Design optimization under uncertainty and speed variability for a piezoelectric energy harvester powering a tire pressure monitoring sensor

The TPMS EH considered in this study is configured as a beam that is clamped at both ends with the PZT material attached on the top surface (figure 1(a)). The design parameters for the EH are shown in figure 1(b) and table 1. The EH module is encased in a housing and is attached to the inner surface of tire (figure 1(c)). For the sake of electrical power increment, a seismic mass is attached to the lower surface of substrate. This model for EH is initially proposed by Keck [27] and investigated in more details by Eshghi et al [10]. The model used in this study has several advantages over the cantilever beam that is fixed at one side and the seismic mass is attached to the other end. First, it is a more robust and durable design by restraining excessive deformation. Second, due to reduced vibrational amplitude, compact integration into the housing is possible (e.g., no need of stopper). A finite element analysis (FEA) model is implemented using ANSYS to evaluate the performance of the EH design. Four types of elements are used to model the EH: (1) SOLID226 (a second order brick element for piezoelectric analysis) for the piezoelectric material, (2) SOLID186 (a second order brick element for elastic analysis), for the substrate (3) MASS21 (a point mass element) for the seismic mass, and (4) CIRCU94 (a two node circuit element for use in piezoelectric-circuit analyses) for the external resistor where power is evaluated. The resistor is connected between top and bottom electrode of PZT patch. Overall, the FEA model contains 3237 nodes and 1456 elements.

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Table 1 summarizes the design variables d = [d₁, d₂, ... d₅] and constant parameters in the design optimization study. The design variable set consists of three geometrical design variables, the load resistance, and the seismic mass. The thicknesses of both substrate and PZT layers are considered as constants due to the limited standard dimensions from manufacturers. A preliminary parametric study found that the length of seismic mass (L_m: length of the mass in contact with the substrate) has the negligible effect on the EH performance. Hence, it is also defined as a constant parameter and the mesh in the seismic mass area (L_m × W) does not change during the optimization process.

In the simulation model, blue steel and PZT-5H is considered as the substrate and the piezoelectric material, respectively. The material properties are provided in table 2 including transverse strain constant (d₃₁) that indicates how effectively bending strain along direction '1' (or x) is converted to electrical voltage along direction '3' (or z).

The contact between a rotating tire and road surface generates a complex dynamic acceleration in three different directions: circumferential, lateral, and radial. The focus of this study is on the radial acceleration, which has the largest magnitudes among those. Several previous studies have investigated the acceleration characteristics in a rotating tire in contact with the road surface [8, 29–31]. In this paper we use the input radial acceleration model in [29]. Figure 2 depicts the radial acceleration for a single rotation of tire. 'a' is the radial acceleration in steady-state conditions, and can be expressed as a function of the speed (v) of the vehicle and radius (r) of the tire, i.e., a = v²/r. The acceleration will suddenly increase (a + b) at the moment before the tire contacts the ground and it gets to zero when in contact with the ground. Then the acceleration instantly increases by 'c' because the radius of curvature increases when the tire becomes flat, but this amount is less than 5% of the peak acceleration (a + b). 'd' is the duration between two instants when there is a significant acceleration change due to the tire deformation right before and after the contact. 'e' denotes the duration when the tire is in contact with the ground and is flat.

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This dynamic acceleration is used as a transient loading in ANSYS to estimate the electrical power and mechanical stresses in the EH. The loading acceleration varies depending on the car speed (velocity) as demonstrated in figure 3 which shows two examples when V = 50 and 100 km h⁻¹. Given an input car speed, the acceleration loading history is updated accordingly and the transient analysis calculates dynamic response of the EH (power and stresses) in time domain.

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Power requirement varies depending on the amount of transmitted information needed per wheel revolution, so it usually increases proportional to the car speed. We will use energy per revolution (in J/rev) as an energy requirement measure [32], which is an appropriate choice to define the energy requirement per rotation regardless of time-varying speed. Energy per revolution is a normalized measure of power by tire angular velocity (W/(rev s⁻¹) = J/rev). Table 3 summarizes the minimum requirements for energy per revolution based on a data transmission rate of 15 bytes per revolution at different car speeds.

