Xuelin Wang

Computation of Eigensolution Derivatives for Nonviscously Damped Systems Using the Algebraic Method

T HE computation of the eigensolution derivatives plays a significant role in dynamic model updating, structural design optimization, structural dynamic modification, damage detection andmany other applications. Themethods to calculate eigensolution derivatives are well established for undamped and viscous damped systems. These common methods can be divided into the modal method, Nelson’s method, and the algebraic method. Fox and Kapoor [1] first proposed the modal method for symmetric undamped systems by approximating the derivative of each eigenvector as a linear combination of all undamped eigenvectors. Later, Adhikari and Friswell [2] and Adhikari [3] extended the modal method to the more general asymmetric systems with viscous and nonviscous damping, respectively. To simplify the computation of eigensolution derivatives, Nelson [4] proposed a method, which requires only the eigenvector of interest by expressing the derivative of each eigenvector as a particular solution and a homogeneous solution for symmetric undamped systems. Later, Friswell and Adhikari [5] extended Nelson’s method to symmetric and asymmetric systemswith viscous damping. Recently, Adhikari and Friswell [6] extended Nelson’s method to symmetric and asymmetric nonviscously damped systems. However, Nelson’s method is lengthy and clumsy for programming. Lee et al. [7] derived an efficient algebraic method, which has a compact form to compute the eigensolution derivatives by solving a nonsingular linear system of algebraic equations for symmetric systems with viscous damping. Later, Guedria et al. [8] extended the algebraic method to general asymmetric systems with viscous damping. Recently, Chouchane et al. [9] wrote an excellent review of the algebraic method for symmetric and asymmetric systems with viscous damping and extended their method to the second-order and high-order derivatives of eigensolutions. In this note, the algebraic method will be extended to symmetric and asymmetric systems with nonviscous damping. The equations of motion describing free vibration of anN-degreeof-freedom (DOF) linear system with nonviscous (viscoelastic) damping can be expressed by [3,6]:

Eigensensitivity Analysis for Asymmetric Nonviscous Systems

T HE eigensensitivities of mechanical systems with respect to structural design parameters have become an integral part of many engineering design methodologies including optimization, structural health monitoring, structural reliability, model updating, dynamic modification, reanalysis techniques, and many other applications. Fox and Kapoor [1] computed the derivative of each eigenvector as a linear combination of all of the undamped eigenvectors. Later, Adhikari and Friswell [2] and Adhikari [3] extended the modal method to the more general asymmetric systems with viscous and nonviscous damping, respectively. Nelson [4] presented a method, which requires only the eigenvector of interest by expressing the derivative of each undamped eigenvector as a particular solution and a homogeneous solution. Later, Friswell and Adhikari [5] extended Nelson’s method to symmetric and asymmetric systems with viscous damping. Recently, Adhikari and Friswell [6] extendedNelson’s method to symmetric and asymmetric nonviscously damped systems. Fox and Kapoor [1] also suggested a direct algebraic method to calculate the eigensensitivity for symmetric undamped systems by solving a nonsingular linear system of algebraic equations. Lee et al. [7] derived an efficient algebraic method, which has a compact linear system with a symmetric coefficient matrix for symmetric systems with viscous damping. Later, Guedria et al. [8] extended the algebraic method to general asymmetric viscous damped systems. Chouchane et al. [9] reviewed the algebraic method and extended their method to the second-order and high-order derivatives of eigensolutions. Li et al. [10] extended the algebraic method to symmetric and asymmetric nonviscously damped systems. Xu andWu [11] proposed a new normalization and presented a method for the computation of eigensolution derivatives of asymmetric systemswith viscously damping. Recently,Mirzaeifar et al. [12] proposed a new method based on a combination of algebraic and modal methods for generally asymmetric viscously damped systems. More recently, Li et al. [13] proposed a method of design sensitivity analysis of asymmetric viscously damped systems with distinct and repeated eigenvalues, which can compute the left and right eigenvector derivatives separately and independently. All of the methods mentioned previously compute the eigensensitivities of asymmetric damped systems by using the left eigenvector. However, these methods have disadvantages in computational cost and storage capacity for the left eigenvector should be calculated. To avoid using the left eigenvector, an algebraic method is presented [14], which does not require the left eigenvector for asymmetric damped systems, but this method is restricted to the case of viscous damping. It should be noted that the coefficientmatrices of the algebraicmethodmay be ill conditioned due to the components of the additional constraints, and system matrices in the coefficient matrices are not all of the same order of magnitude. In addition [15], the normalization adapted in [14] and [12] will fail in some cases because it can equal zero or a very small number. This Note will present a method, which is well conditioned and can calculate the eigensensitivity of asymmetric nonviscous damped systems without using the left eigenvector. Considering an N-degree-of-freedom linear system with nonviscous (viscoelastic) damping [3,6,10]

AN ANALYTICAL MODEL OF OBLIQUE CUTTING WITH APPLICATION TO END MILLING

A new analytical cutting force model is presented for oblique cutting. Orthogonal cutting theory based on unequal division shear zone is extended to oblique cutting using equivalent plane approach. The equivalent plane angle is defined to determine the orientation of the equivalent plane. The governing equations of chip flow through the primary shear zone are established by introducing a piecewise power law distribution assumption of shear strain rate. The flow stress is calculated from Johnson-cook material constitutive equation. The predictions were compared with test data from the available literature and showed good correlation. The proposed model of oblique cutting was applied to predict the cutting forces in end milling. The helical flutes are decomposed into a set of differential oblique cutting edges. To every engaged tooth element, the differential cutting forces are obtained from oblique cutting process. Experiments on machining AISI 1045 steel under different cutting conditions were conducted to validate the proposed model. It shows that the predicted cutting forces agree with the measurements both in trends and values.

