Active vibration control is implemented using multiple piezoelectric actuators and sensors bonded to the top and bottom surfaces of a cantilever beam. The control is exercised using closed-loop displacement feedback. The objective of the study is to determine the optimal locations of patch actuators and sensors such that the frequency gap between higher frequencies of the beam is maximized. Maximizing the frequency gaps is useful in those cases where the excitation frequency can be placed in between two higher order frequencies to avoid the resonance. In these cases the design requirement is to maximize the difference between the two higher order frequencies such as between the first and second frequencies or between the second and third frequencies, etc. In the present study the frequency gaps between the higher order frequencies are investigated with respect to actuator and sensor locations with a view towards determining the optimal locations for largest frequency gaps. The differential equation governing the vibrations of a beam/piezo patch system is solved using an integral equation approach. The equivalent integral equation formulation of the problem avoids the discontinuities which arise due to partial length piezo patches. The solution is approximated using the eigenfunctions of the freely vibrating structure which leads to a system of algebraic equations. The numerical results are given for various patch combinations and the optimal locations of the actuators and the sensors are determined. It is observed that the optimal locations of the piezo patches depend on the specific frequency gap as well as the patch configurations.
Use of piezoelectric patches as sensors and actuators for the vibration control of beams is a well established technology. Various techniques developed to analyze these problems range from analytical to computational ones, each with a different level of complexity and accuracy. In the present paper three techniques, namely, Laplace Transform, integral equation and assumed modes are applied to the vibration control problem involving a cantilever beam with piezoelectric patches attached to the top and bottom surfaces. These patches act as sensors and actuators providing a feedback control mechanism for the damping of vibrations. The Laplace Transform involves the transform of the space part of the partial differential equation governing the motion of the beam and inverse transform to find the exact solution. The integral equation approach transforms the differential equation formulation to an integral equation formulation which, in turn, is replaced by an infinite system of equations. As such this method provides an approximate solution, the accuracy of which depends on the size of the system of linear equations involved. The assumed modes method is quite widely used because of its ease of application, and its accuracy depends on the number of terms in the series approximation used to express the solution. The above solution methods are summarized and difficulties, drawbacks and advantages associated with each method are discussed. The accuracy of each technique is compared and assessed in the context of a vibrating cantilever beam with patches. The results are given in a comparative manner which also includes the exact solutions.
The Euler-Bernoulli model of transverse vibration of a cantilever beam is extended to include strain rate (Kelvin-Voigt) damping to study active vibration control under internal damping. A piezo patch sensor is bonded onto the top of the beam, while an actuator patch is bonded onto the bottom of the beam. Displacement and velocity feedback are considered as the control mechanisms. The resulting partial differential equation is solved using an integral equation approach and investigated for control effectiveness in terms of changes in the natural frequencies and damping ratios for different gains, damping coefficients, and patch locations. Results from the integral equation approach for patch sizes extended to the boundary are compared to results of the boundary control method.
KEYWORDS: Sensors, Actuators, Feedback control, Signal processing, Vibration control, Process control, Mechanical engineering, Mathematics, Digital signal processing, Distance measurement
An elastic beam is sandwiched between two thin layers of mono-axially oriented piezoelectric material which act as a distributed vibration control. One layer acts as a distributed sensor while the other behaves as an actuator. A velocity feedback control with time delay is implemented in which the distributed sensor signal which is proportional to the time derivative of the strains is amplified and applied to the actuator after a time delay. Because of the nature of the problem the control action enters as a boundary control. The control effectiveness, changes in natural frequency and damping ratios, is analyzed for the case of a cantilever beam. Comparisons are also made with the uncontrolled case and the zero-delay system.
An analytical method is given for the determination of the eigenfunctions and eigenfrequencies for two-dimensional structural vibration problems in the presence of patch sensors and patch actuators. The method is based on converting the differential equation formulation of the problem to an integral equation. The conversion is accomplished by introducing an explicit non-symmetric kernel. The kernel consists of two parts, one taking account of the stiffness and the other taking account of the control moments induced by the distributed actuators. The control moments involve piezoelectric constants and feedback voltages made up of gains times the sensor signals. Eigenfrequencies are obtained for a representative example. The results presented in the study can be used for benchmarking solutions based on numerical or approximation approaches.
An analysis of the solutions for various feedback control laws applied to vibrating cantilever beams is evaluated. The control is carried out via piezoelectric patch sensors and actuators. By considering an integral equation formulation, which is equivalent to the differential equation formulation, the analytical results are investigated. The conversion is accomplished by introducing an explicit Green's Function. The feedback controls implemented include displacement, velocity, acceleration and combinations of these. A numerical comparison of eigenvalues will be presented to illustrate the efficacy of the method and to contrast the effects of the controls. The results presented in the study can be used for benchmarking solutions based on numerical or approximation approaches.
An analytical method is given for the determination of the eigenfunctions and eigenfrequencies for one-dimensional structural vibration problems in the presence of patch sensors and patch actuators. The method is based on converting the differential equation formulation of the problem to an integral equation formulation. The conversion is accomplished by introducing an explicit Green's function. The Green's function consists of two parts, one taking account of the stiffness and the other taking account of the control moments induced by the distributed actuators. The control moments involve piezoelectric constants and feedback voltages made up of gains times the sensor signals. Obtaining the eigenvalues and eigenfunctions of this integral equation gives the solution to the piezo-control problem.
Distributed vibration control of an elastic beam is analyzed. The elastic beam is sandwiched between two thin layers of mono-axially oriented piezoelectric material. One layer acts as a distributed sensor while the other behaves as an actuator. A displacement feedback control with a time delay is implemented. In this feedback, the distributed sensor signal which is proportional to strains is amplified and applied to the actuator after a time delay. Because of the nature of the problem the control action enters as a boundary control. An evaluation of the control effectiveness, natural frequency and damping is made for the case of a cantilever beam. Comparisons are given with the uncontrolled cases and zero-delay systems.
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