Vibration control of a string using a scabbard-like actuator

The dynamics of a vibrating string subjected to a constraint at one boundary is investigated in this paper. The constraint is applied by a scabbard that moves a small distance along the mean position of the string. The scabbard is moved instantaneously such that the position and the velocity of the string outside the scabbard is unaffected immediately after application of the constraint, whereas the length of the string covered by the scabbard is brought to rest. The constraint is removed by moving the scabbard back to its original position and the change in energy of the string is investigated for different values of scabbard travel distance and time of application of the constraint. Analytical and numerical simulation results are first provided for the string vibrating in the first mode, and then for a more general case where the string has arbitrary initial conditions. The results show that the energy of the string can increase or decrease depending on the time of application of the constraint for a given distance of travel of the scabbard. This provides the opportunity for active control of vibration of the string through direct physical interaction, using the scabbard as an actuator. A simple feedback control strategy is proposed and numerical simulation results are presented. These results indicate that although removal of the constraint does not change the energy of the string, the effectiveness of the control strategy depends on the time of removal of the constraint.     

Optimizing parametric oscillators with tunable boundary conditions

Parametric excitation or pumping is an effective method to create large oscillations by periodically altering a physical parameter of the governing dynamics. Precisely tuned pumping frequencies can lead to exponentially growing oscillations limited only by nonlinear effects like axial stretching of transversely vibrating string. It is demonstrated that a tuned passive dynamical system amplifies the otherwise limited transverse vibrations amplitudes of a nonlinear string considerably and thus increasing the selectivity of the system. This effect becomes more noticeable for shorter wavelengths where nonlinear stretching limits the obtainable vibration amplitudes severely. The present work analyses a passive dynamical system connected to one end of a taught string which parametrically couples its axial motion to transverse vibration. Analysis shows that a specific selection of parameters can reduce the limiting effect of nonlinear stretching thus allowing one to excite high-order modes with small external forces. The result can possibly affect other disciplines where effective parametric amplification is necessary.     

Modal analysis of a continuous gyroscopic second-order system with nonlinear constraints

A method for the modal analysis of continuous gyroscopic systems with nonlinear constraints is developed. This method assumes that the nonlinear constraint can be expressed as a piecewise linear force-deflection profile located at an arbitrary position within the domain. Using this assumption, the mode shapes and natural frequencies are first found for each state, then a mapping method based on the inner product of the mode shapes is developed to map the displacement of the system between the in-contact and out-of-contact states. To illustrate this method, a model for the vibration of a traveling string in contact with a piecewise-linear constraint is developed as an analog of the interaction between magnetic tape and a guide in data storage systems. Five design parameters of the guide are considered: flange clearance, flange stiffness, symmetry of the force-deflection profile in terms of flange stiffness and offset, and the guide's position along the length of the string. There are critical bifurcation thresholds, below which the system exhibits no chaotic behavior and is dominated by period one, symmetric behavior, and above which the system contains asymmetric, higher periodic motion with windows of chaotic behavior. These bifurcation thresholds are particularly pronounced for the transport speed, flange clearance, symmetry of the force deflection profile, and guide position. The stability of the system is sensitive to the system's velocity, and, compared to stationary systems, more mode shapes are needed to accurately model the dynamics of the system.     

Analysis of energy dissipation in an elastic moving string with a viscous damper at one end

In this paper transverse vibration of an axially moving viscoelastic string with a viscous damper at one end is investigated analytically. The string is assumed to be travelling with constant velocity and the length of string is constant or time varying. The linear and nonlinear mathematical models are derived using the Lagrangian function and implemented using a finite element method. The method considers a time varying state space function applied to the linear model, the Newmark-Beta method is used to solve the response for the nonlinear problem numerically. The case of energy dissipated by a viscoelastic damper at one end of the string for different axial string velocities is considered. When a disturbance arrives at the boundary an exact value for the damper which provides maximum energy dissipation is investigated. Finally, numerical simulations are presented to establish the feasibility of the method.     

Reconstruction of bowing point friction force in a bowed string

A method is presented for reconstructing the friction force and the velocity at the bowing point of a string excited by a rosined bow sliding transverse to the string. Two versions of the method of reconstruction are presented, each approximate in different ways, but both capable of sufficient accuracy to allow useful application to problems of understanding frictional interactions in this dynamical system. The method is illustrated with simulated data to verify its accuracy, and results are shown for two contrasting cases of observed stick-slip string motion. As has been found in other investigations, the friction force during sliding is not determined by the instantaneous sliding speed. The results seem to be compatible with a thermally based model of rosin friction.     

