In the context of automotive noise and vibration design, the issue of road-induced noise arising from electrification has emerged as a significant concern. The noise is generated when tire vibrations, excited by the road surface, are transmitted through a frame structure to panels. Large vibrations in the frame structure are caused by waves traveling through the frame structure, reflecting and transmitting at discontinuities, such as T-junctions, and eventually resonating. It is acknowledged that effective methods of reducing resonance include controlling wave reflection/transmission at discontinuities. In particular, it may be advantageous to utilize the structural damping of a frame structure to adequately dissipate the propagating energy of the traveling waves. In order to utilize this principle, it is essential to extend the propagation distance before the formation of standing waves. The present study proposes a methodology that aims to reduce resonance by suppressing wave propagation in a specific direction to extend the propagation distance and inducing wave attenuation. To achieve directional control, neutralizers were utilized, which can fully reflect waves at a designated frequency. The validity of the proposed methodology was demonstrated through its implementation on a finite element model simulating the frame structure of an automobile. This numerical validation demonstrated the method’s capacity to reduce resonance within a specified frequency band.

To deform ODSs (Operating Deflection Shapes) with small structural modifications is useful for reducing vibrations at the designated frequency. In this study, we developed a method for achieving structural modifications to deform ODSs with as few structural modifications as possible by utilizing sparse modeling techniques. This method is designed to devise doubly sparse structural modifications, in which the initial mass and stiffness matrices are maintained in their sparsity while only some of the elements with values in the matrices are changed, when a finite element model of the target structure exists. The first sparsity was achieved using the Constrained Eigenstructure Assignment Method (CEAM). Furthermore, to ensure the second sparsity, we used the Least Absolute Shrinkage and Selection Operator (LASSO) to search for the weighting coefficients in CEAM. LASSO makes it possible to appropriately minimize the squared error of the ODSs with as few weighting coefficients, that is regularization, as possible. By combining CEAM and LASSO, we were able to obtain a heuristic solution search method that can achieve the desired operating mode by changing only the inertia and stiffness of some of the elements of the design.

NV (Noise and vibration) performance of a mechanical system is determined by the contributions of all subsystems that make up the mechanical system. However, complex mechanical systems such as automobiles are concurrently developed, in which the supplier subsystem is mounted onto the OEM (Original Equipment Manufacturer) subsystem, it is not easy to share intellectual property such as shape information between companies. As a result, NV problems often occur in the final stages of product development. This leads to extensive redevelopment, resulting in longer development times and higher development costs. Therefore, it is necessary to be able to design the NV performance of a whole structure from the supplier's viewpoint at the upstream product development stage. In this paper, we proposed a method to solve the inverse problem of allocating the single resonant frequency of a whole structure, which occurs due to a large contribution from the supplier subsystem, by modifying the structure of the supplier subsystem. This method is composed of kCA (kernel Compliance Analysis) and CEAM (Constrained Eigenstructure Assignment Method), and is realized by using the compliance-FRF matrix of the OEM subsystem provided by the OEM in the upstream design stage. Finally, numerical verification of the proposed method was demonstrated.

NV (Noise and Vibration) performance is one of the key product qualities of a mechanical system. Isolating all resonant frequencies from the frequency band where excitation forces are high is an important strategy for improving NV performance. However, a challenge in designing NV performance is that the resonant frequency is determined by the contributions of all the subsystems that make up the mechanical system. Complex mechanical systems such as automobiles are concurrently developed, in which the supplier subsystem is mounted onto the OEM (Original Equipment Manufacturer) subsystem. As a result, it is not easy to share intellectual property, such as shape information, between companies. For this reason, NV performance, including the assignment of resonant frequencies, is often evaluated in the final stage of product development. However, if NV performance targets are not met at this stage, extensive redevelopment will be required. Therefore, it is necessary to be able to design the NV performance of a whole structure from the supplier's viewpoint at the upstream product development stage. In this paper, we proposed a method to solve the underdetermined inverse problem of appropriately allocating the multiple resonant frequencies of a whole structure, which occurs due to a large contribution from the supplier subsystem, by modifying the structure of only the supplier subsystem. This method is composed of the kCA (kernel Compliance Analysis) and the CMCM (Cross-Model Cross-Mode) method and is realized by using the compliance-FRF matrix of the OEM subsystem provided by the OEM to the supplier in the upstream design stage. Finally, numerical verification of the proposed method was demonstrated.

