Creating a Meaningful Interpretation of Existing Abstract Inequalities and Linking It to Real Applications

Meaningful Interpretations of Abstract Algebraic Inequalities with Applications to Real Physical Systems

There is an alternative way of using inequalities for reliability improvement and risk reduction. This method consists of moving in the opposite direction: it starts with an existing abstract inequality and moves towards the real system or a process. An important step in this process is creating relevant meaning for the variables entering the algebraic inequality, followed by a meaningful interpretation of the left and right parts of the inequality which links it with a real physical system or process.

This process has been outlined in Figure 7.1.

Attaching meaning to the variables in the abstract inequality and a correct and meaningful interpretation links the abstract inequality with reality and the abstract inequality now expresses a physical property.

While the proof of an inequality does not normally pose problems, the meaningful interpretation of an inequality is not a straightforward process. The meaningful interpretation of the variables and the different parts of an inequality usually brings deep and non-trivial insights, some of which stand at the level of a physical law.

Consider an example of this approach. The abstract algebraic inequality

is valid for any set of n non-negative quantities x,. This inequality has been proved rigorously in Chapter 3 by using different techniques.

Appropriate meaning can be attached to the variables entering this inequality, and the two sides of the inequality can be interpreted in various meaningful ways.

Improving reliability and reducing risk by creating relevant meaning and interpretation for an existing abstract inequality

FIGURE 7.1 Improving reliability and reducing risk by creating relevant meaning and interpretation for an existing abstract inequality.

A relevant meaning for the variables in the inequality can be created, for example, if each x, stands for ‘electrical resistance of element V. From physics (Tipler and Mosca, 2008), the equivalent resistances Re s and Re p of n resistors arranged in series and parallel are given by:

where x, is the resistance of the /'th resistor (/ = 1,.,.,/г).

In this case, expression (7.2) on the left side of inequality (7.1) can be meaningfully interpreted as the equivalent resistance of n resistors arranged in series (Figure 7.2a). On the right side of inequality (7.1), expression (7.3) can be meaningfully interpreted as the equivalent resistance of n resistors arranged in parallel (Figure 7.2b).

Inequality (7.1) now expresses a physical property: the equivalent resistance of n resistors arranged in series is at least n2 times larger than the equivalent resistance of the same resistors arranged in parallel, irrespective of the individual values of the resistors. It needs to be pointed out that for resistors of equal values, the equivalent resistance in parallel is exactly n2 times smaller than the equivalent resitance of the resistors in series. This is a simple result, easily derived and known for a long period of time (Rozhdestvenskaya and Zhurovskii, 1968). The bound provided by inequality (7.1) is much deeper. It is valid for any possible values of the resistances.

If inequality (7.1) is written as

this property can also be formulated as: ‘the equivalent resistance of n resistors arranged in parallel is at least n times smaller than the average resistance x = (x, + x2 +... + x„)/n of the resistors, irrespective of the individual values x, of the resistors’. This bound is much stronger than the well-known bound: ‘The equivalent resistance of resistors arranged in parallel is smaller than the least resistance’.

Meaningful interpretations of inequality

FIGURE 7.2 Meaningful interpretations of inequality (7.1) involving (a) resistors arranged in series; (b) resistors arranged in parallel; (c) thermal elements arranged in series and (d) thermal elements arranged in parallel.

The meaning created for the variables in the inequality is not unique and can be altered.

Identical reasoning applies to the problem related to the equivalent thermal resistance of n thermally conducting slabs of different materials arranged in series (Figure 7.2c) and the equivalent thermal resistance of the n slabs arranged in parallel (Figure 7.2d) (Tipler and Mosca, 2008).

Inequality (7.1) now expresses a different physical property: the equivalent thermal resistance of n slabs in series is at least n2 times larger than the equivalent thermal resistance of the same slabs arranged in parallel, irrespective of the individual thermal resistances of the slabs.

Now suppose that x, means capacitance. From physics (Tipler and Mosca, 2008), the equivalent capacitance Ce,p and Ces of n capacitors arranged in parallel and series is given by:

where x, is the capacitance of the ith capacitor (i = 1,...,»).

Expression (7.4) on the left side of inequality (7.1) can now be meaningfully interpreted as the equivalent capacitance of n capacitors arranged in parallel (Figure 7.3a). On the right side of inequality (7.1), expression (7.5) can be meaningfully interpreted as the equivalent capacitance Ce !i of n capacitors arranged in series (Figure 7.3b). The inequality now' expresses a different physical property: the equivalent capacitance of n capacitors arranged in parallel is at least ir times larger than the equivalent capacitance of the same capacitors arranged in series, irrespective of the capacitance of the individual capacitors.

