Natural Frequency Sensitivity Research Articles

THE TRANSFER MATRIX METHOD FOR MODAL ANALYSIS OF CRACKED MULTISTEP BEAM

The present study addresses the modal analysis of multistep beam with arbitrary number of cracks by using the transfer matrix method and modal testing technique. First, there is conducted general solution of free vibration problem for uniform beam element with arbitrary number of cracks that allows one to simplify the transfer matrix for cracked multistep beam. The transferring beam state needs to undertake only at the steps of beam but not through crack positions. Such simplified the transfer matrix method makes straightforward to investigate effect of cracks mutually with cross-section step in beam on natural frequencies. It is revealed that step-down and step-up in beam could modify notably sensitivity of natural frequencies to crack so that the analysis provides useful indication for crack detection in multistep beam. The proposed theory was validated by an experimental case study

Structure Optimization of Wheel Force Transducer Based on Natural Frequency and Comprehensive Sensitivity

The current research of wheel force transducer (WFT) mainly focuses on test signal processing and decoupling methods based on signal itself, while the WFT structure optimization research related to decreasing the mass and increase the natural frequency and comprehensive sensitivity is not enough. In order to improve the WFT test accuracy, a structure optimization method based on natural frequency and comprehensive sensitivity indicators is put forward. The WFT with 8-beam elastic body is used for the finite element modeling (FEM), in which the following variations are taken into consideration: the connection type of elastic body with modified rim, the number of connection holes, and the respects of strain beam including the shape, the cross sectional area and the length, etc.. The test results shows that the natural frequency of the connecting block type is increased by 65.5% compared with the connecting seat type of elastic body & modified rim, and the main channel sensitivity is improved as well. The results show that the connecting block type will achieve the best comprehensive performance when the number of connecting holes between the elastic body and the modified rim is 20. And the thinner and longer strain beam with smaller cross section area is preferable within the scope of elastic body mechanical strength. This research proposes a novel structure optimization method for WFT which contributes to improve the measurement performance of WFT.

Influence of manufacturing defects on modal properties of composite pyramidal truss-like core sandwich cylindrical panels

Defects can easily appear in composite lattice truss core sandwich structures during the complex preparation process, which may significantly affect the structural response and decrease the load-carrying capability. The purpose of this paper is to investigate the manufacturing defect sensitivity of modal vibration responses of carbon fiber composite pyramidal truss-like core sandwich cylindrical panels by modal experiments and finite element analysis. Defects including debonding between face sheets and truss cores (DFT), truss missing (DTM), face sheet wrinkling (DFW) and gap reinforcing (DGR) are introduced into the present intact specimen artificially and modal testing is conducted to study their dynamic behavior under free-free boundary conditions. Finite element models consistent with the experiments are then developed to further study the effect of defect extents, locations and forms on the modal parameters of the present sandwich cylindrical panels. Results indicate that the degree of sensitivity of natural frequencies of the present sandwich cylindrical panels mainly depends on the vibration modes, defect extents, locations and forms. In addition, damping loss factors are much more sensitive than their corresponding frequencies. Some conclusions and essential mechanisms are summarized, which is helpful to vibration-based non-destructive evaluation (NDE) of such kind of composite lattice sandwich structures.

FREQUENCY ANALYSIS OF CRACKED FUNCTIONALLY GRADED CANTILEVER BEAM

In this paper, a functionally graded cantilever beam with an open crack is investigated on the base of Timoshenko beam theory; power law of functionally graded material (FGM) and taking into account actual position of neutral axis instead of the central one. The open and edge crack is modeled by coupled translational and rotational springs stiffness of which is calculated by the formulas conducted accordingly to fracture mechanics. Using the frequency equation obtained in the framework of the theory natural frequencies of the beam are examined along the crack parameters and material properties. This analysis demonstrates that sensitivity of natural frequencies of FGM beam to crack is strongly dependent on the material constants of FGM

Sensitivity Analysis of Dynamic Characteristics of Composite Beams with Viscoelastic Layers

In this paper, a beam consisting of elastic layers and a viscoelastic layer located between them is considered. The Euler-Bernoulli theory is used to describe the elastic layer and the Timoshenko theory is used to describe the viscoelastic layer. The fractional Zener model is used to express the viscoelastic material properties. The equation of motion of the considered system in the frequency domain is derived using the finite element method, the virtual work principle and the Laplace transform. Eigenproblem is solved by means of the continuation method. The sensitivity analysis is carried out by means of the direct method and allows for determination of sensitivities of eigenvalue and its corresponding eigenvector with respect to the chosen design parameter. Formulae, which enable the calculation of the sensitivities of natural frequencies and non-dimensional damping ratios, are also derived. In order to illustrate the presented method, the numerical example is calculated.

