FRF-based Substructuring Method Research Articles

Vibration analysis of a rigid body with a flexible appendage supported by elastic bodies using FRF-based substructuring method

Vibration analysis of a rigid body with a flexible appendage supported by elastic bodies using FRF-based substructuring method

Vibration analysis of a submarine elastic propeller-shaft-hull system using FRF-based substructuring method

A dynamic model to study the coupled longitudinal and transverse vibrations of a submarine elastic propeller-shaft-hull system is developed using the FRF-based substructuring method (FBSM). The total system is firstly modeled as two substructures: the elastic propeller-shaft subsystem and the hull shell. For the former substructure, the elastic propeller is modeled by using harmonic blade array elements and the shafts are assumed to be Timoshenko beams, while the latter one is modeled using traditional finite element method. After that, the two substructures are synthesized using FBSM. The modes, the natural frequencies and the coupled longitudinal and transverse vibration characteristics of the propeller-shaft subsystem, the hull shell, and the total system are analyzed. An experiment studying the dynamic characteristics of a large-scale submarine experimental setup is processed and compared with the numerical results, which shows great consistency. Finally, a further discussion is carried out focused on how the bearing stiffness affects the coupled vibration characteristics of the total system.

Effects of modal truncation and condensation methods on the Frequency Response Function of a stage reducer connected by rigid coupling to a planetary gear system

The present paper is aimed at the application of a substructure methodology, based on the Frequency Response Function (FRF) simulation technique, to analyze the vibration of a stage reducer connected by a rigid coupling to a planetary gear system. The computation of the vibration response was achieved using the FRF-based substructuring method. First of all, the two subsystems were analyzed separately and their FRF were obtained. Then the coupled model was analyzed indirectly using the substructuring technique. A comparison between the full system response and the coupled model response using the FRF substructuring was investigated to validate the coupling method. Furthermore, a parametric study of the effect of the shaft coupling stiffness on the FRF was discussed and the effects of modal truncation and condensation methods on the FRF of subsystems were analyzed.

Stiffness optimization of marine propulsion shafting system by FRF-based substructuring method and sensitivity analysis

The present work is concerned with a dynamic modeling and optimization method to minimize the vibration transmission through the marine propulsion shafting system by the FRF-based substructuring method and the sensitivity analysis. The dynamic model of the coupled propeller - propulsion shafting - hull system under propeller exciting forces and excitation forces from the propulsion electric machine is established. Vibration response characteristics for the coupled system are analyzed. Mean-square forces and power flow transmitted to the foundation is employed as objective functions to evaluate the optimal design of propulsion shafting system. The proposed method can take into consideration of real propellers as well as the fluid-structure interaction between the propeller, hull and the surrounding water. It is shown that propeller flexibility contributes obvious peaks in the vibration responses. For the studied model, the stiffness of the bearings and the isolators should be lowered; the stiffness of the coupling should be increased to suppress vibration transmission. The mean-square force and the power flow transmitted to the foundation in the concerned frequency range is decreased to 0.712 and 0.525 of the initial value, respectively. The proposed dynamic modeling and optimization method is capable of performing optimization with greatly improved efficiency.

Design scheme of a passive isotropic multi-strut vibration isolation platform constructed by three-parameter isolators based on the optimum damping frequency concept

A design procedure based on the optimum damping frequency of the three-parameter isolator is proposed to construct a passive isotropic multi-strut vibration isolation platform (VIP) with three-parameter isolators. Two steps are necessary to implement the proposed design scheme. The first step is determination of the optimal damping frequency of the three-parameter isolator; the second step is designing the isofrequency VIP with natural frequencies tuned as the optimal damping frequency in three orthogonal directions. Three methods are summarized to identify the optimal damping frequency theoretically or experimentally. A frequency response function (FRF)-based substructuring method is proposed to predict natural frequencies of the VIP in an isolation system, which accounts for the coupling between different directions and structural flexibilities. Analytical expressions of configurations to implement isofrequency VIP are derived from FRF-based substructuring method for special cases. The three-parameter isolator by employing bellows is manufactured and a hexapod VIP is built. The proposed design scheme is verified by identifying the optimal damping frequency and comparing acceleration transmissibility under vertical and horizontal base displacement excitation for varying loaded mass. The study can shed some light on the design of a passive multi-strut VIP with three-parameter isolators.

Modeling and Optimization of Octostrut Vibration Isolation Platform by FRF-Based Substructuring Method

This paper addresses dynamic modeling and performs optimization of a passive multistrut vibration isolation platform (VIP) in vibration isolation for an assembly of control moment gyroscopes (CMGs). Each strut of the VIP is composed of a metal rod and a three-parameter isolator in series. First, the frequency response function (FRF)–based substructuring method was developed to establish the dynamic model of a whole satellite system. This method can take into account the influence of the flexibility of the satellite bus and the CMG assembly as well as the gyroscopic effects of CMGs. Then, an optimization model was developed, and the FRF-based sensitivity analysis was employed to optimize the characteristic parameters of a strut under two objective functions. Finally, a numerical model was developed to verify the validity and effectiveness of the proposed method. The dynamic characteristics of an octostrut VIP were analyzed. The isolation performance was greatly enhanced after optimization. This study shows that the method is an effective and efficient tool that can be used in the design phase of VIPs.

