Measurement of the natural frequencies of a uniform rod loaded with centrifugal forces using a laser Doppler vibrometer

An experimental investigation of transverse vibrations in a rotating uniform plate, possessing double symmetry in the transverse cross section, is described. A method of modeling the centrifugal force acting on a uniform plate (under static conditions) and measurement details are presented. Test data are given. The results obtained agree well with calculations of the natural frequencies for the second and third harmonics of natural oscillations. __________ Translated from Izmeritel'naya Tekhnika, No. 1, pp. 30–33, January, 2006.     

Determination of spurious eigenvalues and multiplicities of true eigenvalues using the real-part dual BEM

The complex-valued dual BEM has been employed by Chen and Chen (1998) to solve the acoustic modes of a cavity with or without a thin partition. A novel method using only the real part of the complex-valued dual BEM was found by Chen (1998) to be equivalent to the dual MRM. However, spurious eigenvalues occur. In this paper, we propose the singular value decomposition technique to filter out spurious eigenvalues and to determine the multiplicity of true eigenvalues by combining the real-part dual equations. Also, the role of the real-part dual BEM for problems with a degenerate boundary is examined. Four examples, including a square cavity with multiple eigenvalues, a rectangular cavity, a rectangular cavity with a zero thickness partition and a rectangular cavity with a partition with finite thickness, are presented to demonstrate the validity of the proposed method. Also, the analytical solution if available, the FEM results obtained by Petyt et al. and obtained using ABAQUS and the experimental measurements are compared with those of the proposed method, and it is found that agreement between them is very good.     

Recursive second order convergence method for natural frequencies and modes when using dynamic stiffness matrices

When exact dynamic stiffness matrices are used to compute natural frequencies and vibration modes for skeletal and certain other structures, a challenging transcendental eigenvalue problem results. The present paper presents a newly developed, mathematically elegant and computationally efficient method for accurate and reliable computation of both natural frequencies and vibration modes. The method can also be applied to buckling problems. The transcendental eigenvalue problem is first reduced to a generalized linear eigenvalue problem by using Newton's method in the vicinity of an exact natural frequency identified by the Wittrick–Williams algorithm. Then the generalized linear eigenvalue problem is effectively solved by using a standard inverse iteration or subspace iteration method. The recursive use of the Newton method employing the Wittrick–Williams algorithm to guide and guard each Newton correction gives secure second order convergence on both natural frequencies and mode vectors. The second order mode accuracy is a major advantage over earlier transcendental eigenvalue solution methods, which typically give modes of much lower accuracy than that of the natural frequencies. The excellent performance of the method is demonstrated by numerical examples, including some demanding problems, e.g. with coincident natural frequencies, with rigid body motions and large‐scale structures. Copyright © 2003 John Wiley & Sons, Ltd.     

Efficient evaluation of weakly/strongly singular domain integrals in the BEM using a singular nodal integration method

In many analyses of engineering problems based on boundary element methods, a large number of regular and/or singular domain integrals must be accurately evaluated over a single domain. Evaluation of such domain integrals is very time-consuming and is frequently the main source of errors and loss of accuracy in the solutions. Previous efforts have been constantly made in order to facilitate or overcome such shortcomings. In this article, we propose novel and efficient approaches in the framework of Cartesian transformation method (CTM) and the radial integration method (RIM) that can be used for fast evaluation of numerous weakly/strongly singular two-dimensional domain integrals over a single domain. The domain integrals essentially are expressed in terms of some coefficient matrices and vectors, most of which are independent of the integrand of the domain integrals and are dependent only on the geometry. Several examples for the evaluation of weakly/strongly singular domain integrals and two examples for the flow field analysis in micro-channels are presented and the accuracy and convergence of the proposed approaches are investigated.     

