Engineering Transactions,

**11**, 2, pp. 235-252, 1963### Metoda Krakowianowa w Rozwiazywaniu Równań Ruchu Układów Dynamicznych

This paper is devoted to the application of the Cracovian calculus lo the integration of Lagrange's equations of motion. The Cracovian calculus gives considerable simplicity and lucidity to familiar arguments and schemes.

The discussion proper is preceded by a section devoted to eigenvectors and eigenvalues of a Cracovian. The notions of eigenvector and eigenvalue are introduced similarly to those of the matrix algebra. Some fundamental theorems concerning these notions are given. In particular the case of symmetric Cracovian is discussed in more detail. The section is concluded by a description of the iteration method for determining the eigenvectors and eigenfunctions showing the simplicity of the computation.

The second section is devoted to a discussion of Lagrange's equations for dynamic systems in the neighbourhood of' stable equilibrium, A free conservative system is considered, for which Lagrange's equations are replaced with a Cracovian differential equation of the second order. Its solution is reduced to the determination of the eigenvectors of a certain Cracovian. A few theorems express the properties of the solutions.

Next, the results are generalized to the case of a conservative system subject to an excitation. This case is described by a non-homogeneous linear Cracovian equation of the second order. To solve it the method of the Cracovian root may be used as well as a convenient procedure proposed in the present paper.

The work is concluded by a section devoted to dissipative systems. This case is described by a linear equation of the first order with coefficients constituting block Cracovians. Its solution reduces again to the determination of the eigenvectors of a certain Cracovian.

The discussion proper is preceded by a section devoted to eigenvectors and eigenvalues of a Cracovian. The notions of eigenvector and eigenvalue are introduced similarly to those of the matrix algebra. Some fundamental theorems concerning these notions are given. In particular the case of symmetric Cracovian is discussed in more detail. The section is concluded by a description of the iteration method for determining the eigenvectors and eigenfunctions showing the simplicity of the computation.

The second section is devoted to a discussion of Lagrange's equations for dynamic systems in the neighbourhood of' stable equilibrium, A free conservative system is considered, for which Lagrange's equations are replaced with a Cracovian differential equation of the second order. Its solution is reduced to the determination of the eigenvectors of a certain Cracovian. A few theorems express the properties of the solutions.

Next, the results are generalized to the case of a conservative system subject to an excitation. This case is described by a non-homogeneous linear Cracovian equation of the second order. To solve it the method of the Cracovian root may be used as well as a convenient procedure proposed in the present paper.

The work is concluded by a section devoted to dissipative systems. This case is described by a linear equation of the first order with coefficients constituting block Cracovians. Its solution reduces again to the determination of the eigenvectors of a certain Cracovian.

**Full Text:**PDF

Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

#### References

S. BANACH, Mechanika, tom II, Czytelnik 1947.

T. BANACHIEWICZ, Rachunek krakowianowy, PWN Warszawa 1959.

J. P. DEN HARTOG, Mechanical vibrations, New York 1933.

W.N. FADDIEJEWA, Metody numeryczne algebry liniowej, PWN Warszawa 1955.

E. KAMKE, Differentialgleichungen, Lösungsmethoden u. Lösungen. Band I., Leipzig 1943.

T. KARMÁN i M.A. BIOT, Metody matematyczne w technice, PWN Warszawa 1958.

A. MOSTOWSKI i M. STARK, Algebra wyższa, tom I, III, PWN Warszawa 1953 i 1954.