Amplitude Magnification Factor Research Articles

Enhanced suppression of vibrational wave transmission in structures with periodically attached absorbers exploiting amplitude magnification mechanisms

Large-span spatial structures can exhibit excessive vibrations caused by dynamic loads. Different types of vibration suppression devices have been proposed and there has been much research on the optimization and improvement of traditional dynamic vibration absorbers. Nevertheless, it is still of great significance to develop ways of comprehensively improving the working capacity of vibration suppression devices. This article proposes a design idea where an amplitude magnification mechanism is used to enhance significantly the working capacity of the vibration suppression device. The proposed amplification of the amplitude can not only strengthen the inertia of the vibration suppression device, but also simultaneously enhance its damping and stiffness characteristics. In-depth analysis is carried out, revealing the relationship between the amplitude magnification factor and the working capacity of the vibration suppression device, and providing important design guidelines of the vibration suppression device. From the perspective of controlling the propagation characteristics of bending vibration waves in structures, a beam attached with the proposed periodically arranged vibration suppression devices with amplitude magnification effect is taken as an example. The energy band structure and vibration transmission characteristics are investigated using the energy method, and the reliability of the proposed model is verified from the energy band structure perspective through the finite element method. The results reveal that the working capacity of the vibration suppression device is increased by α2 times when the amplitude is amplified by α times. Consequently, the attenuation ability of the vibration wave in the bandgap is significantly enhanced. On this basis, the principles of anti-resonance and dynamic vibration absorption are combined to design a new type of vibration suppression device with an amplitude magnification effect, achieving ultra-wideband vibration reduction.

Effect of Inerter in Traditional and Variant Dynamic Vibration Absorbers for One Degree-of-Freedom Systems Subjected to Base Excitations

Abstract In this paper, closed-form optimal parameters of inerter-based variant dynamic vibration absorber (variant IDVA) coupled to a primary system subjected to base excitation are derived based on classical fixed-points theory. The proposed variant IDVA is obtained by adding an inerter alone parallel to the absorber damper in the variant dynamic vibration absorber (variant DVA). A new set of optimum frequency and damping ratio of the absorber is derived, thereby resulting in lower maximum amplitude magnification factor than the inerter-based traditional dynamic vibration absorber (traditional IDVA). Under the optimum tuning condition of the absorbers, it is proved both analytically and numerically that the proposed variant IDVA provides a larger suppression of resonant vibration amplitude of the primary system subjected to base excitation. It is demonstrated that adding an inerter alone to the variant DVA provides 19% improvement in vibration suppression than traditional IDVA when the mass ratio is less than 0.2 and the effective frequency bandwidth of the proposed IDVA is wider than the traditional IDVA. The effect of inertance and mass ratio on the amplitude magnification factor of traditional and variant IDVA is also studied.

Exact Optimization of a Three-Element Dynamic Vibration Absorber: Minimization of the Maximum Amplitude Magnification Factor

In this study, the maximum amplitude magnification factor for a linear system equipped with a three-element dynamic vibration absorber (DVA) is exactly minimized for a given mass ratio using a numerical approach. The frequency response curve is assumed to have two resonance peaks, and the parameters for the two springs and one viscous damper in the DVA are optimized by minimizing the resonance amplitudes. The three-element model is known to represent the dynamic characteristics of air-damped DVAs. A generalized optimality criteria approach is developed and adopted for the derivation of the simultaneous equations for this design problem. The solution of the simultaneous equations precisely equalizes the heights of the two peaks in the resonance curve and achieves a minimum amplitude magnification factor. The simultaneous equations are solvable using the standard built-in functions of numerical computing software. The performance improvement of the three-element DVA compared to the standard Voigt type is evaluated based on the equivalent mass ratios. This performance evaluation is highly accurate and reliable because of the precise formulation of the optimization problem. Thus, the advantages of the three-element type DVA have been made clearer.

