1.1. Introduction to the design problem

Designing a gearbox is a complex process. The American Gear Manufacturers Association (AGMA) is an important authority responsible for disseminating knowledge pertaining to the design and analysis of gearing. The methods presented by this organisation are in general use in the United States when strength and wear are of primary concern. Although AGMA provides a systematic procedure for designing gears, they also offer numerous correction factors associated with modelling the geometry, materials, manufacturing process and operating conditions. These correction factors have a huge impact on the design. It is critical to accurately ascertain their values to ensure that the gears are safe and reliable but not overdesigned. However, this requires comprehensive knowledge about operating conditions, materials, design and manufacturing processes which are often challenging. The presence of uncertainties involved at various stages of making gearboxes (see Table 2) adds to the challenge of designing safe and reliable gearboxes. The two major decisions in the design of gearboxes are decisions about the dimensions and the material. In addition, a gearbox designer is also expected to fulfil a number of design constraints while also fulfilling multiple functional requirements.

Table 2. Critical review of literature

By relying on the AGMA guidelines and procedure, through a robust design method, we offer an extension on the AGMA design procedure to support robust decision-making for designing gearboxes using coupled decisions that has the following salient features:

1. (i) Consideration of multiple design criteria.

2. (ii) Coupled decision approach for simultaneous material selection and geometry exploration.

3. (iii) Robust decision-making to manage uncertainties.

4. (iv) Consideration of decision coupling for modelling concurrent and hierarchical decisions.

The simplified flowchart to illustrate AGMA procedure (AD) and the proposed robust design method for designing gears (RD) is shown in Figure 1. The design procedure implemented in the two methods can be explained in a simplified way with the following steps:

Steps in the AGMA design procedure (AD)

• Step 1: Design input. In Step 1, the design inputs required to solve the design problem are identified and defined. The information about power requirement (torque and speed), design variables and their bounds, and constraints are clearly defined.

• Step 2: Select material. In Step 2, a suitable material is selected. This selection involves making judgments using the information gathered in Step 1. Then, all necessary correction factors as defined by AGMA are applied to the material.

• Step 3: Determine design variables. In Step 3, suitable values for the design variables are ascertained. This may involve some calculations and iterations.

• Step 4: Satisfy design constraints. The values ascertained for the design variables and the selected material are checked to determine whether the design constraints identified in Step 1 are satisfied. If the constraints are satisfied, a design solution is obtained, otherwise, Steps 1, 2 and/or 3 are iterated until a satisfactory solution is obtained.

Steps in the proposed robust design method (RD):

• Step 1: Design input. In Step 1, design inputs required to solve the design problem are identified and defined. The power requirement (torque and speed), design variables and their bounds, and constraints are clearly defined.

• Step 2: Coupled DSP. In Step 2, a coupled DSP is formulated. The coupled DSP consists of two decisions defined in Steps 2a and 2b, which are implemented concurrently. Step 2a is a compromise decision, and Step 2b is a selection decision modelled as cDSPs and sDSPs. The implementation of the cDSP helps determine the correct values for design variables, that is, the gear geometry. At the same time, the sDSP is used to select a suitable gear material from a given pool of material alternatives. The two decisions in Figure 1 are used for illustrative purposes, while there could be multiple interacting decisions. In the gearbox example used in this article, there are three interacting decisions. Two coupled decisions of gear design interact with one decision about shaft design (see Section 2).

  Uncertainties (A) and Multiple Design Criteria (B) are inputs to the coupled DSP. The control factors, noise factors, responses and bounds are specified for the given design problem for uncertainty management. In the example implemented in this article, three types of uncertainties (Type I, Type II and Type III) are considered (see Tables 3 and 4). For consideration of multiple design criteria, goals are defined and implemented within the coupled DSP construct. By assigning different weights for the multiple goals, several design scenarios are generated.

• Step 3: Satisfy design constraints. The values ascertained for the design variables and the material selected in Step 2 using coupled DSP are checked to determine whether the design constraints identified in Step 1 are satisfied. If the constraints are satisfied, a design solution is obtained; otherwise, Steps 1 and/or 2 are repeated until a satisfactory solution is obtained.

Figure 1. Comparison of proposed design method with American Gear Manufacturers Association (AGMA) for gear design.

