A Suitable Active Control for Suppression the Vibrations of a Cantilever Beam

In our consideration, a comparison between four different types of controllers for suppression the vibrations of the cantilever beam excited by an external force is carried out. Those four types are the linear velocity feedback control, the cubic velocity feedback control, the non-linear saturation controller (NSC) and the positive position feedback (PPF) controller. The suitable type is the PPF controller for suppression the vibrations of the cantilever beam. The approximate solution obtained up to the first approximation by using the multiple scale method. The PPF controller effectiveness is studied on the system. We used frequency-response equations to investigate the stability of a cantilever beam. We notified that, there is a good agreement between the analytical solution and the numerical solution.     

Efficient modal control strategies for active control of vibrations

Some efficient strategies for the active control of vibrations of a beam structure using piezoelectric materials are described. The control algorithms have been implemented for a cantilever beam model developed using finite element formulation. The vibration response of the beam to an impulse excitation has been calculated numerically for the uncontrolled and the controlled cases. The essence of the method proposed is that a feedback force in different modes be applied according to the vibration amplitude in the respective modes i.e., modes having lesser vibration may receive lesser feedback. This weighting may be done on the basis of either displacement or energy present in different modes. This method is compared with existing methods of modal space control, namely the independent modal space control (IMSC), and modified independent modal space control (MIMSC). The method is in fact an extension of the modified independent space control with the addition that it proposes to use the sum of weighted multiple modal forces for control. The proposed method results in a simpler feedback, which is easy to implement on a controller. The procedure is illustrated for vibration control of a cantilever beam. The analytical results show that the maximum feedback control voltage required in the proposed method is further reduced as compared to existing methods of IMSC and MIMSC for similar vibration control. The limitations of the proposed method are discussed.     

VIBRATION-BASED DIAGNOSTICS OF FATIGUE DAMAGE OF BEAM-LIKE STRUCTURES

The analytical investigation of vibration of damaged structures is a complicated problem. This problem may be simplified if a structure can be represented in the form of a beam with corresponding boundary and loading conditions. In this connection, free vibrations of an elastic cantilever Bernoulli-Euler beam with a closing edge transverse crack is considered in the present work as a model of a structure with a fatigue crack. The modelling of bending vibrations of a beam with a closing crack is realized based on the solutions for an intact beam and for a beam with an open crack. The algorithm of consecutive (cycle-by-cycle) calculation of beam mode shapes amplitudes is presented. It is shown that at the instant of crack opening and closing, the growth of the so-called concomitant mode shapes which differ from the initially given mode shape takes place. Moreover, each of the half-cycles is characterized by a non-recurrent set of amplitudes of concomitant modes of vibration and these amplitudes are heavily dependent on the crack depth.The vibration characteristics of damage based on the estimation of non-linear distortions of the displacement, acceleration and strain waves of a cracked beam are investigated, and the comparative evaluation of their sensitivity is carried out.     

Free vibration and stability of a cantilever beam attached to an axially moving base immersed in fluid

Free vibration and stability are investigated for a cantilever beam attached to an axially moving base in fluid. The equations of motion of the slender cantilever beam affiliated to an axially moving base at a known rate while immersed in an incompressible fluid are derived first. An “axially added mass coefficient” is taken into account in the obtained equations. Then, a coordinate transformation is introduced to fix the boundaries. Based on the Galerkin approach, the natural frequencies of the beam system are numerically analyzed. The effects of moving speed of the base and several other system parameters on the dynamics and stability of the beam are discussed in detail. It is found that when the moving speed exceeds a certain value the beam becomes unstable and the instability type is sensitive to the system parameters. When the values of system parameters, such as mass ratio and axially added mass coefficient, are big enough, however, no instabilities are detected. The variations of the lowest unstable critical moving speed with respect to several key parameters are also investigated.     

