Abstract
Lagrange and Hamilton formalisms derived from variational calculus can be applied nearly in all engineering sciences. In this study, the reader is introduced using tensorial variables in covariant and contravariant forms, to the extended Lagrangian \(\mathcal{L}\) and herewith to the modified momentum \({p}_{{k}}^{*}\). Through both, the extended Hamiltonian \(\mathcal{H}\) of a dissipative engineering system is derived to analyze the engineering system in an analytical way. In addition, a nonconservative Hamiltonian H ∗n for systems with elements of higher order is introduced in a similar manner. Moreover, different forms of extended Hamiltonian are represented. How these forms are achieved and how to derive the equations of generalized motion in different forms is also explained. As an example, a coupled electromechanical system in different formulations is given on behalf of the reader. The example is even extended to a case including some elements of higher order.
Zusammenfassung
Auf Variationsrechnungen basierende Lagrange- und Hamilton-Berechnungsformalismen lassen sich in nahezu allen Ingenieursanwendungen einsetzen. In dieser Arbeit wird der Leser in die Verwendung von Tensorvariablen in kovarianter und kontravarianter Form für die erweiterte Lagrangefunktion \(\mathcal{L}\) in Bezug auf die modifizierte Impulsgleichung \({p}_{{k}}^{*}\). Die erweitere Hamiltonfunktion \(\mathcal{H}\) eines dissipativen technischen Systems wird für die Durchführung seiner Analyse abgeleitet. Hierzu wird zusätzlich eine nichtkonservative Hamiltonfunktion H ∗n für Systeme mit Elementen höherer Ordnung eingeführt und erweiterte Formen vorgestellt, und die Ableitung generalisierter Bewegungsgleichungen beschrieben. Die Anwendung der beschriebenen Systematik auf ein gekoppeltes elektromechanisches System wird dargestellt und auch auf einen Fall mit Elementen höherer Ordnung erweitert.
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Abbreviations
- q i :
-
ith generalized coordinate in tensorial contravariant form
- \(\dot{q}^{i}\) :
-
ith generalized velocity in tensorial contravariant form
- q i :
-
ith generalized coordinate in tensorial covariant form
- \(\dot{q}_{{i}}\) :
-
ith generalized velocity in tensorial covariant form
- p i :
-
ith generalized momentum in tensorial covariant form
- p i :
-
ith generalized momentum in tensorial contravariant form
- \({p}_{{i}}^{{*}}\) :
-
ith (generalized) modified momentum
- f −1( ):
-
inverse function of
- \(\delta_{j}^{i}\) :
-
Kronecker delta function
- t :
-
time variable
- L :
-
Lagrange function or Lagrangian for short
- D :
-
(generalized) velocity proportional Rayleigh dissipation function
- H :
-
Hamilton function or Hamiltonian for short
- \(\mathcal{L}\) :
-
extended Lagrangian
- \(\mathcal{H}\) :
-
extended Hamiltonian
- [g ij ]:
-
metric tensor
- [g ij]:
-
inverse metric tensor
- K :
-
constant of proportionality
- \(\mathop{q}\limits^{{j}}\!{}^{{i}}\) :
-
jth derivative of ith generalized coordinate in contravariant form
- \(\mathop{{p}}\limits^{{j}}\!{}_{{i}}\) :
-
jth derivative of ith generalized momentum in covariant form
- f :
-
degree of freedom
- H ∗n :
-
nonconservative Hamiltonian for systems containing elements of higher order
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Süsse, R., Civelek, C. Analysis of coupled dissipative dynamic systems of engineering using extended Hamiltonian \(\mathcal{H}\) for classical and nonconservative Hamiltonian H ∗n for higher order Lagrangian systems. Forsch Ingenieurwes 77, 1–11 (2013). https://doi.org/10.1007/s10010-012-0158-7
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DOI: https://doi.org/10.1007/s10010-012-0158-7