Table 3. Uncertainties involved in making of a gearbox

Table 4. Utility of three types of uncertainties in gearbox design

Some distinctive advantages of RD over AD for designing gears are as follows:

1. (i) Exploration of gear design space against multiple conflicting goals,

2. (ii) Simultaneous exploration of gear dimensions against available material alternatives and

3. (iii) Consideration of uncertainties enables the design of reliable and robust gears.

Although mechanical gearboxes are optimised for torque and speed, there are numerous other functional requirements to be satisfied, such as efficiency (Höhn et al. Reference Höhn, Michaelis and Hinterstoißer2009), noise, vibration (Bozca Reference Bozca2010), volume and size (Osyczka Reference Osyczka1978). The available literature that addresses gearbox design as multicriteria design problem. Höhn et al. (Reference Höhn, Michaelis and Hinterstoißer2009), Bozca (Reference Bozca2010) and Goharimanesh et al. (Reference Goharimanesh, Akbari and Tootoonchi2014) present a method for designing gearboxes with a single criterion while Osyczka (Reference Osyczka1978), Deb ' Jain (Reference Deb and Jain2003) and Stefanović-Marinović et al. (Reference Stefanović-Marinović, Troha and Milovančević2017) present a method for designing gearboxes with multiple criteria. Suitable material selection is another critical aspect of designing gearboxes. Multicriteria decision-making methods are common approaches applied in material selection. Terán et al. (Reference Terán, Martínez-Gómez and Milla2020) present multicriteria decision methods for selecting material by optimising surface fatigue in gearbox while also increasing its resistance to wear. Das et al. (Reference Das, Bhattacharya and Sarkar2016) showcase an approach for simultaneous selection of material and geometric variables in gear design using decision-based design. Designing gearboxes in the presence of uncertainty has also received due attention among researchers. Alemayehu ' Ekwaro-Osire (Reference Alemayehu and Ekwaro-Osire2014) offer a probabilistic, multibody dynamic analysis (PMBDA) design approach for designing wind turbine gearboxes by addressing the loading and design parameter uncertainties while considering five performance functions. Gautham et al. (Reference Gautham, Kulkarni, Panchal, Allen and Mistree2017) propose a robust design method for designing gears by addressing the uncertainties associated with the introduction of numerous ‘correction factors’ by the AGMA. The authors introduce a method to eliminate two correction factors from the AGMA design standards for designing a spur gear: the factor of safety in contact and the reliability factor by introducing uncertainty in the magnitude of the load and material properties. On the other hand, Ognjanovic ' Milutinovic (Reference Ognjanovic and Milutinovic2013) adopt a reliability-based methodology to design an automotive gearbox to operate under varying and random operation conditions. Salomon et al. (Reference Salomon, Avigad, Purshouse and Fleming2016) extend the gearbox design optimisation to consider uncertainties in load demand. The authors utilise active robust (AR) optimisation to optimise the number of transmissions and their gearing ratios for an uncertain load demand. AR optimisation is a methodology to design products that attain robustness to uncertain or changing environmental conditions through adaptation.

It has been established that uncertainties degrade a designer’s ability to make an informed decision. When applied, the design decisions do not account for uncertainties induced by noise factors (uncontrollable parameters), design variables (controllable parameters), model approximations used in the simulation models and uncertainty propagation across different length and time scales. Therefore, uncertainty management methods are critical to enable robust decision-making. The body of literature on available robust design methods is discussed by Eifler, Ebro ' Howard (Reference Eifler, Ebro and Howard2013). The robust design method presented in this article is based on the sources of uncertainties. In this article, three types of uncertainty are managed, Types I, II and III. Type I is related to uncertainties due to noise factors; Type II is related to uncertainties due to design variables and Type III is related to uncertainties due to mathematical models. To manage these uncertainties, three robust design methods are utilised, they are, Types I, II and III robust design methods. The details are discussed in Chen et al. (Reference Chen, Allen, Tsui and Mistree1996) and McDowell et al. (Reference McDowell, Panchal, Choi, Seepersad, Allen and Mistree2009). In context of designing gearboxes, uncertainties and methods to deal with the uncertainties are tabulated in Tables 3 and 4, respectively.

In Section 2, we describe the types of decisions required and the interactions among these decisions. In Section 3, the robustness metrics, DCI and EMI, are described in detail. In Section 4, we present the robust formulation for the coupled decision/problem in the context of designing a gearbox. The exploration of the solution space to identify satisficing robust design solutions is covered in Section 5. In Section 5, we present ternary plots as an approach to explore the robust solution space. Finally, we close the article with our remarks in Section 6.