Meaningful interpretations of inequality (7.1) involving (a) capacitors arranged in parallel; (b) capacitors arranged in series

FIGURE 7.3 Meaningful interpretations of inequality (7.1) involving (a) capacitors arranged in parallel; (b) capacitors arranged in series: (c) elastic elements arranged in parallel and (d) elastic elements arranged in series.

If inequality (7.1) is written as

this property can also be formulated as: ‘the equivalent capacitance of n capacitors arranged in series is at least n times smaller than the average capacitance x = (xi +x2 + ... + x„)/n of the capacitors, irrespective of the individual values x, of the capacitance’. This bound is much stronger than the well-known upper bound: ‘The equivalent capacitance of capacitors arranged in series is smaller than the least capacitance’.

Suppose that new meaning for the variables x, in inequality (7.1) is created, for example, each x, now stands for stiffness of an elastic element i. Consider the equivalent stiffness k,,:S of n elastic elements arranged in series and the equivalent stiffness k,. p of n elastic elements arranged in parallel. The stiffness of the separate elastic elements denoted by xbx2,.. .,x„ is unknown. The equivalent stiffness of n elastic elements in series is given by the well-known relationship:

and for the same elastic elements in parallel, the equivalent stiffness is

Now, the two sides of inequality (7.1) can be meaningfully interpreted in the following way. Expression (7.7) on the left-hand side of inequality (7.1) can be interpreted as the equivalent stiffness of n elastic elements arranged in parallel (Figure 7.3c). The right side of inequality (7.1) can be interpreted as the equivalent stiffness of n elastic elements arranged in series (Figure 7.3d). The inequality (7.1) expresses a different physical property: the equivalent stiffness of n elements arranged in parallel is at least n2 times larger than the equivalent stiffness of the same elements arranged in series, irrespective of the individual stiffness values characterising the separate elements.

These are examples of physical properties derived from a meaningful interpretation of an abstract algebraic inequality.

Inequality (7.1) is truly domain independent. It provides a tight bound in electrical engineering, theory of heat transfer and strength of components. At the same time, the uncertainty associated with the relationship between the equivalent parameters characterising elements arranged in series and parallel (due to the epistemic uncertainty related to the values of the building elements) has been reduced.

These examples also demonstrate that the best results in reducing uncertainty are obtained from combining domain-specific knowledge and the domain- independent method based on algebraic inequalities. Domain-specific knowledge alone is not sufficient to achieve uncertainty reduction. These simple bounds have never been suggested in standard textbooks and research literature related to the specific domains from which they have been taken. This clearly demonstrates that the lack of knowledge of the domain-independent method based on algebraic inequalities made these simple results invisible to the domain experts.

Consider the abstract inequality

valid for any sequence al,a2,...,a„ of real numbers and any sequence Ь{2,...,Ьп for positive real numbers. Inequality (7.8) has been proved by induction in Chapter 3.

A relevant meaning can be created for the variables ah h, entering the inequality. Inequality (7.8) creates the unique opportunity to segment a factor a = £"=| a, and a factor h = £"=|h, in order to achieve a larger effect (the sum on the left-hand side of inequality 7.8). Factor a could, for example, be voltage and factor h could, for example, be resistance.

Consider, a source of voltage V applied to n elements in series, with resistances rbr2,...,r„. The source of voltage V could be segmented into n smaller sources of voltage v, such that V = v, +...+v„. Let a, = v„ i = 1,...,« be the segmented sources applied to the separate elements with resistances If h, = inequality (7.8) can be rewritten as

Both sides of inequality (7.9) can be interpreted in the next new theorem, which is relevant to electrical circuits:

Theorem. The power output from a source with voltage V, on elements in series, never exceeds the total power output from the sources V/, £, v, = V, resulting from segmenting the source V and applying the voltages v, to the separate elements.

This theorem holds irrespective of the individual resistances /; of the elements and the individual voltages v, into which the voltage V is segmented.

According to the rearrangement inequality, to maximise the left-hand side of inequality (7.9), the squared voltages arranged in descending order v2 > vf >... > v2 must correspond (be applied) to the resistances arranged in ascending order (/; < r2 <... < r„).

