The Instability and Vibration of Rotating Beams With Arbitrary Pretwist and an Elastically Restrained Root

2000 ◽  
Vol 68 (6) ◽  
pp. 844-853 ◽  
Author(s):  
S. M. Lin
Keyword(s):  
Differential Equations ◽  
Governing Equation ◽  
Transition Matrix ◽  
Coupling Effect ◽  
General System ◽  
Solution Procedure ◽  
Frequency Equation ◽  
Rotating Speed ◽  
Mass Moment ◽  
Elastically Restrained

The governing differential equations and the boundary conditions for the coupled bending-bending-extensional vibration of a rotating nonuniform beam with arbitrary pretwist and an elastically restrained root are derived by Hamilton’s principle. The semianalytical solution procedure for an inextensional beam without taking account of the coriolis forces is derived. The coupled governing differential equations are transformed to be a vector characteristic governing equation. The frequency equation of the system is derived and expressed in terms of the transition matrix of the vector governing equation. A simple and efficient algorithm for determining the transition matrix of the general system with arbitrary pretwist is derived. The divergence in the Frobenius method does not exist in the proposed method. The frequency relations between different systems are revealed. The mechanism of instability is discovered. The influence of the rotatory inertia, the coupling effect of the rotating speed and the mass moment of inertia, the setting angle, the rotating speed and the spring constants on the natural frequencies, and the phenomenon of divergence instability are investigated.

Bending Vibrations of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root

1994 ◽  
Vol 61 (4) ◽  
pp. 949-955 ◽  
Author(s):  
Sen Yung Lee ◽  
Shueei Muh Lin
Keyword(s):  
Differential Equations ◽  
Coupling Effect ◽  
Fundamental Solutions ◽  
Variable Coefficients ◽  
Bending Vibrations ◽  
Rotating Speed ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Elastically Restrained ◽  
Angle Of Rotation

Without considering the Coriolis force, the governing differential equations for the pure bending vibrations of a rotating nonuniform Timoshenko beam are derived. The two coupled differential equations are reduced into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The explicit relation between the flexural displacement and the angle of rotation due to bending is established. The frequency equations of the beam with a general elastically restrained root are derived and expressed in terms of the four normalized fundamental solutions of the associated governing differential equations. Consequently, if the geometric and material properties of the beam are in polynomial forms, then the exact solution for the problem can be obtained. Finally, the limiting cases are examined. The influence of the coupling effect of the rotating speed and the mass moment of inertia, the setting angle, the rotating speed and taper ratio on the natural frequencies, and the phenomenon of divergence instability (tension buckling) are investigated.

Dynamic Analysis of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root

1999 ◽  
Vol 66 (3) ◽  
pp. 742-749 ◽  
Author(s):  
S. M. Lin
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Differential Equations ◽  
Excitation Frequency ◽  
Solution Procedure ◽  
Variable Coefficients ◽  
The Laplace Transform ◽  
Governing Differential Equation ◽  
Systematic Solution ◽  
Elastically Restrained

A systematic solution procedure for studying the dynamic response of a rotating nonuniform Timoshenko beam with an elastically restrained root is presented. The partial differential equations are transformed into the ordinary differential equations by taking the Laplace transform. The two coupled governing differential equations are uncoupled into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The general solution and the generalized Green function of the uncoupled system are derived. They are expressed in terms of the four corresponding linearly independent homogenous solutions, respectively. The shifting relations of the four homogenous solutions of the uncoupled governing differential equation with constant coefficients are revealed. The generalized Green function of an nth order ordinary differential equation can be obtained by using the proposed method. Finally, the influence of the elastic root restraints, the setting angle, and the excitation frequency on the steady response of a beam is investigated.

scholarly journals Semi-Analytical Solution for Free Vibration Differential Equations of Curved Girders

2017 ◽  
Vol 63 (1) ◽  
pp. 115-132
Author(s):  
Y. Song ◽  
X. Chai
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Free Vibration ◽  
Coupling Effect ◽  
Longitudinal Vibration ◽  
Synthesis Method ◽  
Variable Separation ◽  
Element Analysis ◽  
Plane Vibration ◽  
Curved Girders

Abstract In this paper, a semi-analytical solution for free vibration differential equations of curved girders is proposed based on their mathematical properties and vibration characteristics. The solutions of in-plane vibration differential equations are classified into two cases: one only considers variable separation of non-longitudinal vibration, while the other is a synthesis method addressing both longitudinal and non-longitudinal vibration using Rayleigh’s modal assumption and variable separation method. A similar approach is employed for the out of- plane vibration, but further mathematical operations are conducted to incorporate the coupling effect of bending and twisting. In this case study, the natural frequencies of a curved girder under different boundary conditions are obtained using the two proposed methods, respectively. The results are compared with those from the finite element analysis (FEA) and results show good convergence.

scholarly journals Contact Symmetries of a Model in Optimal Investment Theory

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 217
Author(s):  
Daniel J. Arrigo ◽  
Joseph A. Van de Grift
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Governing Equation ◽  
Optimal Investment ◽  
Original Equation ◽  
Case Scenario ◽  
Investment Theory ◽  
Symmetries Of Differential Equations ◽  
Contact Symmetry ◽  
Terminal Value Problem

It is generally known that Lie symmetries of differential equations can lead to a reduction of the governing equation(s), lead to exact solutions of these equations and, in the best case scenario, lead to a linearization of the original equation. In this paper, we consider a model from optimal investment theory where we show the governing equation possesses an extensive contact symmetry and, through this, we show it is linearizable. Several exact solutions are provided including a solution to a particular terminal value problem.