This paper addresses a method for devising an undamped periodic structure that suppresses energy propagation due to a specific wave mode based on the Wave and Finite Element (WFE) method. Then, structural modifications to reduce the active transmitted power are derived by solving an inverse eigenvalue problem of the WFE method. In solving the inverse problem, it is important that the solution is feasible in the sense that it is low-cost and easy to fabricate as an actual mechanical structure. Therefore, in the proposed method, the designer specifies the regions in which structural modifications may be made before solving the inverse problem. As a result, the solution of the inverse problem is approximated within these regions. Finally, the method allows us to suggest feasible structural modifications necessary to reduce the amount of active transmitted power for a given wave mode and a given frequency. However, this method cannot consider the changes in wave numbers due to structural modification. It should therefore be noted that this method cannot be used to achieve significant reductions in active transmitted power due to large-scale structural modification.

Vibration related to several performances such as durability and noise performance is the one of the important performances on automotive product. The technique of reshaping eigenmode is a practical way to improve the performance on the product design phase. However, it is difficult to design desired mode shapes efficiently because the mode shapes are determined by the balance between mass and stiffness distribution on a whole structure. In this paper, a method for designing desired eigenmodes through changing internal forces between subsystems is proposed. Firstly, according to frequency based substructuring, the dynamic stiffness matrix of a whole structure could be divided into the matrices of subsystems and internal forces occurred from coupling components. Then, on a specified eigenfrequency as a design target, it can be derived that the mode shapes are controllable by changing to desired internal forces. Secondly, desired internal forces are calculated through designing spring constants between subsystems based on the kernel compliance analysis method which can analyze vibration when several subsystems are coupled on multiple degrees of freedom. Furthermore, when desired internal forces are calculated, a method to change a component of the internal force vector one by one is also proposed. This method can visualize the range which can be designed all spring constants as positive values in advance and can avoid selecting negative spring constants. Finally, this proposed method is applied to a numerical case study to reduce vibration responses with allocating modal strain energy to subsystems through reshaping an eigenmode.

In the development of mechanical structures, it has been required for a long time to establish a modular design method of resonance by controlling the vibration coupling among subsystems. From the analytical perspective of coupled vibration among modules, the kernel compliance analysis (kCA) method exists. The kCA method has the advantage of being able to analyze vibration coupling when modules, which will be called subsystems below, are connected with multiple degrees of freedom. However, the kCA method is not directly applicable to damped systems. This is because the kernel compliance matrix, which is the core of the kCA method, of the damped system is a complex symmetric matrix. To avoid the difficulty, in this paper, the Autonne-Takagi factorization is employed instead of the general eigenvalue decomposition to decompose the complex symmetric matrix. Furthermore, the old-type frequency-based substructuring was introduced so that vibration coupling analysis can be performed among multi-subsystems. With these extensions of the kCA, a method to analyze the energy distribution ratio between subsystems was derived. By using the analysis results, it is possible to shift the resonance frequency and determine the appropriate placement of damping materials.

In modern vehicle development, concurrent modular design, which is a productive manufacturing approach that uses pre-made and interchangeable modules to build vehicles and subsystems, has become mainstream to improve design efficiency and strengthen market responsiveness along with the increasing trend towards electrification and electronic systematization. The achievement of the modular design method is also desired in the design of vibration related to several performances such as durability and noise performance. However, developing the modular design method is a challenging task as the performance is influenced by all the subsystems constituting the whole structure. In this paper, a method for reshaping the eigenmode of only focused subsystem to the desired one is proposed as one of the modular design methods. The formulations to calculate a desired eigenmode shape of the focused subsystem are derived based on the dynamic stiffness matrix-based sub-structuring approach and the perspective of maintaining the relationship of vibration coupling between subsystems, which is constituted of internal forces and the eigenmode vector on coupling region where the forces act. Furthermore, the one-on-one correspondence between the reshaping amount of eigenmode and the dynamic stiffness matrix deviation for the reshaping is built involving the finite element model of the focused subsystem and the boundary conditions by expanding the formulation above. Therefore, reshaping to the desired eigenmode with realizable structural modification is achieved. Finally, this proposed method is applied to a numerical case study to reduce vibration responses by allocating modal strain energy to a focused subsystem from the perspective of increasing damping effect.

Vibration performance is the one of the important performances in automotive product because of related to not only ride comfort and control performance of vehicle but also durability and noise performance. However, it is difficult to design efficiently because the performance is determined by the influence of all components constituting of a whole structure. In previous paper, we presented an effective way to design vibration performance based on an inverse method for structural modification that keeps one specified eigenfrequency and its eigen vector of the whole structure the same. In this paper, we propose a new method to keep the specified multiple eigenfrequencies and their eigen vectors the same. Firstly, the matrices for structural modification are calculated as dynamic stiffness matrix variation from the zero-divisors of the specified multiple eigen vectors. The design of dynamic stiffness matrix variation is simplified by representing the matrix variation as just stiffness matrix variation which is independence on eigen frequency. Furthermore, it is possible to represent the calculated matrices redundantly by using an arbitrary non-zero weighting matrix. However, the matrices are generally calculated as fully populated and non-symmetric matrices. Therefore, secondly, we transform the matrices calculated as the zero-divisors into the sparse matrix of reduced row echelon form in advance. This transformation simplifies building symmetric and sparse matrices for realizable structural modification. This sparse matrix, redundantly represented by a weighting matrix, allows a value analysis to select among alternative structural changes to simplify structural complexity while keeping several performances originated from vibration performance. Finally, the proposed method was applied to a numerical case study.

