On Geometric Nonlinear Vibrations of Nonuniform Beams

2007 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Multiple Scales ◽  
Rectangular Cross Section ◽  
Constant Width ◽  
Factorization Method ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Bending Vibrations ◽  
Nonlinear Bending ◽  
Nonlinear Modes ◽  
Weakly Nonlinear

Nonlinear bending vibrations in the case of moderately large curvature are reported for a nonuniform cantilever beam of rectangular cross section and a sharp end. This is a beam of constant width and parabolic thickness variation. The method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. The linear modes are obtained in terms of hypergeometric functions by using the factorization method. In the absence of internal resonance (weakly nonlinear systems) the nonlinear modes are taken to be perturbed versions of the linear modes. The nonlinear mode shapes and frequencies of the beam are reported.

On Subharmonic Resonances of Geometric Nonlinear Vibrations of Nonuniform Beams

2008 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Multiple Scales ◽  
Rectangular Cross Section ◽  
Constant Width ◽  
Phase Evolution ◽  
Factorization Method ◽  
Thickness Variation ◽  
Bending Vibrations ◽  
Order Problem ◽  
First Order ◽  
Subharmonic Resonances

Subharmonic resonances of nonlinear forced bending vibrations in the case of moderately large curvature of nonuniform cantilever beams of rectangular cross section and a sharp end are reported. Cantilevers of constant width and parabolic thickness variation are considered in this research. The method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. Two problems, zeroth- and first-order, result. Using factorization method, the linear modes of the zeroth-order problem are obtained in terms of hypergeometric functions. The first-order problem provides the amplitude and phase evolution equation and consequently the regions where subharmonic responses exist.

On Superharmonic Resonances of Nonlinear Nonuniform Beams

2008 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Multiple Scales ◽  
Rectangular Cross Section ◽  
Constant Width ◽  
Phase Evolution ◽  
Factorization Method ◽  
Thickness Variation ◽  
Bending Vibrations ◽  
Order Problem ◽  
First Order ◽  
Paper Method

Superharmonic resonances of nonlinear forced bending vibrations of moderately large curvature of nonuniform cantilever beams of rectangular cross section and a sharp end are reported. Cantilevers of constant width and parabolic thickness variation are considered in this paper. Method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. Two problems, zero- and first-order problem, result. Solving the zero-order problem, the linear modes are obtained in terms of hypergeometric functions by using the factorization method. The first-order problem provides the amplitude and phase evolution equation and consequently the superharmonic frequency response of the nonlinear system.

Simultaneous Resonances of Geometric Nonlinear Nonuniform Beams

2009 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Multiple Scales ◽  
Constant Width ◽  
Phase Evolution ◽  
Factorization Method ◽  
Thickness Variation ◽  
Bending Vibrations ◽  
Order Problem ◽  
Nonlinear Bending ◽  
First Order ◽  
Simultaneous Resonances

Simultaneous resonances, superharmonic and subharmonic, of two-term excitation nonlinear bending vibrations in the case of moderately large curvature of nonuniform cantilever beams are reported. Cantilevers of constant width and parabolic thickness variation are considered in this research. The method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. Two problems, zero- and first-order, result. Using factorization method, the linear modes of the zero-order problem are obtained in terms of hypergeometric functions. The first-order problem provides the amplitude-phase evolution relationship and consequently the simultaneous resonances response.

On Nonlinear Forced Response of Nonuniform Beams

2008 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Multiple Scales ◽  
Rectangular Cross Section ◽  
Constant Width ◽  
Nonlinear Partial Differential Equation ◽  
Primary Resonance ◽  
Single Mode ◽  
Forced Response ◽  
Thickness Variation ◽  
Softening Effect ◽  
Bending Vibration

This paper reports the primary resonance of single mode forced, undamped, bending vibration of nonuniform sharp cantilevers of rectangular cross-section, constant width, and convex parabolic thickness variation. The case of nonlinear curvature, moderately large amplitudes, is considered. The method of multiple scales is applied directly to the nonlinear partial-differential equation of motion and boundary conditions. The frequency-response is analytically determined, and numerical results show a softening effect of the geometrical nonlinearities.