Stability improvement and regenerative chatter suppression in nonlinear milling process via tunable vibration absorber

In this paper, a tunable vibration absorber set (TVAs) is designed to suppress regenerative chatter in milling process (as a semi-active controller). An extended dynamic model of the peripheral milling with closed form expressions for the nonlinear cutting forces is presented. The extension part of the cutting tool is modeled as an Euler–Bernoulli beam with in plane lateral vibrations (x–y directions). Tunable vibration absorbers in x–y directions are composed of mass, spring and dashpot elements. In the presence of regenerative chatter, coupled dynamics of the system (including the beam and x–y absorbers) is described through nonlinear delay differential equations. Using an optimal algorithm, optimum values of the absorbers' position and their springs' stiffness in both x–y directions are determined such that the cutting tool vibration is minimized. Results are compared for both linear and nonlinear models. According to the results obtained, absorber set acts effectively in chatter suppression over a wide range of chatter frequencies. Stability limits are obtained and compared with two different approaches: a trial and error based algorithm and semi-discretization method. It is shown that in the case of self-excited vibrations, the optimum absorber improves the process stability. Therefore, larger values of depth of cut and consequently more material removal rate (MRR) can be achieved without moving to unstable conditions.     

硅微机械谐振器特性与激励光斑位置关系的研究

阐述了对硅悬臂梁谐振子进行光激励研究的结果，结果表明，谐振子的振幅与激励光斑的位置以及所激励的模态有关，在模态的两节点的中心位置进行激励，可得到最大的振幅，而且振幅随着激励光斑偏离子中心位置而下降基节点处激励时，振幅达到最小。上述结论也适用于其它点热源激励的微机械谐振器，并为点热激激励的微机械谐振器的设计和制作提供了依据。     

Nonlinear vibration of micromachined asymmetric resonators

In this paper, the nonlinear dynamics of a beam-type resonant structure due to stretching of the beam is addressed. The resonant beam is excited by attached electrostatic comb-drive actuators. This structure is modeled as a thin beam-lumped mass system, in which an initial axial force is exerted to the beam. This axial force may have different origins, e.g., residual stress due to micro-machining. The governing equations of motion are derived using the mode summation method, generalized orthogonality condition, and multiple scales method for both free and forced vibrations. The effects of the initial axial force, modal damping of the beam, the location, mass, and rotary inertia of the lumped mass on the free and forced vibration of the resonator are investigated. For the case of the forced vibration, the primary resonance of the first mode is investigated. It has been shown that there are certain combinations of the model parameters depicting a remarkable dynamic behavior, in which the second to first resonance frequencies ratio is close to three. These particular cases result in the internal resonance between the first and second modes. This phenomenon is investigated in detail.     

Dynamic behavior of time-delayed acceleration feedback controller for active vibration control of flexible structures

This paper addresses the design problem of the controller with time-delayed acceleration feedback. On the basis of the reduction method and output state-derivative feedback, a time-delayed acceleration feedback controller is proposed. Stability boundaries of the closed-loop system are determined by using Hurwitz stability criteria. Due to the introduction of time delay into the controller with acceleration feedback, the proposed controller has the feature of not only changing the mass property but also altering the damping property of the controlled system in the sense of equivalent structural modification. With this feature, the closed-loop system has a greater logarithmic decrement than the uncontrolled one, and in turn, the control behavior can be improved. In this connection, the time delay in the acceleration feedback control is a positive factor when satisfying some given conditions and it could be actively utilized. On the ground of the analysis, the developed controller is implemented on a cantilever beam for different controller gain–delay combinations, and the control performance is evaluated with the comparison to that of pure acceleration feedback controller. Simulation and experimental results verify the ability of the controller to attenuate the vibration resulting from the dominant mode.     

Tuneable vibration absorber design to suppress vibrations: An application in boring manufacturing process

Dynamic vibration absorbers are used to reduce the undesirable vibrations in many applications such as electrical transmission lines, helicopters, gas turbines, engines, bridges, etc. Tuneable vibration absorbers (TVA) are also used as semi-active controllers. In this paper, the application of a TVA for suppression of chatter vibrations in the boring manufacturing process is presented. The boring bar is modeled as a cantilever Euler–Bernoulli beam and the TVA is composed of mass, spring and dashpot elements. In addition, the effect of spring mass is considered in this analysis. After formulation of the problem, the optimum specifications of the absorber such as spring stiffness, absorber mass and its position are determined using an algorithm based on the mode summation method. The analog-simulated block diagram of the system is developed and the effects of various excitations such as step, ramp, etc. on the absorbed system are simulated. In addition, chatter stability is analyzed in dominant modes of boring bar. Results show that at higher modes, larger critical widths of cut and consequently more material removal rate (MRR) can be achieved. In the case of self-excited vibration, which is associated with a delay differential equation, the optimum absorber suppresses the chatter and increases the limit of stability.     

