Closed-Form Solutions of Axisymmetrical Transverse Vibrations of Concave Parabolic Thickness Annular Plates

2009 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
Partial Differential ◽  
First Order ◽  
Annular Plates ◽  
Transverse Vibrations

This paper deals with transverse vibrations of axisymmetrical annular plates of concave parabolic thickness. A closed-form solution of the partial differential equation of motion is reported. An approach in which both method of multiple scales and method of factorization have been employed is presented. The method of multiple scales is used to reduce the partial differential equation of motion to two simpler partial differential equations that can be analytically solved. The solutions of the two differential equations are two levels of approximation of the exact solution of the problem. Using the factorization method for solving the first differential equation, which is homogeneous and includes a fourth-order spatial-dependent operator and second-order time-dependent operator, the general solution is obtained in terms of hypergeometric functions. The first diferential equation and the second differential equation (nonhomogeneous) along with the given boundary conditions give so-called zero-order and first-order approximations, respectively, 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 could be further studied using the first-order approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

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.

On Transverse Vibrations of Rectangular Plates of Unidirectional Parabolic Thickness Variation

2005 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Natural Frequencies ◽  
Order Approximation ◽  
Second Order ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Partial Differential ◽  
First Order ◽  
Transverse Vibrations

This paper presents an approach for finding the solution of partial differential equation describing the motion of transverse vibrations of rectangular plates of unidirectional convex parabolic varying thickness. The partial differential equation consists of three operators: fourth-order spatial-dependent, second-order spatial-dependent, and second-order time-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. The first partial differential equation was a homogeneous equation and consisted of two operators, the fourth-order spatial-dependent and second-order time-dependent. Using the factorization method, so-called zero-order approximation of the exact solution has been found. The second partial differential equation was an inhomogeneous equation. Its solution, so-called first-order approximation of the exact solution has been found. This way the first-order approximations of the natural frequencies and mode shapes are found. Various boundary conditions can be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

Transverse Vibrations of Rectangular Plates of Linearly Varying Thickness

2007 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Ion Stroe
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Natural Frequencies ◽  
Order Approximation ◽  
Second Order ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Partial Differential ◽  
First Order ◽  
Transverse Vibrations

This paper presents an approach for finding the solution of partial differential equation describing the motion of transverse vibrations of rectangular plates of unidirectional linear varying thickness. The original partial differential equation consists of three operators: fourth-order spatial-dependent, second-order spatial-dependent, and second-order time-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. The first partial differential equation was a homogeneous equation and consisted of two operators, the fourth-order spatial-dependent and second-order time-dependent. The solution of this equation was found using the factorization method. This solution was zeroth-order approximation of the exact solution. The second partial differential equation was an inhomogeneous equation. The solution of this equation was also found and led to first-order approximation of the exact solution of the original problem. This way the first-order approximations of the natural frequencies and mode shapes are found. Various boundary conditions can be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

Casimir Effect on Voltage Response of Superharmonic Resonance of NEMS Resonators

2017 ◽  
Author(s):  
Martin Botello ◽  
Christian Reyes ◽  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Equation Of Motion ◽  
Second Order ◽  
Mode Shapes ◽  
Voltage Response ◽  
Casimir Forces ◽  
Superharmonic Resonance ◽  
Reference Length

This paper investigates the voltage response of superharmonic resonance of the second order of electrostatically actuated nano-electro-mechanical system (NEMS) resonator sensor. The structure of the NEMS device is a resonator cantilever over a ground plate under Alternating Current (AC) voltage. Superharmonic resonance of second order occurs when the AC voltage is operating in a frequency near-quarter the natural frequency of the resonator. The forces acting on the system are electrostatic, damping and Casimir. To induce a bifurcation phenomenon in superharmonic resonance, the AC voltage is in the category of hard excitation. The gap distance between the cantilever resonator and base plate is in the range of 20 nm to 1 μm for Casimir forces to be present. The differential equation of motion is converted to dimensionless 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 Method of Multiple Scales (MMS) and Reduced Order Model (ROM) are used to model the characteristic of the system. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. ROM, based on the Galerkin procedure, uses the undamped linear mode shapes of the undamped cantilever beam as the basis functions. The influences of parameters (i.e. Casimir, damping, fringe, and detuning parameter) were also investigated.

