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IN THIS ISSUE

### TECHNICAL PAPERS

J. Vib. Acoust. 2005;128(3):269-281. doi:10.1115/1.2172254.

This paper reports on the modeling and experimental identification of a high speed rotor-magnetic bearing test rig. An accurate nominal model and an uncertainty representation are developed for robust controller synthesis and analysis. A combination of analytical modeling, model updating, and identification is employed for each system component and for the system as a whole. This approach takes advantage of both the behavior modeling and input/output modeling methods. Analytical models of the rotor and the magnetic bearings are first developed from physical laws and refined by comparison with the experimental data. The substructure model is directly identified from the experimental data by a structured identification approach. Models of the electronic systems, such as the filters, amplifiers, sensors, and digital controller, are developed through experimental identification. These component models are then assembled to obtain the overall system model. Closed-loop tests are conducted to identify parameters in the model. Advanced control techniques based on $H∞$ control and $μ$ synthesis are developed and successfully implemented on the test rig, which further validates the model.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2005;128(3):282-293. doi:10.1115/1.2172255.

This paper studies the dynamics of a self-sustained electromechanical transducer. The stability of fixed points in the linear response is examined. Their local bifurcations are investigated and different types of bifurcation likely to occur are found. Conditions for the occurrence of Hopf bifurcations are derived. Harmonic oscillatory solutions are obtained in both nonresonant and resonant cases. Their stability is analyzed in the resonant case. Various bifurcation diagrams associated to the largest one-dimensional (1-D) numerical Lyapunov exponent are obtained, and it is found that chaos can appear suddenly, through period doubling, period adding, or torus breakdown. The extreme sensitivity of the electromechanical system to both initial conditions and tiny variations of the coupling coefficients is also outlined. The experimental study of̱the electromechanical system is carried out. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the electromechanical system. Correspondences are established between the coefficients of the electromechanical system model and the components of the electronic circuit. Harmonic oscillatory solutions and phase portraits are obtained experimentally. One of the most important contributions of this work is to provide a set of reliable analytical expressions (formulas) describing the electromechanical system behavior. These formulas are of great importance for design engineers as they can be used to predict the states of the electromechanical systems and respectively to avoid their destruction. The reliability of the analytical formulas is demonstrated by the very good agreement with the results obtained by both the numeric and the experimental analysis.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2005;128(3):294-302. doi:10.1115/1.2172256.

When a conductive material experiences a time-varying magnetic field, eddy currents are generated in the conductor. These eddy currents circulate such that they generate a magnetic field of their own, however the field generated is of opposite polarity, causing a repulsive force. The time-varying magnetic field needed to produce such currents can be induced either by movement of the conductor in the field or by changing the strength or position of the source of the magnetic field. In the case of a dynamic system the conductor is moving relative to the magnetic source, thus generating eddy currents that will dissipate into heat due to the resistivity of the conductor. This process of the generation and dissipation of eddy current causes the system to function as a viscous damper. In a previous study, the concept and theoretical model was developed for one eddy current damping system that was shown to be effective in the suppression of transverse beam vibrations. The mathematical model developed to predict the amount of damping induced on the structure was shown to be accurate when the magnet was far from the beam but was less accurate for the case that the gap between the magnet and beam was small. In the present study, an improved theoretical model of the previously developed system will be formulated using the image method, thus allowing the eddy current density to be more accurately computed. In addition to the development of an improved model, an improved concept of the eddy current damper configuration is developed, modeled, and tested. The new damper configuration adds significantly more damping to the structure than the previously implemented design and has the capability to critically damp the beam’s first bending mode. The eddy current damper is a noncontacting system, thus allowing it to be easily applied and able to add significant damping to the structure without changing dynamic response. Furthermore, the previous model and the improved model will be applied to the new damper design and the enhanced accuracy of this new theoretical model will be proven.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2005;128(3):303-317. doi:10.1115/1.2172257.

