Potential Power Output from Vehicle Suspension Energy Harvesting Given Bumpy and Random-Surfaced Roads

Abstract

The energy efficiency of vehicles is a crucial challenge relating to sustainable energy preservation and regeneration methods. Regenerative breaking has proven feasible, and there is interest in whether harvesting energy from a vehicle’s suspension is similarly feasible. We here provide methods for estimating the amount of power that can be regenerated from the suspension for given vehicle and road parameters. We show that a reasonable road model is a generalised Gaussian process known as AR(1). Using this model, we can derive the key equation used in the ISO 8608 standard for measuring road roughness, such that the AR(1) parameters can be related to the measured road roughness data. We find that the road roughness coefficient of ISO 8608 and the diffusion coefficient of the AR(1) road are equal up to a factor. We provide an analytical expression for the maximum amount of power that can be generated for given road and car parameters, derived via Fourier analysis. We further model harvesting from large bumps using Simulink. These results help to estimate the potential power output given the measured road data.

1. Introduction

Energy harvesting has been a central concern in human technology historically and continues to be so today. This includes the practice of harvesting energy from motion [1,2]. A basic principle of energy harvesting from motion created by humans, such as our body motion or vehicle motion, is to harvest energy that is otherwise lost as friction [2] so that one gains energy without impeding the desired motion. A potential source for such energy is the undesired vertical motion of vehicle bodies, which is presently typically lost as heat in the suspension system.

There is accordingly significant research into harvesting energy from suspension systems (see Refs. [3,4,5] for reviews). There are proposals involving variable damping coefficients [6], active suspension [7,8], regenerative control [9], and more [5,10,11,12]. There are several companies undertaking research on suspension harvesting (see, for example, Refs. [13,14]).

A crucial question in such research is how much energy can be extracted. The rough amount of energy available in passenger cars is estimated at 100–400 W [15,16]. The amount of energy that can potentially be recovered from the suspension depends on both the car and the road parameters. Understanding this dependence is important for judging the feasibility of suspension harvesting and what the optimal power outputs are.

In this article, we provide several results concerning the possible power output from the suspension as a function of the road and vehicle parameters. We employ numerical and analytical tools, including stochastic processes models and system dynamics methods [17]. We employ the widely used quarter car model which is regarded as accurate while not being too complex [18,19]. In the quarter car model, one models a quarter of the car, as illustrated in Figure 1. We model the road either as having randomly distributed uneven bumps or speed bumps. The model we propose using for randomly distributed bumps is known as the AR(1) process (autoregressive model of order 1) [20]. In this process, there are random changes in the vertical profile of the surface, as well as a degree of memory of the surface at the previous point. We show that this model matches the ISO 8608 standard [21] for road roughness well. We find that the road roughness coefficient of ISO 8608 and the diffusion coefficient of the AR(1) road are equal up to a factor. We model replacing the resistive dampening in the suspension with an energy harvester. This approach is chosen so that the energy harvesting does not impact the motion differently to a normal suspension. We consider both resistive loads and supercapacitor loads. We provide numerical simulations and analytical expressions concerning how much power can be extracted for the given road and vehicle parameters.

2. One Speed Bump

This section concerns how much power can be extracted from a single speed bump. We firstly describe the simulation model and then analyse how much power can be harvested when the car goes over the bump.

2.1. Simulation Model

We model a speed bump by generating a road height $z_r\left(t\right)=Hsin\left({\displaystyle \frac{\pi v}{L}}t\right)$, where H is the maximal height, L is the length, v is the car speed, and t is the time. The bump is depicted in Figure 2. We use this model because it is stated to be a good approximation for the commonly used circular speed bump [22,23].

In the simulation, the harvester replaces the damper c shown in Figure 1. We employ the paradigm that only energy otherwise dissipated as heat may be harvested, such that the motion of the vehicle is not altered relative to the normal suspension when the harvester is introduced. Whilst in some circumstances, such as water wave energy harvesting [24], it can be advantageous to optimise power output by maximising the motion, in the case of vehicle suspension, this is clearly not desired. The damping force from the harvester is modelled as $F_{\mathrm{damp}}=-c{\dot{z}}_{\mathrm{rel}}$, where $z_{\mathrm{rel}}=z_2-z_1$. This $F_{\mathrm{damp}}$ models the scenario of a resistive load connected to an EM harvester as described in Appendix A. The case of a supercapacitor as the load is considered in Appendix C. The damping force conducts infinitesimal work $\mathrm{d}W=F_{\mathrm{damp}}\mathrm{d}{z}_{\mathrm{rel}}$ on the quarter car with associated power $P={\displaystyle \frac{\mathrm{d}W}{\mathrm{d}t}}=F_{\mathrm{damp}}{\dot{z}}_{\mathrm{rel}}=-c{\dot{z}}_{\mathrm{rel}}^2$. Energy conservation then means that $c{\dot{z}}_{\mathrm{rel}}^2$ is the maximal power that can be generated by the harvester in the process. The energy generated E is the instantaneous power P integrated from the moment of hitting the bump to when the motion induced by the bump stops.

