A novel vibration control method utilizing magnetically mounted piezoelectric elements is described. Piezoelectric elements are bonded to permanent magnets, termed here as control mounts, which are attached to the surface of a steel beam through their magnetic attraction. The magnetic-piezoelectric control mounts are an alternative to traditional epoxy attachment methods for piezoelectric elements which allows for easy in-the-field reconfiguration. In model and laboratory measurements, the beam is driven through base excitation and the resonant shunt technique is utilized to demonstrate the attenuation characteristics of two magnetic-piezoelectric control mounts. The coupled system is discretized using a Galerkin finite element model that incorporates the tangential and vertical contact stiffnesses of the beam-magnet interface. The vibration reduction provided by the control mounts using a single magnet are compared to those designed with a magnetic array that alternates the magnetic dipoles along the length of the mount. Even though each design uses the same magnet thickness, the alternating magnetic configuration's interfacial contact stiffness is over 1.5 and 4 times larger in the tangential and vertical directions, respectively, than that of the single magnet, resulting in increased vibration reduction. Measured and simulated results show that the magnetic-piezoelectric control mounts reduced the beam's tip velocity by as much as 3.0 dB and 3.1 dB, respectively. The design tradeoffs that occur when replacing the traditional epoxy layer with a magnet are also presented along with some methods that could improve the vibration reduction performance of the control mounts. This analysis shows that the control mounts attenuate significant vibration despite having an imperfect bond with the beam, thus providing a viable and adaptable alternative to traditional piezoelectric attachment methods.

and excitation, and that maximizes dissipation through synchronized switching is described. In the model and laboratory measurements, a cantilever beam is driven through base excitation and two piezoelectric elements are attached to the beam for vibration control purposes. The distributed-parameter model for the beam-element system is discretized by using Galerkin’s method, and time histories of the system’s response describe the controller’s attenuation characteristics. The system is piecewise linear, and a stateto-state modal analysis method is developed to simulate the coupled dynamics of the beam and piezoelectric circuit by mapping the generalized coordinates between the sets of modes for the open-switch and closed-switch configurations. In synchronized switching control, the elements are periodically switched to an external resonant shunt, and the instants of optimal switching are identified through a filtered velocity signal. The controller adaptively aligns the center frequency of a bandpass filter to the beam’s fundamental frequency through a fuzzy logic algorithm in order to maximize attenuation even with minimal a priori knowledge of the excitation or the system’s mass and stiffness parameters. In implementation, the controller is compact owing to its low inductance and computational requirement. The adaptive controller attenuates vibration over a range of excitation frequencies and is robust to variations in system parameters, thus outperforming traditional synchronized switching. DOI: 10.1115/1.3117189

In magnetic tape drives, lateral in-plane vibration of the tape leads to misalignment between data tracks and read/write head's position resulting in reduced storage capacity. To attenuate this lateral tape motion (LTM), surface guides-which include grooved, porous or roughened rollers-are used. The axial motion of the tape over the roller surface switches between two states: a sticking state when the axial force is smaller than the static frictional force; and a slipping state when the axial force is larger than the frictional force. A good understanding of the physical phenomena involved in this frictional interaction between the magnetic tape and surface friction guides will allow the appropriate and optimal choice of roller characteristics. We conduct a parametric study of frictional interaction between roller surface and the traveling magnetic tape by systematically varying the axial tension and transport velocity of the tape. An experimental setup is used to independently control these parameters and obtain lateral vibration measurements at two equidistant points-upstream and downstream-from the tape-roller interface. Techniques from spectral analysis are applied to the two signals to analyze and isolate the effect of stick-slip friction on LTM. It is noticed that the coherence function between the LTM signals provides valuable insight into the nature of stick-slip friction at the interface. We subsequently use it as a metric to construct and understand the “dynamic phase diagram,” i.e., to demarcate regions in the tension-velocity phase-space where predominance of stick or slip occurs.