Recovery of equilibrium on star patches using a partition of unity technique

A technique for recovering equilibrated element stresses is developed for finite element models of structural mechanics problems. The data for the method consist of the prescribed loading and the stress and displacement fields resulting from a conventional compatible finite element model. Local problems are defined for each star of elements, via the introduction of fictitious body forces and strains. These problems can be solved independently for equilibrium in a process that can be easily parallelized. The quality of the solutions is assessed, for two‐dimensional linear elastic problems, by using them to compute bounds of the error of the finite element solutions, in terms of both global and local quantities of interest. Copyright © 2009 John Wiley & Sons, Ltd.     

Vibration isolation by a row of piles using a 3‐D frequency domain BEM

The problem of vibration isolation by a row of piles is numerically solved in a three‐dimensional context by an advanced frequency domain boundary element method (BEM). Both the piles and the soil are modelled by boundary elements and coupled together through equilibrium and compatibility at their interfaces. Linear elastic or viscoelastic material behaviour is assumed for both the piles and the soil. The piles can be tubular or solid and have circular or square cross‐section. The vibration source is a vertical harmonically varying with time force and the row of piles acts as a wave barrier in a passive way. The boundary element method is first validated for accuracy by solving two three‐dimensional wave diffraction problems dealing with spheres and trenches as scatterers for which there are analytical and highly accurate numerical solutions available in the literature. Numerical examples dealing with passive vibration isolation by a row of piles are then solved and the screening effectiveness of these wave barriers is assessed and compared against that of trenches. Copyright © 1999 John Wiley & Sons, Ltd.     

Variation of natural frequencies of beams using the active stiffening effect

The present study investigates the stiffening effects of a simply supported and clamped–free symmetric piezolaminated composite type beams. The structure consists of PZT layers or two sets of patches bonded to the surface of the beam. The analysis considers the linear piezoelectric constitutive relations and a first order shear deformation theory (FSDT). The influence of the actuators is evaluated by means of the pin-force model and their size and location along the beam are taken into account. Two coupled equations of motion for the lateral displacement and bending rotation and one uncoupled equation for the axial displacement of symmetric piezolaminated composite beam are solved numerically to obtain the natural frequencies and mode shapes. Numerical experiments demonstrate that the natural frequencies and mode shapes of the beam can be actively altered using the piezoelectric bonded actuators. The results of the present degenerated model are compared to results presented in the literature. The comparisons yielded excellent matches.     

Simulation of ductile tearing by the BEM with 2D domain discretization and a local damage model

A numerical procedure based on the Boundary Element Method with internal cells and dedicated to the simulation of the ductile tearing of thin metal sheets is presented. Plasticity is handled with an integral formulation based on the initial strain approach involving a discretization of the planar domain. Time integration is performed in an implicit way for the local strain-stress relationships while the global algorithm relies on an explicit formulation. Damage is represented by the scalar parameter of the uncoupled local damage model of Rice and Tracey. Within the scope of our applications, the cracks propagate along paths a priori known. As damage spreads, boundary elements are gradually released. Elastoplastic problems with large yielding zones are solved and compared to reference solutions. At last, the ductile tearing of a specimen is addressed. The calibration of the critical damage parameter leads to numerical results in good agreement with the experimental ones.     

Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using an efficient mixed-part dual BEM

 In this paper, we develop an efficient mixed-part dual BEM to solve the eigensolutions of a circular cavity analytically and numerically. The method is proposed by choosing a fewer number of equations from the dual BEM instead of all of the equations in the dual BEM developed by Chen and his coworkers. To solve this problem analytically, the spurious solution can be filtered out by adding constraints from the dual boundary integral equations. The proposed method is superior to the complex-valued BEM not only for half effort in constructing the influence matrix, but also for its fewer size of dimension. Also, numerical experiments are performed to compare with the analytical results and the true eigensolutions can be easily extracted out in conjunction with the singular value decomposition technique (SVD). The optimum number of collocation point and appropriate collocating positions for the additional constraints are discussed.     

