Applications of Holography to Dynamics: High-Frequency Vibrations of Beams

1970 ◽  
Vol 37 (2) ◽  
pp. 287-291 ◽  
Author(s):  
R. Aprahamian ◽  
D. A. Evensen
Keyword(s):  
Cantilever Beam ◽  
Timoshenko Beam ◽  
Holographic Interferometry ◽  
High Frequency ◽  
Beam Theory ◽  
Mode Shapes ◽  
Resonant Frequencies ◽  
Transverse Vibrations ◽  
High Frequency Vibrations ◽  
Time Average

The relatively new experimental technique of holographic interferometry is described, and time-average holography is discussed. Time-average holography has been applied to study high-frequency transverse vibrations of a uniform cantilever beam. Modes from the 5th through the 34th were identified and recorded on holograms, the corresponding resonant frequencies ranged from 500–25,665 cps. In addition, the 77th mode was recorded at 99,395 cps, which demonstrates that the technique is workable at frequencies on the order of 100 kc. The experimental mode shapes and frequencies show good correlation with the Timoshenko beam theory. Other applications of holography to dynamics are briefly discussed.

Applications of Holography to Dynamics: High-Frequency Vibrations of Plates

1970 ◽  
Vol 37 (4) ◽  
pp. 1083-1090 ◽  
Author(s):  
R. Aprahamian ◽  
D. A. Evensen
Keyword(s):  
High Frequency ◽  
Plate Theory ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Classical Plate Theory ◽  
Simply Supported ◽  
Transverse Vibrations ◽  
Shear Effects ◽  
High Frequency Vibrations ◽  
Very High

The techniques of holographic interferometry are applied to study the high-frequency transverse vibrations of a simply supported rectangular plate. Over 110 vibration modes were identified using stored beam holography, at frequencies ranging from 162 cps to 20,000 cps. In addition, three very high modes were identified at frequencies up to 76.8 kcps. The corresponding modal numbers were m = 17, n = 31, for the highest mode identified. Time average holograms were made of these and other modes, and photographs made from the holograms are included herein. The experimental mode shapes and frequencies were generally in agreement with classical plate theory, except for the three highest modes. The latter agreed with Mindlin’s plate theory, which includes rotary inertia and shear effects. Other applications of holography to dynamics are briefly discussed.

scholarly journals Natural Frequency Characteristics of the Beam with Different Cross Sections Considering the Shear Deformation Induced Rotary Inertia

2020 ◽  
Vol 10 (15) ◽  
pp. 5245
Author(s):  
Chunfeng Wan ◽  
Huachen Jiang ◽  
Liyu Xie ◽  
Caiqian Yang ◽  
Youliang Ding ◽  
...  
Keyword(s):  
Shear Deformation ◽  
Cross Sections ◽  
Timoshenko Beam ◽  
High Frequency ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Timoshenko Beam Theory ◽  
Rotary Inertia ◽  
Cross Sectional

Based on the classical Timoshenko beam theory, the rotary inertia caused by shear deformation is further considered and then the equation of motion of the Timoshenko beam theory is modified. The dynamic characteristics of this new model, named the modified Timoshenko beam, have been discussed, and the distortion of natural frequencies of Timoshenko beam is improved, especially at high-frequency bands. The effects of different cross-sectional types on natural frequencies of the modified Timoshenko beam are studied, and corresponding simulations have been conducted. The results demonstrate that the modified Timoshenko beam can successfully be applied to all beams of three given cross sections, i.e., rectangular, rectangular hollow, and circular cross sections, subjected to different boundary conditions. The consequence verifies the validity and necessity of the modification.

scholarly journals Mode shape analysis of multiple cracked functionally graded beam-like structures by using dynamic stiffness method

2017 ◽  
Vol 39 (3) ◽  
pp. 215-228 ◽  
Author(s):  
Tran Van Lien ◽  
Ngo Trong Duc ◽  
Nguyen Tien Khiem
Keyword(s):  
Timoshenko Beam ◽  
Dynamic Stiffness ◽  
Beam Theory ◽  
Functionally Graded ◽  
Theoretical Development ◽  
Mode Shapes ◽  
Functionally Graded Beam ◽  
Stiffness Method ◽  
Dynamic Stiffness Method ◽  
Fgm Beam