Design of Multiple-level Parallel Dynamic Vibration Absorber Based on the Flexible Beam

The amplitude magnification factor dimensionless expression is gained through modeling multi-level parallel dynamic vibration absorber mathematical model combined with the impedance of elastic foundation based on the assumption of elastic foundation. It is found that the influence of elastic foundation on the main system cannot be ignored and the necessity is proposed to optimize the multiple-level parallel system combining the advantage of decaying vibration. The influence of the distance u between the connection and the end on the main system is also analyzed: the less u, the smaller the influence of the beam elastic vibration on the main system. The first frequency will not occur coupling vibration with the main system when u  1m, and u=2m has the most influence. The numerical optimization method is used to design the parameters of 2-grade parallel and 4-grade parallel system. The result shows: the result of rigid connects system is not applicable to the elastic system. It will effectively reduce the influence of the beam elastic vibration on the primary system through optimizing the whole system. At last, the connection dynamic stiffness is researched, it shows that the first natural frequency of beam has no effect on the first three nature frequencies when 40 d K  k , the influence can be neglected when 70 d K  k .

二重動吸振器の最大振幅倍率最小化設計（並列型および直列型における最適性規準法の適用）

二重動吸振器の最大振幅倍率最小化設計（並列型および直列型における最適性規準法の適用）

Analytically optimal parameters of fractional-order dynamic vibration absorber

In this paper the optimal parameters of the fractional-order Voigt type dynamic vibration absorber (DVA) are analytically studied for two cases, named as H∞ and H2 optimization criteria. At first the approximately analytical solution is obtained by the averaging method when the primary system is subjected to harmonic excitation. Then the optimal fractional coefficient and order are obtained based on H∞ optimization criterion, which is designed to minimize the maximum amplitude magnification factor of the primary system. Based on H2 optimization criterion, the optimal fractional parameters are obtained to reduce the total vibration energy of the primary system over the whole-frequency range. The comparisons of the approximate solutions with the numerical ones in the two cases are fulfilled, and the results verify that the approximately analytical solutions are correct and satisfactorily precise. At last the control performance of the fractional-order Voigt type DVA is compared with the classical integer-order counterpart, and it could be concluded that the fractional-order DVA has superiority in vibration engineering, and fractional-order element could replace the traditional damper and spring simultaneously in some cases.

Vibration Characteristics of a Mistuned Bladed Disk considering the Effect of Coriolis Forces

To investigate the influence of Coriolis force on vibration characteristics of mistuned bladed disk, a bladed disk with 22 blades is employed and the effects of different rotational speeds and excitation engine orders on the maximum forced response are discussed considering the effects of Coriolis forces. The results show that if there are frequency veering regions, the largest split of double natural frequencies of each modal family considering the effects of Coriolis forces appears at frequency veering region. In addition, the amplitude magnification factor considering the Coriolis effects is increased by 1.02% compared to the system without considering the Coriolis effects as the rotating speed is 3000 rpm, while the amplitude magnification factor is increased by 2.76% as the rotating speed is 10000 rpm. The results indicate that the amplitude magnification factor may be moderately enhanced with the increasing of rotating speed. Moreover, the position of the maximum forced response of bladed disk may shift from one blade to another with the increasing of the rotational speed, when the effects of Coriolis forces are considered.

二重動吸振器の最大振幅倍率最小化設計

The optimal design method of double-mass dynamic vibration absorbers (DVAs) is discussed with respect to the minimization of the maximum amplitude magnification factor. The performance of a double-mass DVA is superior to a single-mass DVA with the same mass ratio, although the design methods are still the subject of studies. The design optimizations of double-mass DVAs that are arranged in parallel or series are formulated precisely using an optimality criteria approach in which the optimal parameters are obtained as the numerical solutions of simultaneous algebraic equations. The primal equations are derived using Vieta's formula with the assumption that the optimal design is realized with three resonant points of equal height. The additional equations are derived as the determinants of Jacobian matrices that are defined using the primal equations. After rearrangement, these formulations realize the direct numerical solution of the design optimization via the solution of simultaneous algebraic equations. Examples are provided that prove the effects of double-mass DVAs. The formulations used in this study are variants of the algebraic approach developed by O. Nishihara that realized the closed-form algebraic exact solutions to a popular design optimization of a single-mass DVA. The well-known design formulae that use the fixed points by J. P. Den Hartog and J. E. Brock correspond to an approximate solution to this problem.

