A New Differential Operator Method to Study the Mechanical Vibration

Affiliation(s)

Shandong Iron and Steel Company Ltd., Jinan, China.

Department of Mathematical Sciences, Yeshiva University, New York, USA.

The State Key Laboratory of Tribology, Tsinghua University, Beijing, China.

Shandong Iron and Steel Company Ltd., Jinan, China.

Department of Mathematical Sciences, Yeshiva University, New York, USA.

The State Key Laboratory of Tribology, Tsinghua University, Beijing, China.

ABSTRACT

In this paper, we propose a unified differential operator method to study mechanical vibrations, solving inhomogeneous linear ordinary differential equations with constant coefficients. The main advantage of this new method is that the differential operator*D* in the numerator of the fraction has no effect on input functions (*i.e.*, the derivative operation is removed) because we take the fraction as a whole part in the partial fraction expansion. The method in various variants is widely implemented in related fields in mechanics and engineering. We also point out that the same mistakes in the differential operator method are found in the related references [1-4].

In this paper, we propose a unified differential operator method to study mechanical vibrations, solving inhomogeneous linear ordinary differential equations with constant coefficients. The main advantage of this new method is that the differential operator

Cite this paper

Q. Ji, X. Ji, L. Ji and Y. Zheng, "A New Differential Operator Method to Study the Mechanical Vibration,"*Modern Mechanical Engineering*, Vol. 2 No. 3, 2012, pp. 65-70. doi: 10.4236/mme.2012.23009.

Q. Ji, X. Ji, L. Ji and Y. Zheng, "A New Differential Operator Method to Study the Mechanical Vibration,"

References

[1] Department of Mathematics, Shanghai Tongji University, “Advanced Mathematics I,” 6th Edition (in Chinese), Publishing House of High Education, Beijing, 2007, pp. 350-352.

[2] C. H. Edwards and D. E. Penney, “Elementary Differential Equations,” 6th Edition, Pearson Prentice Hall, Upper Saddle River, 2008, pp. 340-345.

[3] S. E. Kerufunikefu, “Mechanical Dynamics with Elastic Key Ring,” (in Russian, No Further Version), Kiev, 1961, pp. 86-89.

[4] “Mathematics Handbook,” (in Chinese), Publishing House of People Education, Beijing, 1979, pp. 675-677 (reprinted by Publishing House of High Education, Beijing, 1999).

[5] R. P. Agnew, “Differential Equations,” 2nd Edition, McGRAW-HILL, BOOK COMPANY, INC., New York, Toronto, London, 1960, pp.216-250.

[6] L. E. El’sgol’ts, “Differential Equations,” in the Series International Monographs on Advanced Mathematics and Physics, Gordon and Breach Publishers, Inc., New York, and Hindustan Publishing Corporation, Delhi, 1961, pp. 145-155.

[7] C.-S. Ji, “A New Solution of Constant Differential Equation Group by Differential Operator and Application in Calculating Rolling-Mill Torsion Vibration,” Proceedings of the 6th International Modal Analysis Conference, Kissimmee, 1-4 February 1988, pp. 598-602.

[8] Z. Zheng, “Mechanical Vibration I,” (in Chinese), Publishing House of Mechanical Industry, Beijing, 1980.

[9] “Mechanical Design Handbook,” (in Chinese), Publishing House of Chemical Industry, Beijing, 2009.

[1] Department of Mathematics, Shanghai Tongji University, “Advanced Mathematics I,” 6th Edition (in Chinese), Publishing House of High Education, Beijing, 2007, pp. 350-352.

[2] C. H. Edwards and D. E. Penney, “Elementary Differential Equations,” 6th Edition, Pearson Prentice Hall, Upper Saddle River, 2008, pp. 340-345.

[3] S. E. Kerufunikefu, “Mechanical Dynamics with Elastic Key Ring,” (in Russian, No Further Version), Kiev, 1961, pp. 86-89.

[4] “Mathematics Handbook,” (in Chinese), Publishing House of People Education, Beijing, 1979, pp. 675-677 (reprinted by Publishing House of High Education, Beijing, 1999).

[5] R. P. Agnew, “Differential Equations,” 2nd Edition, McGRAW-HILL, BOOK COMPANY, INC., New York, Toronto, London, 1960, pp.216-250.

[6] L. E. El’sgol’ts, “Differential Equations,” in the Series International Monographs on Advanced Mathematics and Physics, Gordon and Breach Publishers, Inc., New York, and Hindustan Publishing Corporation, Delhi, 1961, pp. 145-155.

[7] C.-S. Ji, “A New Solution of Constant Differential Equation Group by Differential Operator and Application in Calculating Rolling-Mill Torsion Vibration,” Proceedings of the 6th International Modal Analysis Conference, Kissimmee, 1-4 February 1988, pp. 598-602.

[8] Z. Zheng, “Mechanical Vibration I,” (in Chinese), Publishing House of Mechanical Industry, Beijing, 1980.

[9] “Mechanical Design Handbook,” (in Chinese), Publishing House of Chemical Industry, Beijing, 2009.