Cantilever Beam System Research Articles

Structural Damage Detection through Chaotic Interrogation and Attractor Analysis

In this paper, a new approach for damage detection using a chaotic signal as an input excitation and steady state attractor-based measures as diagnostic parameters is investigated by means of numerical simulations. The method utilizes the deterministic, extreme sensitive properties of the chaotic signal to give rise to a low-dimensional response for feature extraction. This approach is applied to two numerical examples, the 4 DOF spring-mass-damper and a cantilevered beam system, where the damage is produced by varying the structural damping and stiffness, respectively. Lyapunov dimension is calculated as a “feature” for detecting the damage. Results show that this approach is feasible to detecting structural damage.

High performance open loop control of scanning with a small cylindrical cantilever beam

The steady-state response motion of a base excited cantilever beam with circular cross-section excited by a unidirectional displacement will fall along a straight line. However, achieving straight-line motion with a real cantilever beam of circular cross-section is difficult to accomplish. This is due to the fact that nonlinear effects, small deviations from circularity, asymmetric boundary conditions, and actuator cross coupling can induce whirling. The vast majority of previous work on cantilever beam whirling has focused on the effects of system nonlinearities. We show that whirling is a much broader problem in the design of resonant beam scanners in that the onset of whirling does not depend on large amplitude of motion. Rather, whirling is the norm in real systems due to small system asymmetries and actuator cross coupling. It is therefore necessary to control the growth of the whirling motion when a unidirectional beam motion is desired. We have developed a novel technique to identify the two eigen directions of the beam. Base excitation generated by virtual electrodes along these orthogonal eigen axes of the cantilever beam system generates tip vibration without whirl. This leads to accurate open loop control of the motion of the beam through the combined actuation of two pairs of orthogonally placed actuator electrodes.

The analysis of a piezoelectric bimorph beam with two input base motions for power harvesting

The exploitation of usable power from vibration environments shows potential benefit for recharging batteries and powering wireless transmission. In this paper, we present a novel technique for simulating the electromechanical cantilevered piezoelectric bimorph beam system with two input base transverse and longitudinal motions for predicting power harvesting. The piezoelectric bimorph beam with tip mass was modelled using the Euler-Bernoulli beam assumptions. The strain fields from transverse bending and longitudinal forms can affect the physical behaviour of the polarity and electric field in terms of the series and parallel connections of the piezoelectric bimorph, in such way that each connection has two vector configurations of X-poling and Y-poling due to input base motions. This situation must be correctly identified to form the piezoelectric couplings. The piezoelectric couplings can create the electrical force and moment of each piezoelectric layer in the mechanical domain. At this point, we introduce a new method of modelling the piezoelectric bimorph beam under two input base-motions using coupling superposition of the elastic-polarity field for predicting power harvesting. The constitutive dynamic equations were derived using the weak form from the Hamiltonian theorem, with Laplace transforms being used to obtain the multi-mode frequency response functions (FRFs) relating the input mechanical vibrations with the output dynamic displacement, velocity and power harvesting. The power harvesting predictions under parallel connection at frequencies close to the fundamental bending frequency demonstrate a possibility of being able to produce around 0.4 mW per unit input base transverse acceleration of 3 m/s2. Furthermore, it is shown that varying the load resistance from 20 kΩ to 200 kΩ affects the amplitude of power harvesting as well as resulting in a shift of the first natural frequency from 76 Hz to 79 Hz.

An adaptive harmonic balance method for predicting the nonlinear dynamic responses of mechanical systems—Application to bolted structures

Aeronautical structures are commonly assembled with bolted joints in which friction phenomena, in combination with slapping in the joint, provide damping on the dynamic behavior. Some models, mostly nonlinear, have consequently been developed and the harmonic balance method (HBM) is adapted to compute nonlinear response functions in the frequency domain. The basic idea is to develop the response as Fourier series and to solve equations linking Fourier coefficients. One specific HBM feature is that response accuracy improves as the number of harmonics increases, at the expense of larger computational time. Thus this paper presents an original adaptive HBM which adjusts the number of retained harmonics for a given precision and for each frequency value. The new proposed algorithm is based on the observation of the relative variation of an approximate strain energy for two consecutive numbers of harmonics. The developed criterion takes the advantage of being calculated from Fourier coefficients avoiding time integration and is also expressed in a condensation case. However, the convergence of the strain energy has to be smooth on tested harmonics and this constitutes a limitation of the method. Condensation and continuation methods are used to accelerate calculation. An application case is selected to illustrate the efficiency of the method and is composed of an asymmetrical two cantilever beam system linked by a bolted joint represented by a nonlinear LuGre model. The practice of adaptive HBM shows that, for a given value of the criterion, the number of harmonics increases on resonances indicating that nonlinear effects are predominant. For each frequency value, convergence of approximate strain energy is observed. Emergence of third and fifth harmonics is noticed near resonances both on vibratory responses and on approximate strain energy. Parametric studies are carried out by varying the excitation force amplitude and the threshold value of the adaptive algorithm. Maximal amplitudes of vibration and frequency response functions are plotted for three different points of the structure. Nonlinear effects become more predominant for higher force amplitudes and consequently the number of retained harmonics is increased.