The energy harvested E depends on the bump and car parameters. Unless stated otherwise, the simulations for the speed bump employ the following parameters: $c_1=1000$ N$·$s/m, $c_2=1400$ N$·$s/m, $k_1=5700$ N/m, $k_2=$ 135,000 N/m, $m_1=466.5$ kg, $m_2=49.8$ kg, $H=0.1$ m, and $L=5.2$ m. These were chosen to match a low-speed electric vehicle such as an E-Trike (a three-wheeled low-speed electric vehicle).

In this section on speed bumps, we use Simulink [25] to calculate the energy harvested from the relative motion of the car body and the wheel. The associated state-space model is described in Appendix A.

2.2. Most Power Is Available When Passing the Bump

In Figure 3, we see a typical output of a simulation, divided into ‘being over the bump’ (region I) and ‘having passed the bump’ (region II), respectively. The wheel is initially accelerated quickly in region I; the suspension translates this into a smoother motion of the car chassis, consistent with a significant relative motion between the chassis and wheel. It is from this relative motion that one can gain power by damping the motion with a transducer.

Figure 4 shows how most energy is harvested at two moments associated with entering and exiting the bump. It moreover shows that there is a band of damping coefficients from $c=3000\mathrm{N}·\mathrm{s}/\mathrm{m}$ to $c=8000\mathrm{N}·\mathrm{s}/\mathrm{m}$, with higher and lower values relative to that band yielding less. Intuitively, those extreme values of c lead to the energy being dissipated in the wheel instead of being harvested.

2.3. How Harvested Power Depends on Speed, Damping Constant, and Bump Parameters

Very different values, ranging from 3000 N$·$ s /m to 11,000 N$·$ s /m, of the damping coefficient c give rise to maximal power for different speeds, as shown in Figure 5. For example, if in the 40 km/h case c was assigned to be 3110 Ns/m, for which the harvested energy from a quarter car model moving at 15 km/h is maximal, the harvested energy would be 229.27 J, taking up 72.15 % of that harvested by setting $c=11,050\mathrm{N}·\mathrm{s}/\mathrm{m}$, which is 317.75 J. The speed bump simulations show that the height of the bump has little to no effect on which c maximises the power harvested, whereas the length of the bump does (see Appendix B).

Next, we turn towards a road with randomly distributed bumps of varying heights and a more analytical treatment.

3. Random Road Surface

We now consider suspension energy harvesting given a road model corresponding to imperfections rather than well-structured speed bumps. A simple approach to model a random road is to pick the height $z_{\mathrm{r}}\left(x\right)$ from a Gaussian distribution (also called the Bell curve or Normal distribution) at each position x (for some discretisation of x). However, physical surfaces tend to have smoother changes, so we consider a generalisation of a Gaussian road modelled by an autoregressive process, as will be explained below. We use the generalised model to derive an expression for how much power can be extracted as a function of the parameters involved. The derivation involves, for technical convenience, Fourier transforming the road surface and then working in the frequency domain.

3.1. Model of Generalised Gaussian Road AR(1)

We assume the road surface can be modelled by the autoregressive process AR(1) [26]. The term ‘autoregressive’ stems from the fact that the current value of the process is regressed upon (or predicted from) its own past values. The number 1 indicates that only the most recent time value is used. More specifically,

$$z_r(x+\Delta x)=(1-d)z_r\left(x\right)+\xi (0,\sigma ),$$

(1)

where $z_r\left(x\right)$ is the height of road at position x, $\Delta x$ is the distance between road height samples, d is a decay parameter, and $\xi$ is a normal distribution function independent of x, with the mean 0 and standard deviation $\sigma$.