Natural frequency optimization of braided bistable carbon/epoxy tubes: Analysis of braid angles and stacking sequences

The natural frequency of cantilevered bistable carbon/epoxy reeled composite (BRC) slit tubes constructed from combinations of braided and unidirectional (UD) plies is optimized with respect to fiber orientation angles and laminate stacking sequences. BRC tubes have the same geometry as a carpenter’s tape; however, they also have a second stable configuration in the coiled state, and it is considered likely that the coiled state diameter will be fixed by the geometry of the deployment mechanism or its housing. The optimization process uses the BRC coiled diameter as a constraint, and the maximum and minimum physically achievable braid angles as bounds. Both individual tubes, and a simple deployable solar array concept are analyzed. It is observed that the braid angle, rather than ply location in the stack is of greater importance when optimizing long slender or shallow BRCs, whereas both factors must be considered in shorter BRCs. The sensitivity of natural frequency and coiled diameter to braid angle perturbations indicates the importance of precision during manufacture.

Analysis of surface mechanical properties effects on the dynamic characteristics of a circular micro-diaphragm with a distributed mass

The influences of surface mechanical properties effects on the dynamic characteristics of a circular micro-diaphragm with a partially uniformly distributed mass are investigated. Surface mechanical properties effects on two sides are considered to be different as one side of the diaphragm contacts with biochemical media. A size-dependent analytical model is developed based on the thin plate theory and surface elasticity theory for the micro-diaphragm with a partially uniformly distributed mass. The Galerkin procedure is used to solve the governing equation. The size-dependence of the natural frequency and mass sensitivity of the micro-diaphragm with different surface mechanical properties effects is discussed. Results show that the influences of different surface mechanical properties effects on the natural frequency and mass sensitivity are more significant for thinner micro-diaphragms. The influences depend on the differences in the surface mechanical properties effects of the two surfaces.

Nonlinear normal modes for damage detection

The sensitivity of the nonlinear dynamic response to damage is investigated in simply supported beams by computing the nonlinear normal modes (NNMs) and exploiting the rich information they unfold about the nonlinear system behavior. Damage is introduced as a reduction of the flexural stiffness within a small segment of the beam span. The problem is formulated in a piece-wise fashion and analytically tackled by using the method of multiple scales to compute the NNMs and the backbone curves. The latter describe the frequency versus free oscillation amplitude for each mode and as such they represent the skeleton of the nonlinear frequency response around each beam modal frequency. The bending of the backbones is regulated by the so-called effective nonlinearity coefficients associated with each mode. The comparison between the damaged and undamaged beams shows a sensitivity of the effective nonlinearity coefficients higher than the sensitivity of the linear natural frequencies. Moreover, the nonlinear frequency trends unfold an interesting dependence of the nonlinear free response on the stiffness reduction at the damage site. Such dependence yields useful information about the damage position. An effective identification strategy for the damage position is proposed by computing the damage-induced discontinuities in the second derivative of the NNMs (i.e., first order estimate of the bending curvature), without resorting to a baseline model of the undamaged beam. This goal is achieved by computing a high order derivative of the Nadaraya–Watson kernel estimator, which is evaluated from a finite set of sampled beam deflections associated with each individual NNM. The results show that the second derivatives of higher NNMs are more sensitive to damage-induced discontinuities than the linear modal curvatures. The robustness against noise of the damage identification process is also discussed.

Dynamics Simulation Analysis of Column for NC Bevel Gear Grinder

To avoid the influence of column vibration on machining precision, it is necessary to simulate the stiffness characteristics of the columns for CNC grinding machine, and to optimize the structure of the column by using sensitivity method. Based on structural characteristics of the grinding machine and structural characteristics of the columns, a finite element model is established to simulate and calculate the first-four models of the columns. In order to optimize the structure of the column, a sensitivity analysis is made to calculate the sensitivity of natural frequencies and mass of the column to each panel and ribbed plate. Considering a weighting factor and taking integrated natural frequency of the column as an optimization objective, the mass of the column as a constraint, an optimization equation is established. By a quadratic programming method, the parameter optimization of the column is got as follows: the mass decreased by 4%, integrated natural frequency increased by 2%. Dynamic stiffness of the column has been optimized by harmonic response analysis, the results showed that the largest column in the Z resonance peak reduced by 8.44%.

