The concept of negative mass in vibration analysis
There may not be a “negative mass” in the universe but just as the concept of “imaginary number” helps to solve a vast range of mathematical problems the concept of negative mass can be used to solve some vibration problems.
The natural frequencies of an object depend on constraints such as the requirement that the movement of a bridge be zero at the supports. In some computational methods, the vibratory displaced form of the object is expressed as a series of assumed shapes, each conforming to the support constraints. The natural frequencies and vibration modes of the object are then determined by applying the physical law governing the motion. In the popular energy method known as the Rayleigh-Ritz method, this is achieved by minimising an energy function. This procedure allows the contribution from each assumed shape to be adjusted in such a way as to produce the best possible estimate of the natural frequencies and modes.
What is the limitation of this method?
However, for many practical problems, selecting shapes that satisfy all such conditions is a challenging task. One common trick employed by physicists and engineers is to model rigid supports as elastic springs supports but with very high stiffness. This allows more freedom in the choice of the shapes to be used in the analysis as each shape is allowed to violate the support conditions. However, the final shape consisting of a combination of the assumed shapes will be forced to have only very small movements at the supports. This is because the energy required to stretch or compress the artificial springs is included in the analysis. The high stiffness would effectively force the structure to have only very small displacements at the supports. The idea of replacing rigid supports with stiff spring supports was put forward at least as early as in 1943, by Professor Courant. Although it has enhanced the Rayleigh-Ritz method, there are still some problems with this approach.
What are the problems with this trick?
The main problem is in choosing the value for the spring stiffness. If it is not large enough, the error due to the movement at the support may be large. If it is too large there may be computational problems. Generally a suitable value is selected by trial and error until the solutions show some convergence. It is well known that the natural frequencies of an object would increase with the stiffness. For example, tightening a guitar string would make the string stiffer (it would be harder to pluck it) and increase its pitch (the frequency). So, with increasing stiffness of the artificial springs, the natural frequencies will increase but they can never exceed the frequencies of the object with properly constrained conditions at the support (which is like using springs with infinite stiffness). The frequencies approach the frequencies of the system that is being modelled from below. However, the actual error due to this modelling remained unknown. Recently, Ilanko showed that by using negative values for the stiffness, the frequencies of the actual system can be approached from above. Therefore by using positive and negative stiffness values for the springs, one can calculate and control the error due to modelling but the use of negative stiffness values introduces some critical regions which must be calculated and avoided.
What is the new solution?
To suppress the vibration of an object at a support where it ought to be zero, an artificial mass or inertia could also be attached to the object and the kinetic energy required to vibrate the mass included in the analysis. If the mass is very large, the mathematical process will ensure that the assumed shape will be combined in such a way to produce only very small movements at the support. The natural frequencies decrease with mass and the frequencies of the object with proper support conditions can be approached from above. What is interesting is that if one were to use negative values for the mass or inertia in calculations, the natural frequencies increase and for very large magnitudes, they approach the required frequencies from below. Unlike artificial stiffness, the use of positive and negative artificial mass in calculations does not introduce critical regions and the required natural frequencies may be calculated more conveniently.
To read some news behind this development visit my university chronicle - read page 6
To read this article and another article on using negative stiffness for modelling visit the Royal Society Publications page and search for "negative inertial functions" and "asymptotic modelling theorems"
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