# Centrifugal Modeling

## Centrifugal Modeling

a method of physical modeling used for scientific research and for modeling studies of the properties or characteristics of structures acted on by gravity. Centrifugal modeling is used most often in the study of earthen structures or of structures that interact with soil, such as slopes, embankments, dams, foundations, and underground structures. The purpose of such studies is to determine the stresses and strains occurring in a structure, that is, to determine the conditions under which the structure is safe from failure or to establish the causes and nature of structural failure.

The methods of centrifugal modeling may be used to study the effect on human beings and objects of the *g* loads experienced in atmospheric and space flights (see SPACE FLIGHT, SIMULATION OF). The concept of centrifugal modeling may also be employed to create artificial gravity aboard a spacecraft (see WEIGHTLESSNESS).

In modeling, equality of similarity criteria must be obtained. When the primary load on a structure is due to gravity and the model is made of the same material as the full-scale object, the

condition for modeling is

(1) *g*_{1}*l*_{1} = *gl*

where *g* is the acceleration of gravity, *l* is the linear size of the full-scale object, *g*_{1} is the model acceleration, and *l*_{1}, is the linear size of the model. Since the model is usually smaller than the full-scale object, that is, since *l*_{1} < *l*, conditions under which *g*_{1} > *g* must be provided for in the model. Such conditions may be approximated by placing the model in a centrifuge. This is the idea of centrifugal modeling.

In the centrifuge, a chamber that contains the model rotates about the vertical axis with an angular velocity of ω. A centrifugal force acts on every particle of the model; the force is exerted outward from the axis of rotation and is equal to *m _{k}h_{k}* ω

^{2}, where

*m*is the mass of a particle and

_{k}*h*is the particle’s distance from the axis of rotation. The dimensions of the centrifuge are such that the distances

_{k}*h*are large in comparison with the dimensions of the model. Then, the approximation may be made that all

_{k}*h*=

_{k}*h*, where

*h*is the distance of the model’s center of gravity from the axis of rotation. In addition, the forces acting on the particles of the model may be considered equal to

*m*ω

_{k}h^{2}, that is, analogous to the gravitational forces

*m*

_{k}g_{1}. where

*g*

_{1}=

*h*ω

^{2}. As a result, condition (1) takes on the form

(2) *h*ω^{2}*l*_{1} = *gl* or ω^{2} = *gl*/*l*_{1}*h*

The angular velocity at which centrifugal modeling may be carried out for a model of a given size is determined on the basis of condition (2), since the smaller *l*_{1}, the higher should be ω.

If the model and the full-scale object are made of materials with different densities and with different strength characteristics—as given, for example, by Young’s modulus *E*—the condition for modeling changes. In this case, centrifugal modeling may be performed when

(3) ω^{2} = *gl* ρ *E*_{1}/*l*_{1}ρ_{1}*Eh*

An annular chute in the form of a closed water-filled ring that rotates about the vertical axis passing through the center of the ring is used in the centrifugal modeling of the motion of bodies near the surface in water or of wave formation and motion. In such modeling, equality of the Reynolds numbers and of the Froude numbers may be obtained simultaneously.

The general concept of centrifugal modeling was proposed by the French scientist E. Phillips in 1869. In the USSR, the concept was developed in detail and applied by G. I. Pokrovskii and I. S. Fedorov in 1932.

### REFERENCES

Pokrovskii, G. I., and I. S. Fedorov.*Tsentrobezhnoe modelirovanie v stroitel’nom dele*. Moscow, 1968.

Pokrovskii, G. I., and I. S. Fedorov.

*Tsentrobezhnoe modelirovanie v gornom dele*. Moscow, 1969.

Ramberg, H.

*Modelirovanie deformatsii zemnoi kory s primeneniem tsentrifugi*. Moscow, 1970. (Translated from English.)

G. I. POKROVSKII