One challenge in designing web conveyance systems is controlling the displacement and vibration of the webs by guides without introducing instabilities or higher frequency disturbances from flange impacts. A solution to this problem is to use an actively or passively tilted guide or roller to steer the web. In this paper, a model of tilted guides with friction is developed, and it is shown that tilted guides produce a change in the web's displacement, slope, bending moment, and shear force. When the web is conceptually unwrapped from its path, the normal force between the web and a tilted guide has a component that acts in the direction of the web's lateral displacement, resulting in an equivalent force and bending moment acting on the web. The model is validated by measurements, and is compared to a previously existing model of guide tilt. In the configurations studied, the displacement of the web near the guide is linearly dependent on the tilt angle and tension and it increases exponentially with the web's span length. When the guide's tilt is oriented towards the center of the web's wrap around the guide, the equivalent bending moment is zero in the absence of friction, and there is good agreement between the model developed in this paper and the previously existing model. However, when the center of the web's wrap is oriented 90° away from the guide's tilt orientation, the equivalent force is zero in the absence of friction, and measurements demonstrate the necessity of the equivalent bending moment.

Magnetic tape is a flexible mechanical structure having dimensions that are orders of magnitude different in its thickness, width, and length directions. In order to position the tape relative to the read/write head, guides constrain the tape’s lateral motion, but even the modest forces that develop during guiding can cause wear and damage to the tape’s edges. This paper presents a tensioned axially-moving viscoelastic Euler–Bernoulli beam model used to simulate the tape’s lateral dynamics, the guiding forces, and the position error between the data tracks and the read/write head. Lateral vibration can be excited by disturbances in the form of pack runout, flange impacts, precurvature of the tape in its natural unstressed state, and spiral stacking as tape winds onto the take-up pack. The guide model incorporates nonlinear characteristics including preload and deadbands in displacement and restoring force. A tracking servo model represents the ability of the read/write head’s actuator to track disturbances in the tape’s motion, and the actuator’s motion couples through friction with the tape’s vibration. Low frequency excitation arising from pack runout can excite high frequency position error because of the nonlinear characteristics of the guides and impacts against the pack’s flanges. The contact force developed between the tape and the packs’ flanges can be minimized without significantly increasing the position error by judicious selection of the flanges’ taper angle.

A method for the modal analysis of continuous gyroscopic systems with nonlinear constraints is developed, which assumes that the nonlinear constraints can be expressed with piecewise-linear force-deflection profiles. Using this assumption, the mode shapes and natural frequencies are found for each state, and a mapping method based on the inner product of the mode shapes is developed to map the displacement of the system between 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. Several design parameters of the guide are considered: flange clearance, stiffness, symmetry, and the guide's position. Critical bifurcation thresholds exist, 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.

A novel vibration control method utilizing magnetically mounted piezoelectric actuators is described. Piezoelectric actuators are bonded to permanent magnets, which are attached to the surface of a steel cantilever beam through their magnetic attraction. The magnetic-piezoelectric control mounts are an alternative to traditional epoxy attachment methods for piezoelectrics which allows easy in-the-field reconfiguration. In model and laboratory measurements, the beam is driven through base excitation and the resonant shunt and synchronized switching techniques are applied to two magnetic-piezoelectric control mounts to attenuate vibration. The coupled system is discretized using a Galerkin finite element model that incorporates relative axial motion between the beam and the mounts, which is governed by the sticking contact stiffness per unit length of the beam-magnet interface. The control mounts are designed using a magnetic array configuration which increases the attraction force for a given magnet thickness. Results show that the magnetic-piezoelectric control mounts provide attenuation, while also providing the flexibility to easily adjust the actuators along the length of the beam.

An autonomous vibration controller that adapts to variations in a system's mass, stiffness, and excitation, and maximizes dissipation through synchronized switching, is described. In the model and laboratory measurements, a cantilever beam is driven through base excitation and two piezoelectric elements are attached to the beam for vibration control purposes. The distributed-parameter model for the beam-element system is discretized by using Galerkin's method, and time histories of the system's response describe the controller's attenuation characteristics. The system is piecewise linear and a state-to-state modal analysis method is developed to simulate the coupled dynamics of the beam and piezoelectric circuit by mapping the generalized coordinates between the sets of modes for the open-switch and closed-switch configurations. In synchronized switching control, the elements are periodically switched to an external resonant shunt, and the instants of optimal switching are identified through a filtered velocity signal. The controller adaptively aligns the center frequency of a bandpass filter to the beam's fundamental frequency through a fuzzy logic algorithm in order to maximize attenuation even with minimal a priori knowledge of the excitation or the system's mass and stiffness parameters. In implementation, the controller is compact owing to its low inductance and computational requirement. The adaptive controller attenuates vibration over a range of excitation frequencies, is robust to variations in system parameters, and can outperform traditional synchronized switching.