Measurement of the longitudinal and transverse vibration frequencies of a rod by speckle interferometry

A heterodyne interferometric system was used to detect the normal and tangential components of the displacement at one point of a sample when it is vibrating freely in its natural modes, arising from transient excitation. A cylindrical aluminium rod was the test sample. An analysis of the spectrum associated with the in-plane and out-of-plane displacements made it possible to determine the values of the longitudinal and transverse natural frequencies.     

Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real‐part dual BEM

It has been found recently that the multiple reciprocity method (MRM) (Chen and Wong. Engng. Anal. Boundary Elements 1997; 20 (1):25–33; Chen. Processings of the Fourth World Congress on Computational Mechanics, Onate E, Idelsohn SR (eds). Argentina, 1998; 106; Chen and Wong. J. Sound Vibration 1998; 217 (1): 75–95.) or real‐part BEM (Liou, Chen and Chen. J. Chinese Inst. Civil Hydraulics 1999; 11 (2):299–310 (in Chinese)). results in spurious eigenvalues for eigenproblems if only the singular (UT) or hypersingular (LM) integral equation is used. In this paper, a circular cavity is considered as a demonstrative example for an analytical study. Based on the framework of the real‐part dual BEM, the true and spurious eigenvalues can be separated by using singular value decomposition (SVD). To understand why spurious eigenvalues occur, analytical derivation by discretizing the circular boundary into a finite degree‐of‐freedom system is employed, resulting in circulants for influence matrices. Based on the properties of the circulants, we find that the singular integral equation of the real‐part BEM for a circular domain results in spurious eigenvalues which are the zeros of the Bessel functions of the second kind, Y (kρ), while the hypersingular integral equation of the real‐part BEM results in spurious eigenvalues which are the zeros of the derivative of the Bessel functions of the second kind, Y_n′(kρ). It is found that spurious eigenvalues exist in the real‐part BEM, and that they depend on the integral representation one uses (singular or hypersingular; single layer or double layer) no matter what the given types of boundary conditions for the interior problem are. Furthermore, spurious modes are proved to be trivial in the circular cavity through analytical derivations. Numerically, they appear to have the same nodal lines of the true modes after normalization with respect to a very small nonzero value. Two examples with a circular domain, including the Neumann and Dirichlet problems, are presented. The numerical results for true and spurious eigensolutions match very well with the theoretical prediction. Copyright © 2000 John Wiley & Sons, Ltd.     

Fluorescent imaging of pH with optical sensors using time domain dual lifetime referencing

We present a referenced scheme for fluorescence intensity measurements that is useful for imaging applications. It is based on the conversion of the fluorescence intensity information into a time-dependent parameter. A phosphorescent dye is added in the form of approximately 10-microm particles to the sample containing the pH-sensitive fluorescent indicator. Both the reference dye and the pH probe are excited simultaneously by a blue LED, and an overall luminescence is measured. In the time-resolved imaging method presented here, two images taken at different time gates were recorded using a CCD camera. The first image is recorded during excitation and reflects the luminescence signal of both the fluorophore (pH) and the phosphor (reference). The second image, which is measured after a certain delay (after switching off the light source), is solely caused by the long-lived phosphorescent dye. Because the intensity of the fluorophore contains the information on pH, whereas phosphorescence is pH-independent, the ratio of the images displays a referenced intensity distribution that reflects the pH at each picture element (pixel). The scheme is useful for LED light sources and CCD cameras that can be gated with square pulses in the microsecond range. The fundamentals and potential of this new method, to which we refer as time domain dual lifetime referencing (t-DLR), are demonstrated.     

Improved measurements of neutron dose equivalent rates using a dual instrument technique

A dual instrument technique, employing a neutron area survey meter and a tissue-equivalent proportional counter, has been developed to improve the measurement of neutron dose equivalent rates in workplace fields. A simple algorithm estimates the ambient dose equivalent response of the area survey meter from the ratio of the dose equivalent rates measured by the two instruments. This estimate can then be used to adjust the reading from the area survey meter, providing a better measure of the neutron ambient dose equivalent rate in the workplace. The technique was tested on 225 spectra from the neutron spectrum catalogue SPKTBIB.