Dynamic analysis of an optical fiber coupler in telecommunications

This paper studies the vibration of an optical fiber coupler which is used in telecommunications subjected to a half sine shock. The emphasis is focused on analyzing the vibration response of the optical fibers inside the coupler and examining the influence of various coupler parameters on the vibration of the optical fibers, since their dynamic behavior is a critical factor in optical fiber communications. A simplified model of the optical fiber coupler is proposed, which consists of a beam and a string representing the substrate and the bundle of the optical fibers of the coupler, respectively. The beam and the string are bonded at four points using adhesive material, and therefore the boundary conditions for their equations of motion are coupled, which increases the complexity of the problem. For the string, two models are developed— the linear model assumes that the tension in the string is constant, while the non-linear one takes into account large transverse deflection and tension variation. With each model, both analytical study and numerical simulations for the vibration of the system under a half shock are carried out. Furthermore, numerical results are compared between the two models. Finally, parametric study leads to conclusions which are of practical importance to the design of optical fiber couplers.     

Vibration of a string wrapping and unwrapping around an obstacle

The problem of a string vibrating against a smooth obstacle is investigated in this paper. The obstacle is located at one of the boundaries and the string is assumed to wrap and unwrap around the obstacle during vibration. The wrapping of the obstacle is modeled by a series of perfectly inelastic collisions between the obstacle and adjacent segments of the string and unwrapping is assumed to be energy conserving. The geometry of the string is determined iteratively starting from an initial configuration where the string is vibrating in a single mode and is not in contact with the obstacle. The obstacle can be regarded as a passive mechanism for vibration suppression in which the energy lost during each cycle of oscillation depends on the energy content of the string at the beginning of the cycle. Numerical simulation results are provided for the string vibrating in different modes for circular- and elliptic-shaped obstacles. The loss of energy is found to be greater for higher modes of oscillation and for obstacles that induce greater length of wrapping.     

Vibration of strings with nonlinear supports

The dynamic string motion, which displacement is unilaterally constrained by the rigid termination condition of an arbitrary geometry has been simulated and analyzed. The treble strings of a grand piano usually terminate at a capo bar, which is situated above the strings. The apex of a V-shaped section of the capo bar defines the end of the speaking length of the strings. A numerical calculation based on the traveling wave solution is proposed for modeling the nonlinearity inducing interactions between the vibrating string and the contact condition at the point of string termination. It was shown that the lossless string vibrates in two distinct vibration regimes. In the beginning the string starts to interact in a nonlinear fashion with the rigid terminator, and the resulting string motion is aperiodic. Consequently, the spectrum of the string motion depends on the amplitude of string vibrations, and its spectral structure changes continuously with the passage of time. The duration of that vibration regime depends on the geometry of the terminator. After some time of aperiodic vibration, the string vibrations settle in a periodic regime where the resulting spectrum remains constant.     

Layout optimization methodology of piezoelectric transducers in energy-recycling semi-active vibration control systems

An optimization methodology is proposed for the piezoelectric transducer (PZT) layout of an energy-recycling semi-active vibration control (ERSAVC) system for a space structure composed of trusses. Based on numerical optimization techniques, we intend to generate optimal location of PZTs under the constraint for the total length of PZTs. The design variables are set as the length of the PZT on each truss based on the concept of the ground structure approach. The transient problems of the mechanical and electrical vibrations based on the ERSAVC theory are considered as the equations of state. The objective is to minimize the integration of the square of all displacement over the whole analysis time domain. The sensitivity of the objective function is derived based on the adjoint variable method. Based on these formulations, an optimization algorithm is constructed using the fourth-order Runge–Kutta method and the method of moving asymptotes. Numerical examples are provided to illustrate the validity and utility of the proposed methodology. Using the proposed methodology, the optimal location of PZTs for the vibration suppression for multi-modal vibration is studied, which can be benchmark results of further study in the context of ERSAVC systems.     

