THE INFLUENCE OF TEMPERATURE ON THE DRIFT OF GYROSCOPE

1995 ◽  
Vol 19 (4) ◽  
pp. 363-382
Author(s):  
Zheng-Ming Ge ◽  
Ming-Yen Liu ◽  
Sann-Chie Lee
Keyword(s):  
Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Approximate Model ◽  
Analytical Approximation ◽  
Precession Angle ◽  
Model Study ◽  
Nutation Angle ◽  
Influence Of Temperature On

The dynamics, especially the drift, of a gyroscope with slowly varying axisymmetrical mass distribution are studied. By the data of temperature field of the inner gimbal, the changes of the position of center of mass and moments of inertia of the inner gimbal of gyro with time are obtained through the finite element method and approximate model study. The drift of a gyroscope can be analytically investigated according to the variation of the nutation angle and the precession angle. Since the temperature varies with time, the equations of motion are nonautonomous nonlinear differential equations. The analytical approximation is obtained by multiple scales method. The nutation angle and the precession angle in some given time intervals are calculated and are compared with the numerical results, obtained by using the MATLAB software package, with satisfactory consistency.

Nonlinear Free Vibration of a Beam With Off-Axis Attached Mass

2013 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Free Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Beam Theory ◽  
Governing Equations ◽  
Attached Mass ◽  
Lumped Mass ◽  
Nonlinear Free Vibration ◽  
Euler Bernoulli Beam

A model of a clamped-clamped beam with an attached lumped mass is presented in this paper. The system is modeled using the Euler-Bernoulli beam theory. In the models presented in literature, it is assumed that the center of mass of the attached mass is located on the neutral axis of the beam. In this paper, this assumption is relaxed. The governing equations of motion are derived. It has been shown that the off-axis center of mass of the attached mass generates an amplitude-dependent transverse force in the beam, which introduces a quadratic nonlinearity. The nonlinear governing equations of motion are solved using the Multiple Scales method. The nonlinear free vibration frequencies are determined.

Nonlinear Dynamics of Rotors due to Large Deformations and Shear Effects

2011 ◽  
Vol 110-116 ◽  
pp. 3593-3599
Author(s):  
Muhammad Rizwan Shad ◽  
Guilhem Michon ◽  
Alain Berlioz
Keyword(s):  
Nonlinear Dynamics ◽  
Large Deformations ◽  
Multiple Scales ◽  
Slenderness Ratio ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Circular Cross Section ◽  
Spring Type ◽  
Shear Effects ◽  
Linear And Nonlinear

An analysis of linear and nonlinear dynamics of rotors is presented taking into account the shear effects. The nonlinearity arises due to the consideration of large deformations in bending. The rotor system studied is composed of a rigid disk and a circular shaft. In order to study the combined effect of rotary inertia and shear effects the shaft is modeled as a Timoshenko beam of circular cross section. A mathematical model is developed consisting of 4th order coupled nonlinear differential equations of motion. Method of multiple scales is used to solve these nonlinear equations. Linear and nonlinear dynamic behavior is studied numerically for different values of slenderness ratio r. Resonant curves are plotted for the nonlinear analysis. Due to nonlinearity these curves are of hard spring type. This spring hardening effect is more visible for lower values of r. Also the nonlinear response amplitude is higher when shear deformations are taken into account.

On nonlinear perturbation analysis of a structure carrying a circular cylindrical liquid tank under horizontal excitation

2018 ◽  
Vol 25 (5) ◽  
pp. 1058-1079 ◽  
Author(s):  
N. K. A. Attari ◽  
F. R. Rofooei ◽  
Z. Waezi
Keyword(s):  
Differential Equations ◽  
Perturbation Analysis ◽  
Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Structural Response ◽  
Single Mode ◽  
Excitation Amplitude ◽  
Combined System ◽  
The Stability

The lateral response of a single degree of freedom structural system containing a rigid circular cylindrical liquid tank under harmonic and earthquake excitations at a 1:2 autoparametric resonance case is considered. The governing nonlinear differential equations of motion for the combined system are solved by means of a multiple scales method considering the first three liquid sloshing modes (1,1), (0,1), and (2,1), under horizontal excitation. The fixed points of the gyroscopic type of governing differential equations are determined and their stability is investigated employing the perturbation method. The obtained results reveal an increase in the stability region for a single-mode response with respect to the excitation amplitude. The saturation phenomenon is observed in the decoupled modes of the system; however, the structural mode and the first anti-symmetric mode of liquid are a combination of the saturated mode and another mode whose scale factor is crucial for the structural response. The results of perturbation analysis are in good agreement with results obtained from numerical methods.