Comparison of Alternative Techniques for Characterizing Magnetostriction and Inverse Magnetostriction in Magnetic Thin Films

We compare the direct and inverse techniques of measuring magnetostriction in magnetic thin films. We chose a set of four magnetic thin film samples (Co <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">95</sub> Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sub> , Co <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">60</sub> Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">20</sub> B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">20</sub> , Ni <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">65</sub> Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">15</sub> Co <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">20</sub> , and Ni <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">80</sub> Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">20</sub> ) for the measurements, representing positive and negative magnetostriction and having saturation magnetostriction of magnitudes ranging from 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-7</sup> to 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> . We made the direct measurements on a high-precision optical cantilever beam system, and we carried out the inverse magnetostriction measurements on a nondestructive inductive <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> looper with three-point bending stage.

Adaptive importance sampling for risk analysis of complex infrastructure systems

Complex civil infrastructure systems are typically exposed to random loadings and have a large number of possible failure modes, which often exhibit spatially and temporally variable consequences. Monte Carlo (level III) reliability methods are attractive because of their flexibility and robustness, yet computational expense may be prohibitive, in which case variance reduction methods are required. In the importance sampling methodology presented here, the joint probability distribution of the loading variables is sampled according to the contribution that a given region in the joint space makes to risk, rather than according to probability of failure, which is the conventional importance sampling criterion in structural reliability analysis. Results from simulations are used to intermittently update the importance sampling density function based on the evaluations of the (initially unknown) risk function. The methodology is demonstrated on a propped cantilever beam system and then on a real coastal dike infrastructure system in the UK. The case study demonstrates that risk can be a complex function of loadings, the resistance and interactions of system components and the spatially variable damage associated with different modes of system failure. The methodology is applicable in general to Monte Carlo risk analysis of systems, but it is likely to be most beneficial where consequences of failure are a nonlinear function of load and where system simulation requires significant computational resources.

Cumulative effect of structural nonlinearities: chaotic dynamics of cantilever beam system with impacts

The nonlinear analysis of a common beam system was performed, and the method for such, outlined and presented. Nonlinear terms for the governing dynamic equations were extracted and the behaviour of the system was investigated. The analysis was carried out with and without physically realistic parameters, to show the characteristics of the system, and the physically realistic responses. Also, the response as part of a more complex system was considered, in order to investigate the cumulative effects of nonlinearities. Chaos, as well as periodic motion was found readily for the physically unrealistic parameters. In addition, nonlinear behaviour such as co-existence of attractors was found even at modest oscillation levels during investigations with realistic parameters. When considered as part of a more complex system with further nonlinearities, comparisons with linear beam theory show the classical approach to be lacking in accuracy of qualitative predictions, even at weak oscillations.

Cumulative effect of structural nonlinearities: chaotic dynamics of cantilever beam system with impacts

Abstract The nonlinear analysis of a common beam system was performed, and the method for such, outlined and presented. Nonlinear terms for the governing dynamic equations were extracted and the behaviour of the system was investigated. The analysis was carried out with and without physically realistic parameters, to show the characteristics of the system, and the physically realistic responses. Also, the response as part of a more complex system was considered, in order to investigate the cumulative effects of nonlinearities. Chaos, as well as periodic motion was found readily for the physically unrealistic parameters. In addition, nonlinear behaviour such as co-existence of attractors was found even at modest oscillation levels during investigations with realistic parameters. When considered as part of a more complex system with further nonlinearities, comparisons with linear beam theory show the classical approach to be lacking in accuracy of qualitative predictions, even at weak oscillations.

VIBRATION CONTROL OF A SMART STRUCTURE USING PERIODIC OUTPUT FEEDBACK TECHNIQUE

ABSTRACTActive vibration control is an important problem in structures. One of the ways to tackle this problem is to make the structure smart, adaptive and self‐controlling. The objective of active vibration control is to reduce the vibration of a system by automatic modification of the system's structural response. This work features the modeling and design of a Periodic Output Feedback (POF) control technique for the vibration control of a smart flexible cantilever beam system for a Single Input Single Output case. A POF controller is designed for the beam by bonding patches of piezoelectric layer as sensor/actuator to the master structure at different locations along the length of the beam. The entire structure is modeled in state space form using the Finite Element Method by dividing the structure into 3, 4, 5 elements, thus giving rise to three types of systems, viz., system 1 (beam divided into 3 finite elements), system 2 (4 finite elements), system 3 (5 finite elements).POF controllers are designed for the above three types of systems for different sensor/actuator locations along the length of the beam by retaining the first two vibratory modes. The smart cantilever beam model is developed using the concept of piezoelectric bonding and Euler‐Bernouli theory principles. The effect of placing the sensor/actuator at various locations along the length of the beam for all the three types of systems considered is observed and the conclusions are drawn for the best performance and for the smallest magnitude of the control input required to control the vibrations of the beam. The tip displacements with the controller is obtained. Performance of the system is also observed by retaining the first 3 vibratory modes and the conclusions are drawn.

