W. D. Zhu

Energetics and Stability of Translating Media with an Arbitrarily Varying Length

The linear dynamics of a class of translating media with an arbitrarily varying length is investigated. The tension in the media arising from their longitudinal accelerations is incorporated. The dynamic stability of the continuous media relative to the inertial and moving coordinate systems is studied from the energy standpoint. The exact expressions for the rates of change of energies of media are derived and interpreted from both control volume and system viewpoints. The stability analyses relative to the inertial and moving coordinate systems result in the same predictions. Examples including a robotic arm through a prismatic joint and an elevator cable in a high-rise building illustrate the analysis. In particular, the results explain an inherent unstable shortening cable behavior encountered in elevator industry.

Experimental and Numerical Investigation of Structural Damage Detection Using Changes in Natural Frequencies

A robust iterative algorithm is used to identify the locations and extent of damage in beams using only the changes in their first several natural frequencies. The algorithm, which combines a first-order, multiple-parameter perturbation method and the generalized inverse method, is tested extensively through experimental and numerical means on cantilever beams with different damage scenarios. If the damage is located at a position within 0-35% or 50-95% of the length of the beam from the cantilevered end, while the resulting system equations are severely underdetermined, the minimum norm solution from the generalized inverse method can lead to absolution that closely represents the desired solution at the end of iterations when the stiffness parameters of the undamaged structure are used as the initial stiffness parameters. If the damage is located at a position within 35-50% of the length of the beam from the cantilevered end, the resulting solution by using the stiffness parameters of the undamaged structure as the initial stiffness parameters deviates significantly from the desired solution. In this case, a new method is developed to enrich the measurement information by modifying the structure in a controlled manner and using the first several measured natural frequencies of the modified structure. A new method using singular value decomposition is also developed to handle the ill-conditioned system equations that occur in the experimental investigation by using the measured natural frequencies of the modified structure.

Active Control of Translating Media With Arbitrarily Varying Length

The general control laws for pointwise controllers to dissipate vibratory energies of translating beams and strings with arbitrarily varying length are presented. Special domain and boundary control laws that can be easily implemented result as a special case. Sufficient conditions for uniform stability and uniform exponential stability of controlled systems are established via Lyapunov stability criteria. Numerical simulations demonstrate the effectiveness of the active controllers in stabilizing translating media during both extension and retraction. Optimal gains leading to the fastest rates of decay of vibratory energies of controlled systems are identified. It is shown that under the optimal control gains, translating media can be completely stabilized during extension and retraction.

Theoretical and Experimental Investigation of Elevator Cable Dynamics and Control

The vibratory energy of a moving cable in an elevator increases in general during upward movement. A control method is presented to dissipate the energy associated with the lateral vibration of the cable. A novel experimental method is developed to validate the theoretical predictions for the uncontrolled and controlled lateral responses of a moving cable in a high-rise elevator. This includes the design and fabrication of a scaled elevator, experimental setup, and development of measurement and parameter estimation techniques. Experimental results show good agreement with the theoretical predictions.

On an iterative general-order perturbation method for multiple structural damage detection

A general order perturbation method involving multiple perturbation parameters is developed for eigenvalue problems with changes in the stiffness parameters. The perturbation solutions and eigenparameter sensitivities of all orders are derived explicitly. The perturbation method is used iteratively in conjunction with an optimization method to identify the stiffness parameters of structures. The generalized inverse method is used efficiently with the first order perturbations, and the gradient and quasi-Newton methods are used with the higher order perturbations. Numerical simulations on discrete and continuous structural systems demonstrated the robustness of the algorithm in detecting the locations and extent of small to large levels of damage. The effects of measurement noise and reduced measurements on the performance of the algorithm are evaluated.

Free and Forced Vibration of an Axially Moving String With an Arbitrary Velocity Profile

The exact response of an axially moving string with an arbitrary velocity profile is determined under general initial conditions and external excitation. In terms of the curvilinear characteristic coordinates, the time-dependent governing equation reduces to one with constant coefficients. The domain of interest in the characteristic coordinate plane is bounded by two monotonic curves with the same shape corresponding to the spatial boundaries of the string. Solutions of the transformed equation are derived for the free and forced responses. The amplitude of the free response is shown to be bounded under arbitrary variation of the transport speed in the subcritical regime. The unbounded displacement of the string predicted by Floquet theory for the spatially discretized models does not occur. The method is demonstrated on two illustrative examples with piecewise constant and sinusoidal accelerations.