Applying Equation (1) iteratively n times obtains

$$z_r(x+n\Delta x)={(1-d)}^n{z}_r\left(x\right)+\sum _{i=0}^{n-1}{(1-d)}^i{\xi }_i,$$

(2)

where each ${\xi }_i$ is picked independently with the mean 0 and standard deviation $\sigma$.

We shall later encounter ratios of the parameters d, $\sigma$, and $\Delta x$ associated with the AR(1) road. We will mainly focus on the regime where d is close to 0, since we will show that the AR(1) model matches the ISO standard for characterising road roughness in that regime. We note that, for small d, two ratios of AR(1) parameters are expected to directly correspond to physical properties of the road and not be artefacts of the sampling spacing $\Delta x$. Consider replacing $\Delta x$ with $n\Delta x$. Then, (i) d changes to $nd$ for small d from Equation (2) and (ii) ${\sigma }^2$ changes to $n{\sigma }^2$. Therefore, the ratios $\lambda ={\displaystyle \frac{d}{\Delta x}}$ and $D={\displaystyle \frac{{\sigma }^2}{\Delta x}}$ remain constant under such sample interval replacement. Keeping in line with common terminology of stochastic processes in physics, we term D the diffusion constant and $\lambda$ the decay rate.

The impact of the road surface on the quarter car relative velocity ${\dot{z}}_{rel}={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}(z_1-z_2)$ can be more conveniently calculated in terms of sinusoidal decompositions. The derivatives in the Newtonian equations of motion are easy to calculate for sinuisodal functions of time. We shall therefore make use of the Fourier transform of the road surface. To distinguish it from other heights, we shall label the road height $z_{\mathrm{r}}$ (and the heights of the two masses as $z_1$ and $z_2$ where $z_{\mathrm{rel}}=z_1-z_2$). Since the car travels with speed v, for time t, $z_i\left(x\right)=z_i\left(vt\right)$ whether i is 1, 2, r, or $rel$. There is accordingly one Fourier transform of $z_i$ associated with the x parametrisation,

$${\widehat{z}}_i(\Omega ):={\int }_{-\infty }^{\infty }{\mathrm{e}}^{-i\Omega x}{z}_i\left(x\right)\mathrm{d}x$$

(3)

and another associated with t as follows:

$$Z_i\left(\omega \right):={\int }_{-\infty }^{\infty }{\mathrm{e}}^{-i\omega t}{z}_i\left(t\right)\mathrm{d}t.$$

(4)

We shall analyse how each Fourier component of the road is associated with the Fourier component of the relative motion ${\dot{z}}_{\mathrm{rel}}$ of the same frequency. A Fourier component of the road with weight ${\widehat{z}}_{\mathrm{r}}(\Omega )$ in the Fourier decomposition will turn out to be required by the equations of motion to have a fixed ratio with a Fourier component ${\widehat{\dot{z}}}_{\mathrm{rel}}(\Omega )$ of the relative motion. This ratio is termed the transfer function $H(\Omega )$ of the two given Fourier components. The potential power output will be proportional to $\int |{\widehat{\dot{z}}}_{\mathrm{rel}}{(\Omega )|}^{2}d\Omega ={\int \left|H(\Omega )\right|}^2{\left|{\widehat{z}}_r(\Omega )\right|}^{2}d\Omega$. We next calculate $|{\widehat{z}}_{\mathrm{r}}{(\Omega )|}^2$, known as the spectral power density of the road, followed by calculating the transfer function $H(\Omega )$.

3.2. Spectral Power Density of Generalised Road

We will derive the expression of the spectral power density [27] $S_{z_r}(\Omega ):={\left|{\widehat{z}}_r(\Omega )\right|}^2$ for the generalised Gaussian road AR(1), where the notation “$:=$” denotes a definition.

We start with the fact that the spectral power density $S_{z_r}(\Omega )$ is the Fourier transform of the autocorrelation function:

$$S_{z_r}(\Omega )={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }E\left(z_r\left(x\right)z_r(x+X)\right){\mathrm{e}}^{-i\Omega X}\mathrm{d}X,$$

(5)

where $E\left(z_r\left(x\right)z_r(x+X)\right)$ is termed the autocorrelation of $z_r\left(x\right)$. Equation (5) follows from noting that $E\left[S_{z_r}(\Omega )\right]=E\left[{\widehat{z}}_r{\widehat{z}}_r^{\ast }\right]={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\mathrm{e}}^{-i\Omega x}{\mathrm{e}}^{i\Omega x^{\prime }}E\left[z_r\left(x\right)z_r\left(x^{\prime }\right)\right]dxd{x}^{\prime }$ and from letting $x=x^{\prime }+X$. (See, for example, Ref. [26] as a background reference on such calculations.)