An identification method for joint structural parameters using an FRF-based substructuring method and an optimization technique

A new method is proposed to identify the joint structural parameters of complex systems using a frequency response function (FRF)-based substructuring method and an optimization technique. The FRF method is used to estimate the joint parameters indirectly by minimizing the difference between the reference and calculated responses using a gradient-based optimization technique with analytical gradient information. To assess the robustness of the identification method with respect to noisy input data, FRFs contaminated by uniformly distributed random noise were tested in a numerical example. The effects of the random noise and the magnitude of the connection stiffness values on the accuracy of the method were investigated while identifying the joint parameters. When the FRFs were contaminated with random noise, the proposed procedure performed well when used to identify the stiffness values, but the accuracy of identification is deteriorative when used to identify the damping coefficients. The joint parameters of a real bolted structure were also identified by the proposed method. The results show that it can be applied successfully to real structures, and that a hybrid approach using both calculated and measured FRFs in the substructure model can enhance the quality of the identification results.

전달함수 다중합성법을 이용한 진동시스템의 결합부 특성값 추정

A method of identifying structural parameters such as stiffness and damping coefficients at interfacial points of vibro-acoustic systems is suggested using an optimization technique. To identify the parameters using a numerical optimization algorithm, cost functions are defined. The cost function should be zero at the correct parameter values. To minimize the cost functions using an optimization technique, a design sensitivity analysis procedure is developed in the framework of the multi-domain FRF-based substructuring method. As a numerical example, a ladder-like structure problem is introduced. With known parameter values and different initial guesses of the parameters, convergence characteristics to the exact value are compared for the three cost functions. Investigating the contours of the cost functions, we find the first cost function has the largest convergent region to the correct value. As another practical problem, the stiffnesses of engine mounts and bushings in a passenger car are identified. The numerical examples show that the proposed method is efficient and accurate for realistic problems.

Parametric optimization of complex systems using a multi-domain FRF-based substructuring method

A general procedure for a parametric sensitivity analysis of a complex system has been presented using a multi-domain FRF-based substructuring formulation. The direct differentiation of reaction forces on the interface boundaries with respect to the design vector is used to obtain a design sensitivity formula of responses in any substructure. A simple example for a ladder structure is introduced to verify the formulation numerically. The proposed method is applied to an optimization problem of an engine mount system to minimize the interior sound over the concerned rpm range. The optimized interior noise in a passenger car shows that the proposed design sensitivity formulation is efficient and accurate.

DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION OF AN ENGINE MOUNT SYSTEM USING AN FRF-BASED SUBSTRUCTURING METHOD

The FRF-based substructuring method is one of the most powerful methods in analyzing the responses of complex built-up structures with high modal density. In this paper, a general procedure for the design sensitivity analysis of a vibro-acoustic system has been presented using the FRF-based substructuring formulation. For an acoustic response function, the proposed method gives a parametric design sensitivity expression in terms of the partial derivatives of the connection element properties and the transfer functions of the substructures. The derived noise sensitivity formula is combined with a non-linear programming module to obtain the optimal design for the engine mount system of a passenger car. The objective function is defined as the area of the interior noise graph integrated over a concerned r.p.m. range. The interior noise variations with respect to the dynamic characteristics of the engine mounts and bushings have been calculated using the proposed sensitivity formulation and transferred to a non-linear optimization software. To obtain the FRFs, a finite element analysis was used for the engine mount structures and experimental techniques were used for the trimmed body including the cabin cavity. The optimization based on the sensitivity analysis gives the ideal stiffness of the engine mount and bushings. The resultant interior noise in the passenger car shows that the proposed method is efficient and accurate.

주파수 응답함수를 이용한 부분구조 합성에서 모드자름 오차 보정에 관한 수치적 연구

A FRF-based substructuring method attempts to predict the dynamic characteristics of a complex structure from predetermined FRFs of the comprising uncoupled substructures. Although this method has the advantage of being able to incorporate experimental component FRFs directly, it is prone to errors : measurement errors, coordinate incompleteness, modal incompleteness, etc. Among the various sources of errors, this paper deals with the problem of modal incompleteness (or residual problem) of which importance is underestimated compared to others. It is a well-known rule of thumb that such a problem can be overcome by including modes up to 2 or 3 times the upper frequency of interest. Using a simulated case study, it is demonstrated that even including modes up to 20 times the upper frequency of interest does not guarantee a satisfactory result. A method to compensate the residual errors is introduced. This method requires the whole FRF matrices of substructures which is practically impossible for a complex structure. An applicable alternative is suggested and applied successfully to the case study. Finally, the effects of measurement errors on the residual compensation are also discussed.

Sensitivity Analysis of Engine Mount System using FRF-based Substructuring Method

A general procedure for the design sensitivity analysis of structural dynamic problems has been presented in frame of the FRF-based substructuring formulation. For a system response function, the proposed method gives a parametric design sensitivity formula in terms of the partial derivatives of the connection element properties and the transfer matrix of the subsystems. The derived design sensitivity formula is applied to an engine mount system. An interior noise problem in the passenger car is analyzed using the FRF-based substructuring method and the proposed formulation is adopted to study the response variations with respect to the dynamic characteristics of the engine mounts and the bushes. To obtain the FRFs, a finite element model is built for the engine mount structures, and test data is used for the trimmed body including cabin cavity. The comparison of sensitivities derived by the proposed method and the finite difference method shows that the proposed method is efficient and accurate. The proposed sensitivity analysis method indicates effectively the most sensitive location to the interior noise among the engine mounts and the bushes.