Coupled Bending-Bending Vibrations of Pre-Twisted Cantilever Blading Treated by the Rayleigh-Ritz Energy Method

1968 ◽  
Vol 10 (5) ◽  
pp. 381-388 ◽  
Author(s):  
B. Dawson
Keyword(s):  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Ritz Method ◽  
Computing Time ◽  
Good Choice ◽  
Thickness Ratio ◽  
Mode Shapes ◽  
Characteristic Functions ◽  
Bending Vibrations

The Rayleigh-Ritz method is used to determine the natural frequencies and mode shapes of vibration of pre-twisted rectangular cross-section beams. The method is dependent upon a good choice of approximating functions for the dynamic deflection curves. In the present analysis, series of the characteristic functions representing the normal modes of vibration are taken as the approximating functions for the bending displacements in the directions of the co-ordinate axes. The choice of this particular series leads to a considerable reduction in the number of elements in the final matrix equation and also considerably reduces the computing time. The natural frequencies of vibration are obtained for various width-to-thickness ratio beams with pre-twist angle in the range 0-90°, and the mode shapes of vibration are presented for one particular width to thickness ratio beam. The results are compared to results obtained by other methods and to experimental results, and good agreement is shown to exist.

Crack detection of a beam-like bridge using 3D mode shapes

2014 ◽  
Vol 36 (1) ◽  
pp. 13-25
Author(s):  
Khoa Viet Nguyen
Keyword(s):  
Crack Detection ◽  
Rectangular Cross Section ◽  
Mode Shapes ◽  
Crack Model ◽  
Coupling Mechanism ◽  
Cracked Beam ◽  
Bending Vibrations ◽  
Space Curves ◽  
Beam Height ◽  
Beam Bridge

In this paper, mode shapes of a 3D cracked beam with a rectangular cross section are analyzed for crack detection. The influence of coupling mechanism between horizontal and vertical bending vibrations due to the 3D crack model on the mode shapes is investigated. Due to the coupling mechanism the mode shapes of a beam are twisted in space. They change from plane curves to space curves. This phenomenon can be used for crack detection. The existence of the crack can be detected when the mode shapes are space curves. Also, the mode shapes of a cracked beam bridge have distortions or sharp changes at the crack position. Therefore, the position of the crack can be determined as a position at which the mode shapes exhibit such distortions or sharp changes. Moreover, using the mode shapes in 3D crack model, a crack with depth as small as 1% of the beam height can be detected, while in previous studies using 2D crack model, distortions in the mode shapes caused by a small crack cannot be detected. These results are new and can be used for crack detection of a beam bridge. The stiffness matrix of a 3D cracked element obtained from fracture mechanics is presented and numerical simulations are provided in this paper. 

Nonlinear Normal Modes of a Cantilever Beam

1995 ◽  
Vol 117 (4) ◽  
pp. 477-481 ◽  
Author(s):  
A. H. Nayfeh ◽  
C. Chin ◽  
S. A. Nayfeh
Keyword(s):  
Cantilever Beam ◽  
Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Mode Shapes ◽  
Nonlinear Modes ◽  
Linear Mode ◽  
Complete Set ◽  
Nonlinear Normal

Two approaches for determination of the nonlinear planar modes of a cantilever beam are compared. In the first approach, the governing partial-differential system is discretized using the linear mode shapes and then the nonlinear mode shapes are determined from the discretized system. In the second approach, the boundary-value problem is treated directly by using the method of multiple scales. The results show that both approaches yield the same nonlinear modes because the discretization is performed using a complete set of basis functions, namely, the linear mode shapes.

