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The x,y,z axis are body fixed axis, rotating with the body; the solutions for ω x(t),ω y (t) and ω z give the components of ω following these moving axis. If angular velocity transducers were mounted on the body to measure the components of ω, ω x(t),ω y(t) and ω z from the solution to the Euler equations would be obtained,
Therefore the problem can be formulated as finding a minimal norm solution to an underdetermined system of equations: min δ A(ω)δ=B(ω) By inspection we can see that matrix A has full row rank for any value of ω unequal to zero. If ω = 0 the solution is δ3 = −a3, and the rest of the δk equal to zeros. For all other values of ω the
The solution for a pressurized cylinder in plane strain was given above, i.e. where zz was enforced to be zero. There are two other useful situations: (1) The cylinder is free to expand in the axial direction. In this case, zz is not forced to zero, but allowed to be a constant along the length of the cylinder, say zz. The zz
As an example of the theorem, consider the situation depicted in Fig. 13.3, where a cylin-drically symmetric mass distribution is rotated about is symmetry axis, and about an axis tangent to its side. The component Izz of the inertia tensor is easily computed when the origin lies along the symmetry axis:
The general body is shown in the figure. We fix the x, y, z axis to the body and instantaneously align them with x, y, z. Referring to the figure, we see the components of ω,—- ω1, ω2 and ω3—- and the components of the angular moment vector H, which in general is not aligned with the angular velocity vector.
Derive the Lagrangian for a system of interconnected particles and rigid bodies. Use the Euler-Lagrange equation to derive equations of motions given a Lagrangian. Use the method of Lagrange multipliers to add constraints to the equations of motions. Determine the generalized momenta of a system.
16 Ιαν 2022 · In a two-dimensional problem, the body can only have clockwise or counterclockwise rotation (corresponding to rotations about the \(z\) axis). This means that a rigid body in a two-dimensional problem has three possible equilibrium equations; that is, the sum of force components in the \(x\) and \(y\) directions, and the moments about the \(z ...
