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§ DYNAMICS
Dynamics is concerned with studying the motion of particles and rigid bodies. By studying the motion
of a single hypothetical particle, one can discern the motion of a system of particles. This in turn leads to
the study of the motion of individual points in a continuous deformable medium.
Particle Movement
The trajectory of a particle in a generalized coordinate system is described by the parametric equations
xi = xi(t),i=1,...,N ()
where t is a time parameter. If the coordinates are changed to a barred system by introducing a coordinate
transformation
xi = xi(x1,x2,...,xN ),i=1,...,N
then the trajectory of the particle in the barred system of coordinates is
xi = xi(x1(t),x2(t),...,xN (t)),i=1,...,N. ()
The generalized velocity of the particle in the unbarred system is defined by
dxi
vi = ,i=1,...,N. ()
dt
By the chain rule differentiation of the transformation equations () one can verify that the velocity in
the barred system is
dxr ∂xr dxj ∂xr
vr = = = vj,r=1,...,N. ()
dt ∂xj dt ∂xj
Consequently, the generalized velocity vi is a first order contravariant tensor. The speed of the particle is
obtained from the magnitude of the velocity and is
2 i j
v = gij v v .
The generalized acceleration f i of the particle is defined as the intrinsic derivative of the generalized velocity.
The generalized acceleration has the form
i n i 2 i m n
i δv i dx dv i m n d x i dx dx
f = = v,n = + v v = + ()
δt dt dt mn dt2 mn dt dt
and the magnitude of the acceleration is
2 i j
f = gij f f .
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Figure -1 Tangent, normal and binormal to point P on curve.
-Serret Formulas
The parametric equations () describe a curve in our generalized space. With reference to the figure
-1 we wish to define at each point P of the curve the following orthogonal unit vectors:
T i = unit tangent vector at each point P.
N i = unit normal vector at each point P.
Bi = unit binorm