Nonlinear Lumped System Research Articles

Buckling dynamics of imperfect Elastica subject to various loading conditions

Buckling dynamics of a pinned-pinned flexible imperfect beam attached to a sliding mass is investigated using nonlinear Elastica theory. Initially straight flexible buckling beam having pinned end boundary conditions loaded at one of its end with curved imperfection is considered. Large deflection analysis of flexible beam is studied using nonlinear Elastica theory. Imperfection analysis of the flexible beam is investigated considering the imperfection as an initial curvature. The governing differential equations are expressed in terms of nonlinear functions that are typical of flexible beams, generally leading to highly implicit relationships involving elliptic integrals and functions. Dynamic simulation of the flexible beam is studied using numerical simulation procedures with various types of loading (step, ramp, and sinusoidal) assuming this member buckles in its first mode. Dynamic response of the imperfect buckling Elastica has been obtained by using numerical Runge-Kutta methods. Load deflection characteristics of flexible beams are presented in polynomial curve fits. The polynomial curve fits obtained from nonlinear inextensible exact beam theory may then be used as the nonlinear lumped system stiffness. The buckling Elastica may find applications in compliant mechanism design. The motivation behind this research is not only to present the dynamic behavior of the buckling beam considering the magnitude of the imperfection but also to provide a tool to design new types of compliant mechanisms. Original compliant mechanism designs are presented demonstrating where the buckling dynamics of imperfect Elastica or flexible curved beams might be needed in mechanism design and synthesis.

The Effect of Lumping and Expanding on Kinetic Differential Equations

Let us consider the differential equation $ \dot{\bf y}(t)={\bf f}({\bf y}(t)) $ with an $\bf f$ from ${\bf R}^N$ to ${\bf R}^N$ and suppose that there exists a transformation $\bf h$ from ${\bf R}^N$ to ${\bf R}^{\hat N}$ ($\hat N\le N$) such that $ \hat{\bf y}:={\bf h}\circ{\bf y} $ obeys a differential equation $ \dot{\hat{\bf y}}(t)={\hat{\bf f}}({\hat{\bf y}}(t)) $ with some function $\hat{\bf f}$; then the first equation is said to be lumpable to the second by $\bf h$. Here mainly the case is investigated when the original differential equation has been induced by a complex chemical reaction. We provided a series of necessary and sufficient conditions for the existence of such functions $\bf h$ and $\hat{\bf f}$; some of them are formulated in terms of $\bf h$ and $\bf f$ only. Beyond these conditions our main concern here is how lumping changes those properties of the solutions which are either interesting from the point of view of the qualitative theory of differential equations or from the point of view of formal reaction kinetics. Results show that each eigenvalue of the Jacobian of the nonlinear lumped system at an equilibrium is an eigenvalue of the original system at the corresponding equilibrium. (Invariant sets, equilibria, and periodic solutions are lumped into invariant sets, equilibria, and periodic solutions, respectively.) Under certain conditions a Lyapunov function of the lumped system can be used to create a Lyapunov function for the original one (to test relative stability) and vice versa---both around equilibria and far from equilibria. These general statements do not necessarily imply close qualitative resemblance of lumped and original solutions but provide criteria by which lumping schemes may be selected for this purpose. The precise meaning of the conditions in the general statements is illustrated by examples taken from formal reaction kinetics.

Analysis and identification of non-linear systems via shifted Jacobi series

Based on shifted Jacobi series, algorithms have been established for the analysis and identification of non-linear systems described by a Hammerstein model consisting of a single-valued non-linearity followed by a linear plant. Moreover, the parameter estimation of a certain non-linear lumped system is also presented. By using the shifted Jacobi expansion for the analysis, the solution of a non-linear state equation is reduced to the solution of a linear algebraic matrix equation. For the identification, by expanding the measured input-output data into the shifted Jacobi series the unknown parameters of the non-linear system are estimated through the least-squares method. Examples are given to demonstrate the usefulness of this approach.

A characterization of the linear and non-linear dynamic performance of a practical piezoelectric actuator part 2: Theory

Lumped parameter equations that describe the linear and non-linear operation of a practical piezoelectric actuator are developed from the continuum equations of non-linear electroelasticity. Exact solutions of this non-linear lumped system are developed for sinusoidally varying inputs. Theoretical results are compared to the measurements of Part 1: agreement between theory and experiment is good.

Liapunov Type Design of Lumped System in Conflict with Environment

After introducing a few propositions on a two-player differential game-against-nature, we discuss synthesis (design) of a fairly general, nonlinear lumped system with storage-accumulative and dissipative characteristics under random-but-bounded environmental input. The discussion is made in terms of the above game via the Liapunov Direct Method. Sufficient criteria for qualitative and optimal Payabilities are proposed. A storage-oriented strategy for specific local behaviour of the system in the state space is then derived from the above criteria.

Dynamics of High-Speed Cam-Driven Mechanisms—Part 2: Nonlinear System Models

The dynamic responses of the cam-driven mechanism are investigated, based on a non-linear lumped system model. The nonlinearity is an energy-dissipating element which consists of viscous, quadratic, Coulomb and static frictions combined. The nonlinear equation of motion of a single degree of freedom is first analyzed using a numerical method and the results of time responses are presented and characterized in the phase-plane. The primary and residual shock response spectra in nondimensional form for a number of typical cam input excitations are presented and compared with those of the associated linear cases.

A Direct Numerical Formulation of Galerkin’s Method for Nonlinear Lumped Systems

Use of Galerkin’s method for the approximate analysis of nonlinear lumped systems is considered. A numerical procedure for finding Galerkin solutions of initial value problems is developed and illustrated by two examples. The approach employs Galerkin’s method formulated as a two-point boundary-value problem. While solving the two-point problem, a recursion algorithm is obtained; the algorithm is applied to the Galerkin integral directly, and therefore analytical integration of the weighted equation residuals and solution of specific algebraic equations are not required. The numerical formulation renders Galerkin’s method systematic, and it extends the utility of the method.