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An effective methodology is proposed to evaluate the boundary stiffnesses of beams with generally restrained edges. The boundary conditions at the ends of the beams are implemented by the penalty method. The unified Jacobi polynomials are introduced to represent the assumed mode shape functions of the beam. Hamilton’s principle with the assumed mode method is employed to derive the equation of motion of the beam with generally restrained boundaries. Subsequently, a novel approach is proposed to evaluate the boundary stiffnesses of the beam using the measured natural frequencies of the beam by vibration experiments. The accuracy of this method is verified by comparing the results with the previously published literature, the finite element method (FEM), and the vibration experiments. The effects of the types and values of the boundary spring stiffnesses on the vibration properties of the beam are also discussed. The calculation method in this research can provide an effective prediction technique for the boundary stiffnesses of engineering structures with generally restrained edges. [ABSTRACT FROM AUTHOR]

• An effective method is developed for dynamic modelling of hemispherical shells with fillets. • Effects of fillets on thermoelastic quality factors of hemispherical shells are deeply explored. • The lower fillet has larger influence on the natural frequencies than the upper fillet does. • For the lower fillet, there exists size interval where hemispherical shell has higher energy losses. In engineering applications, hemispherical shell resonators are typically machined with fillets to reduce stress concentration and enhance structure strength. The fillets will inevitably affect the dynamic properties and the mechanical quality factor of hemispherical shell resonators, which has been seldom investigated before. In this paper, an effective analytical method is developed to explore the free vibration and thermoelastic damping (TED) characteristics of the hemispherical shell with fillets. The fillets are characterized by the variation in the thickness of the hemispherical shell during modelling. The first-order shear deformation theory (FSDT) is used to describe the theoretical formulas of the hemispherical shell with fillets. By employing the unified Jacobi polynomials and Fourier series as the assumed mode shape functions, the equation of motion of the structure is established by Hamilton's principle and the assumed mode method. The analytical model for thermoelastic quality factor (Q TED) which is determined by TED is obtained by computing the dissipated energy and the maximum elastic potential energy of the hemispherical shell with fillets. The validity and accuracy of the present method are confirmed by comparing the present solutions with the published results and those obtained from the finite element method (FEM). The influences of fillets on the vibration behaviors and Q TED characteristics of the hemispherical shells are analyzed in detail. The present model can be used to optimize the design of the fillets of the hemispherical shell resonators with high quality factors. [ABSTRACT FROM AUTHOR]

Although some investigations on the vibration properties of multi-span beams have been conducted, studies on multi-span Timoshenko beams by using the assumed mode method have been relatively few. In this paper, the multi-span Timoshenko beams are investigated, and the mode shapes of the beams are modified by the interpolation functions to model the vibration modes of the multi-span beams. Hamilton’s principle is applied to establish the equation of motion of the structure, and the natural circular frequencies and the free vibration responses of the multi-span beams are obtained. The numerical results demonstrate good agreement between the present results and those from the open literature and the ANSYS software. The influences of the length-thickness ratio, the disorder degree and the span number on the free vibration of the structure are also analyzed. It is observed that the displacement amplitude and the vibration period at the midpoint of the three-span Timoshenko beams are reduced when the disorder degree and the length-thickness ratio are reduced. The increase in the span number of the multi-span beams with equal spans leads to the decrease in the displacement amplitude and the period of the structures. Furthermore, some interesting phenomena are found, e.g., the same even-order frequencies of different three-span beams are equal under specific disorder degree. [ABSTRACT FROM AUTHOR]

So far, little attention has been paid to the vibration analysis of multi-span lattice sandwich beams, particularly using the assumed mode method (AMM). In this paper, the mode shapes of multi-span sandwich beams are assumed as those of uniform beams modified by the interpolation functions. The equation of motion of the beam is established using Hamilton’s principle. The natural frequencies of multi-span pyramidal and Kagome sandwich beams so calculated agree well with those determined using the ANSYS software, which indicates that the present methodology is suitable for solving multi-span sandwich beams with lattice truss cores. The effects of Young's modulus, damping and geometric parameters of cores and sheets on the natural frequencies and time domain responses of two kinds of multi-span sandwich beams are analyzed. When the truss radius and sheet thickness are increased, the natural frequencies are increased initially and then decrease, while the vibration amplitudes at the mid-points of both the multi-span pyramidal and Kagome sandwich beams decrease. With the increase of the inclination angle of the truss α c , the natural frequencies of structures experience a slight decline. In contrast, the amplitudes at the mid-points of the two different three-span sandwich beams both rise. [ABSTRACT FROM AUTHOR]

