Mechanics of Solids 2011 Part 5 docx

1-56 Section 1 Equations of Motion in Three Dimensions The equations of motion for a rigid body in three dimensions are extensions of the equations previously stated. (1.3.78) where a C = acceleration of mass center H C = angular momentum of the body about its mass center xyz = frame fixed in the body with origin at the mass center Ω = angular velocity of the xyz frame with respect to a fixed XYZ frame Note that an arbitrary fixed point O may be used for reference if done consistently. Euler’s Equations of Motion Euler’s equations of motion result from the simplification of allowing the xyz axes to coincide with the principal axes of inertia of the body. (1.3.79) where all quantities must be evaluated with respect to the appropriate principal axes. Solution of Problems in Three-Dimensional Motion In order to solve a three-dimensional problem it is necessary to apply the six independent scalar equations. (1.3.80) These equations are valid in general. Some common cases are briefly stated. Unconstrained motion. The six governing equations should be used with xyz axes attached at the center of mass of the body. Motion of a body about a fixed point. The governing equations are valid for a body rotating about a noncentroidal fixed point O. The reference axes xyz must pass through the fixed point to allow using a set of moment equations that do not involve the unknown reactions at O. Motion of a body about a fixed axis. This is the generalized form of plane motion of an arbitrary rigid body. The analysis of unbalanced wheels and shafts and corresponding bearing reactions falls in this category. Fa MH H H ∑ ∑ = == () +× m C CC C xyz C ˙˙ Ω MI II MI II MI II xxx yzyz yyy zxzx zzz xyxy ∑ ∑ ∑ =−− () =−− () =−− () ˙ ˙ ˙ ωωω ωωω ωωω F ma F ma F ma xC yC zC xyz ∑∑∑ === MH H H MH H H MH H H xxyzzy yyzxxz zzxyyx ∑ ∑ ∑ =+ − =+ − =+ − ˙ ˙ ˙ ωω ωω ωω . consistently. Euler’s Equations of Motion Euler’s equations of motion result from the simplification of allowing the xyz axes to coincide with the principal axes of inertia of the body. (1.3.79) where. of moment equations that do not involve the unknown reactions at O. Motion of a body about a fixed axis. This is the generalized form of plane motion of an arbitrary rigid body. The analysis of. 1 -56 Section 1 Equations of Motion in Three Dimensions The equations of motion for a rigid body in three dimensions are extensions of the equations previously stated. (1.3.78) where