3.1. Design capability index

DCIs represent the safety margin against system failure due to uncertainty in design variables. In particular, DCIs represent the degree of reliability by measuring the capability of design decisions:

1. (i) To satisfy the design requirements.

2. (ii) To tolerate the effect of uncertainty in design variables.

In Figure 4, the mean response (μ y) for the model is illustrated by a solid curve. At x, for the variation of +∆x in a design variable, the expected variation in the response given by the mean response model is ∆Y. This lets us calculate the maximum expected deviation in response for any given value of x and ∆x. In Figure 5, we show the mathematical formulations for implementing DCIs as a goal in DSPs. ‘Smaller is better’ signifies that we are looking to minimise the targeted function, while ‘Larger is better’ signifies maximising the targeted function. Further, ‘Nominal is better’ means we are interested in obtaining a value near to the target set. That is, we want to avoid underachievement as well as overachievement.

Steps for formulating goals as DCIs

• Step 1: Using a first-order Taylor series expansion, estimate the response variation due to variation in the design variable vector x = {x₁, x₂, …, x_n}. The response variation (∆y) for small variations in design variables is:

  (1) $$ \Delta Y=\sum_{i=1}^n\left|\frac{\mathrm{\partial}f}{{\mathrm{\partial}x}_i}\right|\cdot \Delta {x}_i. $$

• Step 2: Using the mean response (μ _y) obtained from the mean response model (f ₀(x)) and the response variation due to variation in design variables (ΔY), calculate the DCIs. For a ‘Larger is Better’ case, the DCI is calculated as:

  (2) $$ DCI=\frac{\;{\mu}_y- LRL}{\Delta Y}. $$

The DCI is a mathematical metric representing the safety margin against system failure due to uncertainty in design variables. A DCI < 1 is a mean of the system performance which falls outside the systems requirement range. A DCI ≥ 1 means that the ranged set of design specifications satisfies a ranged set of design requirements, and the system is robust against uncertainty in design variables. Therefore, a designer aims to force DCI to unity so that a more significant portion of performance deviation falls within the design requirements range. The higher the value for DCI, the higher the measure of safety against failure due to uncertainty in design variables (Choi et al. Reference Choi, Austin, Allen, McDowell, Mistree and Benson2005).

Figure 4. Formulation of uncertainty bounds due to variations in a design variable (Choi et al. Reference Choi, Austin, Allen, McDowell, Mistree and Benson2005).

Figure 5. Mathematical constructs for design capability indexes (DCIs).

3.2. Error margin index

EMIs represent the amount of safety margin against system failure due to uncertainty in the design model. In particular, EMIs represent the degree of reliability by measuring the capability of design decisions:

1. (i) To satisfy the design requirements.

2. (ii) To tolerate the effect of uncertainty in design models.

In Figure 6, the model’s mean response (μ _y ) is shown as a solid curve, and the two adjacent dotted curves represent the uncertainty bounds associated with the system model. At X, the variation of +∆X in the design variable, the expected variation in the response given by the mean response model is ∆Y. Similarly, for the same change in design variable at X, the expected variation in response for the two uncertainty bounds are ∆Y ₁ and ∆Y ₂, respectively as shown in Figure 6. This lets us calculate the maximum expected deviation in response for any given value of X and ∆X. In Figure 7, we show the mathematical formulations for implementing EMIs as a goal in DSPs.

Steps for formulating goals as EMIs

• Step 1: If a system model has k uncertainty bounds, the response variation (∆y _j) for each of them for small variation in design variables is calculated as:

  (3) $$ \Delta {y}_j=\sum_{i=1}^n\left|\frac{\mathrm{\partial}{y}_j}{{\mathrm{\partial}x}_i}\right|\cdot \Delta {x}_i, $$
  where j = 0, 1, 2, …, k (number of uncertainty bounds).

• Step 2: After evaluating the multiple response variations of mean response function and the k uncertainty bound functions for variations in design variables, the minimum and maximum responses by considering the variability in design variables and uncertainty bounds around the mean response are calculated as:

  (4) $$ {Y}_{max}=\left[(x)+\varDelta {y}_j\right], $$
  (5) $$ {Y}_{min}=\left[(x)-\varDelta {y}_j\right], $$
  where j = 0, 1, 2, …, k (number of uncertainty bounds), f ₀(x) is the mean response function and f ₁(x) …. (x) are the uncertainty bound functions.

