On the use of negative penalty parameters in aeroelastic divergence analysis of lifting surfaces

This paper shows that a numerical modelling method in which constraints are replaced with positive and negative penalty functions, which may be regarded as artificial elastic restraints of positive and negative stiffness, may be safely used to determine the critical speed associated with aeroelastic divergence. The critical speeds of a beam with restraints of positive and negative stiffness are found to converge to that of the constrained system, from below if the stiffness is positive and from above otherwise. A uniform Euler–Bernoulli beam clamped at the rear end is analysed using an artificial restraint to enforce the constraint of zero rotation at the clamp, and the results are compared with the exact critical speed of the constrained system obtained analytically. The paper shows that, contrary to common belief that the penalty parameter must be positive, the inclusion of a negative penalty parameter enables the determination of errors due to violation of the constraints.