In this study, the average energy demand is estimated based on the US federal drive cycles [33] representing the vehicle speed profiles under two driving conditions: (i) Urban dynamometer driving schedule (city driving conditions), and (ii) highway fuel economy driving schedule (highway driving conditions). Figure 4 shows the histogram of the vehicle speed data. By combining the data for both urban and highway cases using the same weights, the PDF of speed, f(V), is estimated using a normal kernel function and is evaluated at equally spaced points as shown in figure 5.

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Based on this PDF, the energy per revolution is calculated from equation (1):

$$\begin{eqnarray}{P}_{{\rm{r}}{\rm{e}}{\rm{q}}} & = & \displaystyle {\int }_{V}{P}_{\min }(V)f(V){\rm{d}}V\simeq \displaystyle \sum _{V=0}^{V=100}{P}_{\min }(V)\\ & & \times f(V)\times {\rm{\Delta }}V\\ {P}_{{\rm{a}}{\rm{v}}{\rm{e}}}({\rm{d}}) & = & \displaystyle {\int }_{V}P({\rm{d}},V)f(V){\rm{d}}V\simeq \displaystyle \sum _{V=0}^{V=100}P({\rm{d}},V)\\ & & \times f(V)\times {\rm{\Delta }}V,\end{eqnarray} \tag{ 1 }$$

where P_min(V) is the energy requirement at car speed V, P(d, V) is the energy per revolution for a certain EH design (d) at car speed V, and ΔV = 1 km h⁻¹ is the speed increment for numerical integration. Based on the data in table 3, P_min(V) can be approximated as a forth order polynomial and P_req is calculated as 6.1 μJ/rev P_ave calculation in equation (1) is done similarly as P_req, after obtaining another forth order polynomial for P(d, V). Figure 6 demonstrates the examples of power profile per revolution at 50 km h⁻¹ and 100 km h⁻¹, respectively. We evaluate the energy per revolution at four different speeds and build the polynomial for P(d, V). Equation (1) is used to obtain P_ave which is considered in the optimization formulation (section 3).

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A conservative stress evaluation concept is used in this paper, by considering the maximum stresses at the highest speed of vehicle speed profile. That is, the stresses are evaluated at the highest speed in f(V) (100 km h⁻¹) and the maximum stress values (from substrate and PZT) in one rotation cycle are figured out as shown in figure 7. We use these maximum values in the design formulation, constrained to be less than the yield stresses of substrate and PZT material.

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This subsection explains the formulation of the DDO. This method is a typical optimization process where the design variables are found to minimize (or maximize) the objective function while satisfying the constraints. The objective function is chosen to be the envelope volume of EH which is defined as the total length of harvester multiplied by the width and its overall thickness. The envelope volume indicates the smallest box volume that includes the harvester, and it is considered as the objective function for the compactness of the harvester design. Since the design should satisfy minimum power requirement and also durability, three constraints are defined in the DDO formulation as:

$$\begin{eqnarray}&&\min \,{\rm{\Psi }}({\bf{d}})={L}_{s}\times W\times ({t}_{p}+{t}_{s})\\ &&{\rm{s}}{\rm{u}}{\rm{b}}{\rm{j}}.\,{\rm{t}}{\rm{o}}.\,{G}_{1}({\bf{d}})={\sigma }_{{\rm{Max}}}^{{\rm{S}}{\rm{u}}{\rm{b}}}({\bf{d}})-{\sigma }_{{\rm{y}}{\rm{i}}{\rm{e}}{\rm{l}}{\rm{d}}}^{{\rm{S}}{\rm{U}}{\rm{S}}304}\leqslant 0\\ &&\,\,\,{G}_{2}({\bf{d}})={\sigma }_{{\rm{Max}}}^{{\rm{P}}{\rm{Z}}{\rm{T}}}({\bf{d}})-{\sigma }_{{\rm{y}}{\rm{i}}{\rm{e}}{\rm{l}}{\rm{d}}}^{{\rm{P}}{\rm{Z}}{\rm{T}}}\leqslant 0\\ &&\,\,\,{G}_{3}({\bf{d}})={{\rm{P}}}_{{\rm{r}}{\rm{e}}{\rm{q}}}-{P}_{{\rm{a}}{\rm{v}}{\rm{e}}}({\bf{d}})\leqslant 0\\ &&\,\,\,{d}_{i}^{L}\leqslant {d}_{i}\leqslant {d}_{i}^{U},\end{eqnarray} \tag{ 2 }$$

