Horizontal Pendulum With Sudden Changes in Platform Tilt

2011 ◽  
Author(s):  
Clark C. McGehee ◽  
Zach C. Ballard ◽  
Brian P. Mann
Keyword(s):  
Analytical Solutions ◽  
Rotational Velocity ◽  
Equations Of Motion ◽  
Phase Correction ◽  
Method Of Averaging ◽  
Equilibrium Conditions ◽  
Ongoing Work ◽  
Approximate Analytical Solutions ◽  
Horizontal Pendulum ◽  
Platform Tilt

The equations of motion for a horizontal pendulum mounted to a rocking platform are developed. Equilibrium conditions are found as a function of the platform tilt angle. The equations are transformed to describe motion about the shifting equilibrium point, and an approximation for the amplitude and phase correction of the solution is developed using the method of averaging. This approximation is used to construct an iterative map that describes the pendulum angle and rotational velocity between each instance of platform rocking. Approximate analytical solutions are compared to numerical and experimental results, and ongoing work is discussed.

scholarly journals Perturbation Solutions For Magnetohydrodynamics (Mhd) Flow of in a Non-Newtonian Fluid Between Concentric Cylinders

2019 ◽  
Vol 24 (1) ◽  
pp. 199-211
Author(s):  
M. Yürüsoy ◽  
Ö.F. Güler
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Variable Viscosity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Mhd Flow ◽  
Model Space ◽  
Viscosity Model ◽  
Concentric Cylinders ◽  
Approximate Analytical Solutions

Abstract The steady-state magnetohydrodynamics (MHD) flow of a third-grade fluid with a variable viscosity parameter between concentric cylinders (annular pipe) with heat transfer is examined. The temperature of annular pipes is assumed to be higher than the temperature of the fluid. Three types of viscosity models were used, i.e., the constant viscosity model, space dependent viscosity model and the Reynolds viscosity model which is dependent on temperature in an exponential manner. Approximate analytical solutions are presented by using the perturbation technique. The variation of velocity and temperature profile in the fluid is analytically calculated. In addition, equations of motion are solved numerically. The numerical solutions obtained are compared with analytical solutions. Thus, the validity intervals of the analytical solutions are determined.

scholarly journals Active Vibration Control of a Nonlinear Beam with Self- and External Excitations

2013 ◽  
Vol 20 (6) ◽  
pp. 1033-1047 ◽  
Author(s):  
J. Warminski ◽  
M. P. Cartmell ◽  
A. Mitura ◽  
M. Bochenski
Keyword(s):  
Analytical Solutions ◽  
Order Approximation ◽  
Equations Of Motion ◽  
Active Vibration Control ◽  
Harmonic Excitation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Beam ◽  
Approximate Analytical Solutions ◽  
Full Structure

An application of the nonlinear saturation control (NSC) algorithm for a self-excited strongly nonlinear beam structure driven by an external force is presented in the paper. The mathematical model accounts for an Euler-Bernoulli beam with nonlinear curvature, reduced to first mode oscillations. It is assumed that the beam vibrates in the presence of a harmonic excitation close to the first natural frequency of the beam, and additionally the beam is self-excited by fluid flow, which is modelled by a nonlinear Rayleigh term for self-excitation. The self- and externally excited vibrations have been reduced by the application of an active, saturation-based controller. The approximate analytical solutions for a full structure have been found by the multiple time scales method, up to the first-order approximation. The analytical solutions have been compared with numerical results obtained from direct integration of the ordinary differential equations of motion. Finally, the influence of a negative damping term and the controller's parameters for effective vibrations suppression are presented.

The Theory and Application of a Classical Eulerian Multibody Code

2005 ◽  
Vol 33 (2) ◽  
pp. 149-176 ◽  
Author(s):  
D. I. M. Forehand ◽  
M. P. Cartmell ◽  
R. Khanin
Keyword(s):  
Analytical Solutions ◽  
Equations Of Motion ◽  
Automatic Generation ◽  
Subject Area ◽  
Physical Problem ◽  
Symbolic Form ◽  
Worked Example ◽  
Approximate Analytical Solutions ◽  
Difficult Subject ◽  
Selection Of

The topic of multibody analysis deals with the automatic generation and subsequent solution of the equations of motion for a system of interconnected bodies. Many academic and industrial computer programs have been developed to carry out this task. However, although some of these programs can obtain the equations of motion in fully symbolic form, it is believed that all the existing programs for multibody analysis solve these equations numerically. The idea behind the research which underpins this paper is to select and implement a symbolic version of an existing multibody algorithm and then to integrate it with a recently developed solver which can obtain approximate analytical solutions to the equations of motion. The present paper deals with the selection of this multibody method. The theory behind the chosen method (the Roberson and Schwertassek algorithm) is described in some detail but also in a much more concise form than can be found in the literature. In addition, a fully worked example of the application of the multibody algorithm to a practical physical problem is given. Such examples are rare in the literature, and so it is intended that this paper can serve as a basis for enhanced didactic practice in this traditionally difficult subject area.

Vibration of Beams with a Single Delamination under Axial Loading

2011 ◽  
Vol 675-677 ◽  
pp. 477-480
Author(s):  
Dong Wei Shu
Keyword(s):  
Free Vibration ◽  
Natural Frequency ◽  
Analytical Solutions ◽  
Axial Load ◽  
Natural Frequencies ◽  
Axial Loading ◽  
Composite Beams ◽  
Mode Shapes ◽  
Equilibrium Conditions ◽  
Monotonic Relation

In this work analytical solutions are developed to study the free vibration of composite beams under axial loading. The beam with a single delamination is modeled as four interconnected Euler-Bernoulli beams using the delamination as their boundary. The continuity and the equilibrium conditions are satisfied between the adjoining beams. The studies show that the sizes and the locations of the delaminations significantly influence the natural frequencies and mode shapes of the beam. A monotonic relation between the natural frequency and the axial load is predicted.

Approximate analytical solutions of stability problems for skew plates under combined loading

2011 ◽  
Vol 54 (2) ◽  
pp. 115-124 ◽  
Author(s):  
N. I. Akishev ◽  
I. I. Zakirov ◽  
V. A. Ivanov ◽  
V. N. Paimushin ◽  
M. A. Shishov
Keyword(s):  
Analytical Solutions ◽  
Combined Loading ◽  
Stability Problems ◽  
Approximate Analytical Solutions
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