Mode shapes of multiple cracked beam-like structures made of Functionally Graded Material (FGM) are analyzed by using the dynamic stiffness method. Governing equations in vibration theory of multiple cracked FGM beam are derived on the base of Timoshenko beam theory; power law variation of material; coupled spring model of crack and taking into account the actual position of neutral axis. A general solution of vibration in frequency domain is obtained and used for constructing dynamic stiffness matrix of the multiple cracked FGM Timoshenko beam element that provides an efficient method for modal analysis of multiple cracked FGM frame structures. The theoretical development is illustrated by numerical analysis of crack-induced change in mode shapes of multi-span continuous FGM beam.

Free Vibration Analysis of Shafts on Resilient Bearings Using Timoshenko Beam Theory

1999 ◽  
Vol 121 (2) ◽  
pp. 256-258 ◽  
Author(s):  
S. Karunendiran ◽  
J. W. Zu
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Frequency Equation ◽  
Timoshenko Beam Theory ◽  
Mode Shapes ◽  
Complex Eigenvalues ◽  
First Time

This paper presents an analytical method adopted for the free vibration analysis of a shaft, both ends of which are supported by resilient bearings. The shaft is modeled by Timoshenko beam theory. Based on this model exact frequency equation to calculate complex eigenvalues is derived and presented in complex compact form for the first time. Explicit expressions to compute the corresponding mode shapes are also presented.

Application of Holography to the Study of Tire Vibration

1973 ◽  
Vol 1 (3) ◽  
pp. 255-266 ◽  
Author(s):  
G. R. Potts
Keyword(s):  
Real Time ◽  
Mode Shapes ◽  
Resonant Frequencies ◽  
Time Average

Abstract Tire vibrations were studied by applying an oscillatory load to bias, belted bias, and radial ply tires in the radial, lateral, and circumferential directions. Resonant frequencies were noted in each type of tire and the corresponding mode shapes observed with both real time and time average holography. The importance of tire vibrations in affecting vehicle ride is noted and the variables affecting these vibrations discussed.

Modeling the tip-sample interaction in atomic force microscopy with Timoshenko beam theory

2013 ◽  
Vol 2 (1) ◽  
Author(s):  
Julio R. Claeyssen ◽  
Teresa Tsukazan ◽  
Leticia Tonetto ◽  
Daniela Tolfo
Keyword(s):  
Atomic Force Microscopy ◽  
Boundary Conditions ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Surface Effects ◽  
Mode Shapes ◽  
Force Microscopy ◽  
Atomic Force ◽  
Matrix Differential ◽  
Micro Cantilever

AbstractA matrix framework is developed for single and multispan micro-cantilevers Timoshenko beam models of use in atomic force microscopy (AFM). They are considered subject to general forcing loads and boundary conditions for modeling tipsample interaction. Surface effects are considered in the frequency analysis of supported and cantilever microbeams. Extensive use is made of a distributed matrix fundamental response that allows to determine forced responses through convolution and to absorb non-homogeneous boundary conditions. Transients are identified from intial values of permanent responses. Eigenanalysis for determining frequencies and matrix mode shapes is done with the use of a fundamental matrix response that characterizes solutions of a damped second-order matrix differential equation. It is observed that surface effects are influential for the natural frequency at the nanoscale. Simulations are performed for a bi-segmented free-free beam and with a micro-cantilever beam actuated by a piezoelectric layer laminated in one side.

scholarly journals 3D FEM Analysis of High-Frequency AlN-Based PMUT Arrays on Cavity SOI

Sensors ◽  
2019 ◽  
Vol 19 (20) ◽  
pp. 4450 ◽  
Author(s):  
Wenjuan Liu ◽  
Leming He ◽  
Xubo Wang ◽  
Jia Zhou ◽  
Weijiang Xu ◽  
...  
Keyword(s):  
High Frequency ◽  
Electrical Impedance ◽  
Fem Simulation ◽  
Laser Doppler Vibrometer ◽  
Fem Analysis ◽  
3D Models ◽  
Passive Layer ◽  
Mode Shapes ◽  
3D Fem ◽  
Resonant Frequencies

