generalized coordinates

[′jen·rə‚līzd kō′ȯrd·ən·əts]

(mechanics)

A set of variables used to specify the position and orientation of a system, in principle defined in terms of cartesian coordinates of the system's particles and of the time in some convenient manner; the number of such coordinates equals the number of degrees of freedom of the system Also known as Lagrangian coordinates.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Generalized Coordinates

parameters q_i (i = 1, 2, …, s) that have any dimension, are mutually independent, and are equal in number to the number s of degrees of freedom of a mechanical system for which they uniquely determine the position. The law of motion for a system in generalized coordinates is given by s equations of the type q_i = q_i(t), where t is time.

Generalized coordinates are used in the solution of many problems, especially when a system is subject to constraints on its motion. In this case, the number of equations describing the motion of the system is substantially reduced in comparison with, for instance, the equations in Cartesian coordinates. In systems having an infinitely large number of degrees of freedom, such as a continuous medium or a physical field, the generalized coordinates are particular functions of the space and time coordinates and are given special names, such as potentials and wave functions.

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References in periodicals archive

A vector of generalized coordinates that define the position of the mechanical system of bodies in space is:

Forced Vibrations of the Hopper of Fertilizer Applying Machine

We point out that the CL model is a 1D model where the dynamical variables (as the generalized coordinates) are only the function of time-evolution parameter.

Christ-Lee Model: Augmented Supervariable Approach

For the developed kinematic model, it is necessary to perform a calibration by determining the actual values of the geometric parameters ([d.sub.i], [[theta].sub.i], [a.sub.i], [[alpha].sub.i]): displacements of generalized coordinates (initial relative position of links), real lengths of links and displacements between them.

Control system of the Articulated Arm Braking Mechatronic Machine (AABMM)

The generalized coordinates, which we used to study the dynamics of the robot, are shown in Figure 2.

Position Control of the Single Spherical Wheel Mobile Robot by Using the Fuzzy Sliding Mode Controller

An inspection of (1)-(3), we can observe that the Lagrangian L is expressed in terms of three generalized coordinates [rho], [[phi].sub.1], [[phi].sub.2] and three corresponding generalized velocities [mathematical expression not reproducible].

The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

For the extra generalized coordinates [h.sub.1], [h.sub.2], and [h.sub.3], we need extra three constraint equations.

Design of a 3-DOF Parallel Hand-Controller

The generalized coordinates of the system are chosen according to their degrees of freedom.

Suspended Load Path Tracking Control Strategy Using a Tilt-Rotor UAV

where [q.sub.B(n)] (t), [q.sub.S(n)](t), and [q.sub.S(0)](t) are generalized coordinates to be determined in order to satisfy initial conditions.

Forced Vibration of a Timoshenko Beam Subjected to Stationary and Moving Loads Using the Modal Analysis Method

where M is the mass matrix and q are the generalized coordinates and expressed as q = [[[R.sup.T] [[theta].sup.T]].sup.T], such that R = [[[R.sub.x] [R.sub.y] [R.sub.z]].sup.T] are the Cartesian coordinates of the body frame origin and [theta] = [[[[theta].sub.1] [[theta].sub.2] [[theta].sub.3] [[theta].sub.4]].sup.T] is the vector of Euler parameters that represent the rotational coordinates [15].

Computational Design Scheme for Wind Turbine Drive-Train Based on Lagrange Multipliers