Robust multivariable control of a double beam cantilever smart structure

The uncertain surrounding environment in which smart structures are employed meansthat a fixed reference base with which to operate sensing/actuation devices cannot beassumed. The distributed placement of piezoceramic devices has made them the mostpopular choice for such an application. The limited available output force makes it essentialto consider the problems of control signal saturation on the robustness of resultingcontrollers. Much previous work has considered smart structural control in a singledimension. The current work looks at the vibration control of a two-dimensional smartstructure. Actuation is applied by means of two pairs of piezoactuators capable ofapplying moments with a controllable phase and amplitude difference allowing acontrollable ratio of bending and torsion motion. The design process of a multivariable(multi-input multi-output) generalized minimum variance control system is discussed.Application to an experimental double cantilever beam system is shown to result inrobust closed-loop control. Comparisons to the benchmark of multi-loop controldemonstrate numerous advantages of the multivariable system. Results are furthersupported with the use of sensitivity functions showing the response of the closed-loopsystems to frequencies of interest, particularly the important vibrational modes.

Anatomic suitability of the C1-C2 complex for pedicle screw fixation.

A computed tomography (CT) study of 60 consecutive patients (120 sides) was performed to assess suitability for either transarticular or pedicle screw fixation. A C1 lateral mass and C2 pedicle screw fixation with a rigid cantilever beam system has been described. The anatomic constraints relevant for this technique have not. Fifty consecutive patients underwent standard CT of the cervical spine. Pedicle and transarticular screw trajectories were plotted, and the maximum safe diameter for screw placement was determined for each trajectory. Also, trajectories were plotted in 10 additional patients with known craniocervical junction abnormalities using three-dimensional (3-D) imaging and computer-aided navigation tools. Screw placement was considered feasible if a 4-mm diameter trajectory could be plotted without impingement on neural or vascular structures. Four-millimeter diameter pedicle screws could be placed in 91 of 100 C2 pedicles in the CT studies and in 20 of 20 pedicles in the 3-D studies. Four-millimeter diameter C1-C2 transarticular screws could be placed in 94 of 100 sides in the CT study and in 19 of 20 sides in the 3-D study. Four sides could tolerate a C2 pedicle screw and not a transarticular screw; the opposite situation existed in five sides. Placement of screws into C1 was not an issue in any patient. The mean maximum diameter of potential transarticular screws was 6.5 mm, and the mean maximum diameter of the pedicle screws was 5.3 mm (P < 0.01). C1-C2 pedicle screw fixation is a technique that appears to be widely applicable and may represent an alternative fixation technique in selected patients.

Modeling of Passive Constrained Layer Damping as Applied to a Gun Tube

We study the damping effects of a cantilever beam system consisting of a gun tube wrapped with a constrained viscoelastic polymer on terrain induced vibrations. A time domain solution to the forced motion of this system is developed using the GHM (Golla-Hughes-McTavish) method to incorporate the viscoelastic properties of the polymer. An impulse load is applied at the free end and the tip deflection of the cantilevered beam system is determined. The resulting GHM equations are then solved in MATLAB by transformation to the state-space domain.

Dynamic behavior of double cantilever beams subjected to impact

The dynamic plastic behavior of a double cantilever beam system subjected to impact by a rigid mass at their tips is studied. Three approaches are adopted: a rigid, perfectly plastic (r.p.p) complete solution, a mode solution, and an elastic, perfectly plastic (e.p.p) solution based on non-linear beam theory using the Dyna3D finite element code. In the complete solution based on the r.p.p idealization, the dynamic response of the system comprises three successive phases, which are characterized by different sets of plastic hinges in the system and expressed in closed analytical forms. The mode solution based on the r.p.p idealization is presented in a much simpler form. Particular attention is paid to the partitioning of the dissipation of the total input energy between the two beams. Numerical examples are given to demonstrate the influence of some of the structural parameters on the energy partitioning. In particular, the results obtained from the three approaches are compared, and the validity of the r.p.p models in predicting the energy partitioning between the two beams is examined.