A finite element method is developed to treat the steady-state vibration of two axisymmetric structures—a base substructure and an attached damper substructure—that are driven by traveling wave excitation, and that couple through a spatially-distributed hysteretic friction interface. The base substructure is representative of a rotating brake rotor or gear, and the damper is a ring affixed to the base under preload and intended to control vibration through friction along the interface. In the axisymmetric approximation, the equation of motion of each substructure is reduced in order to the number of nodal degrees of freedom through the use of a propagation constant phase shift. Despite nonlinearity and with contact occurring at an arbitrarily large number of nodal points, the response during sticking, or during a combination of sticking and slipping motions, can be determined from a low-order set of computationally tractable nonlinear algebraic equations. The method is applicable to element types for longitudinal and bending vibration, and to an arbitrary number of nodal degrees of freedom in each substructure. In two examples, friction damping of the coupled base and damper is examined in the context of in-plane circumferential vibration (in which case the system is modeled as two unwrapped rods), and of out-of-plane vibration (alternatively, two unwrapped beams). The damper performs most effectively when its natural frequency is well below the base’s natural frequency (in the absence of contact), and also when its natural frequency is well-separated from the excitation frequency.

Multibody systems that undergo a prescribed rotational motion arise in such engineered systems as robots, spacecraft, propulsion and power generation systems, and certain sensors and actuators. The sensitivity of the system's response to changes in the design variables is important for optimization and trade-off studies, as well as for understanding the implications of manufacturing tolerances. A general formulation is developed for analytically calculating the first-order design sensitivities of coordinate values for a multibody system's dynamic equilibrium state during prescribed rotational motions. The method is based upon the use of relative coordinates, and a velocity transformation technique, and it is applicable to multibody systems having open or closed loop configurations. To illustrate effectiveness, accuracy, and computational efficiency, the present method is applied in three examples, and the sensitivities obtained analytically are compared with those obtained by the standard finite difference method. The finite difference approach is particularly sensitive to the choice of step size near a critical speed, and its implementation is generally more costly that the present method. In particular, there is a zero-sensitivity point at which the equilibrium configuration is insensitive to small perturbations in the design parameter's value. That condition can be a useful design point to the extent that manufacturing tolerance and variation in design parameter's values have no effect on dynamic equilibrium positions.

As the density of information stored in automated magnetic tape libraries continues to increase, greater requirements are placed on the precision of mechanical positioning in order to successfully read and write data bits. The location of the read/write head in the direction across the tape’s width (termed the lateral direction) is actively controlled in order to maintain alignment between the head and data tracks, even in the presence of the tape’s lateral vibration. However, during repositioning, vibration is undesirably transmitted from the laterally moving head structure to the axially-moving tape because of frictional contact between the two adjacent surfaces. As an analog of that interaction, a model is developed here to describe frictional vibration transmission from a surface having prescribed lateral motion to a tensioned beam that travels and slides over it. The beam is divided into contiguous regions corresponding to free spans and the beam’s portion that contacts the surface. A critical engagement length between the beam and the surface exists for which vibration transmission at a particular natural frequency can be substantially reduced, and for a given mode, that length depends weakly on the surface’s position along the beam’s span. By contouring the surface to have portions of differing radii of curvature, the extent of vibration transmission can be reduced over a broad range of frequency.

A general formulation for the tolerance analysis of dynamic equilibria in multibody systems having prescribed rotational motion is developed. In a state of dynamic equilibrium, a subset of generalized coordinates assume constant values, while the remaining coordinates vary and respond in time. Applications in which multibody systems exhibit dynamic equilibria include robots, spacecraft, propulsion and power generation systems, and some sensor devices. In the derived approach, manufacturing tolerances are mathematically modeled by probabilistic and statistical variables, through an analytical approach and through Monte Carlo simulation. An efficient computational method based upon direct differentiation is developed to calculate the first order sensitivity of the equilibria with respect to the design and manufacturing variables. To verify the accuracy and effectiveness of the present method, the present analytical method and the companion Monte Carlo approach are applied in examples to a rotating pendulum, a mechanical speed governor, and a model of a rate gyroscope sensor.