This paper presents three-dimensional (3D) models of high-frequency piezoelectric micromachined ultrasonic transducers (PMUTs) based on the finite element method (FEM). These models are verified with fabricated aluminum nitride (AlN)-based PMUT arrays. The 3D numerical model consists of a sandwiched piezoelectric structure, a silicon passive layer, and a silicon substrate with a cavity. Two types of parameters are simulated with periodic boundary conditions: (1) the resonant frequencies and mode shapes of PMUT, and (2) the electrical impedance and acoustic field of PMUT loaded with air and water. The resonant frequencies and mode shapes of an electrically connected PMUT array are obtained with a laser Doppler vibrometer (LDV). The first resonant frequency difference between 3D FEM simulation and the measurement for a 16-MHz PMUT is reasonably within 6%, which is just one-third of that between the analytical method and the measurement. The electrical impedance of the PMUT array measured in air and water is consistent with the simulation results. The 3D model is suitable for predicting electrical and acoustic performance and, thus, optimizing the structure of high-frequency PMUTs. It also has good potential to analyze the transmission and reception performances of a PMUT array for future compact ultrasonic systems.

Local coordinate systems-based method to analyze high-order modes of n-step Timoshenko beam

2016 ◽  
Vol 23 (1) ◽  
pp. 89-102 ◽  
Author(s):  
MS Cao ◽  
W Xu ◽  
Z Su ◽  
W Ostachowicz ◽  
N Xia
Keyword(s):  
Differential Equations ◽  
Timoshenko Beam ◽  
Transverse Vibration ◽  
Dynamic Stiffness ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
High Order ◽  
Mode Shapes ◽  
Coordinate Systems ◽  
K 12

High-frequency transverse vibration of stepped beams has attracted increasing attention in various industrial areas. For an n-step Timoshenko beam, the governing differential equations of transverse vibration have been well established in the literature on the basis of assembling classic Timoshenko beam equations for uniform beam segments. However, solving the governing differential equation has not been resolved well to date, manifested by a computational bottleneck: only the first k modes ( k ≤ 12) are solvable for i-step ( i ≥ 0) Timoshenko beams. This bottleneck diminishes the completeness of stepped Timoshenko beam theory. To address this problem, this study first reveals the root cause of the bottleneck in solving the governing differential equations for high-order modes, and then creates a sophisticated method, based on local coordinate systems, that can overcome the bottleneck to accomplish high-order mode shapes of an n-step Timoshenko beam. The proposed method uses a set of local coordinate systems in place of the conventional global coordinate system to characterize the transverse vibration of an n-step Timoshenko beam. With the method, the local coordinate systems can simplify the frequency equation for the vibration of an n-step Timoshenko beam, making it possible to obtain high-order modes of the beam. The accuracy, capacity, and efficiency of the method based on local coordinate systems in acquiring high-order modes are corroborated using the well-known exact dynamic stiffness method underpinned by the Wittrick-Williams algorithm as a reference. Removal of the bottlenecks in solving the governing differential equations for high-order modes contributes usefully to the completeness of stepped Timoshenko beam theory.

scholarly journals A closed-form solution for free vibration of multiple cracked Timoshenko beam and application

2017 ◽  
Vol 39 (4) ◽  
pp. 315-328
Author(s):  
Nguyen Tien Khiem ◽  
Duong The Hung
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Crack Depth ◽  
Closed Form Solution ◽  
Beam Theory ◽  
Frequency Equation ◽  
Form Solution ◽  
Mode Shapes ◽  
Timoshenko Beams

A closed-form solution for free vibration is constructed and used for obtaining explicit frequency equation and mode shapes of  Timoshenko beams with arbitrary number of cracks. The cracks are represented by the rotational springs of stiffness calculated from the crack depth.  Using the obtained frequency equation, the sensitivity of natural frequencies to crack of the beams is examined in comparison with the  Euler-Bernoulli beams. Numerical results demonstrate that the Timoshenko beam theory is efficiently applicable not only for short or fat beams but also for the long or slender ones. Nevertheless, both the theories are equivalent in sensitivity analysis of fundamental frequency to cracks and they get to be different for higher frequencies.

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