The Research of Straight Bevel Gear Face-Hobbing Method on General Five-Axis Machine Tool

To avoid the influence of column vibration on machining precision, it is necessary to simulate the stiffness characteristics of the columns for CNC grinding machine, and to optimize the structure of the column by using sensitivity method. Based on structural characteristics of the grinding machine and structural characteristics of the columns, a finite element model is established to simulate and calculate the first-four models of the columns. In order to optimize the structure of the column, a sensitivity analysis is made to calculate the sensitivity of natural frequencies and mass of the column to each panel and ribbed plate. Considering a weighting factor and taking integrated natural frequency of the column as an optimization objective, the mass of the column as a constraint, an optimization equation is established. By a quadratic programming method, the parameter optimization of the column is got as follows: the mass decreased by 4%, integrated natural frequency increased by 2%. Dynamic stiffness of the column has been optimized by harmonic response analysis, the results showed that the largest column in the Z resonance peak reduced by 8.44%.

Dynamics of belt-pulley-shaft systems

An analytical model of a pair of pulleys mounted on compliant spinning parallel shafts is developed. These form parts of a belt drive system and are connected by a single belt span, which is modeled as a combination of longitudinal and torsional springs. The pulleys are modeled as rigid cylindrical disks and slippage at the belt-pulley interfaces is ignored. The shafts are modeled as continuous beams having torsional flexibilities as well. The rotation speed is high such that the gyroscopic effect is non-negligible. Natural frequency sensitivity to rotation speed (Campbell diagram) and response due to a sinusoidal excitation force on a shaft is obtained. Modal analysis after reducing the discretized system to a first-order form is used for the response calculation. The study shows that the gyroscopic effect is present even for short lengths of the shafts. Splitting of the natural frequencies as well as the mode shapes is observed at low rotation speeds and critical speeds are observed at high rotation speeds. For coupled frequencies, the response of the system is found to be approximated by modal superposition of a smaller number of modes. The peak responses are explained using the Campbell diagram along with the modal energy plots.

フィードバック制御を用いる絶対変位計に対する出力整形による計測帯域の拡大

One of the authors had proposed a new type of active seismometer measuring such vibration possessing long period and wide dynamic range. In order to lower the natural frequency of the sensor and expand the detectable amplitude, the sensor with the lower natural frequency and lower sensitivity is realized by using feedback control. Based on this idea, authors have been developing an experimental active seismometer with feedback controller. In the previous studies, its performance is reported that the measureable lowest frequency is extended from 4.1 Hz to 0.56 Hz, while the acceptable vibration amplitude is expanded to 141 times bigger than original sensor stroke (or the sensitivity is reduced 43 dB). However, due to the limit of feedback gain, further improvement of the dynamical property solely by using feedback control is not hopeful. In this study, to compensate phase-delay of the sensor output and to expand the dynamic range, third order phase-lag compensator is added to shape physically measured output. Feedback controller and third order phase-lag compensator are designed as analog circuit and built into the sensor. Frequency transfer function of the sensor output is identified through shaking table test. It has been demonstrated that the detectable frequency range of the sensor is extended to 0.15 Hz.

179 固有振動数のおもりに対する感度を利用した両端自由はりの線密度同定

179 固有振動数のおもりに対する感度を利用した両端自由はりの線密度同定

Natural Frequency Sensitivity Analysis of a Road Model Based on the Finite Element Method

Natural Frequency Sensitivity Analysis of a Road Model Based on the Finite Element Method

Theory and FEM simulation study of the double-clamped resonant beam with slit-structure

In this paper, we established the vibration model of double-clamped resonant beam with slit structure, and we theoretically analyzed the effect of rectangular slits with round corners on vibration amplitude and natural frequency of the resonant beam. The effect of rectangular slits on detection sensitivity of resonant beam is also analyzed. According to the theoretical analysis, computational studies of slit size and location influence on vibration amplitude, natural frequency and detection sensitivity of the resonant beam were carried out. Meanwhile, stress concentration of proposed slit structure with rounded corners was calculated by finite element method (FEM) and was compared with the stress concentration of slit with right corners. Finally, for resonant beams with slits of different size and location, theoretical calculation results and the FEM simulation results of natural frequency were compared. Theoretical analysis and FEM simulation are in good agreement.

