scholarly journals On the use of variable-separation method for the analysis of vibration problems with time-dependent boundary conditions

2012 ◽  
Vol 226 (12) ◽  
pp. 2912-2924 ◽  
Author(s):  
Josué Aranda-Ruiz ◽  
José Fernández-Sáez
Keyword(s):  
Boundary Conditions ◽  
Solution Method ◽  
Separation Method ◽  
Time Dependent ◽  
Proper Solution ◽  
Variable Separation ◽  
Harmonic Force ◽  
Vibration Problems ◽  
Transversal Vibrations ◽  
Variable Separation Method

In this article, the axial vibrations of a rod with a clamped end and the transversal vibrations of a cantilever beam, both with a time-dependent and non-harmonic force applied on their free ends, are analysed. These are problems in which the traction and the shear, for the rod and the beam, respectively, prescribed in the boundaries of the bodies vary with time. The problems can be solved by the method proposed by Mindlin and Goodman. However, it is usual to solve this problem by the classic variable-separation method (which does not properly fulfil the time-dependent boundary conditions). The displacements and the forces along the systems are derived from both cited methods, and the results are compared. These results highlight the importance of using the proper solution method for the vibration problems with time-dependent boundary conditions.

scholarly journals Exact solutions of time fractional heat-like and wave-like equations with variable coefficients

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 689-693 ◽  
Author(s):  
Sheng Zhang ◽  
Ran Zhu ◽  
Luyao Zhang
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Variable Coefficients ◽  
Separation Method ◽  
Variable Separation ◽  
Science And Engineering ◽  
Special Solutions ◽  
Initial And Boundary Conditions ◽  
Variable Separation Method

In this paper, a variable-coefficient time fractional heat-like and wave-like equation with initial and boundary conditions is solved by the use of variable separation method and the properties of Mittag-Leffler function. As a result, exact solutions are obtained, from which some known special solutions are recovered. It is shown that the variable separation method can also be used to solve some others time fractional heat-like and wave-like equation in science and engineering.

scholarly journals Variable separation for time fractional advection-dispersion equation with initial and boundary conditions

2016 ◽  
Vol 20 (3) ◽  
pp. 789-792 ◽  
Author(s):  
Sheng Zhang ◽  
Mingying Liu ◽  
Luyao Zhang
Keyword(s):  
Boundary Conditions ◽  
Dispersion Equation ◽  
Separation Method ◽  
Mathematical Tool ◽  
Variable Separation ◽  
Advection Dispersion ◽  
Transfer Equations ◽  
Initial And Boundary Conditions ◽  
Variable Separation Method ◽  
Explicit Exact Solution

In this paper, variable separation method combined with the properties of Mittag-Leffler function is used to solve a variable-coefficient time fractional advection-dispersion equation with initial and boundary conditions. As a result, a explicit exact solution is obtained. It is shown that the variable separation method can provide a useful mathematical tool for solving the time fractional heat transfer equations.

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

Vibration of Beam-Mass Systems With Time-Dependent Boundary Conditions

1959 ◽  
Vol 26 (3) ◽  
pp. 353-356
Author(s):  
T. C. Yen ◽  
S. Kao
Keyword(s):  
Boundary Conditions ◽  
Laplace Transforms ◽  
Time Dependent ◽  
Simply Supported ◽  
Vibration Problems ◽  
Fixed End

Abstract Vibration problems of beams with time-dependent boundary conditions are solved by the method of Laplace transforms. The problems treated include the simply supported, cantilevered, and fixed beams carrying a single arbitrarily placed mass; two equal masses symmetrically placed on a simply supported or fixed-end beam and a beam carrying a mass at the center and two equal masses at tips, striking a spring.

scholarly journals Load Localization and Reconstruction Using a Variable Separation Method

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Jing Zhang ◽  
Fang Zhang ◽  
Jinhui Jiang
Keyword(s):  
Impulse Response ◽  
Time History ◽  
Separation Method ◽  
Location Information ◽  
Engineering Practice ◽  
Variable Separation ◽  
Response Matrix ◽  
Optimization Function ◽  
Load Location ◽  
Variable Separation Method

Load identification is very important in engineering practice. In this paper, a novel method for load reconstruction and localization is proposed. In the traditional load localization method, location information is coupled to the impulse response matrix. The inversion of the impulse response matrix leads the process of load localization to be time-consuming. So we propose a variable separation method to separate the load location information from the impulse response matrix. An error optimization function of load histories in different modes is employed to determine the true load location. After locating the external load, the load time history can be easily reconstructed by the measurement responses and determinate impulse response matrix. This method is verified by simulations of a simply supported beam acted by a sine load and an impact separately. An experiment is also carried out to validate the feasibility and accuracy of the proposed method.

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