An Analysis of Thickness-shear Vibrations of an Annular Plate with the Mindlin Plate Equations

The Mindlin plate equations with the consideration of thickness-shear deformation as an independent variable have been used for the analysis of vibrations of quartz crystal resonators of both rectangular and circular types. The Mindlin or Lee plate theories that treat thickness-shear deformation as an independent higher-order vibration mode in a coupled system of two-dimensional variables are the choice of theory for analysis. For circular plates, we derived the Mindlin plate equations in a systematic manner as demonstrated by Mindlin and others and obtained the truncated two-dimensional equations of closely coupled modes in polar coordinates. We simplified the equations for vibration modes in the vicinity of fundamental thickness-shear frequency and validated the equations and method. To explore newer structures of quartz crystal resonators, we utilized the Mindlin plate equations for the analysis of annular plates with fixed inner and free outer edges for frequency spectra. The detailed analysis of vibrations of circular plates for the normalized frequency versus dimensional parameters provide references for optimal selection of parameters based on the principle of strong thickness-shear mode and minimal presence of other modes to enhance energy trapping through maintaining the strong and pure thickness-shear vibrations insensitive to some complication factors such as thermal and initial stresses.Comment: Paper to be presented to the 2015 IEEE International Frequency Control Symposium and European Frequency and Time Forum, Denver, CO, USA. April 12-16, 201

Thickness-shear Frequencies of an Infinite Quartz Plate with Material Property Variation Along the Thickness

Properties of the quartz crystal blank of a resonator is assumed homogeneous, uniform, and perfect in design, manufacturing, and applications. As end products, quartz crystal resonators are frequently exposed to gases and liquids which can cause surface damage and internal degradation of blanks under increasingly hostile conditions. The combination of service conditions and manufacturing process including chemical etching and polishing can inevitably modify the surface of quartz crystal blanks with changes of material properties, raising the question of what will happen to vibrations of quartz crystal resonators of thickness-shear type if such modifications to blanks are to be evaluated for sensitive applications. Such questions have been encountered in other materials and structures with property variations either on purpose or as the effect of environmental or natural processes commonly referred to as functionally graded materials, or FGMs. Analyses have been done in applications as part of studies on FGMs in structural as well as in acoustic wave device applications. A procedure based on series solutions has been developed in the evaluation of frequency changes and features in an infinite quartz crystal plate of AT-cut with the symmetric material variation pattern given in a cosine function with the findings that the vibration modes are now closely coupled. These results can be used in the evaluation of surface damage and corrosion of quartz crystal blanks of resonators in sensor applications or development of new structures of resonators.Comment: This is to be presented and published with the 2014 IEEE International Frequency Control Symposium, May 19-22, 2014, Taipei International Convention Center, Taipe

Thickness-shear Vibration Frequencies of an Infinite Plate with a Generalized Material Property Grading along the Thickness

For quartz crystal resonators of thickness-shear type, the vibration frequency and mode shapes, which are key features of resonators in circuit applications, reflect the basic material and structural properties of the quartz plate and its variation with time under various factors such as erosive gases and liquids that can cause surface and internal damages and degradation of crystal blanks. The accumulated effects eventually will change the surface conditions in terms of elastic constants and stiffness and more importantly, the gradient of such properties along the thickness. This is a typical functionally graded materials (FGM) structure and has been studied extensively for structural applications under multiple loadings such as thermal and electromagnetic fields in recent years. For acoustic wave resonators, such studies are equally important and the wave propagation in FGM structures can be used in the evaluation and assessment of performance, reliability, and life of sensors based on acoustic waves such as the quartz crystal microbalances (QCM). Now we studied the thickness-shear vibrations of FGM plates with properties of AT-cut quartz crystal varying along the thickness in a general pattern represented by a trigonometric function with both sine and cosine functions of the thickness coordinate. The solutions are obtained by using Fourier expansion of the plate deformation. We also obtained the frequency changes of the fundamental and overtone modes which are strongly coupled for the evaluation of resonator structures with property variation or design to take advantages of FGM in novel applications.Comment: Paper for the proceedings of the 2015 IEEE International Frequency Control Symposium and the European Frequency and Time Forum, Denver, CO, USA. April 12-16, 201

Crystal growth, structure and thermal properties of noncentrosymmetric single crystals PrCa4O(BO3)3+

Noncentrosymmetric praseodymium calcium oxyborate single crystals, PrCa4O(BO3)3 (PrCOB), were grown by the Czochralski technique. The monoclinic unit cell parameters were found to be a = 8.177 Å, b = 16.157 Å, c = 3.629 Å and Z = 2 with space group Cm. Crystal density was measured using the Archimedes method, being on the order of 3.47 g cm-3. Thermal properties of PrCOB were investigated, where the specific heat was found to be 0.63 J g-1 °C-1 at room temperature, increasing to 0.85 J g-1°C-1 at 700°C. The thermal expansion coefficients were measured to be α11 = 7.99, α22 = 4.90 and α33 = 9.46 (10-6/°C), respectively. In addition, thermal diffusivity λ22 and thermal conductivity κ22 as a function of temperature were studied, where λ22 was observed to decrease from 0.89 to 0.58 mm2 s-1, while κ22 was found to maintain the same value, being ∼1.90 W m-1°C-1 over the temperature range of 20-700°C. 2013 The Royal Society of Chemistry

