Nonlinear Vibrations of a Beam-Spring-Mass System

1994 ◽  
Vol 116 (4) ◽  
pp. 433-439 ◽  
Author(s):  
M. Pakdemirli ◽  
A. H. Nayfeh
Keyword(s):  
Nonlinear Response ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Partial Differential Equations ◽  
Primary Resonance ◽  
Fast Time ◽  
Lagrange Equations ◽  
Response Curves ◽  
Mass System

The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. The spring-mass system has also a cubic nonlinearity. The response is found by using two different perturbation approaches. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. In the second approach, the Lagrangian is averaged over the fast time scale, and then the equations governing the modulation of the amplitude and phase are obtained as the Euler-Lagrange equations of the averaged Lagrangian. It is shown that the frequency-response and force-response curves depend on the midplane stretching and the parameters of the spring-mass system. The relative importance of these effects depends on the parameters and location of the spring-mass system.

Primary Resonance of the Current-Carrying Beam in Thermal-Magneto-Elasticity Field

2010 ◽  
Vol 29-32 ◽  
pp. 16-21 ◽  
Author(s):  
Xiao Yan Xi ◽  
Zhian Yang ◽  
Li Li Meng ◽  
Chang Jian Zhu
Keyword(s):  
Response Curve ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Response Curves ◽  
Bending Vibration ◽  
Approximation Solution ◽  
System Parameters ◽  
Practical Engineering

On base of the electro-magneto-elastic theory and the theory of the bending vibration of the electric beam, nonlinear vibration equation of current-carrying beam subjected to thermal-magneto-elasticity field is studied. The Lorentz force and thermal force on the beam are derived. According to the method of multiple scales for nonlinear vibrations the approximation solution of the primary resonance of the system is obtained. Numerical analysis results show that the amplitude changed with the system parameters. With the decrease of magnetic intensity, the amplitude increases rapidly. The response curve occurs bending phenomenon and soft features is increased gradually. Increasing current, the amplitudes increase. With the decrease of temperature, the peak of response curves decrease. With the increase of temperature, natural frequency decreased. It is useful in practical engineering.

Effect of Two-to-One Internal Resonances in a Nonlinearly Restrained Fluid Valve

1999 ◽  
Vol 121 (1) ◽  
pp. 59-63 ◽  
Author(s):  
G. Anlas¸
Keyword(s):  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Distributed Parameter ◽  
Parameter System ◽  
Response Curves ◽  
Internal Resonances ◽  
Relief Valve ◽  
Steady State Solutions

The effect of two-to-one internal resonances on the nonlinear response of a pressure relief valve is studied. The fluid valve is modeled as a distributed parameter system at one end and nonlinearly restrained at the other. The method of multiple scales is used to solve the system of partial differential equation and boundary conditions. Frequency-response curves are presented for the primary resonance of either mode in the presence of a two-to-one internal resonance. Stability of the steady-state solutions is investigated. Parameters of the system leading to two-to-one internal resonances are tabulated.

Nonlinear Vibrations of a Beam-Spring-Large Mass System

2017 ◽  
Author(s):  
Mohammad A. Bukhari ◽  
Oumar R. Barry
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Large Mass ◽  
Mode Shapes ◽  
Resonance Excitation ◽  
Response Curves ◽  
Mass System ◽  
Spring System

This paper presents the nonlinear vibration of a simply supported Euler-Bernoulli beam with a mass-spring system subjected to a primary resonance excitation. The nonlinearity is due to the mid-plane stretching and cubic spring stiffness. The equations of motion and the boundary conditions are derived using Hamiltons principle. The nonlinear system of equations are solved using the method of multiple scales. Explicit expressions are obtained for the mode shapes, natural frequencies, nonlinear frequencies, and frequency response curves. The validity of the results is demonstrated via comparison with results in the literature. Exact natural frequencies are obtained for different locations, rotational inertias, and masses.

scholarly journals Performance Analysis of an Electromagnetically Coupled Piezoelectric Energy Scavenger

Energies ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 845 ◽  
Author(s):  
Abdolreza Pasharavesh ◽  
Reza Moheimani ◽  
Hamid Dalir
Keyword(s):  
Energy Harvesting ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equations ◽  
Low Frequency ◽  
Primary Resonance ◽  
Excitation Amplitude ◽  
Load Resistance ◽  
Response Curves ◽  
Piezoelectric Energy ◽  
Resonance Frequencies

The deliberate introduction of nonlinearities is widely used as an effective technique for the bandwidth broadening of conventional linear energy harvesting devices. This approach not only results in a more uniform behavior of the output power within a wider frequency band through bending the resonance response, but also contributes to energy harvesting from low-frequency excitations by activation of superharmonic resonances. This article investigates the nonlinear dynamics of a monostable piezoelectric harvester under a self-powered electromagnetic actuation. To this end, the governing nonlinear partial differential equations of the proposed harvester are order-reduced and solved by means of the perturbation method of multiple scales. The results indicate that, according to the excitation amplitude and load resistance, different responses can be distinguished at the primary resonance. The system behavior may involve the traditional bending of response curves, Hopf bifurcations, and instability regions. Furthermore, an order-two superharmonic resonance is observed, which is activated at lower excitations in comparison to order-three conventional resonances of the Duffing-type resonator. This secondary resonance makes it possible to extract considerable amounts of power at fractions of natural frequency, which is very beneficial in micro-electro-mechanical systems (MEMS)-based harvesters with generally high resonance frequencies. The extracted power in both primary and superharmonic resonances are analytically calculated, then verified by a numerical solution where a good agreement is observed between the results.