Boundary control of an axially moving string via fuzzy sliding-mode control and fuzzy neural network methods

The objective of this study is to develop intelligent control schemes for the transverse vibration reduction of an axially moving string. The proposed approaches are backboned by the methods of fuzzy sliding-mode control (FSMC) and fuzzy neural network control (FNNC). In practice, the control effort for the system is realized through a typical mass-damper-spring (MDS) system attached at the right-hand side boundary of the moving string. The dynamic coupling between the string and the MDS system provides an actuation force to suppress transverse vibration. In the first phase of this study, the framework of FSMC is designed, in which the techniques of region-wise linear fuzzy logic control design and generic algorithm technique are employed to facilitate FSMC to reduce a large number of fuzzy rule bases and to select optimal control gains, respectively. In the second phase, the FNNC is developed, which is, compared to the FSMC, easier to design the control rule, more robust against environment and capable of on-line learning. Numerical simulations are conducted and the comparison between various controllers is made based on simulations. The simulated results show that the transverse vibration can be well suppressed by both approaches. FSMC offers the capability to regulate the transient response, while FNNC holds advantage of on-line learning capability.     

Application of a version of the Rayleigh technique to problems of bars,beams, columns,membranes, and plates

A review is presented, together with some new material, of the application of a little known version of the Rayleigh technique to a variety of problems in solid and structural mechanics. This method was originally applied by Rayleigh in 1894 to the one-dimensional problem of determining the fundamental frequency of a stretched string undergoing small-amplitude vibrations. The essence of the method is that instead of using a specific trial function with an undetermined coefficient for the deflection as in the ordinary Rayleigh method, here one uses a function containing a power-law term with an undetermined exponent. Application of the usual variational procedure for the quantity of interest (such as the buckling load, fundamental frequency, or deflection) results in a dimensionless quantity which is a function only of the undetermined exponent. Finally, the “best” exponent is the one which minimizes the Lagrangian and thus the buckling load, natural frequency, or reciprocal of deflection. In the past five years, the non-integer exponent idea has been applied to elastic torsion, buckling of columns and plates, vibration of bars, beams, columns, membranes, and plates, and deflection of beams and plates. Also, it has been incorporated into the finite element method. All of these applications are discussed and some additional original work of the present investigator is presented.     

Dynamic analysis and design of air spring mounting system for marine propulsion system

Marine propulsion unit (MPU) is one of the dominant vibration and noise sources onboard ship. Its vibration can be attenuated effectively by isolating MPU with low-frequency mounting system. But this is difficult to implement due to the stringent requirement of MPU alignment with the propulsion shafting. In this paper a novel air spring mounting system (ASMS) for propulsion system is proposed consisting of air spring subsystem, alignment control subsystem and safety protection subsystem. The load distribution optimization method and dynamic model of ASMS are presented. The factors that affect system stability and natural frequencies are analyzed, as well as the design measures to enhance system performance. A theoretical model is presented to estimate the isolation effect of ASMS. The monitoring model of alignment between MPU and propulsion shafting is established, followed by the alignment control algorithm and converge rule which assures the fast and uniform convergence of both air springs? load distribution and alignment control process. Safety protection mechanism is designed to ensure that the MPU can operate safely in case of ASMS failure or other extreme circumstances. A scaled ASMS prototype is manufactured and tested on a special experimental setup. Experimental results validate the effectiveness of theoretical models and show that the performance of ASMS satisfies the operation requirements of MPU.     

弦振动特性研究

在金属弦线上形成驻波时,根据振动模式和弦线的长度可以测出所用弦线的线密度。在频率一定时,测量不同张力下的波长,利用最小二乘法得到波长随张力的实际变化关系。     

Parametric control of an axially moving string via fuzzy sliding-mode and fuzzy neural network methods

This study is dedicated to design effective control schemes to suppress transverse vibration of an axially moving string system by adjusting the axial tension of the string. To this end, a continuous model in the form of partial differential equations is first established to describe the system dynamics. Using an energy-like system functional as a Lyapunov function, a sliding-mode controller (SMC) is designed to be applied when the level of vibration is not small. Due to non-analyticity of the SMC control effort generated as vibration level becoming small, two intelligent control schemes are proposed to complete the task — fuzzy sliding-mode control (FSMC) and fuzzy neural network control (FNNC). Both control approaches are based on a common structure of fuzzy control, taking switching function and its derivative as inputs and tension variation as output to reduce the transverse vibration of the string. In the framework of FSMC, genetic algorithm (GA) is utilized to search for the optimal scalings for the inputs; in addition, the technique of regionwise linear fuzzy logic control (RLFLC) is employed to simplify the computation procedure of the fuzzy reasoning. On the other hand, FNNC is proposed for conducting on-line tuning of control parameters to overcome model uncertainty. Numerical simulations are conducted to verify the effectiveness of controllers. Satisfactory stability and vibration suppression are attained for all controllers with the findings that the FSMC assisted by GA holds the advantage of fast convergence with a precise model while the FNNC is robust to model uncertainty and environmental disturbance although a relatively slower convergence could be present.     