Electrostatically Actuated MEMS Cantilevers of Linear Thickness Variation

2013 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Mostafa M. Fath El-Den
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Constant Width ◽  
Thickness Variation ◽  
Electrostatically Actuated ◽  
Phase Amplitude ◽  
Mems Cantilevers ◽  
Relationship Of ◽  
Steady State Solutions

This paper deals with nonuniform linear thickness variation and constant width MEMS cantilever resonators electrostatically actuated through AC voltage near half natural frequency. The frequency response of the structure is investigated. Nonlinearities in the system arise from the electrostatic force. The electrostatic actuation introduces parametric coefficients in both linear and nonlinear parts of the governing equation. The method of multiple scales (MMS) is used to obtain the phase-amplitude relationship of the system, and the steady-state solutions. Parameters’ influences are reported.

On Non-Axisymmetrical Transverse Vibrations of Circular Plates of Convex Parabolic Thickness Variation

2004 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Factorization Method ◽  
Mode Shapes ◽  
Circular Plates ◽  
Partial Differential ◽  
First Order

This paper presents an approach for finding the solution of the partial differential equation of motion of the non-axisymmetrical transverse vibrations of axisymmetrical circular plates of convex parabolical thickness. This approach employed both the method of multiple scales and the factorization method for solving the governing partial differential equation. The solution has been assumed to be harmonic angular-dependent. Using the method of multiple scales, the partial differential equation has been reduced to two simpler partial differential equations which can be analytically solved and which represent two levels of approximation. Solving them, the solution resulted as first-order approximation of the exact solution. Using the factorization method, the first differential equation, homogeneous and consisting of fourth-order spatial-dependent and second-order time-dependent operators, led to a general solution in terms of hypergeometric functions. Along with given boundary conditions, the first differential equation and the second differential equation, which was nonhomogeneous, gave respectively so-called zero-order and first-order approximations of the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the first-order approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) With a Gap in Mindlin Plates or Panels

2007 ◽  
Author(s):  
U. Yuceoglu ◽  
O. Gu¨vendik ◽  
V. O¨zerciyes
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Shear Stresses ◽  
Lap Joint ◽  
Bending Vibrations ◽  
Joint System ◽  
Adhesive Layers ◽  
Bonded Joint ◽  
Free Bending

In this present study, the “Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap in Mindlin Plates or Panels” are theoretically analyzed and are numerically solved in some detail. The “plate adherends” and the upper and lower “doubler plates” of the “Bonded Joint” system are considered as dissimilar, orthotropic “Mindlin Plates” joined through the dissimilar upper and lower very thin adhesive layers. There is a symmetrically and centrally located “Gap” between the “plate adherends” of the joint system. In the “adherends” and the “doublers” of the “Bonded Joint” assembly, the transverse shear deformations and the transverse and rotary moments of inertia are included in the analysis. The relatively very thin adhesive layers are assumed to be linearly elastic continua with transverse normal and shear stresses. The “damping effects” in the entire “Bonded Joint” system are neglected. The sets of the dynamic “Mindlin Plate” equations of the “plate adherends”, the “double doubler plates” and the thin adhesive layers are combined together with the orthotropic stress resultant-displacement expressions in a “special form”. This system of equations, after some further manipulations, is eventually reduced to a set of the “Governing System of the First Order Ordinary Differential Equations” in terms of the “state vectors” of the problem. Hence, the final set of the aforementioned “Governing Systems of Equations” together with the “Continuity Conditions” and the “Boundary conditions” facilitate the present solution procedure. This is the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials). The present theoretical formulation and the method of solution are applied to a typical “Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap”. The effects of the relatively stiff (or “hard”) and the relatively flexible (or “soft”) adhesive properties, on the natural frequencies and mode shapes are considered in detail. The very interesting mode shapes with their dimensionless natural frequencies are presented for various sets of boundary conditions. Also, several parametric studies of the dimensionless natural frequencies of the entire system are graphically presented. From the numerical results obtained, some important conclusions are drawn for the “Bonded Joint System” studied here.

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