where vector d represents the design variable set as described in table 1. Also, ${d}_{i}^{L}$ and ${d}_{i}^{U}$ are the lower and upper bounds for each design variable. Three geometrical design variables—length of substrate (L_s), length of PZT (L_p), and width of harvester (W)—in addition to load resistance (R) and seismic mass (m) are considered in the DDO. As mentioned in section 2.1, it is not practical to consider the thickness as a continuous design variable, so both substrate and PZT thicknesses are considered to be constant. A formulation in equation (2) considers two kinds of constraints. G₁ and G₂ are the stress constraints where the maximum stresses $({\sigma }_{{\rm{Max}}}^{{\rm{S}}{\rm{u}}{\rm{b}}},{\sigma }_{{\rm{Max}}}^{{\rm{P}}{\rm{Z}}{\rm{T}}})$ are evaluated at the highest speed of vehicle. G₃ is the constraint on the energy per revolution (P_ave) where P_req is set as 6.1 μJ/rev as described in section 2.3.

The dynamics of most vibration harvesters is sensitive to the uncertainty of physical parameters such as geometry and material properties. Even though some studies have been conducted on quantifying the effect of uncertainties in the physical parameters (e.g. manufacturing, materials, and operations) on the power generation of the EH [23, 24, 26, 34, 35], little to no EH design study has been conducted to consider the uncertainties. This has become the motivation of this study to address a question how to design an EH with reliable power generation despite the uncertainties that cannot be handled using classical DDO methods. RBDO, which accounts for the stochastic nature of an engineered system, is considered in this study as a potential solution to the question. In general, RBDO integrates the techniques of reliability analysis and design optimization that offer probabilistic approaches to engineering system design [36–43]. RBDO attempts to find the optimum design that minimizes the cost (e.g. volume, mass, or price), with the consideration of reliability with respect to system performance (e.g., durability, power output) while accounting for various sources of uncertainty (e.g. material properties, geometric tolerances, and loading conditions).

In RBDO, reliability is defined as the probability that a system performance meets its specification limit under various sources of uncertainty. Reliability analysis, as a critical element of RBDO, has a significant impact on the accuracy and efficiency of an RBDO solution. Mathematically, reliability analysis entails the computation of a multi-dimensional integration of a joint PDF over a reliable domain, expressed as ${\rm{R}}{\rm{e}}{\rm{l}}\cong \displaystyle {\int }_{{{\rm{\Omega }}}^{s}}{f}_{{\bf{X}}}({\bf{x}}){\rm{d}}{\bf{x}},$ where Rel is the reliability, X = [X₁, X₂, ..., X_N]^T is a N-dimensional vector of random variables that models sources of uncertainty such as material properties, loads, and geometric tolerances,${f}_{{\bf{X}}}({\bf{x}})$ is the joint PDF of X, and ${{\rm{\Omega }}}^{s}$ is the reliable (or safe) domain and can be defined in terms of a system performance function G(X) as ${{\rm{\Omega }}}^{s}=\{{\bf{x}}\,:G({\bf{x}})\lt 0.$ It is noted that, in this paper, a random variable used in RBDO is denoted by the upper-case letter X, and particular realization of the random variable are denoted by the lower-case letter x. In the context of EH design, the performance function G is defined as the predefined power requirement (P_req) of a TPMS sensor subtracted by the output power (P_ave) of the EH design, i.e. G = P_req − P_ave. Then, it becomes obvious that G(X) < 0 represents the reliable domain where the output power is larger than required. Also, two other performance functions have been defined to ensure that maximum stresses in the PZT and substrate do not exceed the yield stresses.