Vibration related to several performances such as durability and noise performance is the one of the important performances on automotive product. The technique of reshaping eigenmode is a practical way to improve the performance on the product design phase. However, it is difficult to design desired mode shapes efficiently because the mode shapes are determined by the balance between mass and stiffness distribution on a whole structure. In this paper, a method for designing desired eigenmodes through changing internal forces between subsystems is proposed. Firstly, according to frequency based substructuring, the dynamic stiffness matrix of a whole structure could be divided into the matrices of subsystems and internal forces occurred from coupling components. Then, on a specified eigenfrequency as a design target, it can be derived that the mode shapes are controllable by changing to desired internal forces. Secondly, desired internal forces are calculated through designing spring constants between subsystems based on the kernel compliance analysis method which can analyze vibration when several subsystems are coupled on multiple degrees of freedom. Furthermore, when desired internal forces are calculated, a method to change a component of the internal force vector one by one is also proposed. This method can visualize the range which can be designed all spring constants as positive values in advance and can avoid selecting negative spring constants. Finally, this proposed method is applied to a numerical case study to reduce vibration responses with allocating modal strain energy to subsystems through reshaping an eigenmode.

NV (Noise and Vibration) performance is determined by the influence of all components constituting a whole structure. It is difficult to design NV performance efficiently because the structural modification of a certain component affects the performance of the whole structure. As one of the effective ways to design the performance, this paper presents an inverse method for structural modification that keeps the specified eigenfrequency and its modal vector of the whole structure the same. Firstly, the matrices for structural modification are calculated as dynamic stiffness matrix variation from the zero-divisors of a specified modal vector. Furthermore, it is possible to represent the calculated matrices redundantly by using an arbitrary non-zero weighting matrix. This makes it possible to obtain various solutions for structural modification. However, the matrices are generally calculated as fully populated and non-symmetric matrices. Therefore, with these matrices, it is difficult to feasibly find a symmetrical and sparse mass or stiffness matrix to use for structural modifications in applications that focus on designing specified regions of the whole structure. Secondly, we propose how to transform the matrices as the zero-divisors into the sparse matrix of reduced row echelon form in advance. This transformation simplifies building symmetric and sparse matrices for realizable structural modification. The sparse matrix, redundantly represented by a weighting matrix, allows a value analysis to select among alternative structural changes to lighten the product or simplify its complexity. Finally, the proposed method was applied to a numerical case study.

Severe vibration may occur in structures such as high-rise buildings and bridges according to wind and seismic excitations, and in large space structures due to their lightweight and flexible constitution. Generally, when several vibration modes are simultaneously excited, application of the model-based control strategies become difficult. In this study, we propose a wave absorption control method for reducing vibrations in multi-degree-of-freedom systems semi-actively using a magnetorheological elastomer-based dynamic absorber. The stiffness of the absorber is changeable according to the applied magnetic field strength. The value is tuned adaptively so that the mechanical impedance at the boundary meets an absorptive boundary requirement. Analytical investigation clarified relationship between the excitation frequency and absorber stiffness that could eliminate the wave reflection from the boundary and maintain no resonant state in the system. Based on this stiffness condition, numerical simulation and experiment for the wave absorption control were performed. The proposed absorber to be used with the wave control scheme was found to significantly reduce structural vibrations within the stiffness variable range.

This paper addresses a method of controlling the nodal positions of mode shapes as desired. The method is developed to calculate reflection and transmission coefficients at discontinuities and ends required for shaping mode shapes. The method is based on the one-dimensional Ray-Tracing method, which is a method that has been used to obtain eigenfrequencies, mode shapes, frequency response functions of a structure, based on the analysis of wave propagation on three-dimensional beam structures. Natural vibration in an undamped system refers to a state in which waves propagating on a structure are superimposed and become standing waves everywhere on the structure. Therefore, the mode shape indicates the shape of this standing wave. In the paper, a method is derived to estimate the reflection and transmission coefficients at discontinuities and ends, which is necessary to change the nodal points of the mode shape of interest. This method requires a matrix operation that changes only the values of the reflection and transmission coefficients at discontinuities and ends without changing the shape of the structure. Therefore, a rank-one reduction method via similarity transformation of matrices is developed. Finally, the effectiveness of the proposed method is confirmed through numerical examples.