The Nonlinear Response of a Simply Supported Rectangular Metallic Plate to Transverse Harmonic Excitation

2000 ◽  
Vol 67 (3) ◽  
pp. 621-626 ◽  
Author(s):  
O. Elbeyli and ◽  
G. Anlas
Keyword(s):  
Critical Points ◽  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Stability Of Solutions ◽  
Metallic Plate ◽  
Simply Supported

In this study, the nonlinear response of a simply supported metallic rectangular plate subject to transverse harmonic excitations is analyzed using the method of multiple scales. Stability of solutions, critical points, types of bifurcation in the presence of a one-to-one internal resonance, together with primary resonance, are determined. [S0021-8936(00)00603-6]

Nonlinear Dynamics of Electrostatically Actuated Coupled MEMS Parallel Cantilever Resonators

2012 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Kyle N. Taylor
Keyword(s):  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Lagrange Equations ◽  
Electrostatically Actuated ◽  
Frequency Responses ◽  
Beam System ◽  
Ground Plate ◽  
Steady State Solutions

This paper deals with the nonlinear response of a coupled cantilever system composed of two micro beams electrostatically actuated. The AC frequency of actuation is near natural frequency of the cantilevers. The two cantilevers are identical. Lagrange equations are used to develop a mathematical model of the system. These equations of motion are nondimensionalized and subjected to the method of multiple scales in order to find steady state solutions. Alternating Current (AC) and Direct Current (DC) actuation voltages are applied between the first cantilever and ground plate with DC voltage applied between the first and second cantilevers. Amplitude-frequency and phase-frequency responses of the system are provided for typical micro beam system structures.

Electrostatically Actuated Coupled MEMS Resonators

2011 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Kyle N. Taylor
Keyword(s):  
Nonlinear Response ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
System Structure ◽  
Lagrange Equations ◽  
Electrostatically Actuated ◽  
Beam System ◽  
Mems Resonators ◽  
The Mathematical Model ◽  
Steady State Solutions

This paper deals with the nonlinear response of an electrostatically actuated cantilever beam system composed of two micro beam resonators near natural frequency. The mathematical model of the system is obtained using Lagrange equations. The equations of motion are nondimensionalized and then the method of multiple scales is used to find steady state solutions. Both AC and DC actuation voltages of the first beam are considered, while the influence on the system of DC on the second beam is explored. Graphical representations of the influence of the detuning parameters are provided for a typical micro beam system structure.

Reduced Order Model for MEMS Cantilever Resonators Under Soft AC Voltage Near Natural Frequency

2012 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Israel Martinez
Keyword(s):  
Natural Frequency ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
First Order ◽  
Electrostatically Actuated

The nonlinear response of an electrostatically actuated cantilever beam microresonator is investigated. The AC voltage is of frequency near resonator’s natural frequency. A first order fringe correction of the electrostatic force and viscous damping are included in the model. The dynamics of the resonator is investigated using the Reduced Order Model (ROM) method, based on Galerkin procedure. Steady-state motions are found. Numerical results for the uniform microresonator are compared with those obtained via the Method of Multiple Scales (MMS).

scholarly journals Nonlinear Vibrations of Functionally Graded Cylindrical Shells: Effect of the Geometry

2012 ◽  
Author(s):  
Matteo Strozzi ◽  
Francesco Pellicano ◽  
Antonio Zippo
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Linear Analysis ◽  
Finite Amplitude ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Displacement Fields

In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The displacement fields are expanded by means of a double mixed series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. In the linear analysis, after spatial discretization, mass and stiff matrices are computed, natural frequencies and mode shapes of the shell are obtained. In the nonlinear analysis, the three displacement fields are re-expanded by using approximate eigenfunctions obtained by the linear analysis; specific modes are selected. The Lagrange equations reduce nonlinear partial differential equations to a set of ordinary differential equations. Numerical analyses are carried out in order to characterize the nonlinear response of the shell. A convergence analysis is carried out to determine the correct number of the modes to be used. The analysis is focused on determining the nonlinear character of the response as the geometry of the shell varies.

On Electrostatically Actuated CNT Bio-Sensors

2011 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Cone S. Salinas Trevino
Keyword(s):  
Carbon Nanotubes ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Van Der Waals ◽  
Primary Resonance ◽  
Mass Detection ◽  
Electrostatically Actuated ◽  
Sensing Applications ◽  
Ground Plate ◽  
Frequency Phase

This paper deals with electrostatically actuated Carbon NanoTubes (CNT) cantilevers for bio-sensing applications. There are three kinds of forces acting on the CNT cantilever: electrostatic, elastostatic, and van der Waals. The van der Waals forces are significant for values of 50 nm or lower of the gap between the CNT and the ground plate. As both forces electrostatic and van der Waals are nonlinear, and the CNT electrostatic actuation is given by AC voltage, the CNT dynamics is nonlinear parametric. The method of multiple scales is used to investigate the system under soft excitations and/or weakly nonlinearities. The frequency-amplitude and frequency-phase behavior are found in the case of primary resonance. The CNT bio-sensor is to be used for mass detection applications.

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