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.

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.

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 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 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.

Voltage Response of Superharmonic Resonance of Electrostatically Actuated MEMS Resonator Cantilevers Using the Method of Multiple Scales

2016 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christian Reyes
Keyword(s):  
Differential Equation ◽  
Multiple Scales ◽  
Electrostatic Force ◽  
Equation Of Motion ◽  
Zero Order ◽  
Voltage Response ◽  
Order Problem ◽  
Superharmonic Resonance ◽  
First Order ◽  
Mems Resonator

This work investigates the voltage response of superharmonic resonance of second order of electrostatically actuated Micro-Electro-Mechanical Systems (MEMS) resonator cantilevers. The results of this work can be used for mass sensors design. The MEMS device consists of MEMS resonator cantilever over a parallel ground plate (electrode) under Alternating Current (AC) voltage. The AC voltage is of frequency near one fourth of the natural frequency of the resonator which leads to the superharmonic resonance of second order. The AC voltage produces an electrostatic force in the category of hard excitations, i.e. for small voltages the resonance is not present while for large voltages resonance occurs and bifurcation points are born. The forces acting on the resonator are electrostatic and damping. The damping force is assumed linear. The Casimir effect and van der Waals effect are negligible for a gap, i.e. the distance between the undeformed resonator and the ground plate, greater than one micrometer and 50 nanometers, respectively, which is the case in this research. The dimensional equation of motion is nondimensionalized by choosing the gap as reference length for deflections, the length of the resonator for the axial coordinate, and reference time based on the characteristics of the structure. The resulting dimensionless equation includes dimensionless parameters (coefficients) such as voltage parameter and damping parameter very important in characterizing the voltage-amplitude response of the structure. The Method of Multiple Scales (MMS) is used to find a solution of the differential equation of motion. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. In this work, since the structure is under hard excitations the electrostatic force must be in the zero-order problem. The assumption made in this investigation is that the dimensionless amplitudes are under 0.4 of the gap, and therefore all the terms in the Taylor expansion of the electrostatic force proportional to the deflection or its powers are small enough to be in the first-order problem. This way the zero-order problem solution includes the mode of vibration of the structure, i.e. natural frequency and mode shape, resulting from the homogeneous differential equation, as well as particular solutions due to the nonhomogeneous terms. This solution is then used in the first-order problem to find the voltage-amplitude response of the structure. The influences of frequency and damping on the response are investigated. This work opens the door of using smaller AC frequencies for MEMS resonator sensors.

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.

Bending Vibrations of Variable Section Beams

1956 ◽  
Vol 23 (1) ◽  
pp. 103-108
Author(s):  
E. T. Cranch ◽  
Alfred A. Adler
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Cantilever Beams ◽  
Constant Depth ◽  
Bending Vibrations ◽  
Variable Section

Abstract Using simple beam theory, solutions are given for the vibration of beams having rectangular cross section with (a) linear depth and any power width variation, (b) quadratic depth and any power width variation, (c) cubic depth and any power width variation, and (d) constant depth and exponential width variation. Beams of elliptical and circular cross section are also investigated. Several cases of cantilever beams are given in detail. The vibration of compound beams is investigated. Several cases of free double wedges with various width variations are discussed.

First and Second Order Wave-Current Interactions for Floating Bodies

2012 ◽  
Author(s):  
Charles Monroy ◽  
Yann Giorgiutti ◽  
Xiao-Bo Chen
Keyword(s):  
Field Method ◽  
Boundary Integral ◽  
Second Order ◽  
Boundary Integral Method ◽  
Diffraction Radiation ◽  
Floating Bodies ◽  
Order Problem ◽  
First Order ◽  
Wave Drift Damping ◽  
Order Quantities

The influence of current in sea-keeping problems is felt not only for first order quantities such as wave run-ups in front of the structure, but also mainly for second order quantities. In particular, the wave drift damping (which is expressed as the derivative of drift force with respect to the current) is of special interest for mooring systems. The interaction effects of a double-body steady flow on wave diffraction-radiation is studied through a decomposition of the time-harmonic potential into linear and interaction components. A boundary integral method is used to solve the first order problem. Ultimately, a far-field method is proposed to get access to second order drift forces.

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