To evaluate the integral in Equation (5) for our case of an AR(1) road, we firstly calculate the autocorrelation function. Given the AR(1) assumption of Equation (1) and recalling that the assumption implies Equation (2), the autocorrelation of the road can be obtained by

$$E\left[z_r(x+n\Delta x)z_r\left(x\right)\right]={(1-d)}^{\left|n\right|}E\left[z_r{\left(x\right)}^2\right]={(1-d)}^{\left|n\right|}{\displaystyle \frac{{\sigma }^2}{1-{(1-d)}^2}}.$$

(6)

The last equality in Equation (6) can be recovered as follows. $E\left[z_r{(x+\Delta x)}^2\right]=E\left[{((1-d)z_r\left(x\right)+\xi (0,\sigma ))}^2\right]={(1-d)}^{2}E\left[z_r{\left(x\right)}^2\right]+E\left[\xi {(0,\sigma )}^2\right]$. Demand $E\left[z_r{(x+\Delta x)}^2\right]=E\left[z_r{\left(x\right)}^2\right]:=a$. Then, $a={(1-d)}^{2}a+{\sigma }^2$, such that $a={\displaystyle \frac{{\sigma }^2}{1-{(1-d)}^2}}$.

We now wish to use Equation (6) to evaluate Equation (5).

When $d\ne 1$, let $\left|n\right|={\displaystyle \frac{\left|X\right|}{\Delta x}}$ and define ${\left({(1-d)}^{{\displaystyle \frac{1}{\Delta x}}}\right)}^X:={\mathrm{e}}^{-\alpha X}$, where $\alpha :=-{\displaystyle \frac{1}{\Delta x}}ln(1-d)>0$. Then ${(1-d)}^{\left|n\right|}={\mathrm{e}}^{-\alpha \left|X\right|}$, such that the term in Equation (5) is proportional to

$${\int }_{-\infty }^{\infty }{\mathrm{e}}^{-\alpha \left|X\right|}{\mathrm{e}}^{-i\Omega X}\mathrm{d}X={\int }_0^{\infty }{\mathrm{e}}^{-\alpha X}{\mathrm{e}}^{-i\Omega X}\mathrm{d}X+{\int }_{-\infty }^0{\mathrm{e}}^{\alpha X}{\mathrm{e}}^{-i\Omega X}\mathrm{d}X={\displaystyle \frac{2\alpha }{{\alpha }^2+{\Omega }^2}}.$$

(7)

Thus, combining Equation (7), Equation (6) and Equation (5) one sees that

$$S_{z_r}(\Omega )={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{{\sigma }^2}{1-{(1-d)}^2}}{\displaystyle \frac{2\alpha }{{\alpha }^2+{\Omega }^2}}={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{{\sigma }^2}{1-{\mathrm{e}}^{-2\alpha \Delta x}}}{\displaystyle \frac{2\alpha }{{\alpha }^2+{\Omega }^2}}.$$

(8)

When $d=1$, from Equation (1), we have $z\left(x\right)=\xi (0,\sigma )$, which corresponds to white noise. A white noise road profile has distribution $N(0,\sigma )$ and is commonly stated as having autocorrelation ${\sigma }^2\delta \left(X\right)$ (where $\delta \left(X\right)$ is the unit impulse function), such that

$$\begin{array}{cc}\hfill S_z(\Omega )=& {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }E\left(z\left(x\right)z(x+X)\right){\mathrm{e}}^{-i\Omega X}\mathrm{d}X\hfill \\ \hfill =& {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\sigma }^2\delta \left(X\right){\mathrm{e}}^{-i\Omega X}\mathrm{d}X\hfill \\ \hfill =& {\displaystyle \frac{1}{2\pi }}{\sigma }^2{\int }_{-\infty }^{\infty }\delta \left(X\right)(cos(\Omega X)-isin(\Omega X))\mathrm{d}X\hfill \\ \hfill =& {\displaystyle \frac{1}{2\pi }}{\sigma }^2(cos\left(0\right)-isin\left(0\right))\hfill \\ \hfill =& {\displaystyle \frac{{\sigma }^2}{2\pi }}.\hfill \end{array}$$

