On the use of direct implicit integration with numerical Jacobian matrix in helicopter rotor dynamics

In recent years, geometrically exact beam theory is widely used in rotor structural dynamics to model composite blade undergoing large deflections. The corresponding differential equations based on this theory show strong nonlinearity and the aeroelastic terms are normally written in implicit format. It brings most difficulties to implicit integration methods since the analytical Jacobian matrix is hard to derive. In this paper, a trapezoidal integration utilizing numerical Jacobian matrix is introduced to solve the dynamic equations of rotor blade. Central difference method with Romberg extrapolation scheme is adopted to numerically calculate the Jacobian. Accuracy of the methods is evaluated on dynamic problems from one cantilevered beam and the other fully articulated rotor blade. Numerical stability and efficiency are also investigated by changing update policies of Jacobian matrix and mesh configurations of typical rotor blade. Numerical results show that both extrapolation scheme for partial derivatives and the integration method with numerical Jacobian perform well on predicting natural frequencies and transient responses in rotor dynamics.