Kinetics of rigid bodies, Mechanical Engineering

Kinetics of Rigid Bodies:

For the bodies undergoing plane motion, a common scheme for solutions is to apply the equations of dynamic equilibrium as below.

∑ Fx  = m ax ,         ∑ Fy  = m a y

∑ M = I α

Keeping in mind that ax, ay and I are calculated with reference to centroidal axes. Inertia forces and inertia couple should be applied to the body before writing down these equilibrium equations. The relation between ax and α is normally obtained by using kinematic relationships.

When motion of rotation takes place about an axis which is not coinciding with centroidal axis, it is convenient to select x and y axis through the mass centre which are positive in the direction of an and ar, respectively. Also ∑ M A  - I A  α = 0 is the equation to be utilized which normally gets rid of the unknown reaction at the axle A. Then these equations get transformed into

∑ Fx = mr ω2

∑ Fy = mr α

∑ M = I α; ∑ M A = I A α

In case of centroidal rotation A coincides with mass centre, r = 0 and the above equations reduce to

∑ Fx  = 0,         ∑ Fy  = 0,           ∑ M = Iα

For homogeneous rolling bodies in which mass centre has a rectilinear motion parallel to the surface on which it rolls it is suitable to select reference axes at the mass centre with the x axis parallel to the surface and positive in the initial direction of motion. The equations become

∑ Fx = m a, ∑ Fy = 0, ∑ M = Iα

If the body rolls without slipping, an additional equation can be used

∑ M c = I c α

Posted Date: 1/29/2013 6:44:47 AM | Location : United States







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