Neural Network Discrimination in Intelligent Vibration Control

The overall goal of this study lies in the multi-mode structural vibration control for systems, based on spatial information. In order to accomplish this task, a neural network was designed to identify the shape of the dominant vibration mode by using properly located piezoelectric sensors. The network then chose a set of scaling gains for a fuzzy logic controller that had been optimally tuned for that particular mode’s shape. Next, a fuzzy logic controller was employed to control the vibratory system. The novel technique was experimentally verified on an aluminum cantilever beam system that was augmented with two lead zirconate titanate (PZT) actuators and two PZT sensors. The optimized control strategy was then compared to traditional fuzzy logic control. Finally the mass of the plant was increased by over a factor of 1.5 and the same controller was used to adequately stop vibration.

Fuzzy model reference learning control: a new control paradigm for smart structures

The overall goal of this study lies in the synergistic combination of intelligent control and smart materials to produce an adaptive and robust controlled system. In order to accomplish this task, steel cantilever beam (the plant) was augmented with a lead zirconate titanate (PZT) actuator pair and a PZT sensor. A controller was developed using algorithms produced from fuzzy logic controls and adaptive fuzzy controls. The expected goal is to dampen the fundamental vibration mode of the system in the presence of modeling uncertainties. Fuzzy model reference learning control (FMRLC) is a learning system with the capability to improve its performance over a period of time when various plant uncertainties are introduced. First, the controller will be employed on unseen plant disturbances. The robustness of the system will be examined when the cantilever beam system properties change. Extra masses will be added to account for variations in system parameters. Controllers such as positive position feedback and direct fuzzy will be developed and compared to the adaptive fuzzy controller.

Modelling and bifurcation analysis of internal cantilever beam system on a steadily rotating ring

Using Hamilton variation principle, a nonlinear dynamic model of the system with a finite deforming Rayleigh beam clamped radially to the interior of a rotating rigid ring, under the assumption that the constitutive relation of the beam is linearly elastic, is discussed. The bifurcation behavior of the simple system with the Euler-Bernoulli beam is also discussed. It is revealed that these two models have no influence on the critical bifurcation value and buckling solution in the steady state. Then we use the assumption model method to analyse the bifurcation behavior of the steadily rotating Euler-Bernoulli beam and get two different types of bifurcation behavior which physically exist. Finite element method and shooting method are used to verify the analytical results. The numerical results confirm our research conclusion.

Self-Tuning Active Vibration Control in Flexible Beam Structures

This paper presents the design and performance evaluation of an adaptive active control mechanism for vibration suppression inflexible beam structures. A cantilever beam system in transverse vibration is considered. First-order central finite difference methods are used to study the behaviour of the beam and develop a suitable test and verification platform. An active vibration control algorithm is developed within an adaptive control framework for broadband cancellation of vibration along the beam using a single-input multi-output (SIMO) control structure. The algorithm is implemented on a digital processor incorporating a digital signal processing (DSP) and transputer system. Simulation results verifying the performance of the algorithm in the suppression of vibration along the beam, using single-input single-output and SIMO control structures, are presented and discussed.

Applying velocity specification techniques to control robotic positioning systems

Abstract Residual vibration limits the terminal positioning precision of robotic manipulators. The residual vibration exhibited by robotic manipulators can be due to low frequency structural modes, self-excited by the manipulator during operations. In these cases, redesigning the structure to modify the resonant frequency(ies) causing residual vibration can be impractical or too costly. In this paper, an intelligent controller scheme is developed to measure and control residual vibration caused by the principal structural resonance. The controller tests the robot structure for critical resonant modes, calculates a drive profile using motion specification techniques to control the residual vibration, and then evaluates its effectiveness. A frequency domain interpretation is used to select the preferred input driver waveform. A breadboard controller is also developed for a single-degree-of-freedom drive system and it is used to validate the proposed strategy. Test results for a simple cantilever beam system and for a desktop articular-arm robot are presented.

Controlling residual vibration of robotic structures

Residual vibration, which limits the terminal positioning precision of robotic manipulators, can be due to low frequency structural modes that are self-excited by the manipulator during operation. Redesigning the structure to modify the resonant frequency(s) causing residual vibration can be impractical or too costly. An intelligent controller scheme is being developed at the Advanced Manufacturing Center of Cleveland State University to measure and control residual vibration caused by the principal structural resonance. Test results for a simple cantilever beam system and for a desktop articular-arm robot are presented.

Development of a simple instrument for determination of salt concentration profile in the operation of solar ponds

The development and construction details of an instrument to measure densities of the brine at different depths in a solar pond are presented and discussed. Using a vertically moving sampling tube, brine is drawn from the pond and then is fed to the bottom of a conical container in which a sealed conical glass vessel is fully submerged. The bouyancy force exerted by the water on the sealed glass vessel is measured by a system of cantilever beam and strain gauges. The output of the strain gauge bridge is proportional to the density of the fluid passing through the conical container. It is shown that this instrument can be effectively used to determine the salt concentration profile in salt gradient solar ponds.