In order to complete
mathematical model of physical system you need to select a set of variables
that describe the physical behavior of the system. Dependent variables are the
variables that describe the physical behavior of the system. Example of
dependent variables are displacement of a particle in a dynamic system, the
components of the velocity vector in a fluid flow problem, the temperature in a
heat transfer problem or the electric current in an AC circuit problem. Independent variables are the variables with
which the dependent variables change. That is. The dependent variables are
functions of the independent variables. Independent variables for most dynamic
systems are electric circuit problems is time. The temperature distribution in
a heat transfer problem may be a function of spatial position as well as time. The
dependent variables in most vibration problems are displacement of specified
particles from the system’s equilibrium position while time is the independent
variable.
The number of degrees of
freedom for a system is the number of kinematically independent variables
necessary to completely describe the motion of every particle in the system.
Any set of n kinematically independent coordinates for a system with n degrees
of freedom is called a set of generalized coordinates. There is not unique
choice of generalized coordinates. The generalized coordinates are the
dependent variables for a vibration problem and are functions of the
independent variable, time. If the time history of the generalized coordinates
is known, the displacement, velocity and acceleration of any particle in the
system can be determined by using kinematics.
A single particle free to
move in space has three degrees of freedom, and a suitable choice of
generalized coordinates is the Cartesian coordinates (x,y,z) of the particle with
respect to a fixed reference frame. As the particle moves in space its position
is a function of time. An unrestrained rigid body has six degrees of freedom. A
suitable choice for a system of generalized coordinates is the Cartesian coordinates
of the body’s center of mass and the angular measure of an axis fixed to the
body with respect to each of the Cartesian coordinate axes fixed in space. The number
of degrees of freedom is reduced if a particle or a rigid body is subject to
constraints. A particle constrained to move in a plane has at most two degrees
of freedom, while a rigid body undergoing planar motion has at most three
degrees of freedom.
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