# Scale model

A scale model of the Tower of London. This model can be found inside the tower.
Scale model of water powered turbine
File:ScaleCarComparison.jpg
L to R with 12 inch ruler at bottom: 1:64 Matchbox Chevrolet Tahoe, 1:43 Ford F-100, 1:25 Revell Monogram 1999 Ford Mustang Cobra, 1:18 Bburago 1987 Ferrari F40
Model ships and castle

A scale model is most generally a physical representation of an object, which maintains accurate relationships between all important aspects of the model, although absolute values of the original properties need not be preserved. This enables it to demonstrate some behavior or property of the original object without examining the original object itself. The most familiar scale models represent the physical appearance of an object in miniature, but there are many other kinds.

Scale models are used in many fields including engineering, architecture, film making, military command, salesmanship and even fun in hobby model building. While each field may use a scale model for a different purpose, all scale models are based on the same principles and must meet the same general requirements to be functional. The detail requirements vary depending on the needs of the modeler.

To be a true scale model all relevant aspects must be accurately modeled, such as material properties, so the scale model's interaction with the outside world is reliably related to the original object's interaction with the real world.

## Requirements for scale models

In general a scale model must be designed and built primarily considering similitude theory. However, other requirements concerning practical issues must also be considered.

### Similitude requirements

Similitude is the theory and art of predicting prototype (original object) performance from scale model observations.[1] The main requirement of similitude is all dimensionless quantities must be equal for both the scaled model and the prototype under the conditions the modeler desires to make observations. Dimensionless quantities are generally referred to as Pi terms, or π terms. In many fields the π terms are well established. For example, in fluid dynamics, a well known dimensionless number called the Reynolds number comes up frequently in scale model tests with fluid in motion relative to a stationary surface.[2] Thus, for a scale model test to be reliable, the Reynolds number, as well as all other important dimensionless quantities, must be equal for both scale model and prototype under the conditions that the modeler wants to observe.

An example of the Reynolds number and its use in similitude theory satisfaction can be observed in the scale model testing of fluid flow in a horizontal pipe. The Reynolds number for the scale model pipe must be equal to the Reynolds number of the prototype pipe for the flow measurements of the scale model to correspond to the prototype in a meaningful way. This can be written mathematically, with the subscript m referring to the scale model and subscript p referring to the prototype, as follows:

where

Observing the equation above it is clear to see that while the Reynolds numbers must be equal for the scale model and the prototype, this can be accomplished in many different ways. For example, in this problem by altering the scale of the dynamic viscosity of the model to work with the scale of the length. This means, the scales of different quantities, for example a material's elasticity in the scale model verses the prototype, are governed by equating the dimensionless quantities and the other quantity's scaling within the dimensionless quantity to ensure the dimensionless quantity of interest is of equal magnitude for the scale model and prototype.

#### Scaling

With the above understanding of similitude requirements, it becomes clear the scale often reported in scale models refers only to the geometric scale, ${\displaystyle \mathrm {S} _{L}}$ (L referring to length), and not the scale of the parameters potentially important to consider in the scale model design and fabrication. In general the scale of any quantity i, perhaps material density or viscosity, is defined as:

where

This relationship must be applied to all quantities of interest in the prototype, observing similitude requirements—so the scale model can be built using dimensions and materials that make scale model testing results meaningful with respect to the prototype.[3] One method to determine the dimensionless quantities of concern for a given problem is to use dimensional analysis.

### Practical requirements

Practical concerns include the cost to construct the model, available test facilities to condition and observe the model, the availability of certain model materials, and even who will build the scale model. Practical requirements are often very diverse depending on the purpose of the scale model and they all must be considered to have a successful scale model experience.

As an example, perhaps an aerospace company needs to test a new wing shape. According to the similitude requirements the test must be carried out in a wind tunnel that can drop the temperature of the air to -128°C, such as the 0.3-Meter Transonic Cryogenic Tunnel at NASA Langley Research Center.[4] However, if a facility such as this one can't be used, perhaps due to cost constraints, the similitude requirements must be relaxed or the test redesigned to accommodate the limitation.

## Classes

For a scale model to represent a prototype in a perfectly true manner, all the dimensionless quantities, or π terms, must be equal for the scale model during the observational period and the prototype under the conditions the modeler desires to study. However, in many situations, designing a scale model that equates all the π terms to the prototype is simply not possible due to lack of materials, cost restrictions, or limitations of testing facilities. In this case, concessions must be made for practical reasons to the similitude requirements.