Vibration Response and Parametric Instability in Beams With Combined Quadratic and Cubic Material Nonlinearities

1995 ◽  
Author(s):  
David Chelidze ◽  
Kambiz Farhang ◽  
Tyler J. Selstad
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Response Amplitude ◽  
Nonlinear Material ◽  
Mode Shapes ◽  
Cross Sectional

Abstract Parametric stability in beams with combined quadratic and cubic material nonlinearities is examined. A general mathematical model is developed for parametrically excited beams accounting for their nonlinear material characteristic. Second- and forth-order nonlinear differential equations are found to govern the axial and transverse motions, respectively. Expansions for displacements are assumed in terms of the linear undamped free-oscillation modes. Boundary conditions are applied to the expansions for displacements to determine the mode shapes. Multiplying the equations of motion by the corresponding shape functions, accounting for their orthogonal properties, and integrating over the beam length, a set of coupled nonlinear differential equations in the time-dependent modal coefficients is obtained. Utilizing the method of multiple scales, frequency response as well as response versus excitation amplitude are obtained for two beams of different cross sectional areas. Results are presented for three boundary conditions. It is found that, qualitatively, the response is similar for all the boundary conditions. Quantitative comparison of the cases considered indicate that the highest response amplitude occurs for the cantilever beam with the end mass. The bifurcation points for simply supported beam occur at lower excitation parameter value. It is apparent that more slender columns have larger response amplitude.

Nonlinear Forced Vibration of a Beam-Type Resonator With Attached Eccentric Mass

2014 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Forced Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Nonlinear Resonance ◽  
Beam Theory ◽  
Attached Mass ◽  
Beam Type ◽  
Nonlinear Forced Vibration ◽  
Euler Bernoulli Beam

In this paper, the nonlinear model of an asymmetric micro-bridge resonator with an attached eccentric mass is investigated. The resonator is treated using the Euler-Bernoulli beam theory. The attached mass represents the electrostatic comb-drive actuator in micro-electromechanical applications. The center of mass of the actuator is assumed to be off the neutral axis of the beam. The governing equations of motion are derived assuming that a concentrated harmonic force is applied to the attached mass. The nonlinear forced vibration of the system is studied using the method of multiple scales. It has been demonstrated that the eccentricity of the mass may lead to different types of nonlinear resonance, e.g., superharmonic and internal resonance. The end application of the structure under investigation is in resonant sensing and energy harvesting applications.

Nonlinear Vibrations of Two-Span Composite Laminated Plates with Equal and Unequal Subspan Lengths

2017 ◽  
Vol 9 (6) ◽  
pp. 1485-1505
Author(s):  
Lingchang Meng ◽  
Fengming Li
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Laminated Plates ◽  
Laminated Plate ◽  
Composite Laminated Plate ◽  
Vibration Localization ◽  
Composite Laminated Plates ◽  
Localization Phenomenon ◽  
Harmonic Resonance

AbstractThe nonlinear transverse vibrations of ordered and disordered two-dimensional (2D) two-span composite laminated plates are studied. Based on the von Karman's large deformation theory, the equations of motion of each-span composite laminated plate are formulated using Hamilton's principle, and the partial differential equations are discretized into nonlinear ordinary ones through the Galerkin's method. The primary resonance and 1/3 sub-harmonic resonance are investigated by using the method of multiple scales. The amplitude-frequency relations of the steady-state responses and their stability analyses in each kind of resonance are carried out. The effects of the disorder ratio and ply angle on the two different resonances are analyzed. From the numerical results, it can be concluded that disorder in the length of the two-span 2D composite laminated plate will cause the nonlinear vibration localization phenomenon, and with the increase of the disorder ratio, the vibration localization phenomenon will become more obvious. Moreover, the amplitude-frequency curves for both primary resonance and 1/3 sub-harmonic resonance obtained by the present analytical method are compared with those by the numerical integration, and satisfactory precision can be obtained for engineering applications and the results certify the correctness of the present approximately analytical solutions.

