This article generalizes the implementation of higher order corrections to state transition matrices during instantaneous reversals in hybrid dynamical systems impacting a discontinuity boundary transversally. A closed form expression for saltation terms in systems possessing a degree of smoothness zero is derived. The difference in flight times of two closely initiated trajectories in state space to the impacting surface has been estimated up to O(2). A comparison of the times of impact estimated with the first order approximation reveals that higher order corrections lead to a significant improvement of estimates. Next, two new algorithms to estimate the Lyapunov spectrum and Floquet multipliers for piecewise-smooth systems have been presented using the derived second order corrections. Stability analyses are subsequently carried out using the proposed framework for two representative cases i.e., of a hard impact oscillator and a pair impact oscillator. It is established that the obtained Floquet multipliers and Lyapunov spectrum accurately predict the stability of the dynamical states, as validated by their corresponding bifurcation diagrams. 

In this paper, the dynamic response of a damaged double-beam system traversed by a moving load is studied, including passive control using multiple tuned mass dampers. The double-beam system is composed of two homogeneous isotropic Euler–Bernoulli beams connected by a viscoelastic layer. The damaged upper beam is simulated using a double-sided open crack replaced by an equivalent rotational spring between two beam segments, and the lower primary beam is subjected to a moving load. The load is represented by a moving Dirac delta function and by a quarter car model, respectively. Road surface roughness (RSR) is classified as per ISO 8606:1995(E). The effect of vehicle speed of the moving oscillator and variable RSR profiles on the dynamics of this damaged double Euler–Bernoulli beam system for different crack-depth ratios (CDRs) at various crack locations is studied. It is observed that coupling of two beams leads to a vehicular effect on the damaged beam, even when no vehicle on it is present. The effects of single and multiple tuned mass dampers to control the vibrational responses of the primary beam due to damage on the secondary beam is studied next. The performance of tuned mass dampers to reduce the transverse vibrations of the damaged double-beam system and of the quarter car is investigated. The paper links the coupling between the two levels of double beam with the inertial coupling of the vehicle to the double-beam system. 

We approach quantum dynamics in one spatial dimension from a systematic study of moments starting from the dynamics of the mean position. This is complementary to the approach of Brizuela whose starting point was generalized recursion relations between moments. The infinite set of coupled equations is truncated which allows us to use the techniques used in the study of dynamical systems. In particular we predict for what initial variance the purely quartic oscillator will time develop with minimal change in the shape of the initial packet and what the frequency of oscillation of the mean position will be. We show how quantum fluctuations will cause a particle to escape from the well of a volcano potential and how they will cause an oscillation between the two wells of a double well potential. Further, we consider an oscillatory external field in addition to the double well potential and work near the separatrix where the classical system is known to be chaotic. We show how the quantum fluctuations suppresses the chaotic behaviour after a time interval inversely proportional to the strength of the quantum fluctuations. 

We explore analytically the quantum dynamics of a point mass pendulum using the Heisenberg equation of motion. Choosing as variables the mean position of the pendulum, a suitably defined generalised variance and a generalised skewness, we set up a dynamical system which reproduces the correct limits of simple harmonic oscillator like and free rotor like behaviour. We then find the unexpected result that the quantum pendulum released from and near the inverted position executes oscillatory motion around the classically unstable position provided the initial wave packet has a variance much greater than the variance of the well known coherent state of the simple harmonic oscillator. The behaviour of the dynamical system for the quantum pendulum is a higher dimensional analogue of the behaviour of the Kapitza pendulum where the point of support is vibrated vertically with a frequency higher than the critical value needed to stabilize the inverted position. A somewhat similar phenomenon has recently been observed in the non equilibrium dynamics of a spin - 1 BoseEinstein Condensate.