Free-free Boundary Conditions Research Articles

A Study on the Free Vibration of the Prestressed Joined Cylindricalspherical Shell Structures

The free vibration characteristics of the prestressed joined spherical–cylindrical shell with free-free boundary conditions are investigated. The Flügge shell theory and Rayleigh-Ritz energy method are applied in order to analyze the free vibration characteristics of the joined shell. In the modal test, the LMS software is used to calculate mode shapes and natural frequencies of the joined shell structure. The natural frequencies and mode shapes are calculated numerically and they are compared with those of the FEM and modal test to confirm the reliability of the analytical solution. The effects of the shallowness and length of the cylindrical shell to the free vibrational behavior of joined shell structure and the effect of internal pressure on the modal charactristics are investigated.

Dynamic Modulus and Damping Ratio Measurements from Free-Free Resonance and Fixed-Free Resonant Column Procedures

AbstractResonant column procedures may be used to quantify a soil’s shear modulus degradation and damping ratio curves along the small strain (∼10−4 to ∼10−1%) range. However, resonant column procedures often cannot provide measurements of very small strain (<∼10−4%) dynamic mechanical properties. The objective of this study is to determine if free-free resonance (FFR) procedures may be used to provide complementary very small strain, maximum dynamic moduli and minimum damping ratio to small strain shear moduli degradation and damping ratio curves from resonant column testing procedures. A plastic control specimen in six different free-free configurations was used to determine appropriate free-free boundary conditions for axial FFR procedures. Five cohesive soil specimens were then subjected to axial FFR and torsional fixed-free resonant column procedures to evaluate the consistency of the resulting strain-dependent modulus and damping measurements. It can be concluded that: (1) the specific free-free bou...

Identification of inertia properties from the results of output-only modal analysis

Accurate knowledge of the rigid body properties of a structure including the mass, the location of mass center and the moments of inertia is important in machine design, vibration analysis, optimization and modeling of mechanical systems. However, estimation of these properties through theoretical methods is difficult when the structure has a complicated shape. In practice, the inertia properties can be estimated using the conventional modal testing methods by extracting the rigid body modes when the structure is tested in free-free boundary condition. However, all the rigid body modes are not always detectable, due to this fact that the structure is not excited at all degrees of freedom. In order to obtain all of the rigid body modes, many activities have been conducted for selecting the type and location of excitation without much success. In operational modal analysis (OMA), the structure can be excited at any arbitrary point and in different directions. In this paper, a new approach is introduced for estimation of the inertia properties from OMA. The data from OMA are adequate to extract all the rigid body modes of structure. A modal method is used for estimating the inertia properties from the rigid body modes extracted from OMA. The suggested approach is applied to a numerical model of a two-dimensional steel beam as well as a numerical model of a 3D frame and the accuracy of results is evaluated. It is shown that OMA can provide enough data to extract the inertia properties. A real beam is also tested in order to evaluate the performance of the method in practice, needless of a complicated procedure as for conventional methods.

Wave Based Method for Free Vibration Analysis of Ring Stiffened Cylindrical Shell with Intermediate Large Frame Ribs

Wave based method which can be recognized as a semi-analytical and semi-numerical method is presented to analyze the free vibration characteristics of ring stiffened cylindrical shell with intermediate large frame ribs for arbitrary boundary conditions. According to the structure type and the positions of discontinuities, the model is divided into different substructures whose vibration field is expanded by wave functions which are exactly analytical solutions to the governing equations of the motions of corresponding structure type. Boundary conditions and continuity equations between different substructures are used to form the final matrix to be solved. Natural frequencies and vibration mode shapes are calculated by wave based method and the results show good agreement with finite element method for clamped-clamped, shear diaphragm - shear diaphragm and free-free boundary conditions. Free vibration characteristics of ring stiffened cylindrical shells with intermediate large frame ribs are compared with those with bulkheads and those with all ordinary ribs. Effects of the size, the number and the distribution of intermediate large frame rib are investigated. The frame rib which is large enough is playing a role as bulkhead, which can be considered imposing simply supported and clamped constraints at one end of the cabin and dividing the cylindrical shell into several cabins vibrating separately at their own natural frequencies.

