Natural Frequencies and Mode Shapes of Deterministic and Stochastic Non-Homogeneous Rods

2006 ◽  
Vol 5-6 ◽  
pp. 207-216
Author(s):  
S. Nachum ◽  
E. Altus
Keyword(s):  
Grain Size ◽  
Perturbation Method ◽  
Random Field ◽  
Mode Shape ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Homogeneous Random Field ◽  
Geometrical Properties ◽  
Stochastic Case ◽  
Point Correlation

In this study the natural frequencies and mode shapes of the kth order of nonhomogeneous (deterministic and stochastic) rods are found. The solution is based on the functional perturbation method (FPM). The natural frequency and mode shape of the kth order is found analytically to any desired degree of accuracy. In the deterministic it is shown that the FPM accuracy range for the frequency ω and the mode shape is less then 1%. The stochastic case demonstrates the power of this method. The material and geometrical properties will be considered as statistically homogeneous random field with exponential two-point correlation. It is shown that the accuracy depends on the stochastic information used, the correlation distance (roughly the “grain size”), and whether we are interested in the properties of ω or ω2.

Natural Frequencies and Normal Modes of a Spinning Timoshenko Beam With General Boundary Conditions

1992 ◽  
Vol 59 (2S) ◽  
pp. S197-S204 ◽  
Author(s):  
Jean Wu-Zheng Zu ◽  
Ray P. S. Han
Keyword(s):  
Boundary Conditions ◽  
Mode Shape ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Single Mode ◽  
Mode Shapes ◽  
Simulation Studies ◽  
Classical Boundary ◽  
First Time

A free flexural vibrations of a spinning, finite Timoshenko beam for the six classical boundary conditions are analytically solved and presented for the first time. Expressions for computing natural frequencies and mode shapes are given. Numerical simulation studies show that the simply-supported beam possesses very peculiar free vibration characteristics: There exist two sets of natural frequencies corresponding to each mode shape, and the forward and backward precession mode shapes of each set coincide identically. These phenomena are not observed in beams with the other five types of boundary conditions. In these cases, the forward and backward precessions are different, implying that each natural frequency corresponds to a single mode shape.

scholarly journals Damage Identification by Using a Self-Synchronizing Multipoint Laser Doppler Vibrometer

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Chong Yang ◽  
Yu Fu ◽  
Jianmin Yuan ◽  
Min Guo ◽  
Keyu Yan ◽  
...  
Keyword(s):  
Mode Shape ◽  
Natural Frequencies ◽  
Damage Identification ◽  
Reference Beam ◽  
Laser Doppler Vibrometer ◽  
Laser Doppler ◽  
Vibration Measurement ◽  
Mode Shapes ◽  
Interference Signal ◽  
Short Period

The vibration-based damage identification method extracts the damage location and severity information from the change of modal properties, such as natural frequency and mode shape. Its performance and accuracy depends on the measurement precision. Laser Doppler vibrometer (LDV) provides a noncontact vibration measurement of high quality, but usually it can only do sampling on a single point. Scanning LDV is normally used to obtain the mode shape with a longer scanning time. In this paper, a damage detection technique is proposed using a self-synchronizing multipoint LDV. Multiple laser beams with various frequency shifts are projected on different points of the object, reflected and interfered with a common reference beam. The interference signal containing synchronized temporal vibration information of multiple spatial points is captured by a single photodetector and can be retrieved in a very short period. Experiments are conducted to measure the natural frequencies and mode shapes of pre- and postcrack cantilever beams. Mode shape curvature is calculated by numerical interpolation and windowed Fourier analysis. The results show that the artificial crack can be identified precisely from the change of natural frequencies and the difference of mode shape curvature squares.

