## Abstract

This paper shows how the recently introduced concept of positive and negative inertial functions in asymptotic modelling may be used effectively. Noting an almost linear variation of the natural frequency over a wide range of values of the reciprocal of the inertial parameter, the results presented show that both the natural frequencies of constrained systems can be calculated accurately using only moderately large artificial inertial parameters, e.g. which are only about 10 times that of the system mass and also that values about a million times more than this do not precipitate ill-conditioning when using 16 figure computer accuracy. This helps to minimize the chances of numerical problems occurring due to the limitation of machine precision.

## 1. Introduction

A very recent paper (Ilanko 2005) introduced the use of positive and negative inertial functions in asymptotic modelling. The nature and advantages of this new method are as follows. 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 is a convenient alternative to the normal practice of using artificial elastic restraints of large stiffness, which may be taken as positive and negative to obtain bounds on the natural frequencies. However, using 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 (Ilanko 2002). 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.

The present note shows that there are substantial advantages in using the reciprocals of the inertial functions instead of their actual values and gives appropriate numerical results.

## 2. Rationale for improved method of asymptotic modelling

The method proposed in Ilanko (2005) was illustrated by using graphs for four numerical examples. The examples and graphs are briefly described in, respectively, the next two paragraphs.

Examples 1–3 are lumped mass and massless spring systems, which are adequately defined by figure 1*a–c*, if it is noted that (*b*) and (*c*) each include a pulley system, which includes a suspended pulley of mass *m* and which causes the displacements of the two nodes 3 and 5 to approach a common value as the mass *m* approaches ±∞. Example 4 depicted in figure 1*d* is a simple continuous system, which consists of a uniform propped cantilever of length *L*, flexural rigidity *EI* and uniformly distributed mass per unit length *μ*.

Figure 2 shows how the results were plotted in Ilanko (2005). Note that both the lowest curves on each of figures (*a–d*) become zero at *m*=∞ and also that there are two such curves for (*b*) and (*c*) because examples 2 and 3 both have two constraints. The horizontal dashed lines show the correct natural frequencies of the constrained systems, which must be estimated as the lines to which the curves become tangential at *m*=∞ and *m*=−∞. Even with the larger span (−20 to +20) of Ilanko (2005), this is not easy to do. It is also not very easy to follow because the required answer lies outside the boundaries of the graph. In a practical sense these problems do not cause much difficulty in delimiting the natural frequencies of the constrained systems, which is done by increasing the magnitude of the inertial parameter until the difference between the natural frequencies of the system with positive and negative inertia becomes less than the desired tolerance. However, in many finite element applications the use of large penalty terms is known to be a source of ill-conditioning. For this reason, it is important to limit the magnitude of the penalty terms to as small a value as possible. Therefore, it is desirable to seek further measures to minimize the magnitude of artificial inertial parameters.

## 3. The proposed method

To address the issues identified above, it is now proposed that reciprocal inertias should be used as the abscissae, instead of inertias. Hence figure 2 is replaced by figure 3, which has the advantages that the curves which matter are surprisingly near to being straight and also give the required natural frequencies as their intercepts with a vertical line at 1/*m*=0 or *μL*/*m*=0. Hence linear interpolation gives the natural frequencies as the mean of the frequencies given by any pair of numerically equal values of *m* and −*m*.

An upper bound on the error caused by the linear interpolation procedure just described is equal to half the difference between the natural frequencies given by using the positive and negative mass values. This upper bound is shown in table 1, which covers each natural frequency of the four examples, using eight pairs of values (*m*, −*m*), the first seven of which are equally spaced on a logarithmic scale. It can be seen that until the bounds become so close that ill-conditioning, combined with the 16 figure accuracy of the computations, intervenes the results rapidly settle to approximately linear convergence as the bounds become closer. For instance, for all four examples making the bounds 10 times closer reduces the upper bound on the error by a factor of approximately 10 for all 11 natural frequencies of table 1, as can be seen by comparing the rows for bounds of ±10^{−1},±10^{−2},±10^{−3} and ±10^{−4}. The results for ±10^{−9} have been included because they show that even using minutely separated bounds does not result in significant ill-conditioning of the results except in the case of example 4 for which the results are still accurate enough for many purposes.

Table 2 presents the percentage error in the value of the frequencies obtained by the linear interpolation. It also covers each natural frequency of the four examples, but this time the first six of the seven pairs of values (*m*, −*m*) used are equally spaced on a logarithmic scale. The rows denoted by zero in the table were obtained by exact methods for examples 1–3 and by using the Lagrangian multiplier method in the case of example 4. The values in these rows form the datum from which the percentage errors given in the rest of the table were calculated. The extremely high accuracy, which can be obtained without the onset of ill-conditioning is immediately apparent from these percentage errors. The observed linearity of the curves of figure 3 throughout a wide range in the vicinity of interest is powerfully emphasized by the fact that the results in table 2 are reasonably accurate even when *m*=1 (or *μL*/*m*=1 for example 4) which for all four examples is equal to, or of the order of, the mass of the system being studied. It can also be seen that until the bounds become so close that ill-conditioning intervenes the results rapidly settle to approximately quadratic convergence as the bounds become closer. For instance, for all four examples making the bounds 10 times closer reduces the error by a factor of approximately 100 for all 11 natural frequencies of table 2, as can be seen by comparing the rows for bounds of ±10^{−1},±10^{−2} and ±10^{−3}. As with table 1, the results for ±10^{−9} show that even using minutely separated bounds causes either negligible or small amounts of ill-conditioning.

## 4. Concluding remarks

The natural frequencies of constrained systems modelled with artificial inertial parameters appear to be approximately linearly related to the inverse of the inertial parameter over a very wide range of values of the parameter. This enables accurate determination of the natural frequencies using interpolation. The simplest form of interpolation uses the average of a pair of natural frequencies of positive and negative inertias of equal magnitude and is found to give accurate results even for moderate magnitudes of inertia. Half of the difference between the results for systems with positive and negative inertia of equal magnitude gives an upper bound on the error due to violation of the constraint to give even more confidence to this method. Therefore, calculation of this upper bound error is recommended as a way to avoid or minimize the occurrence of numerical problems when using asymptotic modelling. However, the error appears to be very much smaller than this upper bound on its value, because the results given by linear interpolation have been observed to approach the correct answers quadratically as the bound separation is decreased, such that serious ill-conditioning only arises when the inertial parameter is of the order of a million or more times that needed to get reasonably accurate results. Thus there is, at least often, a very wide range of inertial values for which acceptable results will be obtained, which makes it relatively easy to choose values of the inertial parameter which will give acceptable results.

## Footnotes

↵† Formerly at: Department of Mechanical Engineering, University of Canterbury, New Zealand.

- Received October 30, 2005.
- Accepted December 13, 2005.

- © 2006 The Royal Society