Wave Based Method for Free Vibration Analysis of Ring Stiffened Cylindrical Shell with Intermediate Large Frame Ribs

Wave based method which can be recognized as a semi-analytical and semi-numerical method is presented to analyze the free vibration characteristics of ring stiffened cylindrical shell with intermediate large frame ribs for arbitrary boundary conditions. According to the structure type and the positions of discontinuities, the model is divided into different substructures whose vibration field is expanded by wave functions which are exactly analytical solutions to the governing equations of the motions of corresponding structure type. Boundary conditions and continuity equations between different substructures are used to form the final matrix to be solved. Natural frequencies and vibration mode shapes are calculated by wave based method and the results show good agreement with finite element method for clamped-clamped, shear diaphragm – shear diaphragm and free-free boundary conditions. Free vibration characteristics of ring stiffened cylindrical shells with intermediate large frame ribs are compared with those with bulkheads and those with all ordinary ribs. Effects of the size, the number and the distribution of intermediate large frame rib are investigated. The frame rib which is large enough is playing a role as bulkhead, which can be considered imposing simply supported and clamped constraints at one end of the cabin and dividing the cylindrical shell into several cabins vibrating separately at their own natural frequencies.

Electro-thermo nonlocal nonlinear vibration in an embedded polymeric piezoelectric micro plate reinforced by DWBNNTs using DQM

In the present paper, electro-thermo nonlinear vibration of a piezo-polymeric rectangular micro plate made from polyvinylidene fluoride (PVDF) reinforced by zigzag double walled boron nitride nanotubes (DWBNNTs) is studied. This plate is embedded in an elastic medium which is simulated by Winkler and Pasternak foundation models. Using nonlinear strain-displacement relations and nonlocal elasticity plate theory as well as considering charge equation for coupling between electrical and mechanical fields, the motion equations are derived based on energy method and Hamilton’s principle. The differential quadrature method (DQM) is employed to computation of nonlinear frequency for different mechanical and free-free electrical boundary conditions. The results indicate that smart composite and consequently the generated G4 improved sensor and actuator applications in several process industries, because it increases the nonlinear vibration frequency. Furthermore, it can be also found that the nonlinear frequency increases as the values of the elastic medium constants, the geometrical aspect ratios and DWBNNTs volume fraction increase but it decreases as nonlocal parameter increases.

Free and forced nonlinear oscillations of a two-layer composite beam with interface slip

Free and forced flexural nonlinear vibrations of a two-layer beam are investigated. Each beam is assumed to have Euler–Bernoulli kinematics and free-free boundary conditions. The interface allows only nonlinear elastic slip between adjacent sides of the beams, so that the transversal displacement is unique. Free vibrations are considered first by the multiple time scale method, which allows to determine the amplitude dependent nonlinear natural frequencies of the system. It is shown that the nonlinear coefficient of the backbone curve is positive, so that hardening/softening behavior of the interface generates hardening/softening behavior of the whole structure. The modifications of the linear normal modes for moderate excitation amplitudes have been computed. Forced and damped nonlinear oscillations are then considered by the same mathematical method, and the nonlinear frequency response curves are obtained.

An Experimental Approach to Free Vibration Response of Woven Fiber Composite Plates under Free-Free Boundary Condition

The present study involves extensive experimental works to investigate the free vibration of woven fiber Glass/Epoxy composite plates in free-free boundary conditions. The specimens of woven glass fiber and epoxy matrix composite plates are manufactured by the hand-layup technique. Elastic parameters of the plate are also determined experimentally by tensile testing of specimens using Instron 1195. An experimental investigation is carried out using modal analysis technique with Fast Fourier Transform Analyzer, PULSE lab shop, impact hammer and contact accelerometer to obtain the Frequency Response Functions. Also, this experiment is used to validate the results obtained from the FEM numerical analysis based on a first order shear deformation theory. The effects of different geometrical parameters including number of layers, aspect ratio, and fiber orientation of woven fiber composite plates are studied in free-free boundary conditions in details. This study may provide valuable information for researchers and engineers in design applications.