应用长度极化压电陶瓷33模态的 半主动减振技术研究*

本文提出利用长度方向极化的压电材料的33模态来实现半主动减振。论文以横梁为例,通过理论分析和有限元仿真,对比研究了当压电材料分别连接31, 33两种模态对应的最佳分流电路时,压电材料两种模态在横梁的共振频率附近的减振效果。结果表明33模态比31模态具有更高的减振效率。此外,鉴于33模态存在极化长度有限的问题,仿真分析了压电材料的尺寸和位置对减振效果的影响。在此基础之上,提出了一个利用压电材料33模态的多模态减振的组合设计,对横梁的前三个模态起到了很好地减振作用。相对31模态而言,横梁的每个振动模态均有约15dB的减振提升。     

Convective instability of the thermovibrational flow of binary mixture in the presence of the Soret effect

The convective instability of the thermovibration flow in a plane horizontal layer filled with an incompressible binary gaseous mixture is investigated. The study takes into account the effect of thermal diffusion or the Ludwig-Soret effect. Several instability mechanisms are discussed. To determine the instability threshold with respect to cell and long-wave perturbations, the Floquet theory was applied to the linearized equations of convection formulated in the Boussinesq approximation. We found that regime parametric instability can occur owing to the finite frequency vibrations. The evolution of plane, spiral and three-dimensional disturbances is studied. We demonstrated that, because of the properties of the system, the subharmonic response of plane disturbances to the external periodic action cannot be observed. The instability can be associated only with synchronous or quasiperiodic modes. Depending on the vibration parameters, modulations can stabilize or destabilize the base state. For spiral perturbations the stability boundary does not depend on the amplitude and frequency of vibrations. In the case of long-wave instability we apply the regular perturbation approach with the wavenumber as a small parameter in power expansions. The stability boundaries are found.     

Dynamic stability of spinning pretwisted beams subjected to axial random forces

This paper studies the dynamic stability of a pretwisted cantilever beam spinning along its longitudinal axis and subjected to an axial random force at the free end. The axial force is assumed as the sum of a constant force and a random process with a zero mean. Due to this axial force, the beam may experience parametric random instability. In this work, the finite element method is first applied to yield discretized system equations. The stochastic averaging method is then adopted to obtain Ito's equations for the response amplitudes of the system. Finally the mean-square stability criterion is utilized to determine the stability condition of the system. Numerical results show that the stability boundary of the system converges as the first three modes are taken into calculation. Before the convergence is reached, the stability condition predicted is not conservative enough.     

Elasticity in the response of nuclei

From the RPA equations of motion, we derive macroscopic equations for the vibrations of nuclei. These equations imply the Tassie model of form factors and yield an energy for the quadrupole vibration 30% lower than the empirical. We interpret the modes as the classical vibration of an elastic solid.     

ASYMPTOTIC APPROACH FOR NON-LINEAR PERIODICAL VIBRATIONS OF CONTINUOUS STRUCTURES

An asymptotic approach for determining periodic solutions of non-linear vibration problems of continuous structures (such as rods, beams, plates, etc.) is proposed. Starting with the well-known perturbation technique, the independent displacement and frequency is expanded in a power series of a natural small parameter. It leads to infinite systems of interconnected non-linear algebraic equations governing the relationships between modes, amplitudes and frequencies. A non-trivial asymptotic technique, based on the introduction of an artificial small parameter is used to solve the equations. An advantage of the procedure is the possibility to take into account a number of vibration modes. As examples, free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non-linear restoring force are considered. Resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived.     

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.     

