A Limit Model for the Linear Dynamics of a Two-Layer Beam

2013 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Original Problem ◽  
Natural Frequencies ◽  
Rotational Inertia ◽  
Asymptotic Approach ◽  
Shear Deformations ◽  
Development Method ◽  
First Order ◽  
Limit Model ◽  
Asymptotic Development ◽  
Perfect Adherence

The asymptotic development method is used to study the effects of axial and rotational inertia, of shear deformations and of interface normal compliance in the vibrations of a two-layer beam. Starting from a model with Timoshenko kinematics, a model with Euler-Bernoulli kinematics for each layer, and with normal perfect adherence at interface, is obtained as a limit case. The limit model has only 1 unknown, versus the 8 of the original problem, and has only 2 dimensionless parameters, versus the 14 of the original problem, and thus it is much more easy. The asymptotic approach used to obtain the limit model guarantees on its reliability. Both the limit natural frequencies and their first order corrections are computed; the latter, in particular, permit to determine the sensitivity with respect to the considered parameters.

On Flexural Vibrations of Shear Deformable Laminated Beams

2012 ◽  
Author(s):  
Stefano Lenci ◽  
Francesco Clementi
Keyword(s):  
Mechanical Characteristics ◽  
Natural Frequencies ◽  
Normal Direction ◽  
Rotational Inertia ◽  
Shear Deformations ◽  
Various Boundary Conditions ◽  
Elastic Interface ◽  
Shear Deformable ◽  
Perfect Adherence ◽  
Tangential Directions

The natural frequencies of a two-layer beam with an elastic interface are investigated. Each beam is modeled by the Timoshenko kinematics, and the interface coupling is linear in both normal and tangential directions. Attention is focused on the shear deformation, axial and rotational inertia, and interface normal stiffness. The dependence of the natural frequencies on these mechanical characteristics is investigated by solving the associated eigenvalue problem. The convergence of the solution toward that of a simplified problem obtained by neglecting axial and rotational inertia, shear deformations and by considering interface perfect adherence in the normal direction, is studied. Various boundary conditions are investigated to extend the generality of the proposed results.

scholarly journals Higher-order accurate compact difference solutions for vibration problems of one dimensional continuous systems

2011 ◽  
pp. 771-794
Author(s):  
Mondher Yahiaoui
Keyword(s):  
Natural Frequencies ◽  
Continuous Systems ◽  
One Dimensional ◽  
First Order ◽  
Compact Difference ◽  
Transverse Vibrations ◽  
Seventh Order ◽  
One Step ◽  
Vibration Problems ◽  
Difference Methods

In this paper, we present a fourth-order accurate and a seventh-order accurate, one-step compact difference methods. These methods can be used to solve initial or boundaryvalue problems which can be modeled by a first-order linear system of differential equations. It is then shown in detail how these methods can be used to solve vibration problems of onedimensional continuous systems. Natural frequencies of a cantilever beam in transverse vibrations are computed and the results are compared to analytical ones to prove the high accuracy and efficiency of both methods. A comparison was also made to a finite element solution and the results have shown that both compact-difference methods yield more accurate values even with a reduced number of intervals.

Free Vibrations of Composite Shallow Circular Cylindrical Shell Panels With a Bonded Central Stiffening Shell Strip

2001 ◽  
Author(s):  
U. Yuceoglu ◽  
V. Özerciyes
Keyword(s):  
Shallow Shell ◽  
Natural Frequencies ◽  
Circular Cylindrical Shell ◽  
Adhesive Layer ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Order System ◽  
Theoretical Formulation ◽  
First Order ◽  
Stress Resultant

Abstract This study is concerned with the “Free Vibrations of Composite Shallow Circular Cylindrical Shells or Shell Panels with a Central Stiffening Shell Strip”. The upper and lower shell elements of the stiffened composite system are considered as dissimilar, orthotropic shallow shells. The upper relatively narrow stiffening shell strip is centrally located and adhesively bonded to the lower main shell element In the theoretical formulation, a “First Order Shear Deformation Shell Theory (FSDST)” is employed. The complete set of the shallow shell dynamic equations (including the stress resultant-displacement and the constitutive equations) and the equations of the thin flexible, adhesive layer are first reduced to a set of first order system of ordinary differential equations. This final set forms the governing equations of the problem. Then, they are integrated by means of the “Modified Transfer Matrix Method”. In the adhesive layer, the “hard” and the “soft” adhesive effects are considered. It was found that the material characteristics of the adhesive layer influence the mode shapes and the corresponding natural frequencies of the composite shallow shell panel system. Additionally, some parametric studies on the natural frequencies are presented.