Damage Diagnosis in Self-Compacting Concrete Beams Base on Eigenfrequencies and Mode Shape Derivatives

In this paper, modal characteristics of self-compacting concrete (SCC) beams were studied through experiment. 3 self-compacting concrete beams were gradually damaged and then subjected to vibration test in free-free boundary conditions after each load step. From analysis, eigenfrequencies could indicate the existence of damage but it could not represent the damage locations. Modal Assurance Criterion (MAC) is lesser reliable to indicate crack damage compare to eigenfrequencies. Next to these, capacity for damage detection and localization of Coordinate Modal Assurance Criterion (COMAC) and the flexibility matrix method are also examined and compared. This paper introduces in detail the dynamic properties and damage localization methods in SCC, which provide basal databases of damage diagnosis for SCC structures and promote the popularization of SCC in civil infrastructures.

Interferometric Techniques in Structural Damage Identification

This paper describes several interferometric techniques and their applications in structural damage identification. With that objective in mind, damaged aluminum beams with clamped-free and free-free boundary conditions are analyzed. Different damages cases are inflicted by creating small cuts perpendicular to the beams longitudinal axis, being the damage, therefore, characterized by the dimensions of these cuts. The out-of-plane displacement field of modal response is measured with an electronic speckle pattern interferometric system. The static and dynamic rotation fields, defined as the spatial derivative of the displacement field, are measured with two different speckle shearography systems. Second and third order spatial derivatives of the displacement field, which are related to the bending moment and shear force, are computed by differentiation techniques. The cuts locations are determined by looking for maximum values and/or perturbations of damage indicators based on bending moments and shear forces. This method is validated by comparing its results with numerical ones, from which the most suitable interferometric technique is chosen.

Interferometric techniques in structural damage identification

This paper describes several interferometric techniques and their applications in structural damage identification. With that objective in mind, damaged aluminum beams with clamped-free and free-free boundary conditions are analyzed. Different damages cases are inflicted by creating small cuts perpendicular to the beams longitudinal axis, being the damage, therefore, characterized by the dimensions of these cuts. The out-of-plane displacement field of modal response is measured with an electronic speckle pattern interferometric system. The static and dynamic rotation fields, defined as the spatial derivative of the displacement field, are measured with two different speckle shearography systems. Second and third order spatial derivatives of the displacement field, which are related to the bending moment and shear force, are computed by differentiation techniques. The cuts locations are determined by looking for maximum values and/or perturbations of damage indicators based on bending moments and shear forces. This method is validated by comparing its results with numerical ones, from which the most suitable interferometric technique is chosen.

Investigation on the choice of boundary conditions and shape functions for flexible multi-body system

The objective of this investigation is to examine the correctness and efficiency of the choice of boundary conditions when using assumed mode approach to simulate flexible multi-body systems. The displacement field due to deformation is approximated by the Rayleigh-Ritz assumed modes in floating frame of reference (FFR) formulation. The deformations obtained by the absolute nodal coordinate (ANC) formulation which are transformed by two sets of reference coordinates are introduced as a criterion to verify the accuracy of the simulation results by using the FFR formulation. The relationship between the deformations obtained from different boundary conditions is revealed. Numerical simulation examples demonstrate that the assumed modes with cantilevered-free, simply-supported and free-free boundary conditions without inclusion of rigid body modes are suitable for simulation of flexible multi-body system with large over all motion, and the same physical deformation can be obtained using those mode functions, differ only by a coordinate transformation. It is also shown that when using mode shapes with statically indeterminate boundary conditions, significant error may occur. Furthermore, the slider crank mechanism with rigid crank is accurate enough for investigating boundary condition problem of flexible multi-body system, which cost significant less simulating time. The project was supported by the National Natural Science Foundation of China (10872126) and the Research Fund of the Doctoral Program of Higher Education of China (20100073110007).

