Direct Frequency Response Method Research Articles

Structural dynamic responses of building structures with non-viscous dampers under Kanai–Tajimi spectrum excitation

Non-viscous dampers are commonly used in building structures for seismic energy dissipation. In this study, a concise analytical method based on the pseudo-excitation method (PEM) was proposed to obtain closed-form solutions of the structural responses of a non-viscous damper system subjected to the Kanai–Tajimi spectrum excitation. For this purpose, first, the ground motion equation of the non-viscous damper system was decoupled using the complex model method (CMM), and the solution expressions for the dynamic responses were unified using the PEM. Second, the power spectral density function (PSDF) of the Kanai–Tajimi spectrum was quadratically orthogonalised using the residue theorem, which was combined with a quadratically orthogonalised frequency response eigenvalue function (FREF) to obtain a quadratic orthogonalised PSDF (QO-PSDF) of the dynamic responses. Subsequently, a concise closed-form solution of spectral moments and variances of the dynamic responses were successfully obtained using the QO-PSDF. Finally, a comparative study was conducted with three test cases to compare the results of the direct frequency response method, traditional PEM, and proposed method, which proved the applicability of the QO-PSDF and the accuracy of the quadratically orthogonalised PEM (QO-PEM). Moreover, the calculation time required in the proposed method was found to be approximately th one-third at of the traditional PEM.

Study on optimization method of GDI engine combustion noise based on noise-performance map

An optimization of GDI engine combustion noise was evaluated through experimental and simulation methods. First, finite element model of engine was established; the vibration response and combustion noise caused by combustion incentives were calculated by the direct frequency response method and the acoustic transfer vector method. Above of this, the pure combustion noise attenuation curve was defined. According to the combustion spectrum calculated, the optimized frequency range was determined. Then, combining with the analysis result of the combustion noise attenuation curve, the frequency position of cylinder pressure spectrum to be optimized was obtained. Besides this, the combustion parameters were measured through test and its relationship to cylinder pressure spectrum and engine power performance was studied, the noise-performance map was established. Based on optimized cylinder pressure spectrum and noise-performance map, the optimum values of combustion parameter were marked very quickly. The results show that, when the fuel injection timing optimization value was used, combustion noise was reduced most by 6.4 dB in the range of 0â–“1000 Hz, while torque reduced only by 3.8 Nm, which achieved the goals of the favoring NVH performance and engine power performance.

Stationary random response of non-viscously damped polymer matrix composite structure systems

The polymer materials are widely used as the matrix of composite structures in the automobile, aircraft, building, bridge, railway, ship, turbine, and appliance industries. The polymer materials with long-chain molecules exhibit non-viscous damping behaviour. Non-viscous damped composite structure systems in which the damping forces depend on the past history of velocities via convolution integrals over some kernel functions have been raised in many different subjects. The developed analysis methods of dynamic response for such structure systems are almost limited to the deterministic time-history excitation. There is little research report on the random response of such non-viscous damped systems. The author has previously developed the non-viscous damping modelling method for polymer matrix composite structures (Liu, to be published). The goal of this paper is focus on developing two methods, i.e., direct frequency response method and iterative method using real modes, to obtain the power spectral density function (PSDF) of non-viscously damped polymer matrix composite structure systems subjected to stationary stochastic excitation. First, the pseudo excitation method converts the stationary stochastic excitation problem into harmonic excitation problem. Second, the direct frequency response method is derived and proven to get the analytical solution of PSDF. Third, the iterative method using real modes to obtain PSDF matrix is developed based on a harmonic response method. The computational procedure of the iterative method is given in detail. Finally, the random response analyses of two non-viscously damped structure systems, subjected to stationary random excitation, are demonstrated. The results indicate the two methods can achieve the exact solution of PSDF matrix of non-viscously damped structure systems. The iterative method using real modes is more efficient than direct frequency response method.

