Asymptotic stabilization of a nonlinear axially moving string by adaptive boundary control

In this paper, a robust adaptive boundary control for an axially moving string that shows nonlinear behavior resulting from spatially varying tension is investigated. A hydraulic actuator equipped with a damper is used as the control actuator at the right boundary of the string. The Lyapunov redesign method is employed to derive a robust control algorithm employing adaptation laws that estimate three unknown system parameters (mass per unit length of string, lumped mass of hydraulic actuator, and damping coefficient of damper) and an unknown boundary disturbance. The uniform asymptotic stability (when the three parameters are all unknown), the exponential stability (when they are known), and the uniform ultimate boundedness (with a bounded boundary disturbance) of the closed loop system are investigated. The convergence of the parameter estimates to the true values is shown. Numerical simulations are performed to demonstrate the effectiveness of the proposed robust adaptive boundary control.     

Simulation of the nonlinear vibration of a string using the cellular automation method

In this study, the nonlinear dynamic responses of a string were simulated using the cellular automaton (CA) method. The local rules were set for the amplitude of vibration and the decay rate of amplitude. In the case of nonlinear systems, the velocity of wave propagation is not constant and depends on the amplitude. Thus, a new treatment of the time step was proposed, i.e., the time step in the CA method is adjusted to real time by considering the effect of the propagation velocity. As numerical examples, first, the dynamic responses of a string with linear characteristic were simulated and a typical resonance curve could be obtained. Secondly the dynamic responses of a string with nonlinear characteristic were simulated. Some characteristic types of vibration could be obtained. It was concluded that the linear and nonlinear dynamic responses of a string could be simulated using the CA method.     

Simulation of the nonlinear vibration of a string using the Cellular Automata based on the reflection rule

In this study, the nonlinear dynamic responses of a string are simulated using the Cellular Automata method based on the reflection rule. In the case of nonlinear systems, the velocity of wave propagation is not constant and depends on the amplitude. A new treatment of the dynamic time step is proposed for the Cellular Automata method considering the effect of the propagation velocity. As numerical examples, first, the dynamic responses of a string with linear characteristic are simulated using the Cellular Automata method. A typical resonance curve can be obtained. Second, the dynamic responses of a string with nonlinear characteristic are simulated using the proposed method. Some characteristic types of vibration can be obtained. It is concluded that the linear and nonlinear dynamic responses of a string may be obtained by simulation using the Cellular Automata method.     

Simultaneous control of longitudinal and transverse vibrations of an axially moving string with velocity tracking

In this paper, an active control scheme for an axially moving string system that suppresses both longitudinal and transverse vibrations and regulates the transport velocity of the string to track a desired moving velocity profile is investigated. The control scheme utilizes three inputs: one control force at the right boundary, which is exerted by a hydraulic actuator equipped with a damper, and two control torques applied at the left and right rollers. The equations of motion are derived by using Hamilton's principle. Two nonlinear partial differential equations govern the longitudinal and transverse motions, where the variation of the tension of the string due to the transverse and longitudinal vibrations is considered. Among four boundary conditions, two describe the rotational dynamics of the left and right rollers; one determines the dynamics of the hydraulic actuator at the right boundary, and the last one denotes that the left boundary is fixed. The Lyapunov method is employed to generate control laws. Asymptotic stability of the transverse and longitudinal dynamics and the velocity tracking error is achieved. The effectiveness of the proposed control scheme is illustrated via numerical simulations.     

Extending the dynamic range of an energy harvester using nonlinear damping

This paper introduces the use of nonlinear damping for extending the dynamic range of vibration energy harvesters. A cubic nonlinear damper is initially considered and the average harvested power and the throw are obtained for different sinusoidal base excitation amplitudes and frequencies, both numerically and analytically. It is demonstrated that when excited at resonance, at an amplitude below its maximum operational limit, the harvested power using a nonlinear damper can be significantly larger than that of a linear energy harvester, therefore expanding its dynamic range. A potential approach to implementing cubic nonlinearity using a shunted electromagnetic device is also presented.     

On variable length induced vibrations of a vertical string

The purpose of this paper is to study the free lateral responses of vertically translating media with variable length, velocity and tension, subject to general initial conditions. The translating media are modeled as taut strings with fixed boundaries. The problem can be used as a simple model to describe the lateral vibrations of an elevator cable, for which the length changes linearly in time, or for which the length changes harmonically about a constant mean length. In this paper an initial-boundary value problem for a linear, axially moving string equation is formulated. In the given model a rigid body is attached to the lower end of the string, and the suspension of this rigid body against the guide rails is assumed to be rigid. For linearly length variations it is assumed that the axial velocity of the string is small compared to nominal wave velocity and the string mass is small compared to car mass, and for the harmonically length variations small oscillation amplitudes are assumed and it is also assumed that the string mass is small compared to the total mass of the string and the car. A multiple-timescales perturbation method is used to construct formal asymptotic approximations of the solutions to show the complicated dynamical behavior of the string. For the linearly varying length analytic approximations of the exact solution are compared with numerical solution. For the harmonically varying length it will be shown that Galerkin?s truncation method cannot be applied in all cases to obtain approximations valid on long timescales.     

