Fractional order spring/spring-pot/actuator element in a multibody system: Application of an expansion formula

• Fractional order models of a spring/spring-pot and spring/spring-pot/actuator elements are proposed.
• Generalized forces of elements are derived using the principle of virtual work.
• An expansion formula for fractional derivative of a composite function is suggested.
• Numerical results for generalized forces are compared for different values of parameters in the model.

Fractional order models of a spring/spring-pot and spring/spring-pot/actuator element connected into a multibody system are proposed in order to represent smart materials and components in adaptronic systems by introducing new tuning parameter. The models are introduced into dynamic equations via generalized forces and using the Lagrange's equations of the second kind in covariant form. Generalized forces are derived by taking into account fractional order derivatives in force–displacement relations and by using the principle of virtual work. The numerical scheme for solving fractional order differential equations proposed in Atanacković and Stanković (2008) is used in order to approximate fractional order derivative of a composite function appearing in the presented fractional order model. Numerical example for the multibody system with three degrees of freedom is presented. The results obtained for generalized forces are compared for different values of parameters in the fractional order derivative model.