Design sensitivity analysis of dynamic response of nonviscously damped systems

The sensitivity problem of dynamic analysis of linear nonviscously damped systems is considered. The assumed nonviscous damping forces depend on the past history of motion via convolution integrals over some kernel functions. The nonviscous damping model can be alternatively chosen from familiar viscoelastically damping structures and is considered as a further generalization of the familiar viscous damping. The computations of dynamic responses are reviewed for the purpose of design sensitivity analysis development. The dynamic response can be easily calculated using direct frequency response method and modal superposition method when the dynamic equation of motion of nonviscously damped systems is transformed into the frequency domain using the Laplace transform. It is shown that the dynamic response of nonviscously damped systems can be obtained using traditional modal analysis in a familiar manner used in undamped or viscously damped systems. The discrete Fourier transform and inverse discrete Fourier transform algorithms are also suggested to obtain the displacement in the time domain. Based on these expressions of dynamic response, the adjoint variable and direct differentiation methods, originally presented to obtain the dynamic response sensitivity of undamped or viscously damped systems, are both developed for efficiently and accurately calculating the sensitivity of dynamic response of nonviscously damped systems. Finally, some case studies are used to show the application, effectiveness and some characters of the derived formulas. The numerical sensitivity results show the sensitivity obtained using the developed methods are in excellent agreement with the finite difference results. However, the finite difference method suffers from computational inefficiency and possible errors.

Finite element study of multi-modal vibration damping for thermal barrier coating applications

A physically-based computational model is developed to predict the damping behavior of oxide thermal barrier coating systems. The constitutive damping model is derived from the theory of point defect relaxation in crystalline solids and implemented within a finite element framework. While oxide coatings have been primarily employed as thermal barriers for gas turbine blades, there is a growing interest in developing multifunctional coatings combining thermal protection and damping capabilities. The direct frequency response method, as well as the modal strain energy method, have been implemented to evaluate the functional dependance of damping on temperature and frequency. Numerical results are validated through the limited experimental data available in the literature, and new results are presented to illustrate the effects of different topcoat oxides. The paper also illustrates how the developed methodology enables the damping capacity under different vibrational modes to be predicted, and to estimate the sensitivity of the design for varying geometrical parameters. Finally, the computational model is applied to investigate the damping performance of an oxide-coated turbine blade.

Damping analysis of beams covered with multiple PCLD patches

An analytical approach for vibration damping analysis of beams with partial passive constrained layer damping (PCLD) treatments is extended to the case where the structure is covered by multiple patches. The energy-based approach and assumed-mode method are used to derive the equations governing the vibration displacement responses of the system, which is characterized in terms of the identical transverse displacement of all layers, and the longitudinal displacements of the base beam and the constraining layers. Furthermore, the longitudinal displacements of the base beam and constraining layer are independent of each other, which is in contrast to Kerwin's weak core assumption. The direct frequency response method is employed to determine the modal loss factors of the damped beam in a frequency range of interest. A convergence study is made on the damped beam to determine appropriate numbers of the mode functions for frequency response function (FRF) calculations. Parametric studies are performed with the established analytical approach and the determined numbers of mode functions to study the effects of dividing a PCLD patch into two pieces in different length ratios and also the spacing between the patches on the damping performance of a simply supported beam.

Numerical Techniques for Computing Rotordynamic Properties of Mechanical Gas Face Seals

The gas film stiffness and damping properties for a spiral grooved mechanical face seal in a flexibly mounted stator configuration are computed using the step jump method and a novel direct numerical frequency response method. The seal model has three degrees of freedom, including axial displacement of the stator and two stator tilts about mutually perpendicular diametral axes. Results from both methods agree well with previously published results computed using the perturbation method, but the two new methods have the advantage of employing computer programs used in the direct numerical simulation of motion. Based on the linearized analysis, the two angular modes are proven to be coupled together and decoupled from the axial mode. Anomalies in the gas film properties tend to occur at large compressibility numbers. The step jump method requires less computing time than the direct frequency response method but at a sacrifice in accuracy at high excitation frequencies.

Integral finite element method for dynamical analysis of elastic–viscoelastic composite structures

Four types of integral finite elements have been developed and used to estimate the dynamical characteristics of elastic–viscoelastic composite structures, namely sandwich beam, plate and shell structures with viscoelastic materials as core layers. The results of the integral finite element prediction are compared with those from the direct frequency response method and experiment, showing that the numerical stability, convergence and accuracy of the integral finite elements are satisfactory in terms of engineering applications.

Frequency dependent network equivalents for electromagnetic transient studies

In electromagnetic transient studies, only a small part of an AC power system can be modeled in detail. The remaining parts of the system should be reduced to equivalent model. Different methods were developed to generate frequency dependent network equivalents. This paper presents a novel approach for obtaining network equivalents for ac power systems using the QZ algorithm. The proposed method obtains accurate resonant frequencies directly from the system equations to be used in finding the network equivalent parameters. Also, the proposed method is a good tool to predict the system response. The use of the QZ algorithm in the creation of the network equivalent model gives the following advantages: 1. Improved accuracy: as compared to the direct frequency response method. 2. Less complexity: because it obtains the minima and maxima frequencies directly from the system equations. 3. Reliable prediction: it gives the complete information about the behavior of the test system and its peaks and troughs before calculating its frequency response. 4. Good evaluation of network's resistance: the proposed method determines the resistors of the network equivalent model as the real part of the impedance function magnitude at the minima frequency points. In this article, a general algorithm is developed to determine the frequency dependent network equivalent model parameters for any given system. Finally a four-bus test system is introduced to show the accuracy and the validation of the proposed method. For simplicity and convenience a comparison between the direct frequency response method and the proposed method, a two-bus system, is also studied in this work. Good results were obtained for network equivalent models for both the frequency and time domains.