(9)

Equation (9) can, reassuringly, also be recovered as a limit of the following computationally well-defined discrete Fourier transform of Equation (6):

$$\begin{array}{cc}\hfill {\int }_{-\infty }^{\infty }{(1-d)}^{\left|n\right|}{\displaystyle \frac{{\sigma }^2}{1-{(1-d)}^2}}{\mathrm{e}}^{-i\Omega X}\mathrm{d}X\approx & {\displaystyle \frac{{\sigma }^2}{1-{(1-d)}^2}}\sum _{n=\infty }^{-\infty }{(1-d)}^{\left|n\right|}{\mathrm{e}}^{-i\Omega \Delta xn}\hfill \\ \hfill =& \left({\displaystyle \frac{{\sigma }^2}{1-{(1-d)}^2}}\right)\left({\displaystyle \frac{1}{1+{(1-d)}^2-2(1-d)cos(\Omega \Delta x)}}\right)\hfill \end{array}$$

(10)

From Equation (10), when $d=1$, $S_{z_r}={\displaystyle \frac{{\sigma }^2}{2\pi }}$.

3.3. Relation to ISO Standard for Characterising Road Roughness

In the ISO 8608 standard, the road profile is characterised in terms of the power density spectral density $S_{z_r}(\Omega )$ [21] as a function of the spatial angular frequency $\Omega$. If we again let x represent the longitudinal position of the vehicle and $z_x$ represent the road height above the reference line at x, then $\widehat{z}(\Omega ):={\int }_{-\infty }^{\infty }{e}^{-i\Omega x}{z}_r\left(x\right)\mathrm{d}x$ and $S_{z_r}(\Omega ):={\left|\widehat{z}(\Omega )\right|}^2$, where ‘$:=$’ denotes a definition. The ISO 8608 standard [21] involves the curve-fitting assumption that

$$S_{z_r}(\Omega )=C{\left({\displaystyle \frac{\Omega }{{\Omega }_0}}\right)}^{-\mathcal{W}}\mathrm{if}2\pi ·0.011\le \Omega \le 2\pi ·2.83\mathrm{rad}/\mathrm{m},\mathrm{and}0\mathrm{otherwise}.$$

(11)

Here, $\Omega$ is the spatial angular frequency, and often ${\Omega }_0=1$ rad/m. C is a coefficient that can be interpreted as the road roughness. The ‘waviness’ $\mathcal{W}$ is a somewhat controversial value, which may depend on $\Omega$ and is often set as 2 for simplicity [28]. The formula for $S_{z_r}(\Omega )$ then only has one free parameter, which is the roughness coefficient C.

We are now in a position to derive $S_{z_r}(\Omega )$ of Equation (11) from the assumption of the road being generated by AR(1), as defined in Equation (1).

For a small decay rate, $\lambda ={\displaystyle \frac{d}{\Delta x}}$, $\alpha \approx {\displaystyle \frac{d}{\Delta x}}$, and $1-{(1-d)}^2\approx 2d$. Then, from Equation (8),

$$\begin{array}{cc}\hfill S_{z_r}(\Omega )=& {\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{{\sigma }^2}{1-{(1-d)}^2}}{\displaystyle \frac{2\alpha }{{\alpha }^2+{\Omega }^2}}\approx {\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{{\sigma }^2}{2d}}{\displaystyle \frac{2\frac{d}{\Delta x}}{{\left(\frac{d}{\Delta x}\right)}^2+{\Omega }^2}}\approx {\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{{\sigma }^2}{\Delta x}}{\Omega }^{-2}={\displaystyle \frac{D}{2\pi }}{\Omega }^{-2},\hfill \end{array}$$

(12)

where the second approximation assumes that ${\Omega }^2\gg {\displaystyle \frac{d}{\Delta x}}$. D is the diffusion rate of the road. When comparing with Equation (11), we find that the road roughness coefficient of ISO 8608 and the diffusion coefficient of the AR(1) road are equal up to a factor:

$$C={\displaystyle \frac{D}{2\pi }}.$$

(13)

4. Energy Harvested on a Random Road Surface

We firstly derive the transfer function which captures how different the frequency components of the road surface are associated with the relative motion of the car body and wheel. We then use the transfer function to derive the power dissipated as a function of the car and road parameters.

