Equations of Motion Formulation of a Pendulum Containing N-point Masses
This paper presents a general formulation of equations of motion of a pendulum with n point mass by use of two different methods. The first one is obtained by using Lagrange Mechanics and mathematical induction(inspection), and the second one is derived by defining a vector. Today, these equations can be obtained by employing numerous programs; however, this study gives a very compact form of these equations that is more efficient than solving Euler-Lagrange Equations for every pendulum with more complex structures than simple or double pendulum.
A new modeling method based on sample data is described in this paper,through physics and sample date analysis on displacement vs.voltage characteristics of x-y plane of cylindrical piezoelectric scanner used in scanning probe microscope(SPM).Through sample data statistical analysis,and considering the error factors such as scan speed,scan angle,nonlinearity and coupling,a model represented by a binomial plus some error expressions is constructed.A nonlinearity correction expression based on the model is deduced.With the software Matlab this nonlinearity correction is simulated.The result of simulation indicates that it can eliminate the errors effectively.The ascendency of the method is that there are no polarization mechanism analysis and fewer number of model parameters.The method is practical applied to engineering.