Dynamics of a charged Ziegler’s double pendulum with a nonholonomic constraint under the joint action of a dead force and a Lorentz force

A variant of viscoelastic Ziegler’s double pendulum is investigated in which an electric charge is concentrated at one end of the device, at which a structural element imposing a nonholonomic constraint is attached. The obtained device is called ‘charged Ziegler’s double pendulum’ and is subjected to the influence of a dead force and a Lorentz force. The latter results from an electric field and a magnetic induction field generated by an ideal solenoid. The mechanical system models a self-propelled microrobot and, in the absence of damping, is conservative up to the energy variation induced by the Lorentz force in the case of explicitly time dependent electric field. Within this framework, two situations are analyzed: Case I, in which the device is placed inside the solenoid and experiences a Lorentz force featuring both electric and magnetic part; Case II, in which the device is placed outside the solenoid and undergoes a Lorentz force due to the sole electric field. After determining the equilibrium for the system under study, a stability analysis is performed, and the conditions are obtained for flutter instability and Hopf bifurcation. In addition, the post-critical behavior of the structure is examined, focusing on the interaction between the Lorentz force and the other forces characterizing its dynamics. We investigate the possibility of revisiting, in the mechanical context provided by Ziegler’s double pendulum, a phenomenology known in electromagnetism as Maxwell–Lodge effect