C. S. Huang

Identification of Instantaneous Modal Parameter of Time-Varying Systems via a Wavelet-Based Approach and Its Application

This work presents an efficient approach us- ing time-varying autoregressive with exogenous input (TVARX) model and a substructure technique to iden- tify the instantaneous modal parameters of a linear time- varying structure and its substructures. The identified instantaneous natural frequencies can be used to iden- tify earthquake damage to a building, including the spe- cific floors that are damaged. An appropriate TVARX model of the dynamic responses of a structure or sub- structure is established using a basis function expan- sion and regression approach combined with continu- ous wavelet transform. The effectiveness of the proposed approach is validated using numerically simulated earth- quake responses of a five-storey shear building with time- varying stiffness and damping coefficients. In terms of accuracy in determining the instantaneous modal param- eters of a structure from noisy responses, the proposed approach is superior to typical basis function expan- sion and regression approach. The proposed method is further applied to process the dynamic responses of an eight-storey steel frame in shaking table tests to iden- tify its instantaneous modal parameters and to locate

In-plane free vibration and stability of loaded and shear-deformable circular arches

The first known equations governing vibrations of preloaded, shear-deformable circular arches are derived according to a variational principle for dynamic problems concerning an elastic body under equilibrium initial stresses. The equations are three partial differential equations with variable coefficients. The governing equations are solved for arches statically preloaded with a uniformly distributed vertical loading, by obtaining a static, closed-form solution and an analytical dynamic solution from series solutions and dynamic stiffness matrices. Convergence to accurate results is obtained by increasing the number of elements or by increasing both the number of terms in the series solution and the number of terms in the Taylor expansion of the variable coefficients. Graphs of non-dimensional frequencies and buckling loads are presented for preloaded clamped arches. They clarify the effects of opening angle and thickness-to-radius ratio on vibration frequencies and buckling loads. The effects of static deformations on vibration frequencies are also investigated. This work also compares the results obtained from the proposed governing equations with those obtained from the classical theory neglecting shear deformation.

Identifying the Modal Parameters of a Structure from Ambient Vibration Data via the Stationary Wavelet Packet

Ambient vibration tests are conducted widely to estimate the modal parameters of a structure. The work proposes an efficient wavelet-based approach to determine the modal parameters of a structure from its ambient vibration responses. The proposed approach integrates the time series autoregressive (AR) model with the stationary wavelet packet transform. In addition to providing a richer decomposition and allowing for an improved time–frequency localization of signals over that of the discrete wavelet transform, the stationary wavelet packet transform also has significantly higher computational efficiency than the wavelet packet transform in terms of decomposing time-shifted signals because the former has a time-invariance property. The correlation matrices needed in determining the coefficient matrices in an AR model are established in subspaces expanded by stationary wavelet packets. The formulation for estimating the correlation matrices is shown for the first time. Because different subspaces contain signals with different frequency subbands, the fine filtering property enhances the ability of the proposed approach to identify not only the modes with strong modal interference, but also many modes from the responses of very few measured degrees of freedom. The proposed approach is validated by processing the numerically simulated responses of a seven-floor shear building, which has closely spaced modes, with considering the effects of noise and incomplete measurements. Furthermore, the present approach is employed to process the velocity responses of an eight-storey steel frame subjected to white noise input in a shaking table test and ambient vibration responses of a cable-stayed bridge.

Identification of Time‐Variant Modal Parameters Using Time‐Varying Autoregressive with Exogenous Input and Low‐Order Polynomial Function

This work presents an approach that ac- curately identifies instantaneous modal parameters of a structure using time-varying autoregressive with exoge- nous input (TVARX) model. By developing the equiva- lent relations between the equation of motion of a time- varying structural system and the TVARX model, this work proves that instantaneous modal parameters of a time-varying system can be directly estimated from the TVARX model coefficients established from displace- ment responses. A moving least-squares technique incor- porating polynomial basis functions is adopted to ap- proximate the coefficient functions of the TVARX model. The coefficient functions of the TVARX model are rep- resented by polynomials having time-dependent coeffi- cients, instead of constant coefficients as in traditional basis function expansion approaches, so that only low orders of polynomial basis functions are needed. Nu- merical studies are carried out to investigate the effects of parameters in the proposed approach on accurately determining instantaneous modal parameters. Numerical analyses also demonstrate that the proposed approach is superior to some published techniques (i.e., recursive technique with a forgetting factor, traditional basis func- tion expansion approach, and weighted basis function expansion approach) in accurately estimating instanta- neous modal parameters of a structure. Finally, the pro- ∗ To whom correspondence should be addressed. E-mail: cshuang@ mail.nctu.edu.tw. posed approach is applied to process measured data for a frame specimen subjected to a series of base excitations in shaking table tests. The specimen was damaged dur- ing testing. The identified instantaneous modal parame- ters are consistent with observed physical phenomena.

An analytical solution for vibrations of a polarly orthotropic Mindlin sectorial plate with simply supported radial edges

Abstract This paper presents the first known analytical solution for vibrations of a polarly orthotropic Mindlin sectorial plate with simply supported radial edges. The solution is a series solution constructed using the Frobenius method and exactly satisfies not only the boundary conditions along the radial and circular edges, but also the regularity conditions at the vertex of the radial edges. The moment and shear force singularities at the vertex are exactly considered in the solution. The correctness of the proposed solution is confirmed by comparing non-dimensional frequencies of isotropic plates obtained from the present solution with published data obtained from a closed-form solution. This paper also investigates the effects of elastic and shear moduli on the vibration frequencies of the sectorial plates with free or fix boundary conditions along the circumferential edge. A study is also carried out about the influence of elastic and shear moduli on the moment and shear force singularities at the plate origin ( r =0) for different vertex angles.