The generalized momentum balance (GMB) methods, explored chiefly by Shabana and his co-workers, treat slap or collision in linear structures as sequences of impulses, thereby maintaining the linearity of the structures throughout. Further, such linear analysis is facilitated by modal representation of the structures. These methods are discussed here and extended. Simulations on a simple two-rod problem demonstrate how this modal impulse approximation affects the system both directly after each impulse as well as over the entire collision. Furthermore, these simulations illustrate how the GMB results differ from the exact solution and how mitigation of these artifacts is achieved. Another modal method discussed in this paper is the idea of imposing piecewise constant forces over short, yet finite, time intervals during contact. The derivation of this method is substantially different than that of the GMB method, yet the numerical results show similar behavior, adding credence to both models. Finally, a novel method combining these two approaches is introduced. The new method produces physically reasonable results that are numerically very close to the exact solution of the collision of two rods. This approach avoids most of the nonphysical, numerical artifacts of interpenetration or chatter present in the first two methods.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2005;128(3):318-327. doi:10.1115/1.2172258.

Magnetic fields can be used to apply damping to a vibrating structure. Dampers of this type function through the eddy currents that are generated in a conductive material experiencing a time-changing magnetic field. The density of these currents is directly related to the velocity of the change in magnetic field. However, following the generation of these currents, the internal resistance of the conductor causes them to dissipate into heat. Because a portion of the moving conductor’s kinetic energy is used to generate the eddy currents, which are then dissipated, a damping effect occurs. This damping force can be described as a viscous force due to the dependence on the velocity of the conductor. In a previous study, a permanent magnet was fixed in a location such that the poling axis was perpendicular to the beam’s motion and the radial magnetic flux was used to passively suppress the beam’s vibration. Using this passive damping concept and the idea that the damping force is directly related to the velocity of the conductor, a new passive-active damping mechanism will be created. This new damper will function by allowing the position of the magnet to change relative to the beam and thus allow the net velocity between the two to be maximized and thus the damping force significantly increased. Using this concept, a model of both the passive and active portion of the system will be developed, allowing the beams response to be simulated. To verify the accuracy of this model, experiments will be performed that demonstrate both the accuracy of the model and the effectiveness of this passive-active control system for use in suppressing the transverse vibration of a structure.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2005;128(3):328-337. doi:10.1115/1.2172261.

In rotating beams, the Coriolis force acts through the mass and rotary inertias of the beam. A rotating beam simply supported off the axis of rotation is used as an example to study effects of this Coriolis force on vibration of structures. By adopting such a simple model, mass- and rotary inertia-induced terms in the free vibration responses can be obtained in separate, closed forms. The effect of each of these terms on vibration characteristics of the rotating beam is discussed in relation to parameters such as nonrotating natural frequencies, the rotation speed, and the slenderness ratio. Practical implications of these results in analyses of rotating structures of other types are discussed, for example estimating the significance of rotary inertias in relation to the slenderness ratio and the rotation speed.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2005;128(3):338-346. doi:10.1115/1.2166855.

A new approach for suppression and control of mechanical vibration in elastic beams undergoing cyclic motion is presented. The proposed model is based on the idea of generating axial uniform damping forces on the surface of the vibrating structure. Equation of motion and expression for system damping of the new model are derived, where the effectiveness of this model for reducing lateral vibration of a base excited beam is theoretically determined at different force levels. The analysis included the first five mode shapes, and the performance at different boundary conditions is also discussed. The theoretical model is verified experimentally, and the technique used to generate the superficial forces is explained. A comparison between theoretical and experimental results is shown. It is found that the higher the generated superficial force value, the higher the attenuation percentage. The new model is characterized by its simplicity, which enhances its reliability and reduces its cost, as it provides the desired results with higher reliability and reasonable cost, compared with other approaches of active and intelligent structural designs.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2005;128(3):347-356. doi:10.1115/1.2172262.