Both DDO and RBDO try to minimize the objective function while satisfying the constraints. DDO deals with deterministic design constraints while RBDO converses with probabilistic constraints that accounts for various sources of uncertainty. The difference between DDO and RBDO is graphically shown in figure 8 in a two-dimensional design space, where X₁ and X₂ are random design variables. DDO usually pushes the optimum point to the boundaries of reliable domain, i.e. G(X) = 0, and it leaves small room for uncertainty in the design variables (shown as small red dots around the DDO point). That means, a small design perturbation of the DDO design will move the design into the failure region and violate design constraint(s). However, in RBDO, the DDO point is pushed to the reliable region, G(X) < 0. Figure 8 shows the true limit state function (G(X) = 0, solid line) as well as its approximated one (dashed line). Also, the concentric ellipses around the RBDO mean design point d = ${\boldsymbol{\mu }}({\bf{X}})$ illustrates the contours of the joint PDF of the two design variables. And the scattered points around the mean design are random design points which are generated based on the joint PDF.

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Depending on how reliability analysis is performed, existing RBDO methods can be categorized into two groups: (i) the first- or second-order reliability method (FORM/SORM), or FORM/SORM-based methods [36–43], and (ii) dimension reduction (DR) based methods [44–46]. The FORM/SORM-based RBDO methods employ the FORM/SORM to perform reliability analysis. The FORM/SORM uses the first- or second-order Taylor expansion to approximate the limit state function (G = 0) at the most probable point (MPP), where the limit state function separates the reliable region (G < 0) from the failure region (G > 0). Some major challenges of the FORM/SORM include: (i) iterative sensitivity analyses are required for identifying the MPP; and (ii) the high nonlinearity of the limit state function or the existence of multiple MPPs could result in a relatively large error in the reliability estimate. These challenges have stimulated a large amount of research on the so-called DR method. This method approximates a multi-dimensional response function as a hierarchical superposition of low-order component functions so that it enables an efficient yet accurate formulation of this response function. A specialized version of this method is called the UDR method, which simplifies one multi-dimensional response function to multiple one-dimensional component functions [21, 44, 45]. Recent RBDO studies [44, 45] reported superior efficiency and satisfactory accuracy of the UDR method in solving RBDO problems. The next section describes UDR method which has been used to estimate where G(X) = 0.

3.2.1. Univariate dimension reduction

DR is a widely used method for reliability analysis. UDR is a special case of DR which additively decomposes a multi-dimensional performance functions into a finite set of one-dimensional component functions [46]. A univariate approximation of the performance function G(X) with the mean vector of the input random variables ${{\boldsymbol{\mu }}}^{i}$ excluding X_i, N-dimensional random vector X = [X₁, X₂, ..., X_N]^T can be expressed as:

$$\begin{eqnarray}&&{G}_{U}({\bf{X}})=\displaystyle \sum _{i=1}^{N}G({X}_{i},{{\boldsymbol{\mu }}}^{i})-(N-1)G({\boldsymbol{\mu }}).\end{eqnarray} \tag{ 3 }$$

To appraise the accuracy of univariate decomposition method, we can expand the univariate function using Taylor series. Therefore, the Taylor expansion of the ith univariate function $G({X}_{i},{{\boldsymbol{\mu }}}^{i})$ at ${X}_{i}={\mu }_{i}$ can be obtained by:

$$\begin{eqnarray}&&G({X}_{i},{{\boldsymbol{\mu }}}^{i})=G({\boldsymbol{\mu }})+\displaystyle \sum _{j=1}^{\infty }\displaystyle \frac{1}{j!}\displaystyle \frac{{\partial }^{j}G}{\partial {X}_{i}^{j}}({\boldsymbol{\mu }}){({X}_{i}-{\mu }_{i})}^{j},\end{eqnarray} \tag{ 4 }$$

where ${\partial }^{j}G({\boldsymbol{\mu }})/\partial {X}_{i}^{j}$ is the jth partial derivative of $G({\bf{X}})$ with respect to ${X}_{i}$ evaluated at the mean vector μ. Hence, the Taylor series of the univariate decomposition at the mean point (equation (4)) can be rewritten using equation (5):

$$\begin{eqnarray}&&{G}_{U}({\bf{X}})=G({\boldsymbol{\mu }})+\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{\infty }\displaystyle \frac{1}{j!}\displaystyle \frac{{\partial }^{j}G}{\partial {X}_{i}^{j}}({\boldsymbol{\mu }}){({X}_{i}-{\mu }_{i})}^{j}.\end{eqnarray} \tag{ 5 }$$

