor includes a square beam carrying an eccentric end-rigid-body rotating about the longitudinal axis and under an electrostatic force. The mathematical model of the system is reduced by using the method of assumed modes. Using a shooting method and long time integration, the dynamic characteristics of the system are investigated and presented in terms of frequency-response plots and force-response curves. The bifurcation points are discussed and the regions of instability are characterized.

Integrity Diagnosis Method of Bolted Joint Based on Time Fluctuation of Reflection Intensity Caused by Nonlinear Wave Modulation

This study presents the integrity diagnosis method of the bolted joint based on nonlinear wave modulation. When the structure that has the contact interface is vibrating at low-frequency, the contact interface is tapping and clapping due to low-frequency vibration. In this condition, the scatter characteristics, such as wave transmissivity and reflectivity, of high-frequency waves in vicinity of the contact interface are fluctuated in synchronization with low-frequency excitation because of the contact acoustic nonlinearity. The time fluctuation of reflection intensity, which expresses the reflectivity in the specific location, of high-frequency waves at the contact interface is given as the reflection intensity map which plots time-spatial map. In this paper, experiment using the beam specimen which has single bolted joint is conducted to examine the performance of the evaluation index based on the fluctuation amplitude of reflection intensity.

LQG Control and Robustness Study for a Prestressed Membrane With Bimorph Actuators

Linear Quadratic Gaussian (LQG) control is developed for a prestressed square membrane with bimorph actuators attached to it. The membrane is modeled using the finite element method and the membrane is assumed to be clamped on all edges. After obtaining the mass, damping, stiffness and input matrices in second order form using the weak form Finite Element Method (FEM), the problem is represented in first order form to develop the LQG controller. To study the robustness of the system, the control and observer gain matrices developed for the nominal system are applied to systems obtained from the nominal system by modifying material properties and prestress.

Study of Snubbing Mechanism Using Finite Element Method

In the rotating machineries, large vibrations of a blade would result in fatigue crack, which is a great threaten to the safety. Therefore, it is of great importance to reduce the blade vibrations. Snubbing technique is a possible solution to this problem. A tiny gap is left between the shrouds of adjacent blades. While the forced vibration makes the relative displacement between two neighboring blades exceed the gap, the contact happens at the contact face of the shrouds, accompanied with friction and energy dissipation, which restricts the vibration. In this paper, a simplified model for a set of rotor blades is established, by using finite element method. The contact between the adjacent shrouds is considered. In this way, snubbing phenomenon can occur under forced vibration. Based on the model, modal analysis has been conducted. The 8x rev. frequency has been chosen as the excitation frequency. Under a certain amplitude of sine excitation, the circumferential vibration of the blades has been simulated. The vibration has been analyzed in the time domain. As expected, the blade motion is divided into four different states in one period. They are: non-contact, rebounding, sticky and escaping state. The four states had different mechanical and motion characteristics. The motion pattern for the set of blades has been also analyzed and the wave spreading along the bladerow has been described. Because of the snubbing mechanism, the waveform was distorted into serrated shape.

Control Concepts for a Vibration Absorber With an Adaptive Joint Connection

The use of absorbers to reduce vibrations of machines is common in industry and can be found in various applications. In most cases passive absorbers are used to cancel one particular eigenfrequency. The disadvantage of this solution is that due to the introduction of an additional degree-of-freedom two resonance peaks occur next to the absorbed eigenfrequency. Given the case that the machine operates in a wider frequency band these two eigenfrequencies could be excited and feature similarly high amplitudes. To address this concern, in the present case an adaptive absorber is used, which is able to adjust its eigenfrequency to the actually excited frequency. Therefore, the anti-resonance can be shifted such that a full cancellation of the resonance is possible. The absorber consists of a mass and two springs. One spring is fixed to the mass permanently and the second can be coupled to the system by an adaptive joint connection. The normal force in the frictional contact serves as control variable to achieve adaptivity of the dynamic eigenfrequency of the absorber. Two control concepts are presented. Both concepts include isolated curves characterizing the nonlinear relation between the dynamic stiffness and the related normal force based on simulations using the Harmonic Balance Method. Due to the isolation of the nonlinearity, linear control concepts like LQR can be applied, which is done in the present case. Furthermore, a direct control of the eigenfrequency is done. The adaptive absorber is applied to a simplified machine tool carriage.

Multi-Pulse Chaotic Dynamics of Four-Dimensional Non-Autonomous Nonlinear System for a Truss Core Sandwich Plate

In this paper, an extended high-dimensional Melnikov method is used to investigate global and chaotic dynamics of a simply supported 3D-kagome truss core sandwich plate subjected to the transverse and the in-plane excitations. Based on the motion equation derived by Zhang and the method of multiple scales, the averaged equation is obtained for the case of principal parametric resonance and 1:2 sub-harmonic resonance for the first-order mode and primary resonance for the second-order mode. From the averaged equation obtained, the system is simplified to a three order standard form with a double zero and a pair of pure imaginary eigenvalues by using the theory of normal form. Then, the extended Melnikov method is utilized to investigate the Shilnikov-type multi-pulse heteroclinic bifurcations and existence of chaos. The analysis of the extended Melnikov method demonstrates that there exist the Shilnikov-type multi-pulse heteroclinic bifurcations and chaos in the four-dimensional non-autonomous nonlinear system. Finally, the results of numerical simulations also show that for the nonlinear system of simply supported 3D-kagome truss core sandwich plate with the transverse and the in-plane excitations, the Shilnikov-type multi-pulse motion of chaos can happen and further verify the result of theoretical analysis.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

Thermal Effects on Internal and External Resonances of a Nonlinear Timoshenko Beam

Large amplitude vibrations of a Timoshenko beam under an influence of thermal and mechanical loadings are studied in the paper. The structural parameters of the beam are considered enabling internal resonance conditions. Moreover, it is assumed that the beam gets instantly temperature which is distributed along its length and thickness. The mathematical model represented by a set of partial differential equations takes into account coupled mechanical and thermal fields. The problem is transformed to a set of ODEs by the Galerkin method and three modes of a simply supported beam at both ends are studied. The effect of temperature on internal and external resonances is analysed on the basis of the proposed reduced model.

Vibrations of an Axial Bar Experiencing Periodic Unilateral Contact Using the Wavelet Balance Method

Efficiently predicting the vibratory responses of flexible structures which experience unilateral contact is becoming of high engineering importance. An example of such a system is a rotor blade within a turbine engine; small operating clearances and varying loading conditions often result in contact between the blade and the casing. The method of weighted residuals is a effective approach to simulating such behaviour as it can efficiently enforce time-periodic solutions of lightly damped, flexible structures experiencing unilateral contact. The Harmonic Balance Method (HBM) based on Fourier expansion of the sought solution is a common formulation, though it is hypothesized wavelet bases that can sparsely define nonsmooth solutions may be superior. This is investigated herein using an axially vibrating rod with unilateral contact conditions. A distributional formulation in time is introduced allowing periodic, square-integrable trial functions to approximate the second-order equations. The mixed wavelet Petrov-Galerkin solutions are found to yield consistent or better results than HBM, with similar convergence rates and seemingly more accurate contact force prediction.