We develop mathematical backgrounds to design novel chaotic systems based on known ones in our paper. We perform our studies by considering the generalized 2nd-order Lagrangian dynamical system that is driven by an external harmonic signal. This signal requires to study of the system as a time-variant dynamical system. That is why we suggest avoiding explicit dependence on system equations from time by replacing harmonic signal in right-hand expressions of system equations with a signal that is defined as the solution of corresponding differential equations. We append equations of the considered dynamical system with equations of the harmonic exciter and study the obtained system as a completed free motion system. If one transforms the motion equations for this system into a canonical state space that is different from the initial one, some new signals can be produced and novel dynamical system structure and parameters are defined. In our paper, we show that the order of the transformed system can be different from the initial one. Thus, it becomes possible to improve the security features of the designed system. We show the use of our approach by considering well-known Duffing pendulum equations and transforming them into canonical form for different output signals. Our studies show that this approach allows to form the chaotic oscillations that are significantly different from known ones. So, the proposed approach can be recommended to design novel chaotic generators.