And the Taylor series expansion of the performance function $G({\bf{X}})$ can be expressed as:

$$\begin{eqnarray}&&G({\bf{X}})=G({\boldsymbol{\mu }})+\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{\infty }\displaystyle \frac{1}{j!}\displaystyle \frac{{\partial }^{j}G}{\partial {X}_{i}^{j}}({\boldsymbol{\mu }}){({X}_{i}-{\mu }_{i})}^{j}+{\boldsymbol{\varepsilon }}.\end{eqnarray} \tag{ 6 }$$

It should be noted that the univariate decomposed response contains univariate terms of any order in the Taylor series expansion. Hence, there is no limitation in UDR to represent the order of nonlinearity in the performance function. In equation (6), ε refers to the residual error from ignoring the second- and higher-order expansion terms which is far less than the second order Taylor expansion error for reliability analysis [46]. ε has a negligible effect in most engineering applications in which $G({\bf{X}})$ is sufficiently smooth, so is ignored in this paper.

3.2.2. Interpolation method for performance function evaluation

Reliability analysis requires large number of function evaluation to approximate $G({\bf{X}}).$ In order to reduce the computational cost, it is important to obtain performance function values at any arbitrary point efficiently. For the univariate function $G({X}_{i})=G({X}_{i},{{\boldsymbol{\mu }}}^{i})$ in equation (3), function values are evaluated at n + 1 sample points $({X}_{i}={x}_{i}^{j},j=1,\,2,\,\ldots ,\,n+1)$ per each X_i, and nN + 1 sample points for a general N variable case. Cubic spline interpolation (CSI) is used in this paper to estimate performance function values at any arbitrary point. This interpolation evaluates function values accurately at n + 1 sample points and its first and second-order derivatives are continuous. The CSI formulation can be expressed as:

$$\begin{eqnarray}&&{Y}_{i}({X}_{i})={\alpha }_{j}{({X}_{i}-{x}_{i}^{j})}^{3}+{\beta }_{j}{({X}_{i}-{x}_{i}^{j})}^{2}\\ &&\,\,\,\,+{\gamma }_{j}({X}_{i}-{x}_{i}^{j})+{\delta }_{j},\end{eqnarray} \tag{ 7 }$$

where Y_i is an arbitrary function to be interpolated, ${X}_{i}\in [{x}_{i}^{j},{x}_{i}^{j+1}],$ and ${\alpha }_{j},\,{\beta }_{j},\,{\gamma }_{j},$ also ${\delta }_{j}$ are the coefficient of the polynomial for the jth interval, which can be determined by solving four independent linear conditions [47]. Therefore, we can measure the performance function values at any point using equation (7).

3.2.3. Uncertainty quantification

UDR and CSI construct the approximated function $\hat{G}$ of the original performance function G as it is shown in figure 8. Now, it is essential to estimate any probabilistic characteristic of $G({\bf{X}})$ including PDF, statistical moments, and reliability. This estimation is validated by comparing them with MCS. As aforementioned the approximation of the reliable domain for the performance function is defined as ${\hat{{\rm{\Omega }}}}^{s}=\{{\bf{x}}\,:\hat{G}({\bf{x}})\lt 0\}.$ Hence, reliability can be quantified using MCS as the following:

$$\begin{eqnarray}&&{\rm{Rel}}\simeq \displaystyle \int {I}_{{\hat{{\rm{\Omega }}}}^{s}}({\bf{x}}){f}_{{\bf{x}}}({\bf{x}}){\rm{d}}{\bf{x}}=E({I}_{{\hat{{\rm{\Omega }}}}^{s}}({\bf{X}}))=\mathop{\mathrm{lim}}\limits_{{n}_{s}\to \infty }\displaystyle \frac{1}{{n}_{s}}\displaystyle \sum _{j=1}^{{n}_{s}}{I}_{{\hat{{\rm{\Omega }}}}^{s}}({{\bf{x}}}_{j}),\end{eqnarray} \tag{ 8 }$$