4.1. Transfer Function on a Gaussian Road

Transfer functions play a crucial role in analysing and optimising suspension systems (see, for example, Ref. [17]). For a given sinusoidal input with a given frequency, there is a certain output response of the given frequency, and the transfer function in question is the ratio of the output and input amplitudes. In our case, the input is the road surface and the output is the relative motion of the car body and tire.

The relevant transfer function of the quarter car model can be derived as follows. Firstly, write down the Newtonian equations of motion for the quarter car:

$$\left\{\begin{array}{cc}\hfill m_1{\ddot{z}}_1& =k_1(z_2-z_1)+c_1({\dot{z}}_2-{\dot{z}}_1)\hfill \\ \hfill m_2{\ddot{z}}_2& =k_2(z_{\mathrm{r}}-z_2)+c_2({\dot{z}}_{\mathrm{r}}-{\dot{z}}_2)-k_1(z_2-z_1)-c_1({\dot{z}}_2-{\dot{z}}_1),\hfill \end{array}\right.$$

(14)

where the terms are as defined in Figure 1. Fourier transforming both sides, and writing for notational simplicity $\mathcal{F}\left[z_i\left(t\right)\right]=Z_i\left(\omega \right)$, yields

$$\left\{\begin{array}{cc}\hfill -m_1{\omega }^2{Z}_1\left(\omega \right)& =k_1(Z_2\left(\omega \right)-Z_1\left(\omega \right))+i\omega c_1(Z_2\left(\omega \right)-Z_1\left(\omega \right))\hfill \\ \hfill -m_2{\omega }^2{Z}_2\left(\omega \right)& =k_2(Z_r\left(\omega \right)-Z_2\left(\omega \right))+i\omega c_2(Z_r\left(\omega \right)-Z_2\left(\omega \right))\hfill \\ \hfill & -k_1(Z_2\left(\omega \right)-Z_1\left(\omega \right))-i\omega c_1(Z_2\left(\omega \right)-Z_1\left(\omega \right))\hfill \end{array}\right.$$

(15)

Equation (15) can be seen in a few lines (see Appendix E) to yield a transfer function $H\left(\omega \right)$, respecting

$$\begin{array}{cc}\hfill H\left(\omega \right):=& {\displaystyle \frac{Z_2\left(\omega \right)-Z_1\left(\omega \right)}{Z_r\left(\omega \right)}}={\displaystyle \frac{-m_1{\omega }^2}{-m_1{\omega }^2+k_1+i\omega c_1}}{\displaystyle \frac{Z_2\left(\omega \right)}{Z_r\left(\omega \right)}}\hfill \\ \hfill =& {\displaystyle \frac{-m_1{\omega }^2(k_2+i\omega c_2)}{(-m_1{\omega }^2+k_1+i\omega c_1)(-m_2{\omega }^2+k_2+i\omega c_2)-m_1{\omega }^2(k_1+i\omega c_1)}}.\hfill \end{array}$$

(16)

Equation (16) corresponds to a resistive load coupled to the EM harvester. In Appendix C, we moreover derive an alternative transfer function H corresponding to a supercapacitor as a load.

We can now write down the transfer function for a common choice of car parameters. The golden car is a quarter car model used in the international road roughness index (see Appendix D) with specific parameter values given in

Table 1. Parameters of the golden car.

Parameter	Value	Unit
$\mu$ = $m_2$/$m_1$	0.15	-
${\kappa }_1$ = $k_1$/$m_1$	63.3	${\mathrm{s}}^{-2}$
${\kappa }_2$ = $k_2$/$m_1$	653	${\mathrm{s}}^{-2}$
$c_{\mathrm{gold}}$ = $c_1$/$m_1$	6.0	${\mathrm{s}}^{-1}$
$c_0$ = $c_2$/$m_1$	0.0	${\mathrm{s}}^{-1}$

. Taking that case as an example, Equation (16) reduces to

$$H_{\mathrm{gold}}\left(\omega \right)={\displaystyle \frac{-{\omega }^2{\kappa }_2}{({\kappa }_1+i\omega c_{\mathrm{gold}}-{\omega }^2)({\kappa }_2-{\omega }^2\mu )-({\kappa }_1+i\omega c_{\mathrm{gold}}){\omega }^2}},$$

(17)

where the golden car parameters are denoted as $\mu =m_2/m_1$, ${\kappa }_1$ = $k_1$/$m_1$, ${\kappa }_2=k_2$/$m_1$, $c_{\mathrm{gold}}=c_1$/$m_1$, and $c_0=c_2$/$m_1=0$ [29].

