Introducing the use of positive and negative inertial functions in asymptotic modelling

Abstract

The work presented here shows that the natural frequencies of constrained systems may be obtained from asymptotic models of corresponding systems where the constraints are replaced by artificial mass or moment of inertia of very large positive and negative values. This offers a convenient alternative to the current practice of using artificial elastic restraints of large stiffness, a concept introduced by Courant in 1943, to remove a limitation on the choice of admissible functions. Recent publications show that in order to control the error caused by approximating constraints with restraints of large stiffness, it is necessary to use both positive and negative stiffness values. However, the negative stiffness introduces instability near the lower modes of vibration and the magnitude of negative stiffness parameter used must be greater than the highest critical stiffness to ensure bounded results are obtained. The use of positive and negative artificial inertial parameters overcomes this problem as they do not introduce instability near the lower modes, allowing the natural frequencies of constrained systems to be delimited to any desired accuracy.

Wittrick–Williams algorithm proof of bracketing and convergence theorems for eigenvalues of constrained structures with positive and negative penalty parameters

“…In these publications the eigenvalues are the natural frequencies, or alternatively buckling loads, of structures and it has been shown that the use of penalty parameters consisting of positive and negative values for the stiffness of the artificial restraints employed to enforce geometric boundary and continuity conditions gives eigenvalues that bracket the corresponding values of the constrained system. Alternatively, it has also been shown for the vibration case that the natural frequencies of constrained systems are bracketed by the frequencies of the corresponding systems for which the constraints are replaced by positive and negative values of inertial parameters and that plotting the natural frequency against the inverse of these inertial parameters (or alternatively of the artificial restraints when they are used as the penalty functions) enables interpolation to give very accurate and efficient convergence onto the frequencies of the constrained systems [14,15]. This also appears to hold for other eigenvalue problems such as critical speeds associated with aeroelastic divergence [16], although no mathematical proof has been obtained for this case.…”

The bipenalty method for arbitrary multipoint constraints

“…It is well known that stiffness penalties introduce spurious modes with very high eigenfrequencies, increasing monotonically as the penalty parameters are increased [10]. On the other hand, mass penalties introduce spurious eigenmodes with very low eigenfrequencies (tending to zero) [11]. The so-called bipenalty method utilises both kinds of penalties simultaneously in order to control the eigenfrequencies of these spurious modes.…”

A new bipenalty formulation for ensuring time step stability in time domain computational dynamics

“…This technique is known as the large mass, mass penalty or inertia penalty method. Again, constraints imposed using mass penalties are only approximate in nature but, instead of increasing the maximum eigenfrequency of the system, one or more eigenfrequencies are greatly reduced, tending to zero for large penalty parameters [5]. This implies that when used simultaneously, the stiffness and mass penalties can be tuned such that the maximum eigenfrequency (along with the order of the stiffness, S, of the system equations) can be preserved.…”

“…In dynamic analysis, penalty methods can also be applied to the mass matrix of a system [5,6]. This technique is known as the large mass, mass penalty or inertia penalty method.…”