  In Figure 6, we show a mean response function (solid black curve) and two uncertainty bounds (dotted curves in black). At any x, we can calculate the value of maximum (y _max ), minimum (y _min) and mean response (μ _y) arising due to uncertainty bounds. This will let us calculate the maximum expected deviation in response for any given value of x.

• Step 3: Calculate the upper and lower deviation of response at x as:

  (6) $$ \varDelta {Y}_u={Y}_{max}-(x), $$
  (7) $$ \varDelta {Y}_l=(x)-{Y}_{min}. $$

  The deviations ∆y _u and ∆y _l are shown in Figure 5.

• Step 4: Using the mean response (μ _y) obtained from the mean response model (f ₀(x)) and the upper and lower deviations (∆y _upper and ∆y _lower), the EMIs are calculated as shown in Figure 7. For a ‘Larger is Better’ case, the EMI is calculated:

  (8) $$ EMI=\frac{\mu_y-LRL}{\varDelta {Y}_l}. $$

Figure 6. Formulation of uncertainty bounds due to variations in a model (Choi et al. Reference Choi, Austin, Allen, McDowell, Mistree and Benson2005).

Figure 7. Mathematical constructs for EMIs.

The EMI is a mathematical metric representing the safety margin against system failure due to uncertainty in the design models. An EMI < 1 means that a requirement limit may be violated due to uncertainty in the model. An EMI ≥ 1 means that the ranged set of design specifications satisfies a ranged set of design requirements, and the system is robust against uncertainty in design models. Therefore, a designer’s aim is to force EMI to unity so that an uncertainty bound meets a requirement limit. The higher the value for EMI, the higher the measure of safety against failure due to uncertainty in design models (Choi et al. Reference Choi, Austin, Allen, McDowell, Mistree and Benson2005).

4.1. Problem statement

The design of a one-stage reduction gearbox (see Figure 8) with a gear ratio of 4 is required. The torque at input is at least 80 Nm @ 3500 rpm. The gears are required to endure at least 10⁷ fatigue cycles. The gears are cut using a rack cutter arrangement with pressure angle (α) = 20°. The reliability for gears must be at least 99%. The gearbox is to be designed for a uniform power source and moderate shock in loads. Restrictions regarding the maximum allowable stresses on the gears and shafts are specified. The design quality is measured in terms of design goals that are to be achieved as much as possible. Specifically, we need a design that has low weight and smaller height while achieving maximum torque. However, the design must have a robust performance to minimise the impact of uncertainties associated with design variables and material properties on performance. The task is to select gear material from a given pool of materials and dimensions for gears and to recommend shear strength for shaft material and shaft dimensions that give a robust performance with respect to the constraints and the design quality specified. The material properties for shafts are available for selection within the specified bounds, while gear materials are available for selection.

Figure 8. Schematic of a one-stage reduction gearbox.

4.2. Mathematical derivation for converting gearbox design goals into robust design goals

Three goals are formulated as DCIs, these are, weight of gear pair (W_g), size of gearbox (H) and weight of shaft (W_s). One goal is formulated as an EMI goal, that is, Torque. The mathematical derivation for the four compromise goals (three goals for gear design and one goal for shafts design) as robustness goals are shown in Equations (9)–(44) using Equations (1)–(8).

Goal 2: Weight of gear pair (Wg):

(9) $$ {W}_g=13.35(\rho \cdot b\cdot {m}^2\cdot {z}^2)\times {10}^{-9}(\mathrm{k}\mathrm{g}), $$

(10) $$ \frac{\mathrm{\partial}{W}_g}{\mathrm{\partial}b}=13.35(\unicode{x03C1} \cdot {\mathrm{m}}^2\cdot {z}^2)\times {10}^{-9}, $$

(11) $$ \frac{\mathrm{\partial}{W}_g}{\mathrm{\partial}m}=26.7(\rho \cdot b\cdot m\cdot {z}^2)\times {10}^{-9}, $$

(12) $$ \frac{\mathrm{\partial}{\mathrm{W}}_{\mathrm{g}}}{\mathrm{\partial}z}=26.7(\rho \cdot b\cdot {m}^2\cdot z)\times {10}^{-9}, $$