H∞ optimization of dynamic vibration absorber variant for vibration control of damped linear systems

This study focuses on the H∞ optimal design of a dynamic vibration absorber (DVA) variant for suppressing high-amplitude vibrations of damped primary systems. Unlike traditional DVA configurations, the damping element in this type of DVA is connected directly to the ground instead of the primary mass. First, a thorough graphical analysis of the variations in the maximum amplitude magnification factor depending on two design parameters, natural frequency and absorber damping ratios, is performed. The results of this analysis clearly show that any fixed-points-theory-based conventional method could provide, at best, only locally but not globally optimal parameters. Second, for directly handling the H∞ optimization for its optimal design, a novel meta-heuristic search engine, called the diversity-guided cyclic-network-topology-based constrained particle swarm optimization (Div-CNT-CPSO), is developed. The variant DVA system developed using the proposed Div-CNT-CPSO scheme is compared with those reported in the literature. The results of this comparison verified that the proposed system is better than the existing methods for suppressing the steady-state vibration amplitude of a controlled primary system.

Experimental Investigation on Natural Characteristics and Forced Response of Bladed Disks

The results of an experimental investigations on the natural characteristics of tuned bladed disk and forced dynamic responses of mistuned bladed disks are reported. Three experimental bladed disks are discussed: a tuned specimen of periodic symmetry with 12-blades which are nominally identical, and two mistuned specimens, which feature small blade-to-blade variations by adding slight blocks to blade tips. All the specimens are subject to travelling wave excitation produced by piezo-electric actuators sticking on the root of blades. The primary objective of this experiment is to observe the natural characteristics of tuned bladed disk, and to research the impact of mistuning on the forced response blade amplitude magnification. Analytical predictions about the blade amplitude magnification factor are verified by the experimental results.

Mistuning Forced Response Characteristics Analysis of Mistuned Bladed Disks

The problem of determining the worst-case mistuning pattern and robust maximum mistuning forced response of a mistuned bladed rotor is formulated and solved as an optimization problem. This approach is exemplified on a two-degrees-of-freedom per blade disk model, two three-degrees-of-freedom per blade disk models, and a mistuned two-stage bladed rotor. The results of the optimum search of the worst-case mistuning patterns for the lumped parameter models are analyzed, which reveals that the maximum blade forced response in a mistuned bladed disk is associated with mistuning jump, which causes strong localization of the vibration response in a particular blade. The mistuning jump-localization phenomenon has been observed for all of the numerical examples, and it is also demonstrated that the highest response was always experienced by a blade of mistuning value jump. The two- and three-degrees-of-freedom per blade disk models are also for determination of its sensitivity coefficients with respect to mistuning variation. Studies show that there is not a threshold of mistuning beyond which the maximum forced response levels off, or even drops, as the degree of mistuning is increased further. The maximum magnification factor is found to increase as the mistuning level is increased and reaches a maximum value at the upper limit of the mistuning level. The influence of the multistage coupling is revealed by comparing the results of single-stage analysis with that of the multistage case. The computed results have been compared with the Monte Carlo simulation produced, and it is demonstrated that the accuracy and efficiency of the maximum amplitude magnification factor computed by the presented method can be better than that of Monte Carlo simulations.

Closed-Form Solutions to the Exact Optimizations of Dynamic Vibration Absorbers (Minimizations of the Maximum Amplitude Magnification Factors)

A typical design problem for which the fixed-points method was originally developed is that of minimizing the maximum amplitude magnification factor of a primary system by using a dynamic vibration absorber. This is an example of usual cases for which their exact solutions are not obtained by the well-known heuristic approach. In this paper, more natural formulation of this problem is studied, and algebraic closed-form exact solutions to both the optimum tuning ratio and the optimum damping coefficient for this classic problem are derived under assumption of undamped primary system. It is also proven that the minimum amplitude magnification factor, resonance and anti-resonance frequencies are entirely algebraic.

Analytical Solutions to H∞ and H2 Optimization of Dynamic Vibration Absorbers Attached to Damped Linear Systems

AbstractH ∞ and H2 optimization problems of the Voigt type dynamic vibration absorber (DVA) are classical optimization problems, which have been already solved for a special case when the primary system has no damping. However, for the general case including a damped primary system, no one has solved these problems by algebraic approaches. Only the numerical solutions have been proposed until now. This paper presents the analytical solutions for the H∞ and H2 optimization of the DVA attached to the damped primary systems. In the H∞ optimization the DVA is designed such that the maximum amplitude magnification factor of the primary system is minimized; whereas in the H2 optimization the DVA is designed such that the squared area under the response curve of the primary system is minimized. We found a series solution for the H∞ optimization and a closed-form algebraic solution for the H2 optimization. The series solution is then compared with the numerical solution in order to check the accuracy in connection with the truncation error of the series. The exact solution presented in this paper is too complicated to handle by a hand-held calculator, so we proposed an approximate solution for the practical object.

