Kinematics Roadmap

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Kinematics Roadmap
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Dynamics of Mechanisms
Dynamics - The study of systems that change with
time
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Overview Fundamental Equations
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Overview Fundamental Equations
where m total mass (sum over all mass
particles) ac acceleration of center of
mass (cm) of all mass particles Fc sum of
external forces applied to system of particles as
if applied at cm Hi ri x mivi angular
momentum of particle i (also called moment
of momentum) H, Hc
angular momentum summed over all particles,
measured about inertial point,
cm point, respectively M, Mc moment of all
external forces applied to system of particles,
measured about inertial point,
cm point, respectively
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Rigid bodies in general motion(translating and
rotating)
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Rigid bodies in general motion(translating and
rotating)
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Rotating rigid body
By integrating the motion over the rigid body, we
can express the angular momentum relative to
the xyz axes as H Hx i Hy j Hz k
(Jxx wx Jxy wy Jxz wz) i   (Jyx
wx Jyy wy Jyz wz) j   (Jzx wx Jzy
wy Jzz wz) k
products
moments
or in matrix form H J w where J
inertia matrix
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Rotating rigid body
Taking the derivative of the previous equations
for angular momentum H and substituting into the
moment equations, also assuming the body axes to
be aligned with the principal axes where the
products of inertia vanish, we get Eulers moment
equations Mx Jxx ax (Jzz - Jyy) wy wz
  My Jyy ay (Jxx - Jzz) wx wz  
Mz Jzz az (Jyy - Jxx) wx wy , where a is
the angular acceleration and w is the angular
velocity.
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Planar mechanisms
In ME 437 we work in the x-y-q plane only,
reducing Eulers equations to the simpler
form  Mz Jzz az where Jzz is about a
fixed point of rotation or the center of mass (or
gravity). We often use I rather than J for mass
moment of inertia.