Second order generalized linear systems arising in analysis of flexible beams

The analysis of a one-link flexible beam using a power series solution with respect to the spatial variable is discussed. The Euler-Bernoulli beam with payload mass and moment of inertia at the right end is clamped at the left end or driven by a torque applied by an actuator located at the base of the beam. The former case results in a second-order state-space equation, while the latter results in a second-order singular equation. For both models, an extended Leverrier-Faddeev algorithm is derived in order to invert the matrix polynomial R(s)=Es²+A, where sε C, E and A are constant matrices and E max be singular. Extension of the algorithm for beams with energy dissipation yielding a matrix polynomial R( s)=Es²+A₁s+ A₂ is given