The governing equation of the inverted pendulum model can be derived by using Lagrange's equations
. The positions of the lump mass m and the cart are written as
The equations of motion were deduced using Lagrange's equations and solved through the small parameter method to obtain their solutions up to the second order of approximation.
The equations of motion are derived using Lagrange's equation and are considered as a nonlinear system of second-order differential equations.
For this edition they have added 59 new problems, and a new chapter on applying Lagrange's equations
to deriving equations of motion.
Firstly, the dynamic model of the flexible links mounted with multiple PZT transducers is formulated using Lagrange's equations and AMM, and the experimental modal tests of the flexible links are implemented to verify the assumed mode shapes.
The general form of Lagrange's equations for elastic generalized coordinates is given as
Organized according to the steps in a control design project, the text first discusses kinematic and dynamic modeling methods, including programmed constraints, Lagrange's equations
, Boltzmann-Hamel equations, and generalized programmed motion equations.
Dynamic behaviour analysis, respectively the writing of the piece movement governing equations with respect to a fixed coordinate system can be done based on analytical mechanics, namely, Lagrange's equations
. As well as before, these feeding systems with disk having nests, respectively with rotor having horizontal palettes, can be considered particular types of one disk having inclined nests.
Using Lagrange's equations
of the second kind (Ripianu, 1977), the differential equations for the TRT1 and RTT robots with three degrees of freedom were deduced, expressed by the equations (1) and (2).
Next, Lagrange's equations
are derived and their integration is discussed.