(13) $$ \Delta {W}_g=\left|\frac{\mathrm{\partial}{W}_g}{\mathrm{\partial}b}\right|\cdot \varDelta b+\left|\frac{\mathrm{\partial}{W}_g}{\mathrm{\partial}m}\right|\cdot \varDelta m+\left|\frac{\mathrm{\partial}{W}_g}{\mathrm{\partial}z}\right|\cdot \varDelta z, $$

(14) $$ \Delta {W}_g=13.35(\rho \cdot m\cdot z)[(m\cdot \varDelta b)+2(b\cdot z\cdot \varDelta m)+2(b\cdot m\cdot \varDelta z)]\times {10}^{-9}, $$

(15) $$ DCIWg=\frac{\;{URL}_{Wg}-{W}_g\;}{\Delta {W}_g}, $$

(16) $$ DCIWg=\frac{URL_{Wg}-13.35(\rho \cdot b\cdot {m}^2\cdot {z}^2)\times {10}^{-9}}{13.35(\rho \cdot m\cdot z)[(m\cdot z\cdot \varDelta b)+2(b\cdot z\cdot \varDelta m)+2(b\cdot m\cdot \varDelta z)]\times {10}^{-9}}, $$

(17) $$ \Delta H=\left|\frac{\mathrm{\partial}H}{\mathrm{\partial}m}\right|\cdot \varDelta m+\left|\frac{\mathrm{\partial}H}{\mathrm{\partial}z}\right|\cdot \varDelta z, $$

(18) $$ \Delta H=5[(z\cdot \varDelta m)+(m\cdot \varDelta z)]. $$

Goal 3: Size of gearbox (H):

(19) $$ H=5m\cdot z(\mathrm{m}\mathrm{m}), $$

(20) $$ \frac{\partial H}{\partial m}=5z, $$

(21) $$ \frac{\partial H}{\partial z}=5m, $$

(22) $$ DCIH=\frac{\;{URL}_H-H\;}{\Delta H}, $$

(23) $$ DCIH=\frac{URL_H-5mz}{5[(z\cdot \Delta m)+(m\cdot \Delta z)]}. $$

Goal 5: Weight of shaft (Ws):

(24) $$ {W}_s=6.129\left({D}_i^2+{D}_o^2\right)\times {10}^{-3}\;\left(\mathrm{kg}\right), $$

(25) $$ \frac{\partial {W}_s}{\partial {D}_i}=12.26{D}_i\times {10}^{-3}, $$

(26) $$ \frac{\partial {W}_s}{\partial {D}_o}=12.26{D}_o\times {10}^{-3}, $$

(27) $$ \Delta {W}_s=\left|\frac{\mathrm{\partial}{W}_s}{\mathrm{\partial}{D}_i}\right|\cdot \varDelta {D}_i+\left|\frac{\mathrm{\partial}{W}_s}{\mathrm{\partial}{D}_o}\right|\cdot \varDelta {D}_o, $$

(28) $$ \Delta {W}_s=12.26[({D}_i\cdot \varDelta {D}_i)+({D}_o\cdot \varDelta {D}_o)]\times {10}^{-3}, $$

(29) $$ DCIWs=\frac{\;{URL}_{Ws}-{W}_s\;}{\Delta {W}_s}, $$

(30) $$ DCIWs=\frac{URL_{Ws}-6.129({D}_i^2+{D}_o^2)\times {10}^{-3}}{12.26[({D}_i\cdot \varDelta {D}_i)+({D}_o\cdot \varDelta {D}_o)\times {10}^{-3}}. $$

Goal 4: Torque (T_u and T_m):

(31) $$ {T}_u={\left(\frac{S_c}{18,348}\right)}^2(b\cdot {m}^2\cdot {z}^2)(\mathrm{N}\mathrm{m}), $$

(32) $$ {T}_m=\left(\frac{S_t}{10,760}\right)(b\cdot {m}^2\cdot z)(\mathrm{N}\mathrm{m}), $$

(33) $$ \frac{\mathrm{\partial}{T}_u}{\mathrm{\partial}b}={\left(\frac{S_c}{18,348}\right)}^2({m}^2\cdot {z}^2), $$

(34) $$ \frac{\mathrm{\partial}{T}_u}{\mathrm{\partial}m}={\left(\frac{S_c}{18,348}\right)}^2(2\cdot b\cdot m\cdot {z}^2), $$