• An analytical model is developed to evaluate thermoelastic damping in cylindrical shells. • Cylindrical shell with clamped-clamped edges dissipates the minimum thermal energy. • Increasing circumferential and radial spring stiffnesses reduces thermoelastic damping. • Decreasing ratios of length to radius and radius to thickness reduces thermoelastic damping. An effective method is proposed for the dynamic modelling and thermoelastic damping (TED) evaluation of the cylindrical shells with arbitrary boundaries. The arbitrary boundary conditions of the cylindrical shell are simulated by a set of springs, and the degree of the boundary condition is achieved by adjusting the spring stiffnesses. Hamilton's principle with the Rayleigh-Ritz method is employed to establish the equation of motion of the cylindrical shell, and the natural frequencies of the cylindrical shell with arbitrary boundaries are obtained by solving the eigenvalue problem of the equation of motion. An analytical model for the TED in the cylindrical shell is obtained by computing the dissipated energy and the maximum elastic potential energy in the cylindrical shell. The correctness and feasibility of the present theoretical model are verified by comparing the natural frequencies and TED obtained by this analytical method with those from the published literature and the finite element method (FEM). The convergence analysis of the analytical expression of TED is carried out. The influences of geometrical parameters and different types of spring stiffnesses on the TED are studied. In addition, the effects of the boundary conditions and the vibration modes of the structure on the TED characteristics of the cylindrical shells with arbitrary boundaries are analyzed. The present model provides theoretical reference for optimizing the cylindrical shell resonators with high quality factors. [ABSTRACT FROM AUTHOR]

• A novel strategy is proposed for the dynamic modelling of hemispherical shell resonator. • An analytical model is developed to evaluate quality factors of hemispherical shell resonators. • Properly decreasing the radius of support rod reduces thermoelastic damping and anchor loss. • Frequency parameter is sensitive to geometric sizes of support rod for lower order modes. A novel strategy is proposed for the dynamic modelling and quality factor evaluation of the hemispherical shell resonators (HSR) considering the coupling of hemispherical shell and support rod. The artificial spring technique is introduced to realize the coupling connection of substructures and simulate the arbitrary boundary conditions. Hamilton's principle with the Rayleigh-Ritz method is employed to establish the equation of motion of the HSR, and the natural frequencies of the HSR are obtained by solving the eigenvalue problem of the equation of motion. The thermal energy method is applied to evaluate the thermoelastic damping (TED) of the HSR. The model of anchor loss is based on a separation and transfer method, in which the HSR and its substrate are first separated for analysis and then the stress from the clamped region of the support rod is transferred to the substrate for determining the vibration displacement across the clamped region. The energy losses to the substrate are formulated as the integral of the product of the stress and the vibration displacement across the clamped region per cycle of vibration. The results calculated from the present theoretical method are compared with those from the published literature and the finite element method (FEM), which validates the correctness and feasibility of the present theoretical model. The effects of the boundary conditions and geometrical parameters on the vibration behaviors of the HSR are analyzed, and the influences of the boundary conditions, geometrical parameters and material properties of the HSR on the quality factors related to the TED and anchor loss are investigated. The present model can be used to optimize the design of the HSR with a high quality factor. [ABSTRACT FROM AUTHOR]

This paper describes a new methodology capable of analyzing coupled bending and torsional vibrations in non-uniform thin-walled beams. The transfer matrix method (TMM) and the differential transform method (DTM) are combined to create the proposed transfer differential transform method (TDTM) for solving equations coupling bending and torsional vibrations in non-uniform beams. Compared with the finite element method (FEM), the TDTM utilizes a type of changeable mode shape functions so that the number of meshed elements can be reduced greatly when the beam's geometric size is uniform. The equations of motion of the non-uniform thin-walled beams are established using Hamilton's principle. The method considers both the warping and rotary effects of the beam section. The natural frequencies and mode shapes of the bending and torsional components are obtained using the TDTM. The accuracy of solutions can be controlled by the mesh density and the series expansion order of the mode shape whose ranges are also discussed. For illustrative purposes, the natural frequencies and the frequency response curves of a uniform beam and a non-uniform beam are studied respectively and are validated by experiments. The effects of warping and the distance between the centroid of a given section and the corresponding shear center on the vibration properties of the beam are also investigated. [ABSTRACT FROM AUTHOR]