where n_s is the number of samples used in the MCS and ${I}_{{\hat{{\rm{\Omega }}}}^{s}}({{\bf{x}}}_{j})$ is the indicator function for the safe or fail situation as it is described below:

$$\begin{eqnarray}{I}_{{\hat{{\rm{\Omega }}}}^{s}}({{\bf{x}}}_{j})=\left\{\begin{array}{ll}1, & {{\bf{x}}}_{j}\in {\hat{{\rm{\Omega }}}}^{s}\\ 0, & {{\bf{x}}}_{j}\in {\rm{\Omega }}/{\hat{{\rm{\Omega }}}}^{s}\end{array}\right..\end{eqnarray} \tag{ 9 }$$

We used 100 000 samples points in the MCS study that employs the explicit interpolation of $\hat{G}$ instead of real function evaluation G, so the computational cost is not significant.

3.2.4. RBDO formulation

In this paper, RBDO is used to optimize design variables and improve the reliability for EH, while accounting for the uncertainty in geometric tolerances. In order to observe how RBDO approach improves DDO design, its formulation is done based on the DDO formulation in equation (2). That is, the objective function is defined identically to minimize the envelope volume while three performance constraints are revised as follow:

$$\begin{eqnarray}&&\min \,\,{\rm{\Psi }}({\bf{d}})={L}_{s}\times W\times ({t}_{p}+{t}_{s})\,\,{\rm{s}}{\rm{u}}{\rm{b}}{\rm{j}}.\,{\rm{t}}{\rm{o}}:\\ &&{\rm{R}}{\rm{e}}{{\rm{l}}}_{1}={\rm{\Pr }}({G}_{1}({\bf{X}};{\bf{d}})={\sigma }_{{\rm{Max}}}^{{\rm{S}}{\rm{u}}{\rm{b}}}({\bf{X}};{\bf{d}})-{\sigma }_{{\rm{y}}{\rm{i}}{\rm{e}}{\rm{l}}{\rm{d}}}^{{\rm{S}}{\rm{U}}{\rm{S}}304}\leqslant 0)\geqslant {\rm{\Phi }}({\beta }^{t})\\ &&\,\,={R}^{t}\\ &&{\rm{R}}{\rm{e}}{{\rm{l}}}_{2}={\rm{\Pr }}({G}_{2}({\bf{X}};{\bf{d}})={\sigma }_{{\rm{Max}}}^{{\rm{P}}{\rm{Z}}{\rm{T}}}({\bf{X}};{\bf{d}})-{\sigma }_{{\rm{y}}{\rm{i}}{\rm{e}}{\rm{l}}{\rm{d}}}^{{\rm{P}}{\rm{Z}}{\rm{T}}5{\rm{H}}}\leqslant 0))\geqslant {\rm{\Phi }}({\beta }^{t})\\ &&\,\,={R}^{t}\\ &&{\rm{R}}{\rm{e}}{{\rm{l}}}_{3}={\rm{\Pr }}({G}_{3}({\bf{X}};{\bf{d}})={P}_{{\rm{r}}{\rm{e}}{\rm{q}}}-{P}_{{\rm{a}}{\rm{v}}{\rm{e}}}({\bf{X}};{\bf{d}})\leqslant 0)\geqslant {\rm{\Phi }}({\beta }^{t})={R}^{t}\\ &&{d}_{i}^{L}\leqslant {d}_{i}\leqslant {d}_{i}^{U}.\end{eqnarray} \tag{ 10 }$$

In the above equation, vector ${\bf{X}}=[{X}_{1},{X}_{2},{X}_{3}]$ is the random design variable and ${\bf{d}}=[{d}_{1},{d}_{2},{d}_{3}]={\boldsymbol{\mu }}({\bf{X}})$ is the mean design vector; ${d}_{i}^{L}$ and ${d}_{i}^{U}$ are the lower and upper bounds on ${d}_{i}$ for $i=1,\,2,\,3.$ Each G demonstrates the performance functions which enables the definition of reliability constraints; Φ is the cumulative distribution function of the standard normal distribution to a target reliability index β^t(=3), R^t is the corresponding target reliability level (99.87%). Minimum power requirement is equal to the DDO case. Three design variables are assumed to have normal distributions with their standard deviations listed in table 4.