Natural frequencies of parabolic arches with a single crack on opposite cross-section sides

2019 ◽  
Vol 25 (7) ◽  
pp. 1313-1325 ◽  
Author(s):  
U Eroglu ◽  
G Ruta ◽  
E Tufekci
Keyword(s):  
Cross Section ◽  
Natural Vibration ◽  
Natural Frequencies ◽  
Damage Identification ◽  
Crack Opening ◽  
Curved Beams ◽  
Governing Equations ◽  
First Order ◽  
Crack Location ◽  
Original Contribution

We study natural vibration of elastic parabolic arches, modeled as plane curved beams susceptible to elongation, shear, and bending, exhibiting small concentrated cracks. The crack is simulated by springs between regular chunks, with stiffness evaluated following stress concentration in usual crack opening modes. We evaluate and compare the linear dynamic response of the undamaged and damaged arch in nondimensional form. The governing equations are turned into a system of first-order differential equations that are solved numerically by the so-called matricant. The original contribution of this study lies in highlighting the dependence of the variation of the first natural frequencies on the crack location not only along the axis but also on opposite sides of the cross-section. We obtain the relative variations of the first frequencies in terms of the two crack locations. The result of this direct problem provides information on the possibility to detect such locations, and gives indications on structural monitoring and damage identification.

Vibration Characteristics of Straight Blades of Asymmetrical Aerofoil Cross-Section

1969 ◽  
Vol 20 (2) ◽  
pp. 178-190 ◽  
Author(s):  
W. Carnegie ◽  
B. Dawson
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Limited Range ◽  
First Order ◽  
Special Cases ◽  
Theoretical Results ◽  
Theoretical Procedure ◽  
Good Agreement

SummaryTheoretical and experimental natural frequencies and modal shapes up to the fifth mode of vibration are given for a straight blade of asymmetrical aerofoil cross-section. The theoretical procedure consists essentially of transforming the differential equations of motion into a set of simultaneous first-order equations and solving them by a step-by-step finite difference procedure. The natural frequency values are compared with results obtained by an analytical solution and with standard solutions for certain special cases. Good agreement is shown to exist between the theoretical results for the various methods presented. The equations of motion are dependent upon the coordinates of the axis of the centre of flexure of the beam relative to the centroidal axis. The effect of variations of the centre of flexure coordinates upon the frequencies and modal shapes is shown for a limited range of coordinate values. Comparison is made between the theoretical natural frequencies and modal shapes and corresponding results obtained by experiment.

QUANTITATIVE IDENTIFICATION OF ROTOR CRACKS BASED ON FINITE ELEMENT OF B-SPLINE WAVELET ON THE INTERVAL

2009 ◽  
Vol 07 (04) ◽  
pp. 443-457 ◽  
Author(s):  
XUEFENG CHEN ◽  
BING LI ◽  
JIAWEI XIANG ◽  
ZHENGJIA HE
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Beam Element ◽  
Rotational Inertia ◽  
Spline Wavelet ◽  
B Spline ◽  
Rotor Systems ◽  
Crack Location ◽  
Contour Lines ◽  
Quantitative Identification

Based on finite element of B-spline wavelet on the interval (BSWI), the quantitative identification method of transverse crack for rotor systems was studied. The new model of BSWI Rayleigh–Euler rotary beam element considering gyroscopic effect and rotational inertia was constructed to solve the first three natural frequencies of the cracked rotor with high precision, and the first three frequencies solution surfaces of normalized crack location and size were obtained by using surface-fitting technique. Then the first three metrical natural frequencies were employed as inputs of the solution curve surfaces. The intersection of the three frequencies contour lines predicted the normalized crack location and size. The numerical and experimental examples were given to verify the validity of the beam element for crack quantitative identification in rotor systems. The new method can be applied to prognosis and quantitative diagnosis of cracks in the rotor system.