Effects of Flexural Stiffness, Axial Velocity and Internal Support Locations on Translating Beam’s Natural Frequencies

2014 ◽  
Vol 532 ◽  
pp. 316-319 ◽  
Author(s):  
Ferid Köstekci
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Flexural Rigidity ◽  
Equations Of Motion ◽  
Flexural Stiffness ◽  
Transverse Vibrations ◽  
Internal Support ◽  
Simple Support ◽  
Moving Speed

The aim of this paper is to examine the natural frequencies of beams for different flexural stiffness, internal simple support locations and axial moving speed. In the present investigation, the linear transverse vibrations of an axially translating beam are considered based on Euler-Bernoulli model. The beam is passing through two frictionless guides and has an internal simple support between the guides. The governing differential equations of motion are derived using Hamiltons Principle for two regions of the beam. The method of multiple scales is employed to obtain approximate analytical solution. Some numerical calculations are conducted to present the effects of flexural rigidity, mean translating speed and different internal support locations on natural frequencies.

Nonlinear Free Vibrations of a Timoshenko Beam Using Multiple Scales Method

Volume 2 ◽  
2004 ◽  
Author(s):  
Asghar Ramezani ◽  
Mehrdaad Ghorashi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Free Vibration ◽  
Timoshenko Beam ◽  
Large Amplitude ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Free Vibrations ◽  
Partial Differential ◽  
Cantilevered Beam

In this paper, the large amplitude free vibration of a cantilever Timoshenko beam is considered. To this end, first Hamilton’s principle is used in deriving the partial differential equation of the beam response under the mentioned conditions. Then, implementing the Galerkin’s method the partial differential equation is converted to an ordinary nonlinear differential equation. Finally, the method of multiple scales is used to determine a second order perturbation solution for the obtained ODE. The results show that nonlinearity acts in the direction of increasing the natural frequency of the thick-cantilevered beam.

Nonlinear Dynamics of a Travelling Beam Subjected to Multi-Frequency Parametric Excitation

2014 ◽  
Vol 592-594 ◽  
pp. 2076-2080 ◽  
Author(s):  
Bamadev Sahoo ◽  
L.N. Panda ◽  
Goutam Pohit
Keyword(s):  
Parametric Excitation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Reduced Equations ◽  
Stable Periodic ◽  
Bifurcation And Stability ◽  
First Order Differential Equations

This paper deals with two frequency parametric excitation in presence of internal resonance. The cubic nonlinearity is inserted into the equation of motion by considering the mid-line stretching of the beam. The perturbation method of multiple scales is applied directly to the governing nonlinear fourth order integro-partial differential equation of motion. This leads to a set of first order differential equations known as the reduced equations or normalized reduced equations, which are utilized to determine the additional instability zones, appeared in the trivial state stability plot, the bifurcation and stability of fixed-points, periodic, quasi-periodic, mixed mode and chaotic responses. The transition of system behaviour from stable periodic to unstable chaotic occurs through intermittency route

On Nonlinear Modes of Continuous Systems

1994 ◽  
Vol 116 (1) ◽  
pp. 129-136 ◽  
Author(s):  
A. H. Nayfeh ◽  
S. A. Nayfeh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Continuous Systems ◽  
Partial Differential ◽  
Nonlinear Beam ◽  
Nonlinear Modes ◽  
One Dimensional

We use several methods to study the nonlinear modes of one-dimensional continuous systems with cubic inertia and geometric nonlinearities. Invariant manifold and perturbation methods applied to the discretized system and the method of multiple scales applied to the partial-differential equation and boundary conditions are discussed and their equivalence is demonstrated. The method of multiple scales is then applied directly to the partial-differential equation and boundary conditions governing several nonlinear beam problems.

Effects of Rotary Inertia and Shear Deformation on Nonlinear Vibration of Micro/Nano-Beam Resonators

2005 ◽  
Author(s):  
Asghar Ramezani ◽  
Aria Alasty
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Shear Deformation ◽  
Large Amplitude ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Rotary Inertia ◽  
Partial Differential ◽  
Amplitude Vibration ◽  
Large Amplitude Vibration

In this paper, the large amplitude free vibration of a doubly clamped microbeam is considered. The effects of shear deformation and rotary inertia on the large amplitude vibration of the microbeam are investigated. To this end, first Hamilton’s principle is used in deriving the partial differential equation of the microbeam response under the mentioned conditions. Then, implementing the Galerkin’s method the partial differential equation is converted to an ordinary nonlinear differential equation. Finally, the method of multiple scales is used to determine a second order perturbation solution for the obtained ODE. The results show that nonlinearity acts in the direction of increasing the natural frequency of the doubly clamped microbeam. Shear deformation and rotary inertia have significant effects on the large amplitude vibration of thick and short microbeams.

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.

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