A general formulation is developed for the tolerance analysis of dynamic equilibria in a multibody system undergoing prescribed rotational motion, with applications including robots, spacecraft, propulsion and power generation systems, and sensors and actuators. In a state of dynamic equilibrium, a subset of the generalized coordinates assumes constant values while the remaining coordinates vary and respond in time. Manufacturing tolerances can be mathematically represented by probabilistic distributions or statistical variables through either an analytical approach or a Monte Carlo simulation. In the present tolerance work, the tolerances of design parameters including lengths, stiffnesses, inertias, and attachment positions are examined. In order to analytically calculate the statistical response of the dynamic equilibrium positions to such tolerances, the first-order sensitivities of the equilibria with respect to parameters are calculated. To illustrate the method’s accuracy and computational efficiency, two numerical examples are considered, and the statistical results obtained analytically for the equilibria are compared with those calculated through Monte Carlo simulation. In some cases, an equilibrium configuration can have an operating condition for which the response has zero standard deviation to perturbations of a design parameter. That condition can be a useful design point to the extent that typical manufacturing tolerances or other sources of variation would have no effect on the dynamic equilibrium configuration.

The operating positional error of a robot manipulator, which develops inevitably because of manufacturing tolerances and assembly clearances, is preferentially maintained within a certain range in order to achieve an acceptable level of performance and accuracy. Because additional cost is incurred when manufacturing tolerances are tightened, an alternative design strategy maximizes the tolerances (so as to reduce the cost) while minimizing positioning error (to satisfy a performance requirement). In this paper, a new joint clearance model is developed for spatial mechanisms that incorporate revolute joints, which in turn are subjected to specified tolerance or uncertainty in the orientation of their axes. Statistical design parameters related to variations of link length and joint axis orientation are identified from the clearance model. The statistical influence of the design parameters on the robot manipulator's response is investigated through a general multibody dynamics sensitivity formulation. The method offers substantial improvement in computational efficiency when compared to the Monte Carlo procedure. The uncertainty in orientation of a revolute joint's axis influences the positioning accuracy of the robot manipulator's response to a greater degree than does uncertainty in the length of a link.

The parametric excitation of an axially moving plate is examined in an application where a partial foundation moves in the plane of the plate and in a direction orthogonal to the plate's transport. The stability of the plate's out-of-plane vibration is of interest in a magnetic tape data storage application where the read/write head is substantially narrower than the tape's width and is repositioned during track-following maneuvers. In this case, the model's equation of motion has time-dependent coefficients, and vibration is excited both parametrically and by direct forcing. The parametric instability of out-of-plane vibration is analyzed by using the Floquet theory for finite values of the foundation's range of motion. For a relatively soft foundation, vibration is excited preferentially at the primary resonance of the plate's fundamental torsional mode. As the foundation's stiffness increases, multiple primary and combination resonances occur, and they dominate the plate's stability; small islands, however, do exist within unstable zones of the frequency-amplitude parameter space for which vibration is marginally stable. The plate's and foundation's geometry, the foundation's stiffness, and the excitation's amplitude and frequency can be selected in order to reduce undesirable vibration that occurs along the plate's free edge.

The free and forced vibration of a moving medium is examined in an application where distributed friction guiding is used to control lateral position passively. Subambient pressure features formed in the guides intentionally modify the naturally occurring selfpressurized air bearing and increase the contact force between the medium and the guide's surface. These features increase friction to a level beyond that achievable based on the nominal wrap pressure. The moving medium is modeled as a beam that is transported over frictional regions and subjected to prescribed boundary disturbances arising from runout of a supply or take-up roll. For axial transport at a speed that is high compared to the velocity of lateral vibration, Coulomb friction between the guides and the moving medium can be well approximated by a derived expression for equivalent viscous damping. The equation of motion is developed for the cases of a single cylindrical guide and of a multiplicity of guides having arbitrary placement. The level of equivalent damping for each mode decreases with transport speed, and critical speeds exist where each vibration mode transitions between the overdamped and underdamped regimes. Parameter studies in the contact pressure, transport speed, and guide geometry identify preferred design configurations for maximizing dissipation in particular modes and for attenuating high-frequency response.