Optimal Trajectories of Open-Chain Robot Systems: A New Solution Procedure Without Lagrange Multipliers

1998 ◽  
Vol 120 (1) ◽  
pp. 134-136 ◽  
Author(s):  
Sunil K. Agrawal ◽  
Pana Claewplodtook ◽  
Brian C. Fabien
Keyword(s):  
Differential Equations ◽  
Lagrange Multipliers ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Optimal Solution ◽  
Solution Procedure ◽  
Optimal Trajectories ◽  
Point Boundary ◽  
Open Chain ◽  
Robot Systems

For an n d.o.f. robot system, optimal trajectories using Lagrange multipliers are characterized by 4n first-order nonlinear differential equations with 4n boundary conditions at the two end time. Numerical solution of such two-point boundary value problems with shooting techniques is hard since Lagrange multipliers can not be guessed. In this paper, a new procedure is proposed where the dynamic equations are embedded into the cost functional. It is shown that the optimal solution satisfies n fourth-order differential equations. Due to absence of Lagrange multipliers, the two-point boundary-value problem can be solved efficiently and accurately using classical weighted residual methods.

Free Vibration of a Circular Cylindrical Shell Elastically Restrained by Axially Spaced Springs

1983 ◽  
Vol 50 (3) ◽  
pp. 544-548 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
Y. Muramoto
Keyword(s):  
Cylindrical Shell ◽  
Free Vibration ◽  
Circular Cylindrical Shell ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Governing Equations ◽  
Structure Matrix ◽  
The Matrix ◽  
Circular Cylindrical Shells ◽  
Elastically Restrained

An analysis is presented for the free vibration of a circular cylindrical shell restrained by axially spaced elastic springs. The governing equations of vibration of a circular cylindrical shell are written as a coupled set of first-order differential equations by using the transfer matrix of the shell. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrices and the point matrices at the springs, and the frequency equation is derived with terms of the elements of the structure matrix under the boundary conditions. The method is applied to circular cylindrical shells supported by axially equispaced springs of the same stiffness, and the natural frequencies and the mode shapes of vibration are calculated numerically.

scholarly journals Modelling and control of flexible guided lifting system with output constraints and unknown input hysteresis

2019 ◽  
Vol 26 (1-2) ◽  
pp. 112-128
Author(s):  
Naige Wang ◽  
Guohua Cao ◽  
Lei Wang ◽  
Yan Lu ◽  
Zhencai Zhu
Keyword(s):  
Neural Network ◽  
Differential Equation ◽  
Differential Equations ◽  
Governing Equation ◽  
Control Input ◽  
Partial Differential ◽  
Adaptive Neural Network Control ◽  
Output Constraints ◽  
Lifting System ◽  
And Control

Modelling and control vibration is studied for the flexible guided lifting system in the presence of output constraints, input hysteresis, guided rope fault, etc. Flexible guided lifting system, subjected to external disturbances from the boundary disturbance or fluid interaction, is an inherent distributed parameter system with time-varying length and infinite dimensions. According to extended Hamilton’s principle, the governing equation in the form of hybrid partial differential equations and ordinary differential equations is derived to reflect the dynamic response of such multiple ropes under the boundary disturbances and multiple constraints. Adaptive neural network control combining with backstepping technique is subsequently designed to suppress undesirable vibration and stabilise the system, where the neural network is provided as a feedforward compensator for the unknown hysteresis nonlinearities in the control input. Asymptotic stability and uniform boundedness of the system are guaranteed through the Lyapunov theorem, which indicates that all states are uniformly convergent by LaSalle’s invariance principle. The original governing equation is solved numerically by using the finite difference method, where simulation results are illustrated to validate the efficiency of the hybrid partial differential equation and ordinary differential equation model and the adaptive stabilisation design.

scholarly journals Flexural Vibration Test of a Beam Elastically Restrained at One End: A New Approach for Young’s Modulus Determination

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Rafael M. Digilov ◽  
Haim Abramovich
Keyword(s):  
Young Modulus ◽  
Flexural Vibration ◽  
Force Sensor ◽  
Frequency Equation ◽  
Bending Vibrations ◽  
Flexural Vibration Test ◽  
Elastically Restrained ◽  
Fixed End ◽  
Partially Restrained

A new vibration beam technique for the fast determination of the dynamic Young modulus is developed. The method is based on measuring the resonant frequency of flexural vibrations of a partially restrained rectangular beam. The strip-shaped specimen fixed at one end to a force sensor and free at the other forms the Euler Bernoulli cantilever beam with linear and torsion spring on the fixed end. The beam is subjected to free bending vibrations by simply releasing it from a flexural position and its dynamic response detected by the force sensor is processed by FFT analysis. Identified natural frequencies are initially used in the frequency equation to find the corresponding modal numbers and then to calculate the Young modulus. The validity of the procedure was tested on a number of industrial materials by comparing the measured modulus with known values from the literature and good agreement was found.

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