Elastic fractal higher-order topological states

In this work, elastic fractal higher-order topological states are investigated. Bott index is adopted to characterize the topological property of elastic fractal structures. The topological corner and edge states of elastic waves in fractal structures are realized theoretically and experimentally. Different from traditional two-dimension (2D) high-order topological insulators based on periodic structures, the high-order topological states based on elastic fractal structures in this work intuitively reflect the fractal dimension in physics, supporting not only abundant topological outer corner states, but also rich inner corner states. The richness of corner states is much higher than that of topological insulators based on periodic structures. The strong robustness of the topological corner states in the fractal structure are verified by introducing disorders and defects. The topological phenomenon of in elastic fractal structures revealed in this work enriches the topological physics of elastic systems and breaks the limitation of that relies on periodic elastic structures. The results have important application prospects in energy harvesting, information transmissions, elastic energy acquisitions and high-sensitivity detections

Anomalous size effects of effective stiffnesses in bistable counter-rotating mechanical metamaterials

Counter-rotating mechanical metamaterials have previously been found to have anomalous characteristics or functions such as auxetics effects, shape changers, and soliton transports, which are all under monostable conditions. The properties of counter-rotating mechanical metamaterials under bistable conditions have not yet been explored. Here, we found that for a bistable counter-rotating metamaterial chain, the effective stiffnesses of the two steady states are different in the chain with even-numbered nodes. For the chain with odd-numbered nodes, the effective stiffnesses corresponding to the two steady states are exactly the same. This special property is not characterized by the characteristic attenuation lengths of the underlying mechanism, but depends on the different symmetries of the underlying mechanism of the chains with odd and even nodes. In addition, the relationship between the abnormal non-monotonic size effect and equilibrium angle are clarified. More interestingly, for one-dimensional chains with even-numbered nodes, the size effect of effective stiffness bifurcates at a specific equilibrium angle, and the according mechanisms are revealed

Crystal growth, structure and thermal properties of noncentrosymmetric single crystals PrCa4O(BO3)3

Noncentrosymmetric praseodymium calcium oxyborate single crystals, PrCa4O(BO3)3 (PrCOB), were grown by the Czochralski technique. The monoclinic unit cell parameters were found to be a = 8.177 Å, b = 16.157 Å, c = 3.629 Å and Z = 2 with space group Cm. Crystal density was measured using the Archimedes method, being on the order of 3.47 g cm-3. Thermal properties of PrCOB were investigated, where the specific heat was found to be 0.63 J g-1 °C-1 at room temperature, increasing to 0.85 J g-1°C-1 at 700°C. The thermal expansion coefficients were measured to be α11 = 7.99, α22 = 4.90 and α33 = 9.46 (10-6/°C), respectively. In addition, thermal diffusivity λ22 and thermal conductivity κ22 as a function of temperature were studied, where λ22 was observed to decrease from 0.89 to 0.58 mm2 s-1, while κ22 was found to maintain the same value, being ∼1.90 W m-1°C-1 over the temperature range of 20-700°C. 2013 The Royal Society of Chemistry

The influence of macrophytes on sediment resuspension and the effect of associated nutrients in a shallow and large lake (Lake Taihu, China)

A yearlong campaign to examine sediment resuspension was conducted in large, shallow and eutrophic Lake Taihu, China, to investigate the influence of vegetation on sediment resuspension and its nutrient effects. The study was conducted at 6 sites located in both phytoplankton-dominated zone and macrophyte-dominated zone of the lake, lasting for a total of 13 months, with collections made at two-week intervals. Sediment resuspension in Taihu, with a two-week high average rate of 1771 g.m(-2).d(-1) and a yearly average rate of 377 g.m(-2).d(-1), is much stronger than in many other lakes worldwide, as Taihu is quite shallow and contains a long fetch. The occurrence of macrophytes, however, provided quite strong abatement of sediment resuspension, which may reduce the sediment resuspension rate up to 29-fold. The contribution of nitrogen and phosphorus to the water column from sediment resuspension was estimated as 0.34 mg.L-1 and 0.051 mg.L-1 in the phytoplankton-dominated zone. Sediment resuspension also largely reduced transparency and then stimulated phytoplankton growth. Therefore, sediment resuspension may be one of the most important factors delaying the recovery of eutrophic Lake Taihu, and the influence of sediment resuspension on water quality must also be taken into account by the lake managers when they determine the restoration target.Peer reviewe