The nonlinear equations of motion for a silicon cantilever beam, covered by a piezoelectric lead–zirconate–titanate layer, subjected to a Lennard-Jones type boundary condition, are derived for voltage excitation. The Lagrangian of the system is obtained from the electric enthalpy density, including the virtual work of the Lennard-Jones potential, assuming the beam undergoes only small displacements. By application of Hamilton’s principle, the nonlinear equations of motion are consistently derived and truncated to third order for perturbation analysis. The evolution equations are obtained by the multiple scales method and periodic solutions to the equations of motion are determined and discussed with respect to different tip to sample distances. An analytically obtained frequency response function enables determination of the frequency shift of individually piezoactuated microbeams, which are proposed as fundamental elements of parallel atomic force microscopy, undergoing forced vibration in a dissipative environment.
Lyotropic liquid crystalline (LLC) phases of amino acid derived polyarylisocyanides were employed as chiral alignment media for the measurement of residual dipolar couplings (RDCs) of small chiral organic molecules. Anisotropic samples in CDCl3 displayed quadrupolar splittings of the deuterium signal in the range of several hundreds of Hertz. The LLC phases showed excellent orienting properties for a broad range of analytes bearing various functional groups. The precise extraction of RDCs in the range of up to ±40 Hertz from F2‐coupled HSQC spectra was possible. Additionally, the chiral environment offers the opportunity for diastereomorphous interactions with the enantiomers of chiral analytes leading to two different sets of RDCs. This differential order effect was particularly pronounced with ketones and alcohols.
The nonlinear equations of motion for a silicon cantilever beam, covered symmetrically by piezoelectric ZnO layers are derived for voltage excitation. Starting with the nonlinear description of the strain distribution in the beam for finite displacement, the Lagrangeian of the system is obtained from the electric enthalpy density. By application of Hamilton’s principle, the nonlinear equations of motion are consistently derived and truncated to third order for perturbation analysis. The evolution equations are obtained by the multiple scales method and periodic solutions to the equations of motion are determined and discussed with respect to the influence of geometric nonlinearities and nonlinear properties of the piezoelectric layer.
A concept for the suppression of resonant vibration of an elastic system undergoing forced vibration coupled to electroactive polymer (EAP) actuators based on dielectric elastomers is demonstrated. The actuators are utilized to vary the stiffness of the end support of a clamped beam, which is forced to harmonic vibration via a piezoelectric patch. Due to the nonlinear dependency of the elastic modulus of the EAP material, the modulus can be changed by inducing an electrostrictive deformation. The resulting change in stiffness of the EAP actuator leads to a shift of the resonance frequencies of the vibrating beam, enabling an effective reduction of the vibration amplitude by an external electric signal. Using a custom-built setup employing an aluminum vibrating beam coupled on both sides to electrodized strips of VHB tape, a significant reduction of the resonance amplitude was achieved. The effectiveness of this concept compared to other active and passive concepts of vi bration reduction is discussed
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