Influence of Adhesive Dimensions on the Transverse Free Vibration of Single-Lap Adhesive Cantilevered Beams

2011 ◽  
Vol 393-395 ◽  
pp. 149-152
Author(s):  
Bao Ying Xing ◽  
Xiao Cong He ◽  
Mo Sheng Feng
Keyword(s):  
Stress Concentration ◽  
Free Vibration ◽  
Dynamic Response ◽  
Significant Influence ◽  
Mode Shape ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Lap Joint ◽  
Adhesive Layers ◽  
Cantilevered Beams

This paper studies the influence of adhesive dimensions on the transverse free vibration of the single-lap adhesive cantilevered beams. The researches are performed by employing software ansys .Efficient analytic results of natural frequencies and mode shapes of transverse free vibration of the beams are provided, corresponding to different adhesive dimensions of bonded thicknesses and bondlines length. Bondlines length has more significant influence on the transverse natural frequencies and the lap joint’s mode shapes of the beams than bonded thickness. The transverse natural frequencies decrease with a decrease in the bondlines length of adhesive, but do not appear to variation observably with a decrease in the bonded thickness. Bondlines length shorting, the lap joint has a sharper mode shape. Simultaneously, the lap joint of even mode shapes influences the dynamic response of the beams significantly. These results indicate a local crack in adhesive layers because of the existence of stress concentration.

Extraction of Normal Modes from Contaminated Measurement With Noise for Highly Coupled Structures

1996 ◽  
Vol 118 (3) ◽  
pp. 430-435 ◽  
Author(s):  
S. Y. Chen ◽  
M. S. Ju ◽  
Y. G. Tsuei
Keyword(s):  
Normal Mode ◽  
Mode Shape ◽  
Present Method ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Mode Shapes ◽  
Response Functions ◽  
Exact Relation ◽  
Phase Data ◽  
Coupled Structures

A frequency-domain technique to extract the normal mode shapes from the contaminated FRF measurements for highly coupled structures is developed. The relation between the complex FRFs and the normal mode shapes is derived. It is found that the normal mode shape cannot be extracted exactly from the complex mode shape. However, an exact relation between the normal mode shape and the complex FRF does exist. In the present method, only the magnitude and phase data at the undamped natural frequencies are utilized to extract the normal mode shapes. Hence, the effects of measurement noise can be reduced. A numerical example is employed to illustrate the applicability of the technique. The results indicate that this technique can successfully extract the normal modes from the noisy frequency response functions of a highly coupled structure.

Free Vibrations of Symmetric Arch: Boundary Conditions of Stress Resultants Revisited

2020 ◽  
Vol 20 (09) ◽  
pp. 2071008
Author(s):  
Joon Kyu Lee ◽  
Byoung Koo Lee
Keyword(s):  
Boundary Conditions ◽  
Natural Frequency ◽  
Mode Shape ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Mode Shapes ◽  
All Solutions

This paper deals with analyzing free vibrations of the symmetric arch. The boundary conditions of the stress resultants are newly derived, which can be replaced by the conventional boundary conditions of the deflections. All solutions of the natural frequency with the mode shape, using the new boundary conditions, are the same as those of the conventional deflections. The boundary conditions mixed with new and conventional conditions act correctly to calculate natural frequencies. The mode shapes of the stress resultants using the new boundary conditions are reported in two types: symmetric and anti-symmetric modes.

VIBRATION OF DOUBLE BELLOWS TYPE EXPANSION JOINT IN LATERAL AND ROCKING MODES

2006 ◽  
Vol 06 (04) ◽  
pp. 575-588
Author(s):  
M. RADHAKRISHNA ◽  
C. KAMESWARA RAO
Keyword(s):  
Mode Shape ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Expansion Joint ◽  
Modes Of Vibration ◽  
Dimensional Parameters

In this paper, the exact frequency and mode shape expressions are derived for universal bellows type of expansion joint in lateral and rocking modes of vibration. The effect of equivalent support stiffness and mass on the natural frequencies and mode shapes are studied in detail and the results for a range of non-dimensional parameters are presented in graphical forms which should be useful for piping and bellow designers.