Thus, for elements with resistances /; = l OQ, r2 = 15Q. r} = 25Q and r4=50Q and voltage source of 16V segmented into v, = 6V, v2 = 5V, v3 = 3V and v4 = 2V, the maximum possible power that can be obtained from the voltage sources is = v2M + vf/r2 + v32//3 + v42/r4 = 5.71V. Any other permutation of voltages to elements will result in a smaller power. If the source of V = 16V is not segmented, then applying the voltage to the four elements in series delivers power of only P = V2I(> + r2 +1} + r4) = 2.56W which is more than twice smaller than the maximum power of Pmax = 5.7W obtained from the voltage source segmentation. By using inequality (7.8) it can also be shown that the energy stored by a charge Q in a capacitor with capacitance C, never exceeds the total energy stored in multiple capacitors, the sum of whose capacitances is equal to C, by segmenting the initial charge Q over the separate capacitors.

These results have not been stated in any publication in the mature field of electrical engineering and electronics, which demonstrates that the lack of knowledge of the domain-independent method of inequalities for uncertainty made these simple results invisible to domain experts.

Applications Related to Robust and Safe Design

The robust design of clamping devices often requires a small variation of the spring force with the spring length. An application of the formulated relationship about the equivalent stiffness of elastic elements can be found in the robust design of clamping devices.

The same required clamping force F can be provided by n springs arranged in parallel, with a large equivalent spring constant kp (stiffness) and a small initial deflection xp ( f = kpXp) (Figure 7.4a) or by the same n springs arranged in series, with a much smaller equivalent spring constant ks < kp and large initial deflection xs >xp (F = ktxx) (Figure 7.4b).

The initial spring deflections xp and x, are always associated with a variation A.v due to errors in cutting the springs to exact length, imperfections associated with machining the ends of the spring coils, sagging of the springs with time due to stress relaxation, variations in the lengths of the springs associated with pre-setting operations, and so on. This variation gives rise to variations AFp and ДFs of the clamping forces of the springs. Because of the different equivalent spring constants, the variations in the clamping force caused by the same variation of the initial spring deflection are very different in magnitude (Figure 7.4).

The same clamping force F attained by (a) n springs arranged in parallel and (b) the same n springs arranged in series

FIGURE 7.4 The same clamping force F attained by (a) n springs arranged in parallel and (b) the same n springs arranged in series.

Indeed, for springs arranged in parallel, from F + AFP = kp(xp +Ax), it follows that ДFp= kpAx. For springs arranged in series, from F + AF, = ks(xs + Ax) it follows that AF, = ksAx. Since kp > ks, it follows that AF,. > AF.

According to inequality (7.1), for springs (elastic elements) arranged in series, rather than parallel, the equivalent spring constant (the equivalent stiffness) is at least n2 times smaller than the equivalent spring constant (the equivalent stiffness) of the parallel arrangement, irrespective of the uncertainty related to the spring constants of the individual springs (elastic elements).

Consequently, the variation AF, of the clamping force of the springs in series is at least n2 smaller than the variation AFp of the clamping force characterising the same springs arranged in parallel. Selecting springs (elastic elements) arranged in series results in a robust design, for which the clamping force F is not particularly sensitive to variations in the initial deflection.

Another application of the formulated relationship related to the equivalent stiffness of a parallel and series arrangement can be found in increasing the energyabsorbing potential upon impact. By arranging elastic elements in series rather than parallel, the smaller equivalent stiffness ke will reduce the maximum stress max upon impact and with this, the risk of overstress failure will also be reduced.

Indeed, the strain energy U accumulated by elastic elements with equivalent stiffness ke is given by Е/ = -A- (Gere and Timoshenko, 1999) where P is the maximum force acting on the component. For a single prismatic component with length L, cross-sectional area A and material with Young’s modulus E, loaded in ten- sion/compression, the stiffness is к = EA/L and the strain energy takes the form: U = pl£.

2ЕЛ

Consider an impact of a body with mass m and velocity v and an elastic component with stiffness ke. Suppose that, upon the impact, a fraction у of the kinetic energy Ek = of the body at the point of impact is transformed entirely into strain energy of the impacted elastic component. The dynamic force resulting from the impact can then be evaluated by equating the strain energy U and the transformed into strain energy kinetic energy Ek: = yEk, from which the dynamic force P can be obtained:

From the last equation, it is clear that if the equivalent stiffness ke is reduced rr times by altering the parallel arrangement to a series arrangement, the magnitude of the initial dynamic force will be reduced n times:

Consequently, altering the arrangement of the elastic elements from parallel to series significantly reduces the magnitude of the dynamic forces upon impact. Even the rearrangement of two elastic elements from parallel to series is sufficient to halve the magnitude of the dynamic force.

 
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