This paper addresses a method of classifying modal clusters to realize resonance control. Especially in this paper, this method is applied to the simultaneous placement of resonance frequencies of the multiple modes belonging to a specified modal cluster through a structural modification to a limited subsystem. The method is useful for separating the resonance frequency of the modal cluster to be designed from the peak frequency of the excitation spectrum all at once. Therefore, the paper proposes a method to consider the multiple modes of a whole structure formed from the same mode of the subsystem belonging to the same modal cluster by using the modal contribution analysis method which is proposed by the authors. Then, under the above definition of a modal cluster, we introduce a method of moving the resonance frequencies of multiple modes belonging to the same modal cluster together without significantly changing the resonance frequencies of multiple modes belonging to other modal clusters. Finally, its usefulness is shown by a numerical example.

When a rigid and small subsystem is rigidly coupled to a main system of a design target, it would be helpful if the multiple resonance frequencies moved by this coupling could be properly assigned so that they do not coincide with the peak of excitation frequencies to avoid resonance. In this paper, a rigid and small subsystem is regarded as a rigid body, and a visualization design method to optimize the assignment of multiple resonance frequencies is developed through the design of the mass matrix of the rigid and small subsystem. In this method, the mass matrix is diagonalized by using the principal axis of inertia of the rigid body, and then the frequency band that satisfies the resonance formation condition is predicted by the kernel Compliance Analysis. Furthermore, by combining this method with Weyl's inequality theorem, we made it possible to design the assignment of multiple resonance frequencies while looking at a single figure.

This paper addresses the modular concept concurrent design of NVH (Noise, Vibration and Harshness) performance. Recent development of modularization design in car companies requires the realization of the modularization of NVH design process. However, there are difficulties for the modularization design of the strong coupling vibration among each module. Modularization design requires the realization of concurrent design by two or more groups. In this case, a true concurrent design cannot be realized unless the problem that a slight structural modification seriously changes the characteristics of a whole structure due to the strong coupling is solved. In this paper, a concurrent modularization design method of NVH performance which utilizes the kCA (kernel Compliance Analysis) was proposed to overcome the strong coupling problem. The basic concept of the kCA is adopted for both the placement of resonance frequencies and the reduction of resonance responses of a whole structure, by the comprehension of resonance generation mechanism between coupled two subsystems. Therefore, the kCA was adopted in this paper as a basis of the proposed method to overcome the strong coupling problem. In the paper, it is shown that the proposed method provides the advantage of no rework at every stage of the design, from upstream to downstream, if two groups which design two subsystems separately follow the specifications which is decided by the proposed design method. Finally, the method was verified by a numerical case study.

It is necessary to separate the multiple resonance frequencies of a vehicle body from outstanding peaks of excitation spectrum to reduce the road induced noise of cars. A vehicle body is always excited by the transmitted forces from a chassis system, and worse the transmitted forces change in real time with road surfaces. Therefore, we suggest that semi-active control of the vibration characteristics of a car body is useful for reducing road induced noise of cars. In this paper, we derived a design method of joint stiffness for simultaneous placement of multiple resonance frequencies. Specifically, we have focused our efforts to make an eigenvalue of a kernel compliance matrix to zero at the assigned natural frequencies by changing the joint stiffness connecting a main system and a subsystem. Consequently, it is enabled to control the multiple resonance frequencies of a whole structure with low computational cost.

The finite element method is widely used to predict mechanical vibration for the purpose of improving the performance of mechanical products. However, in complex structures such as automobiles, much effort is required to make the accurately entire system model. Therefore, in this research, we propose a method to identify the state transition matrix of the entire system by hybridizing the state transition matrix obtained by finite element modeling only the target component and the state transition matrix identified from the experimental data. In this paper, as a numerical verification, we apply the proposed method to a beam model whether it is possible to predict the vibration and the vibration change due to structural modification. 岐阜大学大学院 Graduate School of Gifu University

When designing NVH (Noise, Vibration and Harshness) performance for complex structures such as automobiles, trains, and aircraft, it is rare to target all components of the whole structure and the subsystem be designed are often fixed. In recent years, the kernel Compliance Analysis (kCA) method has been proposed as a method for designing NVH performance by dividing a whole structure into two subsystems. When modifying the structure of the subsystem to be designed by using the kCA, it is an important index which mode of the subsystem has a large contribution to the resonance of a whole structure. However, in the conventional sensitivity of structural modification, only those that determine the contribution of the subsystem for each mode of a whole structure are found. This paper, therefore, derives a new index for understanding the relative importance of the modes of subsystems to the resonance of a whole structure. As the index is derived based on the kCA, it has good compatibility with the design of NVH performance by the kCA. After derivation of the index, the effectiveness of the index will be verified through a numerical example of the resonance assignment of a whole structure combined with the kCA.