Vibration analysis of combined conical-cylindrical shells based on the dynamic stiffness method

Natural frequencies and harmonic responses of multi-story frames using single variable shear deformation theory

Exact free vibration analysis for membrane assemblies with general classical boundary conditions

Dynamic stiffness analysis of toroidal shells

Dynamic stiffness method and CUF-based component-wise theories applied to free vibration analysis of solid beams, thin-walled structures and reinforced panels

The dynamic responses and vibration suppression of frame structures considering the shear deformation are investigated using the spectral element method (SEM). Vibration suppression of the frame structures is studied through the active control technique using the piezoelectric rods. The local dynamic stiffness matrices of the spectral elements for the Timoshenko beams and longitudinal vibration of the rods are obtained in the Fourier phase space and that for the axial-bending-shear coupled vibration of the piezoelectric rod is also gotten. From the transformation between the local and global coordinates, the global dynamic stiffness matrix for the whole frame structures containing the piezoelectric rods is obtained. From the numerical simulation, the dynamic responses of the frame structures can be effectively and exactly calculated by the SEM comparing with the finite element method. It can be found that the dynamic responses of the frame structures can be effectively suppressed by the active control technique based on the piezoelectric rods. The influences of the structural parameters on the dynamic responses and active control effect are also analyzed.

Application of the dynamic stiffness method in the vibration analysis of stiffened composite plates

An exact modal analysis approach to vibration analysis of structures with mass-spring subsystems and rotational joints

The dynamic stiffness method is introduced to investigate the free vibration of laminated composite beams based on a third-order shear deformation theory which accounts for parabolic distribution of the transverse shear strain through the thickness of the beam. The exact dynamic stiffness matrix is found directly from the analytical solutions of the basic governing differential equations of motion. The Poisson effect, shear deformation, rotary inertia, in-plane deformation are considered in the analysis. Application of the derived dynamic stiffness matrix to several particular laminated beams is discussed. The influences of Poisson effect, material anisotropy, slenderness and end condition on the natural frequencies of the beams are investigated. The numerical results are compared with the existing solutions in literature whenever possible to demonstrate and validate the present method.

Dynamic stiffness analysis for free vibrations of axially loaded laminated composite beams

The dynamic stiffness matrix of a gradient elastic flexural Bernoulli–Euler beam finite element is analytically constructed with the aid of the basic and governing equations of motion in the frequency domain. The simple gradient theory of elasticity is used with just one material constant (internal length) in addition to the classical moduli. The flexural element has one node at every end with three degrees of freedom per node, i.e., the displacement, the slope, and the curvature. Use of this dynamic stiffness matrix for a plane system of beams enables one by a finite element analysis to determine its dynamic response harmonically varying with time external load or the natural frequencies and modal shapes of that system. The response to transient loading is obtained with the aid of Laplace transform with respect to time. A stiffness matrix is constructed in the transformed domain, the problem is formulated and solved by the finite element method, and the time domain response is finally obtained by a time domain inversion of the transform solution. Because the exact solution of the governing equation of motion in the frequency domain is used as the displacement function, the resulting dynamic stiffness matrices and the obtained structural response or natural frequencies and modal shapes are also exact. Examples are presented to illustrate the method and demonstrate its advantages. The effects of the microstructure on the dynamic behavior of beam structures are also determined.

In this study, the dynamic stiffness method is employed for the free vibration analysis of helical springs. This work gives the exact solutions for the natural frequencies of helical beams having arbitrary shapes, such as conical, hyperboloidal, and barrel. Both the cross-section dimensions and the shape of the beam can vary along the axis of the curved member as polynomial expressions. The problem is described by six differential equations. These are second order equations with variable coefficients, with six unknown displacements, three translations, and three rotations at every point along the member. The proposed solution is based on a new finite-element method for deriving the exact dynamic stiffness matrix for the member, including the effects of the axial and the shear deformations and the rotational inertia effects for any desired precision. The natural frequencies are found as the frequencies that cause the determinant of the dynamic stiffness matrix to become zero. Then the mode shape for every natural frequency is found. Examples are given for beams and helical springs with different shape, which can vary along the axis of the member. It is shown that the present numerical results agree well with previously published numerical and experimental results.

Of the two main finite element approaches, the displacement method has progressed more rapidly than the force method in the area of element representation. Many authors have contributed to the displacement method with both static and dynamic stiffness matrices for beam elements, plane stress elements and plate bending elements. The plane stress and bending elements being rectangular, triangular or quadrilateral in form. A dynamic stiffness matrix consists of a static stiffness matrix and a mass matrix. In the force method, .static flexibility matrices have been developed for beam elements and plane stress elements. However, static flexibility matrices for plate bending elements have only recently been published (Kaufman and Hall, Morley, Robinson, Przemieniecki). For the past few years, the author has been investigating the rank force method for structural vibration analysis. In ref. 6 a dynamic flexibility matrix is presented for a beam element. This matrix consists of a static flexibility matrix and an inverse mass matrix. Ref. 5 contains the derivation of a dynamic flexibility matrix for a rectangular plate element in bending, twisting and shear. The author, in collaboration with Petyt, demonstrated how a dynamic stiffness and flexibility matrix can be extended to include material damping.

The exact explicit dynamic stiffness matrix of multi-cracked Euler–Bernoulli beam and applications to damaged frame structures

Free vibrations of spatial Timoshenko arches

Harmonic response analysis of coupled plate structures using the dynamic stiffness method