Resonant vibration control of rotating beams

Rotating structures, like e.g. wind turbine blades, may be prone to vibrations associated with particular modes of vibration. It is demonstrated, how this type of vibrations can be reduced by using a collocated sensor-actuator system, governed by a resonant controller. The theory is here demonstrated by an active strut, connecting two cross-sections of a rotating beam. The structure is modeled by beam elements in a rotating frame of reference following the beam. The geometric stiffness is derived in a compact form from an initial stress formulation in terms of section forces and moments. The stiffness, and thereby the natural frequencies, of the beam depend on the rotation speed and the controller is tuned to current rotation speed to match the resonance frequency of the selected mode. It is demonstrated that resonant control leads to introduction of the intended level of damping in the selected mode and, with good modal connectivity, only very limited modal spill-over is generated. The controller acts by resonance and therefore has only a moderate energy consumption, and successfully reduces modal vibrations at the resonance frequency.     

AN EXACT METHOD FOR THE FREE VIBRATION ANALYSIS OF TIMOSHENKO-KELVIN BEAMS WITH OSCILLATORS

In this paper, a modern exact method is proposed for solving the problem of free vibrations of a Timoshenko-type viscoelastic beam with discrete rigid bodies, connected to the beam by means of viscoelastic constraints. The phenomenon of free vibrations of this discrete-continuous system is described by a set of three partial and two subsystem ordinary differential equations with generalized boundary conditions and initial conditions. Vector notation of the equations allows one to identify the self-adjoint linear operators of inertia, stiffness and damping. In this case, these operators are not homothetic hence a separation of variables in this set of equations is possible only in a complex Hilbert space. Such separation of variables leads to ordinary differential equations of motion with respect to time as well as to a set of three ordinary differential equations with respect to a spatial variable and two subsystem algebraical equations. Solution of the boundary-value problem was carried out in the classical way, but its results are complex conjugated. Using these results and the fundamental principle, describing the orthogonality property of complex eigenvectors, the problem of free vibrations of the system with arbitrary initial conditions has been finally solved exactly.     

Nonlinear finite amplitude vibrations of sharp-edged beams in viscous fluids

In this paper, we study flexural vibrations of a cantilever beam with thin rectangular cross section submerged in a quiescent viscous fluid and undergoing oscillations whose amplitude is comparable with its width. The structure is modeled using Euler–Bernoulli beam theory and the distributed hydrodynamic loading is described by a single complex-valued hydrodynamic function which accounts for added mass and fluid damping experienced by the structure. We perform a parametric 2D computational fluid dynamics analysis of an oscillating rigid lamina, representative of a generic beam cross section, to understand the dependence of the hydrodynamic function on the governing flow parameters. We find that increasing the frequency and amplitude of the vibration elicits vortex shedding and convection phenomena which are, in turn, responsible for nonlinear hydrodynamic damping. We establish a manageable nonlinear correction to the classical hydrodynamic function developed for small amplitude vibration and we derive a computationally efficient reduced order modal model for the beam nonlinear oscillations. Numerical and theoretical results are validated by comparison with ad hoc designed experiments on tapered beams and multimodal vibrations and with data available in the literature. Findings from this work are expected to find applications in the design of slender structures of interest in marine applications, such as biomimetic propulsion systems and energy harvesting devices.     

Dynamic stability of liquid-filled cylindrical shells under horizontal excitation,part I: Experiment

Experimental studies have been carried out on the dynamic stability of a cantilever cylindrical shell partially filled with liquid, under horizontal excitation. The test cylinder was harmonically excited with constant acceleration or displacement amplitude. It was found that a combination instability resonance of sum type could occur, involving two natural vibrations with the same axial mode of vibration number but with the circumferential wave numbers differing by one. By varying the dimensionless water height from 0 to 1·0 stepwise by 0·25 increments, the instability regions and vibration modes were determined for two polyester test cylinders. The response waves, axial and circumferential vibration modes, and behavior of the free liquid surface were also observed.     

CRACK IDENTIFICATION IN VIBRATING BEAMS USING THE ENERGY METHOD

An energy-based numerical model is developed to investigate the influence of cracks on structural dynamic characteristics during the vibration of a beam with open crack(s). Upon the determination of strain energy in the cracked beam, the equivalent bending stiffness over the beam length is computed. The cracked beam is then taken as a continuous system with varying moment of intertia, and equations of transverse vibration are obtained for a rectangular beam containing one or two cracks. Galerkin's method is applied to solve for the frequencies and vibration modes. To identify the crack, the frequency contours with respect to crack depth and location are defined and plotted. The intersection of contours from different modes could be used to identify the crack location and depth.