Qibo Mao

Free vibration analysis of stepped beams by using Adomian decomposition method

Abstract The Adomian decomposition method (ADM) is employed in this paper to investigate the free vibrations of a stepped Euler–Bernoulli beam consisting of two uniform sections. Each section is considered a substructure which can be modeled using ADM. By using boundary condition and continuity condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions, step ratios and step locations are presented. Comparing the results using ADM to those given in the literature, excellent agreement is achieved.

Design of Shaped Piezoelectric Modal Sensor for a Type of Non-Uniform Beams Using Adomian-Modified Decomposition Method

In this article, a solution to the problem of finding the shape of piezoelectric modal sensors for a non-uniform beam with continuously exponential varying width under general boundary conditions is proposed using the Adomian modified decomposition method (AMDM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. Other modes are filtered out. The modal sensor shape is expressed as a linear function of the second spatial derivative of the structural mode shape function and the beam width function. Based on the AMDM and employing some simple mathematical operations, the closed-form series solution of the second spatial derivative of the mode shapes can be determined. Then, the shapes of the designed modal sensors are obtained. Finally, several numerical examples are given to demonstrate the feasibility of the proposed modal sensors with various boundary conditions.

Reduction of Structural Sound Radiation and Vibration Using Shunt Piezoelectric Materials

In this paper, structural sound and vibration control using passive and semi-active shunt piezoelectric damping circuits is presented. A piezoelectric patch with an electrical shunt circuit is bonded to a base structure. When the structure vibrates, the piezoelectric patch strains and transforms the mechanical energy of the structure into electrical energy, which can be effectively dissipated by the shunt circuit. Hence, the shunt circuit acts as a means of extracting mechanical energy from the base structure. First, different types of shunt circuits (such as RL series circuit, RL parallel circuit and RL-C circuit), employed in the passive damping arrangement, are analyzed and compared. By using the impedance method, the general modelling of different shunt piezoelectric damping techniques is presented. The piezoelectric shunt circuit can be seen as additional frequency-dependence damping of the system. One of the primary concerns in shunt damping is to choose the optimal parameters for shunt circuits. In past efforts most of the proposed tuning methods were based on modal properties of the structure. These methods are used to minimize the response of a particular structural mode whilst neglecting the contribution of the other modes. In this study, a design method based on minimization of the sound power of the structure is proposed. The optimal parameters for shunt circuits are obtained using linear quadratic optimal control theory. In general, the passive shunt circuit techniques are an effective method of modal damping. However, the main drawback of the passive shunt circuit is that the shunt piezoelectric is very sensitive to tuning errors and variations in the excitation frequency. To overcome this problem, the pulse-switching shunt circuit, a semi-active continuous switching technique in which a RL shunt circuit is periodically connected to a bonded piezoelectric patch, is introduced as structural damping. The switch law for pulse-switching circuit is discussed based on the energy dissipation technique. Compared with a standard passive piezoelectric shunt circuit, the advantages of the pulse-switching shunt circuit is a small required shunt inductance, a lower sensitivity to environmental changes and easier tuning. Very low external power for the switch controller is required so it may be possible to extract this energy directly from the vibration of the structure itself. Numerical simulations are performed for each of these shunts techniques focusing on minimizing radiated sound power from a clamped plate. It is found that the RL series, RL parallel and pulse-switching circuits have basically the same control performance. The RL–C parallel circuit allows us to reduce the value of the inductance L due to the insertion of an external capacity C. However, the control performance will be reduced simultaneously. The pulse-switching circuit is more stable than RL series circuit with regard to structural stiffness variations. Finally, experimental results are presented using an RL series/parallel shunt circuit, RL-C parallel shunt circuit and pulse-switching circuit. The experimental results have shown that the vibration and noise radiation of a structure can be reduced significantly by using these shunt circuits. The theoretical and experimental techniques presented in this study provide a valuable tool for effective shunt piezoelectric damping.

Vibration identification and sound insulation of triple glazing

This paper presents the results of the experimental vibroacoustical and modal testing of a triple pane insulating glazing structure. The testing was performed in the sound transmission facility using acoustic excitation, force excitation and a laser scanning vibrometer for measurement of frequency response functions and volume velocity. The results of the identification include the modal mass, damping and stiffness, vibratory modes and volume velocity distribution. A new measurement method is presented in order to identify the global vibratory modes of all three combined transparent glass panes simultaneously. These results are essential for updating the analytical model of triple glazing for calculation and prediction of sound transmission loss over a wide frequency range. The calculated and measured sound transmission loss is presented and discussed.

Active Control of a Beam for Generating Points of Zero Displacements and Zero Slopes

Piezoelectric transducers have been used extensively as the distributed actuators and sensors in active control of structural vibrations. Piezoelectric actuator/sensors are distributively bonded on or embedded in the host structure and have the inherent advantage of integrating over their surface area, which leads to potentially more robust implementations as compared to implementations that use shaker/accelerometers. For this reason piezoelectric actuator/sensors have attracted more and more attention in recent years. In this paper, a theoretical analysis is presented of the active control of a vibrating beam using collocated triangular and rectangular piezoelectric actuator/sensor pairs. The aim of this study is to generate points of zero displacements and zero slopes at any designated position. So the control systems impose a virtual clamped boundary condition at the control position on the beam, in which both displacement and slope are driven to zero. Two independent single-input single-output (SISO) control systems similar to direct velocity feedback (DVFB) are implemented, i.e. for the rectangular pair the voltage signal measured by a triangular piezoelectric sensor is electronically multiplied by a fixed gain and fed directly back to a collocated piezoelectric actuator. The triangular and rectangular piezoelectric actuator/ sensor pairs positioned at one end of the beam are used to measure and control the displacement and slope of the structure respectively. The active control systems are unconditionally stable for any type of primary disturbance acting on the structure due to the collocated actuator/sensors. It should be noted that the presented control strategy is different to DVFB. In DVFB, when the control gain is increased, the vibration energy of the beam is initially reduced at resonance frequencies because of the active damping effect. However this effect does not continue. When large control gains are implemented, the overall kinetic energy of the beam is increased to the same or even higher values than those of the beam without control systems because the vibration of the beam is rearranged into a new set of lightly damped resonance frequencies. Imposing a virtual clamped boundary condition at the control position is clearly more complicated than DVFB, because in addition to the zero displacement constraints, the zero slope constraints must also be satisfied. The proposed control system allows for certain points of the structure to remain stationary without using any rigid supports. Furthermore, such control systems have the potential to create a region of nearly zero vibration for any ‘excitation’ frequency. This means that no progressive waves or reflected waves exist in the designated region, thus significantly reducing the vibration level in that region of the beam. The control systems impose a virtual clamped boundary condition at the control position on the beam in which the displacement and slope are driven to zero. As a result, the vibration of the actively controlled beam can be described in terms of two beams clamped at the control position. A numerical analysis is then performed to verify the proposed control system. It is found that the new resonance frequencies and mode shapes seen in the simulations are consistent with the natural frequencies and natural modes of the controlled beam derived analytically. The capability of the proposed method for generating a zero-vibration region is also numerically demonstrated.