Application of Differential Transform Method to Free Vibration of Gabled Frames with Rotational Springs

2017 ◽  
Vol 17 (01) ◽  
pp. 1750012 ◽  
Author(s):  
Mohammad Rezaiee-Pajand ◽  
Ahmad Aftabi Sani ◽  
Seyed Mojtaba Hozhabrossadati
Keyword(s):  
Finite Element Method ◽  
Free Vibration ◽  
Natural Frequencies ◽  
Differential Transform Method ◽  
Mode Shapes ◽  
The Finite Element Method ◽  
Compatibility Conditions ◽  
Transform Method ◽  
Geometrical Properties ◽  
Differential Transform

This paper is concerned with the free vibration of a gabled frame with rotational springs at intersecting joints. Each member of the frame may have different mechanical or geometrical properties. The problem at hand is to solve the gabled frame vibration problem for the natural frequencies by applying the differential transform method (DTM). Special attention is directed towards the derivation of the governing differential equations along with boundary and compatibility conditions of the problem. Some plots of mode shapes are also presented. The problem was also solved by using the finite element method to provide an independent check on the solutions.

scholarly journals Exact Damage Identification of Plate-Like Structure using Mode Shapes

2020 ◽  
Vol 9 (4) ◽  
pp. 1264-1272
Keyword(s):  
Modal Analysis ◽  
Mode Shape ◽  
Natural Frequencies ◽  
Damage Identification ◽  
Mode Shapes ◽  
Absolute Difference ◽  
Virtual Instrumentation ◽  
Crack Orientation ◽  
Labview Software ◽  
Alternative Approaches

In this paper, Mode Shape Based Damaged Detection Technique (MSBDT) has been applied for plate-like structures to recognize the damage location and quantify the damage length. Two alternative approaches are exclusively used to extract damage indexes through mode shapes of undamaged plate (i.e. reference data) and damaged plate. The absolute difference of mode shapes used in first approach and mode shape curvatures used in second approach of undamaged and damaged plates. Healthy Aluminium plate was tested in the laboratory for accurate material properties and considered three different damage cases by changing the crack orientation and location for successfully implementation of above approaches. In order to make certain the sensitivity of the proposed approaches, natural frequencies and corresponding mode shapes for first six modes in transverse direction of a plate are obtained by Finite Element Modal Analysis (FEMA) in ANSYS 18.1 and validated by Experimental Modal Analysis (EMA) in virtual instrumentation environment using LabView software.

Free Vibration Analysis of a Nonlinearly Tapered Cone Beam by Adomian Decomposition Method

2018 ◽  
Vol 18 (07) ◽  
pp. 1850101 ◽  
Author(s):  
Alireza Keshmiri ◽  
Nan Wu ◽  
Quan Wang
Keyword(s):  
Free Vibration ◽  
Mode Shape ◽  
Decomposition Method ◽  
Natural Frequencies ◽  
Adomian Decomposition Method ◽  
Present Approach ◽  
Cone Beam ◽  
Mode Shapes ◽  
Shape Functions ◽  
Adomian Decomposition

In this paper, the free vibration of a nonlinearly tapered cone beam is analyzed based on the Euler–Bernoulli hypothesis. The characteristic/eigenvalue equation and mode shape functions of the nonlinearly tapered cone beam are derived by the Adomian decomposition method for the first time. Using a modified mathematical procedure, the natural frequencies and mode shape functions of a general nonuniform beam are analytically derived. Several numerical examples for the vibration of uniform and linearly tapered cantilever beams are presented and compared with previous results to validate the accuracy and fast convergence of the present approach. The natural frequencies and mode shapes of vibration of exponentially and trigonometrically tapered cone beams with different taper ratios are presented. The present approach enables engineers to analytically analyze tapered beams of nonuniform configurations used as various structural components in a mathematically efficient way.

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