Free Asymmetric Vibrations of Composite Full Circular Cylindrical Shells Stiffened by a Bonded Central Shell Segment

2004 ◽  
Author(s):  
U. Yuceoglu ◽  
V. O¨zerciyes
Keyword(s):  
Cylindrical Shell ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Circular Cylindrical Shell ◽  
Cylindrical Shells ◽  
Adhesive Layer ◽  
Shear Stresses ◽  
First Order ◽  
Stress Resultant ◽  
Circular Cylindrical Shells

This study is concerned with the “Free Asymmetric Vibrations of Composite Full Circular Cylindrical Shells Stiffened by a Bonded Central Shell Segment.” The base shell is made of an orthotropic “full” circular cylindrical shell reinforced and/or stiffened by an adhesively bonded dissimilar, orthotropic “full” circular cylindrical shell segment. The stiffening shell segment is located at the mid-center of the composite system. The theoretical analysis is based on the “Timoshenko-Mindlin-(and Reissner) Shell Theory” which is a “First Order Shear Deformation Shell Theory (FSDST).” Thus, in both “base (or lower) shell” and in the “upper shell” segment, the transverse shear deformations and the extensional, translational and the rotary moments of inertia are taken into account in the formulation. In the very thin and linearly elastic adhesive layer, the transverse normal and shear stresses are accounted for. The sets of the dynamic equations, stress-resultant-displacement equations for both shells and the in-between adhesive layer are combined and manipulated and are finally reduced into a ”Governing System of the First Order Ordinary Differential Equations” in the “state-vector” form. This system is integrated by the “Modified Transfer Matrix Method (with Chebyshev Polynomials).” Some asymmetric mode shapes and the corresponding natural frequencies showing the effect of the “hard” and the “soft” adhesive cases are presented. Also, the parametric study of the “overlap length” (or the bonded joint length) on the natural frequencies in several modes is considered and plotted.

An inertial piezoelectric hybrid actuator with large angular velocity and high resolution

2019 ◽  
Vol 30 (14) ◽  
pp. 2099-2111 ◽  
Author(s):  
Huilu Bao ◽  
Jianming Wen ◽  
Kang Chen ◽  
Jijie Ma ◽  
Dan Lei ◽  
...  
Keyword(s):  
High Resolution ◽  
Angular Velocity ◽  
Natural Frequencies ◽  
Angular Displacement ◽  
High Accuracy ◽  
Laser Vibrometer ◽  
Type B ◽  
First Order ◽  
Offset Distance ◽  
Hybrid Actuator

This article proposes an inertial piezoelectric actuator with hybrid design of asymmetrically clamping structures and a bias unit for the achievement of large angular velocity and high resolution. To investigate the influence of asymmetrical clamp and bias unit on the driving performance, two types of actuators were fabricated and tested. Combined effects from asymmetrical clamp and bias unit contribute to type A, while their subtractive effect is applied to type B. Using a scanning laser vibrometer, experiments were conducted to analyze the characteristics of the angular displacement and corresponding velocity. It is indicated that the measured first-order natural frequencies for above two types are 13.828 and 14.141 Hz, which agrees well with the simulation results of 16.666 and 17.379 Hz, respectively. Besides, compared with the actuators with simple asymmetrical clamping structure or bias unit, this hybrid actuator can obtain an angular velocity 6.87 rad/s at 80 V and 16 Hz and a resolution of 2.80 μrad under a square signal of 20 V and 1 Hz and an offset distance of −22 mm. As a result, the proposed actuators can achieve large angular velocity and high resolution, which is potentially applicable to quick positioning with high accuracy.

Free Vibration of Helicoidal Beams of Arbitrary Shape and Variable Cross Section

2002 ◽  
Vol 124 (3) ◽  
pp. 397-409 ◽  
Author(s):  
Wisam Busool ◽  
Moshe Eisenberger
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Stiffness Matrix ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Rotational Inertia ◽  
Variable Cross ◽  
Dynamic Stiffness Matrix ◽  
Helical Springs

In this study, the dynamic stiffness method is employed for the free vibration analysis of helical springs. This work gives the exact solutions for the natural frequencies of helical beams having arbitrary shapes, such as conical, hyperboloidal, and barrel. Both the cross-section dimensions and the shape of the beam can vary along the axis of the curved member as polynomial expressions. The problem is described by six differential equations. These are second order equations with variable coefficients, with six unknown displacements, three translations, and three rotations at every point along the member. The proposed solution is based on a new finite-element method for deriving the exact dynamic stiffness matrix for the member, including the effects of the axial and the shear deformations and the rotational inertia effects for any desired precision. The natural frequencies are found as the frequencies that cause the determinant of the dynamic stiffness matrix to become zero. Then the mode shape for every natural frequency is found. Examples are given for beams and helical springs with different shape, which can vary along the axis of the member. It is shown that the present numerical results agree well with previously published numerical and experimental results.

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