AN ACCELERATED SUBSPACE ITERATION METHOD FOR GENERALIZED EIGENPROBLEMS

Abstract An accelerated subspace iteration for generalized eigenproblems is proposed by combining the repeated inverse iteration with the over-relaxation technique. Two schemes are developed to obtain an over-relaxation factor. Numerical results show that the proposed acceleration is efficient and numerically stable for eigenproblems with a large number of eigenpairs required or with close eigenvalues.

Finite element modelling of human auditory periphery including a feed-forward amplification of the cochlea

A three-dimensional finite element model is developed for the simulation of the sound transmission through the human auditory periphery consisting of the external ear canal, middle ear and cochlea. The cochlea is modelled as a straight duct divided into two fluid-filled scalae by the basilar membrane (BM) having an orthotropic material property with dimensional variation along its length. In particular, an active feed-forward mechanism is added into the passive cochlear model to represent the activity of the outer hair cells (OHCs). An iterative procedure is proposed for calculating the nonlinear response resulting from the active cochlea in the frequency domain. Results on the middle-ear transfer function, BM steady-state frequency response and intracochlear pressure are derived. A good match of the model predictions with experimental data from the literatures demonstrates the validity of the ear model for simulating sound pressure gain of middle ear, frequency to place map, cochlear sensitivity and compressive output for large intensity input. The current model featuring an active cochlea is able to correlate directly the sound stimulus in the ear canal with the vibration of BM and provides a tool to explore the mechanisms by which sound pressure in the ear canal is converted to a stimulus for the OHCs.

Inclusion of Higher Modes in the Eigensensitivity of Nonviscously Damped Systems

C = viscous damping matrix ck = coefficients for mode basis vectors D s = dynamic stiffness matrix EH = modal-truncated error due to higher modes G s = Laplace transform of g t g t = matrix of kernel functions in the time domain K = stiffness matrix L = number of lower modes M = mass matrix m = order of the characteristic polynomial N = degrees of freedom of the system p = design parameter s = Laplace domain parameter t = time u s = Laplace transform of u t u t = response vector v i = ith approximate vector v i = ith exact vector δ t = Dirac delta function θi = normalization constant for ith mode λi = ith eigenvalue φi = ith eigenvector

Pillared graphene as an ultra-high sensitivity mass sensor

Hybrid structure of graphene sheets supported by carbon nanotubes (CNTs) sustains unique properties of both graphene and CNTs, which enables the utilization of advantages of the two novel materials. In this work, the capability of three-dimensional pillared graphene structure used as nanomechanical sensors is investigated by performing molecular dynamics simulations. The obtained results demonstrate that: (a) the mass sensitivity of the pillared graphene structure is ultrahigh and can reach at least 1 yg (10−24 g) with a mass responsivity 0.34 GHz · yg−1; (b) the sizes of pillared graphene structure, particularly the distance between carbon nanotube pillars, have a significant effect on the sensing performance; (c) an analytical expression can be derived to detect the deposited mass from the resonant frequency of the pillared graphene structure. The performed analyses might be significant to future design and application of pillared graphene based sensors with high sensitivity and large detecting area.

Accurate method for harmonic responses of non-classically damped systems in the middle frequency range

This paper is aimed at eliminating the influence of the problem of the harmonic response analysis of non-classically damped systems with lower-higher-modal truncation. Based on the Neumann expansion theorem and the frequency shifting technique, the relationships satisfied by eigensolutions and system matrices are established and an explicit expression on the lower-higher-modal truncation error of harmonic responses can be expressed as a sum of available modes and system matrices. A correction method, which only involves the available modes and system matrices, is then presented to calculate the harmonic responses. The method maintains original-space without having to involve the state-space equation of motion such that it is efficient in computational effort and storage capacity. The method is convergent if and only if all the modes whose resonant frequencies lie within the range of excitation frequencies are available. Finally, a two-stage floating raft isolation system and a sandwich plate are used to to validate the effectiveness of the presented method.

Thermoelastic damping of graphene nanobeams by considering the size effects of nanostructure and heat conduction

Abstract Thermoelastic damping of nanobeams by considering the size effects of nanostructure and heat conduction is studied herein. The size effect of nanostructure is investigated based on Euler–Bernoulli beam assumptions in the framework of nonlocal strain gradient elasticity, and the size dependence of heat conduction is taken into account by incorporating phase-lagging and nonlocal effects. Closed-form solutions of thermoelastic damping and quality factor characterized by thermoelastic coupling are derived. Graphene nanoribbon is chosen as a nanobeam. The effects of relaxation time, aspect ratio, elastic modulus, thermal expansion, and thermal conductivity on quality factor of graphene nanobeams are discussed in detail.

High intrinsic dissipation of graphyne nanotubes

We utilize molecular-dynamics simulations to report the first investigation of energy dissipation of two different doubly clamped graphyne nanotubes (GNTs), where the point of emphasis is to compare their dissipation characteristics with those of carbon nanotube (CNT). The obtained results demonstrate that: (a) GNTs exhibit significantly higher energy dissipation, and thus lower quality (Q) factor which is generally five times lower than that of CNT; (b) the Q factor of GNT further reduces with the increasing percentage of acetylenic linkages, which originates from the larger vibrational mismatch between the acetylenic linkage (sp C-C bonds) and the hexagonal ring (sp 2 C-C bonds); (c) the application of tensile strain is found to be highly beneficial to improving the Q factor of GNTs, especially for those with higher percentage of acetylenic linkages. These findings enable a first insight into the damping behavior of GNTs and also offer a significant guideline for the future design of GNT-based devices.