A ring damper can be affixed to a rotating base structure such as a gear, an automotive brake rotor or a gas turbine's labyrinth air seal. Depending on the frequency range, wavenumber and level of preload, vibration of the base structure can be effectively and passively attenuated by friction that develops along the interface between it and the damper. The assembly is modelled as two rods that couple in longitudinal vibration through spatially distributed hysteretic friction, with each rod having periodic boundary conditions in a manner analogous to an unwrapped ring and disc. As is representative of rotating machinery applications, the system is driven by a travelling wave disturbance, and for that form of excitation, the base structure's and the damper's responses are determined without the need for computationally intensive simulation. The damper's performance can be optimized with respect to normal preload, and its effectiveness is insensitive to variations in preload or the excitation's magnitude when its natural frequency is substantially lower than the base structure's in the absence of contact.

This paper investigates the forced response dynamics of a clamped–clamped beam to which a rigid body is attached, and in the presence of periodic or non-periodic impacts between the body and a comparatively compliant base structure. The assembly is subjected to base excitation at specified frequency and acceleration, and the potentially complex responses that occur are examined analytically. The two sets of natural frequencies and vibration modes of the beam-rigid body structure (in its in-contact state, and in its not-in-contact state), are used to treat the forced response problem through a series of algebraic mappings among those states. A modal analysis based on extended operators for the (continuous) beam and (discrete) rigid body establishes a piecewise linear state-to-state mapping for transition between the in-contact and not-in-contact conditions. The contact force, impulse, and displacement each exhibit complex response characteristics as a function of the excitation frequency. Periodic responses occurring at the excitation frequency, period-doubling bifurcations, grazing impacts, sub-harmonic regions, fractional harmonic resonances, and apparently chaotic responses each occur at various combinations of damping, excitation frequency, and contact stiffness. The results of parameter studies in structural asymmetry and the eccentricity of the contact point's location are discussed.

This paper investigates the experimental dynamics of a beam structure that supports an attached rigid body and that can impact a comparatively compliant base structure. The problem area is motivated by impact phenomena that are observed in certain structures internal to nuclear reactors. The assembly is subjected to base excitation at specified frequency and acceleration, and the resulting displacement and velocity time histories are recorded and used to obtain spectra, phase diagrams, and Poincaré sections. The measurements validate simulation results obtained by using a constraint and modal mapping method based on the two sets of modes when the structure is in-contact, and when it is not-in-contact. Generalized coordinates are mapped across the impact discontinuities in the modal representation. The forced response simulation predicts the test specimen’s response over a range of excitation frequencies. The specimens are fabricated as single integral structures from acrylnitrile butadene styrene plastic through rapid prototyping technology in order to eliminate the undesirable dissipation and flexibility arising from joints and connections. The experimental system can exhibit complex response characteristics, and the influences on complexity of deadband clearance and of asymmetry in the point of impact are examined in the experiments.

This paper investigates the experimental dynamics of a beam structure that supports an attached rigid body and that can impact a comparatively compliant base structure. The problem area is motivated by impact phenomena that are observed in certain structures internal to nuclear reactors. The assembly is subjected to base excitation at specified frequency and acceleration, and the resulting displacement and velocity time histories are recorded and used to obtain spectra, phase diagrams, and Poincare´ sections. The measurements validate simulation results obtained by using a constraint and modal mapping method based on the two sets of modes when the structure is in-contact, and when it is not-in-contact. Generalized coordinates are mapped across the impact discontinuities in the modal representation. The forced response simulation predicts the test specimen's response over a range of excitation frequencies. The specimens are fabricated as single integral structures from acrylnitrile butadene styrene plastic through rapid prototyping technology in order to eliminate the undesirable dissipation and flexibility arising from joints and connections. The experimental system can exhibit complex response characteristics, and the influences on complexity of deadband clearance and of asymmetry in the point of impact are examined in the experiments.