Modeling and experimentation for three-dimensional dynamics of endmills

This paper presents a model for the three-dimensional (3D) dynamic response of endmills while considering the actual fluted cross-sectional geometry and pretwisted shape of the tools. The model is solved using the spectral-Tchebychev (ST) technique. The bending and the coupled torsional-axial behavior of four different fluted endmills is compared to finite element model (FEM) predictions and experimental results obtained using modal testing under free-free boundary conditions. For the first eight modes, including six bending and two torsional/axial modes, the difference between the 3D-ST and experimental natural frequencies is shown to be 3% or less for all four tools tested during this study. For the same modes, the 3D-ST and FEM predictions agree to better than 1%. To demonstrate its application, the 3D-ST model for the fluted section of a commercial endmill is coupled to the spindle–holder to predict the tool-point dynamics using receptance coupling substructure analysis (RCSA) with a flexible connection. The coupled model is validated through experiments.

Coupled torsional and longitudinal vibrations of piezoelectric fiber actuator with helical electrodes

The vibration characteristics of a piezoelectric fiber actuator with helical electrodes are studied theoretically and experimentally. Its working principle indicates that the torsional, longitudinal, and tangential deformations of the fiber are coupled. A simplified dynamic model is deduced to investigate the properties of the coupled vibrations and their corresponding equivalent circuits are also provided. Resonant frequencies and mechanical coupling coefficients in free-free boundary condition are calculated. The trends of resonant frequencies as functions of the electrode helical angle and fiber length are discussed and validated in experiments.

Modelling of automotive disc brake squeal and its reduction using rotor design modifications

This paper proposes a numerical approach to reducing disc brake squeal through rotor design modifications using the finite element method (FEM). The finite element (FE) model is validated based on the measured data extracted from experimental modal analysis for individual brake components in free-free boundary condition and brake assembly under applied pressure. The FE model is used to investigate brake squeal through two numerical approaches namely complex eigenvalue analysis and dynamic transient analysis. Experimental squeal tests are performed using a brake test rig for verification of the predicted results. It is observed that the results of both the complex eigenvalue analysis and dynamic transient analysis agree well with experimental squeal frequencies. In order to reduce brake squeal, a number of structural modifications on the disc are evaluated. The predicted results show that the squeal noise of disc brake is influenced by the natural frequency of the brake rotor and its mode shape. It is also found that a good choice of rotor geometry in the pre-design stage could help in reducing squeal noise.

웨이블릿 해석을 이용한 콘크리트의 동탄성계수 추정 및 응용

The dynamic modulus of elasticity of concrete can be determined nondestructively using impact echo test as prescribed in KS F 2437. The fundamental longitudinal frequency of the concrete cylinders with free-free boundary condition was estimated by the wavelet transform theory. The advantage of the wavelet transform over either a pure spectral or temporal decomposition of the signal is that the features of the pertinent signals can be characterized in the time-frequency plane. For the concrete mix design utilized in this study, no significant difference between the dynamic and the static moduli of elasticity was observed. This was contrary to the perceived general notion of having the dynamic modulus considerably higher than the static modulus. It has been shown that the modulus from static and dynamic by impact echo test are comparable to each other fairly well, when the effect of strain level was properly taken into account. In this experimental test, it was shown that the dynamic modulus is approximately equal to the tangent modulus at strain level.

Vibration of snowboard decks

Vibrations of snowboards are closely related to their performance. The aim of this study was to investigate the frequencies of bending and torsional modes, the damping ratios and location of node lines in two boards with different torsional stiffness under free-free boundary conditions with a non-contact laser vibrometer. The frequencies of the first three bending modes were at 16, 37, and 65 Hz. The frequencies of the first three torsional modes were at 30, 54 and 86 Hz in one board, and 10% higher in the 2nd board. The damping ratios of the two boards investigated ranged between 0.3 and 0.6% for bending and between 0.6 and 1% for torsion. The location of the node lines was comparable to a free-free beam with constant cross-section. Vibration analysis should be a standard investigation for benchmarking of snowboards, in addition to mechanical and geometrical parameters.

