Linear, Cubic and Quintic Coordinate-Dependent Forces and Kinematic Characteristics of a Spring-Mass System

DOI: 10.4236/wjm.2013.36027   PDF   HTML   XML   2,764 Downloads   4,168 Views   Citations

Abstract

By combining a pair of linear springs we devise a nonlinear vibrator. For a one dimensional scenario the nonlinear force is composed of a polynomial of odd powers of position-dependent variable greater than or equal three. For a chosen initial condition without compromising the generality of the problem we analyze the problem considering only the leading cubic term. We solve the equation of motion analytically leading to The Jacobi Elliptic Function. To avoid the complexity of the latter, we propose a practical, intuitive-based and easy to use alternative semi-analytic method producing the same result. We demonstrate that our method is intuitive and practical vs. the plug-in Jacobi function. According to the proposed procedure, higher order terms such as quintic and beyond easily may be included in the analysis. We also extend the application of our method considering a system of a three-linear spring. Mathematica [1] is being used throughout the investigation and proven to be an indispensable computational tool.

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H. Sarafian, "Linear, Cubic and Quintic Coordinate-Dependent Forces and Kinematic Characteristics of a Spring-Mass System," World Journal of Mechanics, Vol. 3 No. 6, 2013, pp. 265-269. doi: 10.4236/wjm.2013.36027.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Mathematica, “A General Computer Software System and Language Intended for Mathematical and Other Applications,” V9.0, Wolfram Research, 2013.
[2] Tronton and Marion, “Classical Dynamics of Particles and Systems,” 5th Edition, Cengage Learning, 2008.
[3] H. Sarafian, “Static Electric-Spring and Nonlinear Oscillations,” Journal of Electromagnetic Analysis & Applications, Vol. 2, No. 2, 2010, pp. 75-81.
[4] H. Sarafian, “Nonlinear Oscillations of a Magneto Static Spring-Mass,” Journal of Electromagnetic Analysis and Applications, Vol. 3, No. 5, 2011, pp. 133-139. doi:10.4236/jemaa.2011.35022
[5] H. Sarafian, “Dynamic Dipole-Dipole Magnetic Interaction and Damped Nonlinear Oscillations,” Journal of Electromagnetic Analysis & Applications, Vol. 1, No. 4, 2009, pp. 195-204. doi:10.4236/jemaa.2009.14030
[6] Jacobi Elliptic Functions. http://mathworld.wolfram.com/JacobiEllipticFunctions.html
[7] J. Richter-Gebert and U. H. Kortenkamp, “The Interactive Geometry Software,” Cinderella 2.0, Springer, 2012.

  
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