Driven by emission regulations in the US and the EU exhaust systems on new diesel engines are equipped with both a catalytic converter (CC) and a diesel particulate filter (DPF). The CC and DPF are normally placed after each other in an expansion chamber, to create a complete after-treatment device (ATD) to reduce the exhaust pollutants. The ATD unit can also affect the acoustical performance of an exhaust system. In this paper, an acoustic model of a complete ATD for a passenger car is presented. The model is made up of four basic elements: (i) straight pipes; (ii) conical inlet/outlet; (iii) CC unit, and (iv) DPF unit. For each of these elements, a two-port model is used and, with the exception of the DPF unit, known models from the literature are available. For the DPF unit, a new model suggested by the authors has been used. Using the models, the complete acoustic two-port model for the investigated ATD unit has been calculated and used to predict the sound transmission loss. The predictions have been compared to experimental data taken at cold conditions for various flow speeds and show a good agreement.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2005;128(3):357-365. doi:10.1115/1.2172263.

An extension of the Karhunen-Loève decomposition (KLD) specifically aimed at the evaluation of the natural modes of $n$-dimensional structures $(n=1,2,3)$ having nonhomogeneous density is presented. The KLD (also known as proper orthogonal decomposition) is a numerical method to obtain an “optimal” basis, capable of extracting from a data ensemble the maximum energy content. The extension under consideration consists of modifying the Hilbert space that embeds the formulation so as to have an inner product with a weight equal to the density. This yields a modified Karhunen-Loève integral operator, whose kernel is represented by the time-averaged autocorrelation tensor of the ensemble of data multiplied by the density function. The basis functions are obtained as the eigenfunctions of this operator; the corresponding eigenvalues represent the Hilbert-space-norm energy associated with each eigenfunction in the phenomenon analyzed. It is shown under what conditions the eigenfunctions, obtained using the proposed extension of the KLD, coincide with the natural modes of vibration of the structure (linear normal modes). An efficient numerical procedure for the implementation of the method is also presented.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2005;128(3):366-374. doi:10.1115/1.2172264.

A numerical method based on the Rayleigh-Ritz method has been presented for the forced vibration of open cylindrical shells. The equations are derived from the three-dimensional strain-displacement relations in the cylindrical coordinate system. The middle surface of the shell represents the geometry, which is defined by an angle that subtends the curved edges, the length, and the thickness. The displacement fields are generated with a predefined set of grid points on the middle surface using considerably high-order polynomials. Each grid point has five degrees of freedom, viz., three translational components along the cylindrical coordinates and two rotational components of the normal to the middle surface. Then the strain and kinetic energy expressions are obtained in terms of these displacement fields. The differential equation governing the vibration characteristics of the shell is expressed in terms of the mass, stiffness, and the load consistent with the prescribed displacement fields. The transient response of the shell with and without damping is sought by transforming the equation of motion to the state-space model and then the state-space differential equations are solved using the Runge-Kutta algorithm.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2006;128(3):375-384. doi:10.1115/1.2172265.

This paper is to develop a unified algorithm to predict vibration of spinning asymmetric rotors with arbitrary geometry and complexity. Specifically, the algorithm is to predict vibration response of spinning rotors from a ground-based observer. As a first approximation, the effects of housings and bearings are not included in this analysis. The unified algorithm consists of three steps. The first step is to conduct a finite element analysis on the corresponding stationary rotor to extract natural frequencies and mode shapes. The second step is to represent the vibration of the spinning rotor in terms of the mode shapes and their modal response in a coordinate system that is rotating with the spinning rotor. The equation of motion governing the modal response is derived through use of the Lagrange equation. To construct the equation of motion, explicitly, the results from the finite element analysis will be used to calculate the gyroscopic matrix, centrifugal stiffening (or softening) matrix, and generalized modal excitation vector. The third step is to solve the equation of motion to obtain the modal response, which, in turn, will lead to physical response of the rotor for a rotor-based observer or for a ground-based observer through a coordinate transformation. Results of the algorithm indicate that Campbell diagrams of spinning asymmetric rotors will not only have traditional forward and backward primary resonances as in axisymmetric rotors, but also have secondary resonances caused by higher harmonics resulting from the mode shapes. Finally, the algorithm is validated through a calibrated experiment using rotating disks with evenly spaced radial slots. Qualitatively, all measured vibration spectra show significant forward and backward primary resonances as well as secondary resonances as predicted in the theoretical analysis. Quantitatively, measured primary and secondary resonance frequencies agree extremely well with those predicted from the algorithm with mostly $<3.5%$ difference.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2005;128(3):385-391. doi:10.1115/1.2175089.