Design of a Dynamic Vibration Absorber for Minimization of Maximum Amplitude Magnification Factor. Derivation of Algebraic Exact Solution.

Fixed points design methods for dynamic vibration absorbers are very well-known in the field of vibration control and commonly applied for practical absorber design. However, they must be understood to be approximation methods from the point of view of their design criterion that the resonance amplitude magnification factor is minimized. In this study, the exact algebraic expressions of optimum tuning and damping parameters for the minimization of the largest resonance amplitude magnification factor of linear dynamic vibration absorbers have been derived for undamped primary systems. The existence of two equal height resonance points is reduced to a multiple root condition of an algebraic equation. The determinant is handled as an equation with respect to the resonance amplitude. It has became very clear that the fixed points theory design by Brock is highly accurate. In particular, it exhibits very small error in the practical mass ratio range, e.g., below unity. Algebraic solutions also exist for the resonance frequencies and the anti-resonance frequency. A numerical extension of the method is introduced in order to investigate the optimization problem for damped primary systems.

Equivalence between Top and Base Excitations of Pillar-type Structures.

Generally, it is difficult to carry out seismic tests of tall pillar-type structures using a vibration testing table, because the table must resist a huge rotating moment, on the structures are too big to mount on the table. In this paper, we discuss an equivalent test using a top excitation method. At resonant or nearly resonant conditions, it is easy to realize equivalent conditions between top and base excitations. At off-resonant conditions, only one among displacement, moment, or shear force can be equivalent, but other components cannot. From the analysis of transient response using seismic wave, impulse wave and sinusoidal wave of three periods at resonant frequency, it is found that nearly equivalent conditions can be realized, because normal modes are excited and are dominant in the response. The amplitude magnification factor at the top of pillar-type structures is usually expressed as m/2ζ, where m≥1. For a uniform beam, m is 1.56. The value of m changes according to the profiles of the structures, such as diameter or cross-sectional width and length at the top and bottom of the structure.

Accumulated damage evaluation for a piping system by the response factor on non-stationary random process. (1st report. A single-degree-of-freedom system).

This paper shows that the average and variance of the accumulated damage caused by earth-quakes on the piping system attached to a building are related to the seismic response factor λ. The earthquakes refered to in this paper are of a non-stationary random process kind. The average is proportional to λ2 and the variance to λ4. The analytical values of the average and variance for a single-degree-of-freedom system are compared with those obtained from computer simulations. Here the model of the building is a single-degree-of-freedom system. Both average of accumulated damage are approximately equal. The variance obtained from the analysis does not coincide with that from simulations. The reason is considered to be the forced vibration by sinusoidal waves, and the sinusoidal waves included random waves. Taking account of amplitude magnification factor, the values of the variance approach those obtained from simulations.

Accumulated damage evaluation for a piping system by the response factor on non-stationary random process (2nd report, A two-degree-offreedom system).

This paper shows that the average and variance of the accumulated damage caused by earth-quakes on the piping system attached to a building are related to the seismic response factor λ. The earthquakes refered to in this paper are of a non-stationary random process kind. The average is proportional to λ2 and the variance to λ4. The analytical values of the average and variance for a single-degree-of-freedom system are compared with those obtained from computer simulations. Here the model of the building is a single-degree-of-freedom system. Both average of accumulated damage are approximately equal. The variance obtained from the analysis does not coincide with that from simulations. The reason is considered to be the forced vibration by sinusoidal waves, and the sinusoidal waves included random waves. Taking account of amplitude magnification factor, the values of the variance approach those obtained from simulations.

Forced torsional vibration of a two degrees of freedom system including clearance and friction. 2nd report Case of stick-slip motion.

The engine-clutch-manual transmission system of an automobile is modelled by a forced torsional vibration system of two degrees of freedom including two nonlinear terms, namely clearance and friction. In the derivation of the analytical solution, the friction term is linearized based upon the approximated solution of motion. The conclusions are as follows : 1) Experiments, computer simulations and the analytical solution show that the stick-slip motion occurs when the friction torque takes a value within certain limits. 2) When the system approaches resonance, the sum of torque impulse becomes large in the system with small friction, but it remains almost unchanged in the system with large friction. 3) It seems that the effects of system parameters on the magnitude of shock can be estimated by the amplitude magnification factor of an approximated single degree of freedom system.