Table 4.  Random design variables for the EH design.

Design variable	Description	Distribution type	Standard deviation $\sigma$
X1	Length of substrate	Normal	5% of DDO point
X2	Length of PZT
X3	Width of harvester

For solving DDO and RBDO problems (equations (1) and (10)), the sequential quadratic programming (SQP) algorithm is used which can start from infeasible domain where all of the constraints are not satisfied; maximum stresses exceed their yield stresses and/or the generated power is less than required.

In the DDO case the optimization history is converged after 16 iterations. Tables 5 and 6 demonstrate the comparison between the results for the start point and optimum point in the DDO study. Since the maximum stress exceeds the yield stress in the PZT layer (45 MPa in table 2) at the initial point, the harvester width has increased to reinforce the structure and reduces the stresses. The increase of PZT area (12.4 × 12 to 10.29 × 15.52), even though its length decreases, contributes to the power increase by 1.5 times.

Since RBDO requires uncertainty quantification and reliability analysis every iteration, its computational cost is more expensive compared to DDO. To save computational cost, the RBDO study in this paper uses the DDO optimum point as an initial guess in using the same optimization algorithm (SQP). The uncertainty of geometrical tolerances due to manufacturing error is considered for the first three design variables—the length of substrate (L_s), length of PZT (L_p) and the overall width of harvester (W). Resistance and seismic mass are fixed at the DDO optimum point. In the RBDO case, it took 78 iteration to obtain the converged result which means that RBDO requires nearly five times of computational cost compared to the DDO case. Tables 7 and 8 demonstrate the results for the RBDO study compared to the DDO. Three constraints in table 8 indicates the value at the mean design point (μ).

Figure 9 demostrates the sequence of EH design variation from initial point to the DDO and RBDO. Transition from DDO to RBDO means getting a more conservative design due to the presence of uncertainties (detail discussion is available in section 4.2.1). That is, the RBDO design is reinforced compared to the DDO design and less flexible. This change is consistently observed in figure 10 that shows the transient response of EH at the highest car speed (100 km h⁻¹). The stress waveform is very similar that of the input acceleration. In both figures one can observe the high frequency vibration of which the frequency corresponds to the resonant frequency of the DDO (740 Hz) and the RBDO (880 Hz) design, respectively.

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One can observe from figure 10 that that the stresses values for both the DDO and RBDO case are lower than the yield stresses. But, the magnitude of stresses for the RBDO case is less than the DDO due to the increased width (W) and decreased substrate length (L_s). Obviously the amount of generated power is reduced campared to the DDO case. However, the RBDO process finds a satisfactory trade-off point to satisfy the power generation requirement by increasing PZT length (L_p).

Figure 11 depicts the corresponding stresses contours in the FEA model for the substrate and PZT. The maximum stress for the DDO case happens at the both ends of substrate and PZT. Therefore the increased width (W) and the decreased length (L_s) reduce the stresses.

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Based on the uncertainty study results, RBDO study increases the system reliability by 25.5% (from 73.67% to 99.17%) resulted from the satisfactory reliability constraints (Rel₁ to Rel₃ in equation (10)). This increment in the reliability has sacrificed the objective function by only 2.5% which is acceptable. Figure 12 shows the comparison on the PDFs between the RBDO and DDO optimum point which is computed from univariate response surface. In these figures, the area under a PDF curve for $G\leqslant 0$ represents the probability that each performance function is satisfied. For the first (G₁, substrate stress) and second performance functions (G₂, PZT stress), reliability has been increased drastically. The reliability for the third performance function (G₃, power) reduces from R^t = 100% to R^t = 97.53%. However, the minimum power requirement is still satisfied for the RBDO design.