Behavior of nonlinear fluid viscous dampers for control of shock vibrations

Fluid viscous dampers have been widely used for suppression of high velocity shocks. While linear fluid viscous dampers have been used for a long time, nonlinear fluid viscous dampers show considerable promise due to their superior energy dissipation characteristics and significant reduction in the damper force compared to a linear fluid viscous damper for the same peak displacement. This paper presents results from experimental study to characterize fluid viscous dampers when subjected to half-cycle sine shock excitation. The mathematical formulation and a numerical study to evaluate the relative performance of structures with fluid viscous dampers subjected to short-duration shock (impulse) loading are also discussed. The influence of damper nonlinearity (α) and the supplemental damping ratio (ξ_sd) on response has been investigated. The supplemental damping ratio of nonlinear fluid viscous dampers when subjected to shock excitation is found by equivalent linearization using the concept of equal energy dissipation. The paper also presents some design charts, which can be used for preliminary decisions on parameters of nonlinear dampers to be used in design.     

Free vibration analysis of linear particle chain impact damper

Impact dampers have gained much research interest over the past decades that resulted in several analytical and experimental studies being conducted in that area. The main emphasis of such research was on developing and enhancing these popular passive control devices with an objective of decreasing the three parameters of contact forces, accelerations, and noise levels. To that end, the authors of this paper have developed a novel impact damper, called the Linear Particle Chain (LPC) impact damper, which mainly consists of a linear chain of spherical balls of varying sizes. The LPC impact damper was designed utilizing the kinetic energy of the primary system through placing, in the chain arrangement, a small-sized ball between each two large-sized balls. The concept of the LPC impact damper revolves around causing the small-sized ball to collide multiple times with the larger ones upon exciting the primary system. This action is believed to lead to the dissipation of part of the kinetic energy at each collision with the large balls.     

Dynamics of gas bubbles in viscoelastic fluids. II. Nonlinear viscoelasticity

The nonlinear oscillations of a spherical, acoustically forced gas bubble in nonlinear viscoelastic media are examined. The constitutive equation [Upper-Convective Maxwell (UCM)] used for the fluid is suitable for study of large-amplitude excursions of the bubble, in contrast to the previous work of the authors which focused on the smaller amplitude oscillations within a linear viscoelastic fluid [J. S. Allen and R. A. Roy, J. Acoust. Soc. Am. 107, 3167-3178 (2000)]. Assumptions concerning the trace of the stress tensor are addressed in light of the incorporation of viscoelastic constitutive equations into bubble dynamics equations. The numerical method used to solve the governing system of equations (one integrodifferential equation and two partial differential equations) is outlined. An energy balance relation is used to monitor the accuracy of the calculations and the formulation is compared with the previously developed linear viscoelastic model. Results are found to agree in the limit of small deformations; however, significant divergence for larger radial oscillations is noted. Furthermore, the inherent limitations of the linear viscoelastic approach are explored in light of the more complete nonlinear formulation. The relevance and importance of this approach to biomedical ultrasound applications are highlighted. Preliminary results indicate that tissue viscoelasticity may be an important consideration for the risk assessment of potential cavitation bioeffects.     

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.     

A comparison of two nonlinear damping mechanisms in a vibration isolator

The vibration transmissibility characteristics of a single-degree-of-freedom (SDOF) passive vibration isolation system with different nonlinear dampers are investigated in this paper. In one configuration, the damper is assumed to be linear and viscous, and is connected to the mass so that it is perpendicular to the spring (horizontal damper). The vibration is in the direction of the spring. The second configuration is one in which the damper is in parallel with the spring but the damping force is proportional to the cube of the relative velocity across the damper (cubic damping). Both configurations are studied for small amplitudes of excitation, when some analysis can be conducted based on analytical expressions, and for large amplitudes of excitation, where the analysis is based on numerical simulations. It is found that the two nonlinear systems can outperform the linear system when force transmissibility is considered. However, for displacement transmissibility, the system with the horizontal damper exhibits some desirable properties, but the system with cubic damping does not.     

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.     