(35) $$ \frac{\mathrm{\partial}{T}_u}{\mathrm{\partial}z}={\left(\frac{S_c}{18,348}\right)}^2(2\cdot b\cdot {m}^2\cdot z), $$

(36) $$ \Delta {T}_u=\left|\frac{\mathrm{\partial}{T}_u}{\mathrm{\partial}b}\right|\cdot \varDelta b+\left|\frac{\mathrm{\partial}{T}_u}{\mathrm{\partial}m}\right|\cdot \varDelta m+\left|\frac{\mathrm{\partial}{T}_u}{\mathrm{\partial}z}\right|\cdot \varDelta z, $$

(37) $$ \Delta {T}_u={\left(\frac{S_c}{11,612}\right)}^2(m\cdot z)[(m\cdot z\cdot \varDelta b)+2(b\cdot z\cdot \varDelta m)+2(b\cdot m\cdot \varDelta z)], $$

(38) $$ {T}_{max}={T}_u+\Delta {T}_u, $$

(39) $$ {T}_{max}={(\frac{S_c}{18,348})}^2(m\cdot z)[(b\cdot m\cdot z)+(m\cdot z\cdot \varDelta b)+2(b\cdot z\cdot \varDelta m)+(2b\cdot m\cdot \varDelta z)], $$

(40) $$ \Delta {T}_u={T}_{max}-{T}_m, $$

(41) $$ {\displaystyle \begin{array}{l}\Delta Tupper={\left(\frac{S_c}{\mathrm{18,348}}\right)}^2\left(m\bullet z\right)\\ {}\;[(b\bullet m\bullet z\Big)+\left(m\bullet z\bullet \varDelta b\right)+2\left(b\bullet z\bullet \varDelta m\right)+2(b\bullet m\bullet \varDelta z)]-\left(\frac{S_t}{\mathrm{10,760}}\right)\left(b\bullet {m}^2\bullet z\right),\end{array}} $$

(42) $$ EMIT=\frac{\mu_y-LRL}{\varDelta {Y}_l}=\frac{T_m-{LRL}_T}{\Delta {T}_u}, $$

(43) $$ EMIT=\frac{(\frac{S_t}{10,760})(b\cdot {m}^2\cdot z)-{LRL}_T}{\Delta {T}_u}, $$

(44) $$ EMIT=\frac{\left(\frac{S_t}{10,760}\right)(b\cdot {m}^2\cdot z)-{LRL}_T}{{(\frac{S_c}{18,348})}^2m\cdot z(b\cdot m\cdot z+m\cdot z\cdot \varDelta b+2b\cdot z\cdot \varDelta m+2b\cdot m\cdot \varDelta z)-\left(\frac{S_t}{10,760}\right)b\cdot {m}^2\cdot z}. $$

Following the steps outlined in Figure 1, Table 6 is formulated. The information in Table 6 is defined using four keywords: Given, Find, Satisfy and Minimise. The keyword ‘Given’ contains the design input, while ‘Find’ defines the coupled decisions. The determination of compromise decisions (gearbox dimensions) and selection decision (gear material) forms a coupled decision. The selection decision involves the process of choosing among a number of possibilities considering several measures of merit or attributes, while compromise decision involves the determination of the ‘right’ values (or combination) of design variables (or parameters). Compromise and selection decisions are respectively modelled with a cDSP and sDSP. The solution to the coupled cDSP–sDSP is explored by considering multiple design criteria and uncertainties. The multiple design criteria are defined by goals G1 to G5. The uncertainty is managed by modifying the original design goals into robustness metrics (see Equations (9)–(44)). The keyword ‘Minimise’ specifies iterating until a satisfactory result is obtained. A satisfactory result is obtained when the constraints are satisfied, and the deviations from the predefined design targets are minimised. The mathematical foundation for designing a gearbox is available in Budynas ' Nisbett (Reference Budynas and Nisbett2008). The mathematical derivation for the four compromise goals (three goals for gear design and one goal for shaft design) as robustness goals are shown in Equations (9)–(44) and summarised in Table 6. Further information about this formulation is available in the work of Sharma (Reference Sharma2020).

Table 6. Mathematical formulation for robust design of a gearbox (Additional information about variables is available in the Nomenclarure section at the end of the article.)