Modeling of Fillets in Thin-Walled Beams Using Shell/Plate and Beam Finite Elements

Fillets are commonly found in thin-walled beams. Ignoring the presence of a fillet in a finite element (FE) model of a thin-walled beam can significantly change the natural frequencies and mode shapes of the structure. A large number of solid elements are required to accurately represent the shape and the stiffness of a fillet in a FE model, which makes the size of the FE model unnecessarily large for global dynamic and static analyses. In this work the equivalent stiffness effects of a fillet in a thin-walled beam are decomposed into in-plane and out-of-plane effects. The in-plane effects of a fillet are analyzed using the wide-beam and curved-beam theories, and the out-of-plane effects of the fillet are analyzed by modeling the whole fillet section as a slender bar with an irregular cross section. A simple shell/plate and beam element model is developed to capture the in-plane and out-of-plane effects of a fillet on a thin-walled beam. The natural frequencies and mode shapes of a thin-walled L-shaped beam specimen calculated using the new methodology are compared with its experimental results for 28 modes. The maximum error between the calculated and measured natural frequencies for all the modes is less than 2%, and the associated modal assurance criterion values are all over 95%. The methodology is also applied to other thin-walled beams, and excellent agreement is achieved between the natural frequencies from the shell/plate and beam element models and those from the solid element models. While the shell/plate and beam element models provide the same level of accuracy as the intensive solid element models, the degrees of freedom of the shell/plate and beam element models of the thin-walled beams are only about 10% or less of those of the solid element models.

A Stochastic Model for the Random Impact Series Method in Modal Testing

A novel stochastic model is developed to describe a random series of impacts in modal testing that can be performed manually or by using a specially designed random impact device. The number of the force pulses, representing the impacts, is modeled as a Poisson process with stationary increments. The force pulses are assumed to have an arbitrary, deterministic shape function, and random amplitudes and arrival times. The force signal in a finite time interval is shown to consist of a wide-sense stationary part and two nonstationary parts. The expectation of the force spectrum is obtained from two approaches. The expectations of the average power densities associated with the entire force signal and the stationary part of it are determined and compared. The analytical expressions are validated by numerical solutions for two different types of shape functions. A numerical example is given to illustrate the advantages of the random impact series over a single impact and an impact series with deterministic arrival times of the pulses in estimating the frequency response function. The model developed can be used to describe a random series of pulses in other applications.

Stabilization of a Translating Tensioned Beam Through a Pointwise Control Force

A spectral analysis determining asymptotically the distribution of eigenvalues of a constrained, translating, tensioned beam in closed form is the subject of this paper. The constraint is modeled by a spring-mass-dashpot subsystem that is located at any position within the span of the beam. It can represent a feedback controller with a collocated sensor and actuator. The necessary and sufficient condition that ensures a uniform stability margin for all the modes of vibration is determined. Influences of system parameters on the distribution of eigenvalues are identified. The analytical predictions are validated by numerical analyses. The constraint location maximizing the stability margin of the distributed model is predicted through a combined analytical and numerical approach. The implications and utility of the results are illustrated. The methodology developed can be extended to predict stability margins and optimize control parameters for controlled translating beams with other types of boundary conditions and controller structures.

Free vibration analysis of a cantilever beam with a slant edge crack

As one of major failure modes of mechanical structures subjected to periodic loads, edge cracks due to fatigue can cause catastrophic failures in such structures. Understanding vibration characteristics of a structure with an edge crack is useful for early crack detection and diagnosis. In this work, a new cracked cantilever beam model is presented to study the vibration of a cantilever beam with a slant edge crack, which cannot be modeled by previous methods considering a uniform edge crack along the width of the beam in the literature. An equivalent stiffness model is proposed by dividing the beam into numerous uniform independent thin pieces along its width. The beam is assumed to be an Euler–Bernoulli beam. The crack is assumed to be distributed along the width of the beam as a straight line and a parabola. The methodology proposed in this work can also be extended to model a crack with an arbitrary curve. Effects of crack depths on the nondimensional equivalent stiffness at the crack section of the cracked cantilever beam are studied. The first three nature frequencies and mode shapes of the cracked cantilever beam are obtained using compatibility conditions at crack tips and the transfer matrix method. Effects of depths and the location of the crack on the first three natural frequencies and mode shapes of the cracked cantilever beam are studied using the proposed cracked cantilever beam model. Numerical results from the proposed model are compared with those from the finite element method and an experimental investigation in the literature, which can validate the proposed model.