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Decomposition of the Equations of Motion in the Analysis of Dynamics of a 3-DOF Nonideal System
(2014) Awrejcewicz, Jan; Starosta, Roman; Sypniewska-Kamińska, Grażyna
The dynamic response of a nonlinear system with three degrees of freedom, which is excited by nonideal excitation, is investigated. In the considered system the role of a nonideal source is played by a direct current motor, where the central axis of the rotor is not coincident with the axis of rotation. This translation generates a torque whose magnitude depends on the angular velocity. During the system operation a general coordinate assigned to the nonideal source grows rapidly as a result of rotation. We propose the decomposition of the equations of motion in such a way to extract the solution which is directly related to the rotation of an unbalanced rotor. The remaining part of the solution describes pure oscillation depending on the dynamical behaviour of the whole system. The decomposed equations are solved numerically. The influence of selected system parameters on the rotor vibration is examined. The presented approach can be applied to separate vibration and rotation of motions in many other engineering systems.
Double pendulum colliding with a rough obstacle.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Sypniewska-Kamińska, Grażyna; Starosta, Roman; Awrejcewicz, Jan
The externally excited and damped vibrations of the double pendulum in the vertical plane are considered. The pendulum can collide with a rough obstacle many times during its motion. The pendulum is modeled as a piecewise smooth system. The differential equations govern the motion of the system in the relatively long time between the collisions. When a contact with the obstacle occurs, the pendulum exhibits a discontinuous behaviour. The velocities of both parts of the pendulum and the reaction forces are changing stepwise. An important element of the solving algorithm is aimed on the continuous tracking of the position of the pendulum in order to detect the collision with the unilateral constraints and to determine the state vector of the pendulum at the impact time instant. A single collision is described by the Euler’s laws of motion in the integral form. The equations are supplemented by the Poisson's hypothesis and Coulomb’s law of friction. The friction law is formulated for the instantaneous values of the reaction forces. The values of their impulses depend on the existence of a slip between the contacting bodies. Furthermore, during the collision the dynamic behaviour may change. Therefore the Coulomb law cannot be generalized for the linear impulses of the forces in a simple way. We have applied the Routh method in order to solve the problem. The method has a simple geometrical interpretation in the impulse space.
Identification of the model parameters based on the ambiguous branches of resonance response curves
(Wydawnictwo Politechniki Łódzkiej, 2021) Sypniewska-Kamińska, Grażyna; Awrejcewicz, Jan
A concept of the method of determining the parameters describing the damping and the nonlinearity, which can be of physical or geometrical nature, is presented in the paper. The main idea is explained regarding mechanical systems with one degree of freedom, however, the method can be also employed to identification for systems with two DoF provided that the couplings are weak and the resonances do not occur simultaneously. The analysis of stationary resonance states can be reduced only then to the third-degree equation, which is a necessary condition for the applicability of this method. Numerical simulations, which are carried out, confirm the usefulness and accuracy of the method.
Mechanical systems with two nonlinear springs connected in series.
(Politechnika Łódzka. Wydział Mechaniczny. Katedra Automatyki, Biomechaniki i Mechatroniki., 2015) Starosta, Roman; Sypniewska-Kamińska, Grażyna; Awrejcewicz, Jan
The aim of the paper is analysis of dynamical regular response of the nonlinear oscillator with two serially connected springs of cubic type nonlinearity. Behaviour of such systems is described by a set of differential-algebraic equations (DAEs). Two examples of systems are solved with the help of the asymptotic multiple scales method in time domain. The classical approach has been appropriately modified to solve the governing DAEs. The analytical approximated solution has been verified by numerical simulations.
Vibration of the system with nonlinear springs connected in series
(Wydawnictwo Politechniki Łódzkiej, 2021) Awrejcewicz, Jan; Starosta, Roman; Sypniewska-Kamińska, Grażyna
Solution of the problem and qualitative analysis of the forced vibration of the spring pendulum containing nonlinear springs connected in series is made in the paper. The method of multiple scales in time domain (MMS) has been employed in order to carry out the analytical computations. The MMS allows one, among others, to predict the resonances which can appear in the systems. The approximate solution of analytical form has been obtained for vibration at main resonance.