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How Do I Form a Performance Index?

1. the functional requirements (the need to carry loads, transmit heat, store elastic or thermal energy, etc)
2. the geometry, and
3. the properties of the material of which it is made, including its cost.

where

• p describes the aspect of performance of the component that is to be optimised: its mass, volume, cost or life for example
• f( ) means 'a function of'.
• F are the specified Functional Requirements
• G are the specified Geometrical Parameters, and
• M are the Material Properties

Experience shows that the groups are usually separable and the equation can be written as:

where f₃(M) is known as the performance Index or Performance Index,

As an example, consider a material required for the core of a thermal storage heater (one that heats up at night using cheap electricity and releases heat during the day). Low cost is essential. We seek a performance index characterising the materials which can store the most thermal energy per unit cost.

Function	Heat-storing medium
Objective	Maximise thermal energy stored per unit material cost
Constraints	Adequate working temperature - 100 to 150°C

The thermal energy E stored in a mass m of solid when heated through a temperature interval DT is

where Cp is the heat capacity of the solid. The material cost is

where Cm is the cost per kg of the solid. The energy stored per unit cost is therefore

The quantity DT is prescribed - it is a functional requirement, in this case 100 to 150°C. The remaining term on the right (in brackets) is the performance index. The energy per unit cost is maximised by maximising

By using this method, the optimum choice of material becomes independent of the design details. The optimum material is the same for all geometries G, and all values of the functional requirements F. Then the optimum material can be identified without solving the complete design problem or even knowing all the details of F and G. This enables enormous simplification: the performance for all F and G is maximised by maximising the 'performance index'.

Each combination of function, objective and constraint leads to a material index (see figure below). The index is characteristic of the combination. In the figure below it can be seen that selection of a beam of minimum weight subject to a constraint on stiffness requires maximisation of the material index M = E^1/2/r where E is the Young's Modulus and r is the Density.