Corner singularities in bi-material Mindlin plates

An eigenfunction expansion solution is first developed to find stress singualrities for bi-material wedges by directly solving the governing equations of the Mindlin plate theory in terms of displacement components. The singularity orders of moments and shear forces at corners are determined from the corresponding asymptotic solutions having the lowest order in r and satisfying the radial boundary conditions and continuity conditions. The present solution is applied to thoroughly examine the singularities occurring at the interface joint of bonded dissimilar isotropic plates and at the vertex of a bi-material wedge with two simply supported radial edges. The corresponding characteristic equations for determining the singularity orders of moments and shear forces are explicitly given. The singularity orders of moment are shown in graphic form as functions of the flexural rigidity ratio and corner angle, while the shear force singularity orders are given as functions of the corner angle and the shear modulus ratio multiplied by the thickness ratio. The order of moment singularity obtained here for bonded dissimilar plates is also compared with that based on the classical plate theory.

Three-Dimensional Sharp Corner Displacement Functions for Bodies of Revolution

Sharp corner displacement functions have been well used in the past to accelerate the numerical solutions of two-dimensional free vibration problems, such as plates, to obtain accurate frequencies and mode shapes. The present analysis derives such functions for three-dimensional (3D) bodies of revolution where a sharp boundary discontinuity is present (e.g., a stepped shaft, or a circumferential V notch), undergoing arbitrary modes of deformation. The 3D equations of equilibrium in terms of displacement components, expressed in cylindrical coordinates, are transformed to a new coordinate system having its origin at the vertex of the corner. An asymptotic analysis in the vicinity of the sharp corner reduces the equations to a set of coupled, ordinary differential equations with variable coefficients. By a suitable transformation of variables the equations are simplified to a set of equations with constant coefficients. These are solved, the boundary conditions along the intersecting corner faces are applied, and the resulting eigenvalue problems are solved for the characteristic equations and corner functions.

On the singularity induced by boundary conditions in a third-order thick plate theory

This paper thoroughly examines the singularity of stress resultants of the form r?F() for 0<1 as r0 (Williams-type singularity) at the vertex of an isotropic thick plate; the singularity is caused by homogeneous boundary conditions around the vertex. An eigenfunction expansion is applied to derive the first known asymptotic solution for displacement components, from the equilibrium equations of Reddys third-order shear deformation plate theory. The characteristic equations for determining the singularities of stress resultants are presented for ten sets of boundary conditions. These characteristic equations are independent of the thickness of the plate, Youngs modulus, and shear modulus, but some do depend on Poissons ratio. The singularity orders of stress resultants for various boundary conditions are expressed in graphic form as a function of the vertex angle. The characteristic equations obtained herein are compared with those from classic plate theory and first-order shear deformation plate theory. Comparison results indicate that different plate theories yield different singular behavior for stress resultants. Only the vertex with simply supported radial edges (S(I)_S(I) boundary condition) exhibits the same singular behavior according to all these three plate theories. ©2002 ASME

DETECTION OF DAMAGE LOCATION USING A NOVEL SUBSTRUCTURE-BASED FREQUENCY RESPONSE FUNCTION APPROACH WITH A WIRELESS SENSING SYSTEM

This paper intends to detect the damage locations for building structures under an earthquake excitation using a novel substructure-based FRF approach with a damage location index (SubFRFDI). An Imote2.NET-based wireless structural health monitoring system was developed and employed in the experimental studies for the sake of deployment flexibility, low maintenance cost, low power consumption, self-organization capability, and wireless communication capability. The feasibility of the proposed approach for damage detection was examined using the numerical response of a six-storey shear plane frame structure subjected to a base excitation. The results demonstrate that the SubFRFDI can be successfully used to identify the damage of different levels at a single site or multiple sites. The SubFRFDI is independent of the responses to various input earthquake excitations. Even with the addition of noises, the SubFRFDI still functions well. The feasibility and robustness of the proposed Imote2.NET-based wireless structural health monitoring system were assessed using a 1/8-scale three-storey steel-frame model. Following this, the proposed SubFRFDI was further applied to identifying the damage locations in a 1/4-scale six-storey steel structure with the proposed Imote2.NET-based wireless monitoring system. It was confirmed experimentally that good data transportation quality can be achieved via reliable data transmission and sensing protocol in identifying the structural dynamic properties, and the proposed SubFRFDI can be used to identify the damage locations effectively.

ACCURATE FREQUENCIES AND MODE SHAPES FOR MODERATELY THICK, CANTILEVERED, SKEW PLATES

Accurate free vibration frequencies and mode shapes are presented for complete sets of moderately thick, cantilevered skew plates of triangular, trapezoidal and parallelogram shape. These accurate results are obtained by using the Ritz method applied to the Mindlin plate theory. Two sets of functions are employed simultaneously for each of the three dependent variables: transverse displacement (w) and bending rotations (ϕx and ϕy). One set is the widely used algebraic polynomials. The other is the set of corner functions which provide the proper stress singularities in the reentrant clamped-free corner, and accelerates the convergence of the solutions. The extensive frequencies presented are exact to the four digits shown. Corresponding mode shapes are also shown, by means of nodal patterns, most of which are novel in the published literature.