The power density spectrum of the relative motion of the golden quarter car is, for any quarter car, given in terms of the power density spectrum

$$S_{z_{rel}}(\Omega )={\left|{\displaystyle \frac{Z_2\left(\omega \right)-Z_1\left(\omega \right)}{Z_{\mathrm{r}}\left(\omega \right)}}{Z}_{\mathrm{r}}\left(\omega \right)\right|}^2={\left|H(\Omega )\right|}^2{S}_{z_r}(\Omega ).$$

(18)

Next, we will apply Equation (17) and Equation (18) to calculate the energy dissipation in the damper/harvester.

4.2. Potential Power Output

The instantaneous power dissipated in the damper/harvester can be written in terms of the relative motion of the tire and car body as $P=F{\dot{z}}_{\mathrm{rel}}=\left(c_1{\dot{z}}_{\mathrm{rel}}\right){\dot{z}}_{\mathrm{rel}}=c_1{\dot{z}}_{\mathrm{rel}}^2$. Taking the stochastic average yields

$$\begin{array}{cc}\hfill E\left[P\right]& =c_{1}E\left[{\dot{z}}_{\mathrm{rel}}^2\right]=c_1{\sigma }_{{\dot{z}}_{\mathrm{rel}}}^2,\hfill \end{array}$$

(19)

where the second equality assumes that the average relative velocity, without conditioning on known previous values, is 0.

$${\sigma }_{{\dot{z}}_{\mathrm{rel}}}^2={\int }_{-\infty }^{+\infty }{\omega }^2{S}_{z_{\mathrm{rel}}}\left(\omega \right)\mathrm{d}\omega .$$

(20)

Moreover,

$$S_{z_{\mathrm{rel}}}\left(\omega \right)={\left|H_{rel}\left(\omega \right)\right|}^2{S}_{z_r}\left(\omega \right)$$

(21)

Moreover, noting that

$$S_{z_r}\left(\omega \right)={\displaystyle \frac{1}{v}}{S}_{z_r}(\Omega ),$$

(22)

and, as shown earlier, the known fact that

$$\begin{array}{cc}\hfill S_{z_r}(\Omega )=& {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }E\left(z_r\left(x\right)z_r(x+X)\right){\mathrm{e}}^{-i\Omega X}\mathrm{d}X,\hfill \end{array}$$

(23)

one sees that ${\sigma }_{{\dot{z}}_{\mathrm{rel}}}^2$ can be written as

$$\begin{array}{cc}\hfill {\sigma }_{{\dot{z}}_{\mathrm{rel}}}^2& ={\int }_{-\infty }^{+\infty }{S}_{{\dot{z}}_{\mathrm{rel}}}\left(\omega \right)\mathrm{d}\omega \hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi v}}{\int }_{-\infty }^{+\infty }{\omega }^2\left({\int }_{-\infty }^{\infty }E\left(\xi \left(x\right)\xi (x+X)\right){\mathrm{e}}^{-i\Omega X}\mathrm{d}X\right){\left|H_{\mathrm{rel}}\left(\omega \right)\right|}^2\mathrm{d}\omega .\hfill \end{array}$$

(24)

Plugging Equation (8) and Equation (16) into Equation (24) yields

$$\begin{array}{cc}\hfill {\sigma }_{{\dot{Z}}_{\mathrm{rel}}}^2=& {\displaystyle \frac{1}{2\pi v}}{\int }_{-\infty }^{+\infty }{\omega }^2\left({\displaystyle \frac{{\sigma }^2}{1-{(1-d)}^2}}{\displaystyle \frac{2\alpha }{{\alpha }^2+{(\omega /v)}^2}}\right)\hfill \\ \hfill & \left({\displaystyle \frac{\mathrm{d}\omega m_1^2(k_2^2+{\omega }^2{c}_2^2){\omega }^4}{{\left(m_1{m}_2{\omega }^4-(m_1{k}_2+m_2{k}_1+c_1{c}_2+m_1{k}_1){\omega }^2+k_1{k}_2\right)}^2+{\left((-m_1{c}_2-m_2{c}_1-m_1{c}_1){\omega }^3+(c_2{k}_1+c_1{k}_2)\omega \right)}^2}}\right)\hfill \end{array}$$

(25)