Effects of Lining Thickness on Squeal in Drum Brake Assembly: Experimental Investigations

This paper presents the effects of brake lining thickness due to wear on drum brake squeal. Brake lining will be worn out and subsequently its thickness will be reduced after a few number of braking applications. Hence dynamic properties of the lining, such as its natural frequency, might be changed. In this work, two different sets of brake lining, i.e., new and worn lining are used to investigate its effect on squeal generation. First, modal testing is performed to determine natural frequencies of those brake linings at free-free boundary condition. Later, squeal tests are carried out using brake dynamometer and squeal frequency is measured up to 10 kHz. Several squeal results are plotted over brake operating conditions to observe the influence of different lining thickness. In addition to these, squeal mechanisms, i.e., modal coupling due to closeness of the natural frequency between drum brake components and negative damping due to negative friction-velocity slope that contribute to the squeal generation are also investigated and discussed.

스마트무인기 기체구조물 지상진동시험

본 논문에서는 스마트 무인기의 자유 진동 특성 및 로터의 회전으로 인한 기체 구조물의 진동특성 즉 강제진동 특성을 실험적으로 규명하기 위해 수행한 시험방법, 센서 및 장비 설치, 시험 결과 검증 방법 및 시험결과를 수록하였다. 스마트 무인기의 지지 조건은 번지코드를 이용하여 자유-자유 경계조건을 구현하였고, 시험은 3개의 가진기를 사용하여 다점 랜덤 가진법으로 구조물을 가진하였으며 약 100여개의 가속도계로부터 기체 구조물의 응답특성을 측정하였다. 주파수 응답함수를 통하여 다기준 최소 자승 복소지수법을 적용하여 고유 진동수, 감쇠율, 모드 형상등의 모달 매개변수를 산출하였다. 또한 강제 진동 시험은 스마트 무인기의 양쪽 로터가 장착되는 나셀 부위에 x,y,z 각 방향으로 가진기를 장착하여 로터 회전 주파수를 가진함으로써 구조물과 각종 장비의 진동응답 특성을 측정하였다. This paper describes the test procedure, instrumentation, verification methodology and the results of the ground vibration test(GVT) and force vibration test(FVT) of the SmartUAV aircraft to estimate experimentally dynamic characteristics of the aircraft. Bungee cords are used to emulate free-free boundary conditions of the test aircraft. The SmartUAV is excited by three shakers and one-hundred frequency response functions(FRF's) is measured. The FRF's are reduced and analyzed to identify the dynamics parameters of the SmartUAV. To extract modal parameters of the SmartUAV such as, natural frequencies and damping ratios, the poly-reference least square complex exponential method is used in the time domain. The mode shape coefficients are estimated with the least squares frequency domain method to identify the vibration modes. The FVT was performed by fixed sine frequency with three shakers on the x, y and z direction and vibration characteristics of structures and detail equipments are measured.

Identification of Damage in a Beam Structure by Using Mode Shape Curvature Squares

During the last decades a great variety of methods have been proposed for damage detection by using the dynamic structure characteristics, however, most of them require modal data of the structure for the healthy state as a reference. In this paper the applicability of the mode shape curvature squares determined from only the damaged state of the structure for damage detection in a beam structure is studied. To establish the method, two aluminium beams containing different-size mill-cut damage at different locations are tested by using the experimentally measured modal data. The experimental modal frequencies and the corresponding mode shapes are obtained by using a scanning laser vibrometer with a PZT actuator. From the mode shapes, mode shape curvatures are obtained by using a central difference approximation. With the example of the beams with free-free and clamped boundary conditions, it is shown that the mode shape curvature squares can be used to detect damage in the structures. Further, the extent of a mill-cut damage is identified via modal frequencies by using a mixed numerical-experimental technique. The method is based on the minimization of the discrepancy between the numerically calculated and experimentally measured frequencies.