Dynamics of two-dimensional composites of elastic and viscoelastic layers

Models of frequency response, acoustic transmission, and transient wave propagation are presented for a two-dimensional composite of elastic and viscoelastic layers, simply supported at the two boundaries. The three models adopt transfer matrices to relate state variables over the two faces of a layer. In the frequency domain, a viscoelastic constitutive law is derived by nonlinear fitting a Padé series to measured data of complex shear modulus. For an elastic material, the eigenproblem admits positive real eigenvalues and their negatives. For a viscoelastic material, it admits positive complex eigenvalues and their negative conjugates. The imaginary part of the eigenvalue acts as a velocity-dependent viscous damper. Modal analysis solving transient response utilizes the complex eigenquantities and the static-dynamic superposition method.     

Feasibility study of nonlinear tuned mass damper for machining chatter suppression

In order to improve the performance of the tuned mass damper (TMD) for machining chatter suppression, a new-type of nonlinear TMD is proposed in this paper. Compared with the common linear TMD, the nonlinear TMD is equipped with an additional series friction-spring element. The capability of the nonlinear TMD in suppressing machining chatter vibration is investigated in this paper. The harmonic balancing method (HBM) is used to estimate the frequency response function (FRF) of the machining system to which the nonlinear TMD is attached. Considering the special nature of the machining stability problem, the optimal design parameters of this nonlinear TMD are those that minimize the magnitude of the real part of the FRF of the nonlinear TMD damped machining system. This paper also demonstrates the performance of the optimally tuned nonlinear TMD for machining stability improvement by calculating the stability diagrams for the milling of the nonlinear TMD damped workpiece. The calculation results show that more than 30% improvement in the critical limiting cutting depth can be obtained, compared to the optimally tuned linear TMD.     

Some solutions of the equations of motion of the relativistic string with massive ends

The classical theory is discussed for the relativistic string with point masses at its ends. The dynamical equations are solved for the class of motions of this system when the time evolution parameter τ is the proper time of both massive string ends. In this case the solution of the boundary equations is given by the almost periodic functions. Constraints on the normal modes resulting from the orthonormal gauge conditions differ essentially from the Virasoro ones. Incidentally one obtains an exact solution for the half-infinite string with mass at one end. It is also proved that the exact solution for the string with massive ends cannot be a periodic function.     

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.     

VIBRATION ATTENUATION VIA CONSTRAINED LAYER DAMPING AND DASHPOT ON A SPRING-MASS-PLATE SYSTEM

The vibrations and damping characteristics of an annular plate with constrained layer damping (CLD) treatment subject to a traveling spring-mass-damper (SMD) are investigated. The equations of the CLD-treated plate are first derived from the energy principle. These equations are simplified via the Donnell-Mushtari-Vlasov assumptions. The response equations are eventually uncoupled for each mode and are in terms of a single-degree-of-freedom (s.d.o.f.) linear oscillator with hysteretic damping. The receptance method follows to joint the plate and the SMD, and the resulting change of natural frequencies and damping ratios are investigated. Individual effects due to the inertia and the stiffness are illustrated as well. The results shows that the damping ratios resulted from the viscoelastic core are more significant than that from the viscous damper. In addition, there exists a best design on the thickness of the viscoelastic material core to have the maximum damping ratios. The results also show that the attachment of SMD bifurcated the plate's natural frequencies for every mode but n = 0. The bifurcation becomes more obvious with the rotational speed. These results provide useful information for vibration suppression in engineering design.     

Normal forces of magnetorheological fluids under oscillatory shear

The normal forces of magnetorheological fluids under oscillatory shear are investigated by a commercial magneto-rheometer with plate-plate geometry. At the constant strain amplitude and frequency, the normal forces almost keep a steady value with the testing time if the strain amplitude is smaller than the critical value. When a larger strain is applied, they will fluctuate periodically. Under the strain sweep mode, the relationships between normal forces and strain amplitude can be divided into three regions: linear viscoelastic region, nonlinear viscoelastic region and the viscoplastic region. Under the frequency sweep method, it is found that the angular frequency show little influence on the normal forces. At last, the normal forces increase with increasing of the temperature under a low magnetic field, while they decrease under a high magnetic field.     

Measurement of viscoelastic properties from the vibration of a compliantly supported beam

A laboratory method is presented by which the viscoelastic properties of compliant materials are measured over a wide frequency range. The test setup utilizes a flexible beam clamped at one end and excited by a shaker at the free end. A small specimen of a compliant material is positioned to support the beam near its midpoint. The deformation from gravity is minimized since the specimen is not loaded by an attached mass. Forced vibration responses measured at two locations along the beam are used to derive a transfer function from which the dynamic properties of compliant materials are directly determined by use of a theoretical procedure investigating the effects of specimen stiffness on the propagation of flexural waves. Sensitivity of the measured properties to experimental uncertainties is investigated. Young's moduli and associated loss factors are determined for compliant materials ranging from low-density closed-cell foams to high-density damping materials. The method is validated by comparing the measured viscoelastic properties to those from an alternative dynamic test method.