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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]

Process sequence optimization based on constraint matrix

The sequence of manufacturing process should meet process constraints during the process planning optimization. The qualitative constraints are represented implicitly as geometry relationships, process rules and manufacturing environment, it is difficult to apply in the process optimization. A constraint matrix is established to express process constraints; transfer rules and regulation are also designed to ensure the reliability and the veracity when the qualitative constraint message is transferred into matrix. Based on constraint matrix, a process sequence optimization model is set up and genetic algorithm is used to obtain the optimal process sequence. An example is presented to indicate that the constraint matrix not only ensure the process optimization¿s veracity but also enhance the performance efficiency.

The Effect of Material Flow Stress on Analytical Simulation of Machining

The flow stress data of the workpiece are extremely crucial for the cutting simulation. The study shows how the input data affect the analytical predictions of cutting force and temperature. The Johnson-Cook material model is used to represent workpiece flow stress in the primary shear zone. A thermomechanical model of orthogonal cutting is proposed based on the main shear plane divides the primary shear zone into two unequal parts. Five different sets of workpiece material flow stress data used in the Johnson-Cook’s constitutive equation are chosen and make the sensitivity analysis for analytical model. Simulation results were compared to orthogonal cutting test data from the available literature, and find the effects of flow stress on analytical model was different from that for finite element model.