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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,
Find the linearized equation of motion using a small angle approximation for θ (define θ such that the static equilibrium position is θ=0). Find the undamped natural frequency of the system. Solution: This is a planar motion problem. See the free body diagram above.
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
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:
Choose a cylindrical coordinate system with the positive z-axis pointing to the right, the direction the inner cylinder is moving in. Then the fluid flows in the z-direction and varies as a function of radius from the cylinder’s axis. v z= v z(r)
• General Problem: • * Given two coordinate systems, x- y and x' - y', and a stress state defined relative to the first coordinate system xyz:σx σy, τxy • * Find the stress components relative to the second coordinate system x’y’z’ : σ’ x σ’ y, τ’ xy
There are 3 basic rotations an aircraft can make: •Roll = Rotation about x-axis. •Pitch = Rotation about y-axis. •Yaw = Rotation about z-axis. •Each rotation is a one-dimensional transformation. Any two coordinate systems can be related by a sequence of 3 rotations. M. Peet Lecture9: 4/24. Review: ForcesandMoments. Forces.