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4.2.1. Interpretation of RBDO results

In order to understand the physical implication of the RBDO result compared to the DDO, it is essential to investigate the behavior of performance functions (G) in more details. In this section, the variation of three performance functions is further investigated around the DDO optimum point. Figures 13–15 demonstrate the 3D contour plots of performance functions with the first three design variables (L_s, L_p, W). Each plot is centered at the DDO optimum point, and two parameters vary in a span of $\pm 3\sigma$ to construct the 3D contour plot while the other parameter is fixed at the DDO optimum. The border for the limit state function (G(X) = 0) is shown with a black solid line. From these figures, the critical performance function is G₂ as its infeasible region is found (G₂ > 0). Therefore, RBDO tries to evolve the design to move into the safe domain of G₂. Based on these plots, it is obvious that decreasing L_s and increasing W can improve the design to get into the feasible domain and satisfy the reliability constraints for G₂. Moreover, it can be observed from G₃ contour plot in figure 15 that increase of L_p improves power reliability (Rel₃). This observation from the 3D contour plots justifies the results in tables 7 and 8. To summarize, the merits of RBDO design can be distinguished from DDO design as it reduces the stresses magnitude for both PZT and substrate layers and increases the reliability drastically. Also, the power reliability, even slightly reduced from the DDO, still exceeds its requirement.

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The optimized EHs for DDO and RBDO cases in the section 4 was manufactured for the experimental evaluation. Precise and careful machining is required because the dimensional precision is in the order of 0.01 millimeter, and PZT is very brittle. We used laser cutting to manufacture all the pieces. The electric potentials (or voltage) of the substrate and the PZT electrode facing the substrate are all identical. Therefore, a conductive epoxy (8331 s-Silver Conductive Epoxy, MG chemicals) is used to attach the PZT layer to the substrate. The conductive epoxy is spread on the substrate in a very thin layer and the PZT layer is put on the top, and cured using a heavy mass on top of it in the room temperature for 24 h. After curing the epoxy, the proof mass is glued to the bottom layer of substrate using super glue (E-120HP, Loctite). Also, a very thin electric wire is soldered to both substrate and the top electrode of PZT layer for the electrical wiring. The prototyping process is shown in figure 16.

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The prototyped harvesters are glued on the inner surface of the tire as shown in figure 17; DDO and RBDO harvesters are attached 180° apart from each other. Two different jigs for both DDO and RBDO are manufactured and glued using epoxy and cured for another 24 h for secure attachment. After mounting both harvesters to the corresponding jigs, the tire is assembled with the rim.

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Figure 18 shows the entire test setup where the tire is in contact with the roller which is belted to the motor. The spinning speed is detected by the laser speedometer as shown in figure 18(a). A special attention should be paid on wiring and signal acquisition from the harvesters which spins with the tire in a high speed. We use a wireless signal transmitter mounted on the rim depicted in figure 18(b). The external loads (resistors) are connected to the both ends of the wires from each harvester. This way we can prevent wire entangling during the experiment.

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Experimental tests has been done to demonstrate the differences in the performance of two designs (DDO and RBDO). We measured the output voltage of EH for both DDO and RBDO case at two different velocities (10 and 20 km h⁻¹). These measurements are conducted for seven consecutive runs. Figure 19 shows the typical output voltage for the DDO and RBDO samples. Two harvesters are glued at the opposite position in the tire, so the voltage peaks are found 180° apart from each other, and these are the moments when each harvester hits the ground.

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Figure 20 compares the output power between the simulation and experimental results for both DDO and RBDO cases at 10 and 20 km h⁻¹. Due to complex vibration loading not explained in the simulation, the experimental voltage waveform is different from the simulation. Table 9 demonstrates the amount of energy per revolution by integrating power over time per one revolution cycle. As one can review, the energy from the experiment is smaller than the simulation. But one can observe the high impact region of voltage when the tire hits the ground, commonly from simulation and experiment. Also the table shows that the power ratio between DDO and RBDO is very similar between simulation and experiment—RBDO cases show about 40%–50% of power compared to DDO cases. On the other hand, RBDO case has a very satisfactory performance in comparison with DDO. Although we did not have the facility to measure the stress magnitude during the test, we observed that the DDO samples failed (PZT is cracked) as we increased the velocity above 20 km h⁻¹ while the RBDO samples were not broken at higher velocities up to 60 km h⁻¹. To summarize, the benefit of RBDO design is consistently observed in the experiment—excellent durability by lowering power generation to the required value.

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