구조-음향 특성을 고려한 편경의 최적 형상 설계

한국 전통 악기인 편경을 체계적으로 설계하기 위한 최적 형상 설계 알고리즘을 제안한다. 편경은 16 개의 서로 다른 크기의 경편으로 구성되어 있다. 각 경편의 첫 번째 진동 고유 주파수는 전통 소리 튜닝 방법에 의해 결정되는 특정 주파수에 일치하여야 하고, 두 번째와 세 번째 진동 고유 주파수들은 첫 번째 진동 고유 주파수와 단 3 화음 관계(1:1.498:2.378)를 가져야 한다. 제안한 설계 알고리즘의 핵심은 경편의 각 변의 길이 변화에 따른 고유 주파수의 민감도를 구하고, 이를 이용하여 위 두 가지 설계 목적에 부합하는 경편의 치수를 결정하는 것이다. 제안하는 방법으로 5 개의 서로 다른 경편의 형상 치수를 결정하였다. 그리고, 제작 과정에서 발생할 수 있는 오차를 보상하기 위해, 경편의 국부적인 두께 변화에 따른 진동 고유 주파수의 변화에 대해서도 살펴보았다.

Analysis of the natural frequency and phase sensitivity of an optical fiber accelerometer coated with a rigid package

The operation frequency band and the output signal-to-noise ratio of the optical fiber accelerometer are decided by the natural frequency and phase sensitivity of the accelerometer, respectively. Until now, existing theoretical models have been incapable of analyzing the performance of the accelerometer coated with the rigid package. In this paper, an improved model is presented to explain the interreaction between the rigid package and the mass spring assembly inside. The natural frequency and phase sensitivity are affected by both the package mass and the damping of the surrounding medium. With the increase of the package mass, the natural frequency becomes lower and the phase sensitivity becomes higher. The damping of the surrounding medium induces the additional mass added on the package, which increases with the density of the medium. Further, the performance of the three-component vector accelerometer is analyzed by using the improved model. Experiments on the unpackaged and packaged accelerometers in different surrounding mediums indicate that our method is efficient. The improved model is also useful for the traditional piezoelectric accelerometer.

Study on Dynamic Characteristics and Parameter Sensitivity of Three Spans Prestressed Continuous Beam

Established the three spans prestressed continuous beam dynamic equation, obtained analytical expression of frequency equation. To solve the frequency equation for natural frequency, and compared with the finite element numerical analysis results, the frequency both with analytical expression and with finite element are in good agreement. The formula can be obtained accurately the dynamic parameters of three spans prestressed continuous beam such as frequency. At the same time, the paper also studied the natural frequency sensitivity analysis of three spans prestressed continuous beam, and focuses on the impact on the frequency with effective prestress and prestressed eccentricity.

ANALYTICAL MODELING OF PLANETARY GEAR AND SENSITIVITY OF NATURAL FREQUENCIES

This work develops an analytical model of planetary gears and uses it to investigate their natural frequencies and vibration modes. The model admits three planar degrees of freedom for each of the sun, ring, carrier and planets. Vibration modes are classified into rotational, translational and planet modes. The natural frequency sensitivities to system parameters are investigated for tuned (cyclically symmetric) planetary gears. Parameters under consideration include support and mesh stiffnesses, component masses, and moments of inertia. Using the well-defined vibration mode properties of tuned planetary gears, the eigen sensitivities are calculated and expressed in simple exact formulae. These formulae connect natural frequency sensitivity with the modal strain or kinetic energy and provide efficient means to determine the sensitivity to all stiffness and inertia parameters by inspection of the modal energy distribution.

Sensitivity of Natural Frequencies to Composite Effects in Reinforced Concrete Elements

This article attempts to explain the effect of composite action in reinforced concrete (RC) elements on the natural frequencies. Natural frequencies are governed by the relationship between the physical properties of RC elements and its composite action. The softening of concrete in tension, compression, and pre- and post-yielding steel behavior are considered. The bonding action between steel and concrete under loading and unloading conditions is also taken into account. The assumption of the increase in stiffness of RC elements in tension due to bond action is investigated by tests on two RC beams. Both static and modal testing were carried out after inducing different levels of damage in the beams. Results show the validity of the assumption made due to composite action. Mode 1 can be an indicator for bond action behavior and for predicting the yield state of steel in the RC beams. Mode 2 behavior is governed by the amount of steel reinforcement, where for maximum steel ratio it will be affected by concrete softening, while for minimum steel ratio it will be affected by both concrete softening and bond action. Modes 3 and higher can be indicators for concrete softening.