Depending on the phenomena being observed, perhaps some dimensionless quantities aren't of interest and thus can be ignored by the modeler and the results of the scale model can still safely be assumed to correspond to the prototype. An example of this from fluid dynamics is flow of a liquid in a horizontal pipe. Possible π terms to consider in this situation are Reynolds number, Weber number, Froude number, and Mach number. For this flow configuration, however, no surface tension is involved, so the Weber number is inappropriate. Also, compression of the fluid is not applicable, so the Mach number can be disregarded. Finally, gravity is not responsible for the flow, so the Froude number can also be disregarded. This leaves the modeler with only the Reynolds number to worry about in terms of equating its values for the scale model and the prototype.[5]

In general, scale models can be classified into three classes depending on the degree of similitude satisfaction they exhibit. To begin, a true model is one with complete similitude—that is, all π terms are equal for the scale model and the prototype. True models are difficult to realize in reality due to the many possible quantities the modeler must consider. As a result, modelers identify the important dimensionless quantities and construct a scale model that satisfies these. Important dimensionless quantities are called first-order dimensional requirements. A model that satisfies first-order similarity is called an adequate model. Finally, for scale models that fail to satisfy one or more of the first-order requirements, the name distorted model is given.[6]

## Examples

Scale models are used by many fields for many different purposes. Some of the specific uses of scale models by specific fields are explained below in the examples.

### Structural scale model

Although structural engineering has been a field of study for thousands of years and many of the great problems have been solved using analytical and numerical techniques, many problems are still too complicated to understand in an analytical manner or the current numerical techniques lack real world confirmation. When this is the case, for example a complicated reinforced concrete beam-column-slab interaction problem, scale models can be constructed observing the requirements of similitude to study the problem. Many structural labs exist to test these structural scale models such as the Newmark Civil Engineering Laboratory at the University of Illinois, UC.[7]

This is a load confinement box from the University of Illinois, UC Structural engineering lab. It can impart six degrees of freedom on structural scale models.[8]

For structural engineering scale models, it is important for several specific quantities to be scaled according to the theory of similitude. These quantities can be broadly grouped into three categories: loading, geometry, and material properties. A good reference for considering scales for a structural scale model under static loading conditions in the elastic regime is presented in Table 2.2 of the book Structural Modeling and Experimental Techniques.[9]

Structural engineering scale models can use different approaches to satisfy the similitude requirements of scale model fabrication and testing. A practical introduction to scale model design and testing is discussed in the paper "Pseudodynamic Testing of Scaled Models".[10]

### Model aircraft

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Scale model of a Douglas DC-3

Model aircraft are divided into two main groups: static and flying models.

#### Static model aircraft

Static model aircraft are commonly built using plastic, but wood, metal, card and paper can also be used. Models are sold painted and assembled, painted but not assembled (snap-fit), or unpainted and not assembled. The most popular types of aircraft to model are commercial airliners and military aircraft. Fewer manufacturers exist today than in the 1970s, but many of the older kits are occasionally available to purchase. Aircraft can be modeled in many "scales". The scale notation is the size of the model compared to the real, full-size aircraft called the "prototype". 1:8 scale will be used as an example; it is read as: "1 inch (or whatever measurement) on the model is equal (: means equal) to 8 inches on the real (prototype) airplane". Sometimes the scale notation is not used; it is simply stated: "my model is one eighth (1/8) scale", meaning "my model is one eighth the size of the real airplane" or "my model is one eight as large as the real airplane". Popular scales are, in order of size, 1:144, 1:72 (the most numerous), 1:48, 1:32, 1:24, 1:16, 1:8 and 1:4. Some European models are available at more metric scales such as 1:50. The highest quality models are made from injection-molded plastic or cast resin. Models made from Vacuum formed plastic are generally for the more skilled builder. More inexpensive models are made from heavy paper or card stock. Ready-made die-cast metal models are also very popular. As well as the traditional scales, die-cast models are available in 1:200, 1:250, 1:350, 1:400, and 1:600.

## Notes

1. Crowe, C., et al. 2010, p. 259
2. Crowe, C., et al. 2010, p. 262
3. Harris, H., et al. 1999, p. 57
4. http://gftd.larc.nasa.gov/facilities/3-m_tct.html
5. Crowe, C., et al. 2010, p. 263
6. Harris, H., et al. 1999, p. 56
7. http://cee.illinois.edu/portalresearch_facilities
9. Harris, H., et al. 1999, p. 62
10. Kumar, et al. 1997, p. 1
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1. In the Lego community, micro scale can refer to anything smaller than minifig scale (1:48), but 1:192 is occasionally set as a standard micro scale. This ratio is arrived at by scaling a person (6 feet) to the height of a Lego brick (3/8 inches). See {{#invoke:citation/CS1|citation |CitationClass=book }}

## References

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