Smart adaptive structures and structronic systems have been increasingly investigated and developed in the last two decades. Although smart structures made of piezoelectrics, shape-memory materials, electrostrictive materials, and electro-/magnetorheological fluids have been evaluated extensively, studies of magnetostrictive continua, especially generic mathematical model(s), are still relatively scarce. This study is to develop a generic mathematical model for adaptive and controllable magnetostrictive thin shells. Starting with fundamental constitutive magnetostrictive relations, both elastic and magnetostrictive stresses, forces, and moments of a generic double-curvature magnetostrictive shell continuum subject to small and moderate magnetic fields are defined. Dynamic magnetomechanical system equations and permissible boundary conditions are defined using Hamilton's principle, elasticity theory, Kirchhoff-Love thin shell theory and the Gibb's free energy function. Magnetomechanical behavior and dynamic characteristics of magnetostrictive shells are evaluated. Simplifications of magnetostrictive shell theory to other common geometries are demonstrated and magnetostrictive/dynamic coupling and actuation characteristics are discussed.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2006;128(3):392-401. doi:10.1115/1.2166857.

The effect of fluid viscosity on the developing process of the instability in an over-hung flexible rotor partially filled with fluid, the dynamical behavior of the rotor system while the instability occurs, the unstable region of rotational speeds, and the whirl frequency of the rotor system in the unstable region of rotational speeds, are experimentally investigated in this paper. It is shown that when the rotational speed is just over the reduced first critical speed of the fluid-filled rotor system that is less than the first critical speed of empty rotor system, the unstable motion occurs. The rotor system in the unstable speed region does not whirl at either a constant rotational speed or the first critical speed of the empty rotor system, the whirl frequency of the rotor system in the unstable speed region, dominated by the fluid-filled ratio and weakly depending on the viscosity of fluid, linearly increases with the rotational speed. There exists a hysteresis range of rotational speeds at the upper bound of the unstable speed region, which is not caused by the rotor transient motion when passing through the unstable speed region. As the fluid viscosity increases, both the unstable speed region and the hysteresis region narrow. The influence of the fluid viscosity on the unstable speed region of a rotor filled with a high viscosity fluid must be considered.

Topics: Fluids , Viscosity , Rotors , Whirls , Motion
Commentary by Dr. Valentin Fuster

### TECHNICAL BRIEFS

J. Vib. Acoust. 2005;128(3):402-407. doi:10.1115/1.2166856.

A neural network based time optimal control of flexible structures is presented. The implementation is done on a flexible inverted L structure with surface-bonded piezoceramic sensors/actuators. The state-space presentation, from control input voltages to sensor output voltages is established in multivariable form. A variable gain multi-input multi-output linear quadratic regulator controller is designed and implemented. The controller gains are varied as the modal energy of the system decreases. The gains are varied in such a manner that the system utilizes maximum control energy from fixed amplitude of control voltage. The gains are calculated by solving the Riccatti equation with weightage in performance index that varies according to the states of the system. Thus at periodic intervals, the gains are updated to fully utilize the available control voltage. Comparison of the present technique is done with the classical bang-bang controller.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2005;128(3):408-410. doi:10.1115/1.2159042.

This note considers the stability of linear time varying second order systems. It studies the case where the stiffness matrix is a function of time. It provides sufficient conditions for stability and asymptotic stability of the system provided that certain conditions on the stiffness matrix are satisfied.

Commentary by Dr. Valentin Fuster

### DISCUSSION

J. Vib. Acoust. 2006;128(3):411. doi:10.1115/1.2172266.
FREE TO VIEW

This short letter offers a reply to the discussion on our previous paper, “Prediction of Time Varying Vibroacoustic Energy Using a New Energy Approach” (1) made by Savin (2). Generally speaking, the discussion is interesting, and the provided comments (particularly points 1 and 3) help to clarify several points that have not been fully covered in the paper. Having said that, the discussion should be corrected to include the following remarks.

Commentary by Dr. Valentin Fuster