Statics Analysis of the Mechanical Model of Nucleosome

2011 ◽  
Vol 343-344 ◽  
pp. 661-667 ◽  
Author(s):  
Yun Xue ◽  
De Wei Weng ◽  
Gang Ming Gong
Keyword(s):  
Mechanical Model ◽  
The Other ◽  
Elastic Rod ◽  
Equilibrium Equations ◽  
Constraint Force ◽  
Precession Angle ◽  
Nutation Angle ◽  
Analytic Expression ◽  
Set Up ◽  
Calculated Analysis

Mechanical model of nucleoside and its equilibrium equations are set up, and the mechanical properties on the equilibrium position are analyzed. In the case constraint force and electrostatic attraction between cylinder OH and elastic rod are balanced, the analytic expression of nutation angle of the section and its conditions of existence are given. It is show that the cylinder OH can maintain equilibrium at any range of the precession angle. In the other case when unbanced, there is phenomenon of separation of elastic rod from cylinder OH in the spiral wound 2 circles, and numerical solution of the precession angle at separation points are calculated. Analysis of equilibrium of cylinder H1 illustrates that the generatrix of cylinder H1 and OH are not parallel, and the angle between them is obtained

scholarly journals Response of Base-Isolated Liquid Storage Tanks

2004 ◽  
Vol 11 (1) ◽  
pp. 33-45 ◽  
Author(s):  
M.B. Jadhav ◽  
R.S. Jangid
Keyword(s):  
Seismic Response ◽  
Seismic Isolation ◽  
Equations Of Motion ◽  
Approximate Model ◽  
Storage Tanks ◽  
Step Method ◽  
Earthquake Excitation ◽  
System Parameters ◽  
Liquid Storage ◽  
Isolation Systems

Seismic response of liquid storage tanks isolated by elastomeric bearings and sliding system is investigated under real earthquake ground motions. The continuous liquid mass of the tank is modeled as lumped masses known as sloshing mass, impulsive mass and rigid mass. The coupled differential equations of motion of the system are derived and solved in the incremental form using Newmark's step-by-step method with iterations. The seismic response of isolated tank is studied to investigate the comparative effectiveness of various isolation systems. A parametric study is also carried out to study the effect of important system parameters on the effectiveness of seismic isolation for liquid storage tanks. The various important parameters considered are: (i) aspect ratio of the tank and (ii) the time period of the isolation systems. It was observed that both elastomeric and sliding systems are found to be effective in reducing the earthquake forces of the liquid storage tanks. However, the elastomeric bearing with lead core is found to perform better in comparison to other systems. Further, an approximate model is proposed for evaluation of seismic response of base-isolated liquid storage tanks. A comparison of the seismic response evaluated by the proposed approximate method and an exact approach is made under different isolation systems and system parameters. It was observed that the proposed approximate analysis provides satisfactory response estimates of the base-isolated liquid storage tanks under earthquake excitation.

scholarly journals Dynamics of Space Particles and Spacecrafts Passing by the Atmosphere of the Earth

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Antonio Fernando Bertachini de Almeida Prado ◽  
Justyna Golebiewska
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
The Sun ◽  
Three Body Problem ◽  
The Earth ◽  
Before And After ◽  
Three Body ◽  
Body Energy ◽  
Spacecraft Position

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.

An Experimental and Theoretical Investigation of Nonlinear Vibrations of Piezoelectrically-Driven Microcantilevers

2006 ◽  
Author(s):  
S. Nima Mahmoodi ◽  
Nader Jalili
Keyword(s):  
Natural Frequency ◽  
Piezoelectric Layer ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Galerkin Approximation ◽  
Closed Form Solution ◽  
Cubic Form ◽  
Dynamic Representation ◽  
Microcantilever Beam

The nonlinear vibrations of a piezoelectrically-driven microcantilever beam are experimentally and theoretically investigated. A part of the microcantilever beam surface is covered by a piezoelectric layer, which acts as an actuator. Practically, the first resonance of the beam is of interest, and hence, the microcantilever beam is modeled to obtain the natural frequency theoretically. The bending vibrations of the beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties in piezoelectric materials. The nonlinear term appears in the form of quadratic due to presence of piezoelectric layer, and cubic form due to geometry of the beam (mainly due to the beam's inextensibility). Galerkin approximation is utilized to discretize the equations of motion. The obtained equation is simulated to find the natural frequency of the system. In addition, method of multiple scales is applied to the equations of motion to arrive at the closed-form solution for natural frequency of the system. The experimental results verify the theoretical findings very closely. It is, therefore, concluded that the nonlinear approach could provide better dynamic representation of the microcantilever than previous linear models.

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