The equations of motion are derived using Lagrange's equation
and are considered as a nonlinear system of second-order differential equations.
According to Lagrange's equation
, dynamic equation of the beam element can be written as
and from Lagrange's equation
of motion and equation (4.4) it can be shown that
with multipliers has been used to write the equation of movement, together with the nonholonomic couplings leading to a system with 1+s equations having 1+s unknowns: [q.sub.1], ..., [q.sub.s]; [[lambda].sub.1], ..., [[lambda].sub.s].
The pre-mentioned differential formulation of the equations of motion is equivalent to integral formulation, which requires the application of Lagrange's equation
The only strict prerequisite is a working knowledge of intermediate undergraduate dynamics, but some familiarity with simple aspects of Lagrange's equation
would also be helpful.
Therefore, the equations were formulated with generalized coordinates according to the general process of Lagrange's equation
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
For this edition they have added 59 new problems, and a new chapter on applying Lagrange's equations
to deriving equations of motion.
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.
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.
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).