Complex eigensensitivity-based characterization of structures with viscoelastic damping

Damping plays an important role in structural stability and vibration control and different methods have been developed to design structures with maximum possible damping capacity. The most effective measure to date is perhaps the viscoelastic damping treatment. However, the dynamic properties of viscoelastic materials are generally frequency and temperature dependent. This makes the dynamic characterization of viscoelastically damped structures a tremendously difficult task. Conventional approximate methods, such as the direct frequency response method and the modal strain energy method, become quite inaccurate when the damping of a structure to be analyzed becomes high. An iterative complex eigensolution algorithm, though it is exact, requires a tremendous amount of computational effort since, for each mode, repeated complex eigensolutions are required until converged results are obtained. In the present paper, a new iterative complex eigensensitivity-based characterization method is presented. Based on the real eigensolution performed using finite-element analysis, the proposed method successively updates the complex eigenvalues and eigenvectors until required accuracy is achieved. As compared with conventional methods, the new method improves considerably numerical accuracy while it maintains the computational efficiency required for practical applications. A multilayer sandwich beam structure is used to demonstrate the practicality of the proposed method and the results show that the method is numerically very accurate and computationally very efficient.

Frequency response analysis of laminated composite beams

The modeling of laminated composite beams has been derived systematically from the three-dimensional elasticity relations. The correctness of the solution found by using the present finite element model is verified by comparison with the results obtained by analytical solutions and other results presented in the literature. Numerical results indicate that the present technique can given accurate results for frequency response analysis for laminated composite beams. Loss factors of structures obtained by the method of complex eigenvalues and the direct frequency response method exhibit very good agreement. Optimum design of a laminated composite beam by the finite element method and the method of experiment planning has been successfully presented.

Structural topology and shape optimization for a frequency response problem

A topology and shape optimization technique using the homogenization method was developed for stiffness of a linearly elastic structure by Bendsøe and Kikuchi (1988), Suzuki and Kikuchi (1990, 1991), and others. This method has also been extended to deal with an optimal reinforcement problem for a free vibration structure by Diaz and Kikuchi (1992). In this paper, we consider a frequency response optimization problem for both the optimal layout and the reinforcement of an elastic structure. First, the structural optimization problem is transformed to an Optimal Material Distribution problem (OMD) introducing microscale voids, and then the homogenization method is employed to determine and equivalent “averaged” structural analysis model. A new optimization algorithm, which is derived from a Sequential Approximate Optimization approach (SAO) with the dual method, is presented to solve the present optimization problem. This optimization algorithm is different from the CONLIN (Fleury 1986) and MMA (Svanderg 1987), and it is based on a simpler idea that employs a shifted Lagrangian function to make a convex approximation. The new algorithm is called “Modified Optimality Criteria method (MOC)” because it can be reduced to the traditional OC method by using a zero value for the shift parameter. Two sensitivity analysis methods, the Direct Frequency Response method (DFR) and the Modal Frequency Response method (MFR), are employed to calculate the sensitivities of the object functions. Finally, three examples are given to show the feasibility of the present approach.

Finite element analysis of sandwich structures

Sandwich panels composed of metal face sheets and viscoelastic core layers are widely used as structural elements in situations where resonant vibration cannot be avoided. The damping afforded by the viscoelastic material allows significant improvements in fatigue life, noise radiation, etc. The analysis of such structures to predict dynamic properties including damping has recently become practical through the use of finite element methods. In this survey paper, finite element modeling techniques and solution techniques appropriate for that purpose are described. The Properties of various plate and solid elements which have been found useful in modeling of sandwich panels are reviewed. Solution techniques such as complex eigenvalue methods, the direct frequency response method, and the modal strain energy (MSE) method are explained. Advantages and drawbacks of each are cited. Primary emphasis is placed on the MSE method because of its applicability in design work. The method is illustrated in terms of comparison to closed form solutions for various simple structural elements such as beams, plates, rings, etc.