基于滑模鲁棒算法的超低频主动隔振系统

振动噪声的有效隔离是冷原子重力仪的关键技术之一.为了减小冷原子重力仪中拉曼反射镜的振动噪声,研制了一套紧凑型低频主动隔振系统.其原理是利用滑模鲁棒控制系统处理和反馈由地震仪采集到的振动信号,利用音圈电机控制和消除被动隔振平台的运动.在0.1—10 Hz频域范围内,滑模鲁棒控制系统的残余振动噪声功率谱密度比被动隔振平台最大降低了99.9%,比超前滞后补偿控制方法最大降低了83.3%.滑模鲁棒控制算法还具有整定参数少、抗干扰能力强等特点.     

NON-LINEAR FREE VIBRATION ANALYSIS OF A STRING UNDER BENDING MOMENT EFFECTS USING THE PERTURBATION METHOD

In this paper, the non-linear free vibration of a string with large amplitude is considered. The initial tension, lateral vibration amplitude, cross-section diameter and the modulus of elasticity of the string have main effects on its natural frequencies. Increasing the lateral vibration amplitude makes the assumption of constant initial tension invalid. Therefore, it is impossible to use the classical equation of transverse motion assuming a small amplitude. On the other hand, by increasing the string cross-sectional diameter, the bending moment effect will increase dramatically, and it will act as an impressive restoring moment. Considering the effects of the bending moments, the non-linear equation governing the large amplitude transverse vibration of a string is derived. The time-dependent portion of the governing equation has the form of the Duffing equation. Due to the complexity and non-linearity of the derived equation and the fact that there is no established exact solution method, the equation is solved using the perturbation method. The results of the analysis are shown in appropriate graphs, and the natural frequencies of the string due to the non-linear factors are compared with the natural frequencies of the linear vibration of a string without bending moment effects.     

Vibration of continuous systems with compliant boundaries

A theoretical study of two types of continuous systems with a general form of compliant boundary conditions is presented. The systems considered are elastic beams and circular plates with elastic damped edge constraints. Beam studies are restricted to those with identical boundary conditions at each end. The method of solution consists of formulating the edge condition of the system in terms of the impedance of the compliant boundary material and of using classical solution techniques to solve the equations of motion. The result of matching the boundary conditions of the system with constraining conditions is the system frequency equation in terms of the constraint impedances.A discussion is presented giving the influence of the compliant material on the vibration of the structure. The models give numerically the effect of elasticity and damping of the supports on the resonant frequencies of the systems. Parameters are obtained which indicate when one may assume simply supported or clamped boundaries for the actual case of elastic damped constraints without introducing large errors in the natural frequencies.     

基于Matlab GUI的弦振动仿真课件制作

利用Matlab GUI制作弦振动仿真课件,集成多种功能,可以模拟弦线上驻波波长随弦线密度、弦线张力、振源频率的变化。可实现数据的定量测量,自动读取、记录数据。集成数据处理功能,采用对数作图法处理数据,并给出拟合曲线和拟合系数。     

FREE TRANSVERSE VIBRATIONS OF AN ELASTICALLY CONNECTED COMPLEX BEAM-STRING SYSTEM

The present work is devoted to theoretical vibration analysis of a complex mixed continuous system. Undamped free transverse vibrations of an elastically connected beam-string system are considered. Solutions of the problem are formulated by using the modal expansion method. Two infinite sequences of the natural frequencies corresponding to two possible kinds of vibration motions: synchronous and asynchronous are determined. In a numerical example, illustrating the theory presented, the effect of string tension force on the natural frequencies of the system is investigated in detail.     

多源激励下双层隔振浮筏系统的线谱混沌化

采用混沌化技术可以重构水下航行器的水声线谱特征,改善隐声性能.基于这一特殊应用背景,研究了多源激励下双层隔振浮筏系统的非线性时延反馈混沌化问题.在二维简化浮筏系统模型的基础上,完整地呈现了非线性时延反馈控制的线谱混沌化方法,为隔振浮筏的线谱混沌化设计提供了标准流程.仿真结果表明了该方法的可行性,探讨了多源激励条件下控制增益、时延和反馈频率等控制参数对系统混沌化效果的影响,并与单源激励进行了对比分析.