By putting in, for example, the golden car parameters, Equation (25) becomes

$$\begin{array}{cc}\hfill & {\sigma }_{{\dot{z}}_{\mathrm{rel}}}^2\hfill \\ \hfill =& {\displaystyle \frac{1}{2\pi v}}{\int }_{-\infty }^{+\infty }{\omega }^2\left({\displaystyle \frac{{\sigma }^2}{1-{(1-d)}^2}}{\displaystyle \frac{2\alpha }{{\alpha }^2+{(\omega /v)}^2}}\right)\hfill \\ \hfill & \left({\displaystyle \frac{426409s^{-2}{\omega }^4}{{\left(0.15{\omega }^4-\left(725.795s^{-2}\right){\omega }^2+41334.9s^{-4}\right)}^2+{\left(6.9s^{-1}{\omega }^3+\left(3918s^{-3}\right)\omega \right)}^2}}\right)\mathrm{d}\omega \hfill \end{array}$$

(26)

Furthermore, for the golden car speed used in the IRI road roughness index, $v=22.22$ m/s, the integral of Equation (26) can be evaluated analytically with Mathematica to yield

$$\begin{array}{cc}\hfill {\sigma }_{{\dot{z}}_{\mathrm{rel}}}^2=& \left({\displaystyle \frac{{\sigma }^2}{1-{(1-d)}^2}}\right)\left({\displaystyle \frac{2142.66\alpha -25732.7{\alpha }^3+173379{\alpha }^5+16738.4{\alpha }^7}{1.27791+16.4885{\alpha }^2+108.163{\alpha }^4+15.3164{\alpha }^6+{\alpha }^8}}\right)\hfill \\ \hfill \approx & \left({\displaystyle \frac{{\sigma }^2}{2}}\right)1676.69{\displaystyle \frac{1}{\Delta x}}={\displaystyle \frac{D}{2}}1676.69\hfill \end{array}$$

(27)

where $\alpha$ is defined as $\alpha =-{\displaystyle \frac{1}{\Delta x}}ln(1-d)$ and $D:={\displaystyle \frac{{\sigma }^2}{\Delta x}}$ is the diffusion constant of the AR(1) process. For concreteness, suppose $D={\displaystyle \frac{0.{01}^2}{10}}$ and that the golden car parameters are used. For a golden car with $m_1=500$ kg, the energy dissipated from Equation (19) is then

$$\begin{array}{c}\hfill P=c_1{\sigma }_{{\dot{Z}}_{\mathrm{rel}}}^2=c_1{\left({\displaystyle \frac{D}{2}}1676.69\right)}^2=6\ast 500\ast {(9.156\ast 0.01)}^2=25.15\mathrm{W}.\end{array}$$

(28)

The above argument exemplifies how, in general, plugging Equation (25) into Equation (19) yields an analytical expression for the expected power as a function of the parameters of the quarter car model and the AR(1) road model. This is a key technical contribution of this paper.

We can also, as a further example, evaluate the power when the road model reduces to AR(0). From Equation (8), the power spectrum is then ${\displaystyle \frac{{\sigma }^2}{2\pi }}$. Similar to Equation (26),

$$\begin{array}{cc}\hfill & {\sigma }_{{\dot{z}}_{\mathrm{rel}}}^2\hfill \\ \hfill =& {\displaystyle \frac{{\sigma }^2}{2\pi v}}{\int }_{-\infty }^{+\infty }{\omega }^2\left({\displaystyle \frac{426409s^{-2}{\omega }^4\mathrm{d}\omega }{{\left(0.15{\omega }^4-\left(725.795s^{-2}\right){\omega }^2+41334.9s^{-4}\right)}^2+{\left(6.9s^{-1}{\omega }^3+\left(3918s^{-3}\right)\omega \right)}^2}}\right)\hfill \\ \hfill \approx & 91.485{\sigma }^2,\hfill \end{array}$$

(29)

which leads to the power of the golden car of $m_1=500$ kg and $\sigma =0.01$ m to be

$$\begin{array}{c}\hfill P=c_1{\sigma }_{{\dot{z}}_{\mathrm{rel}}}^2=6\ast 500\ast 91.485\ast 0.{01}^2=27.45\mathrm{W}\end{array}$$

(30)

The golden car example can further be used to relate the International Roughness Index IRI for roads [29,30] to the extractable power of that golden car, as detailed in Appendix D.

5. Summary and Outlook

We investigated how much power can be harvested from the suspension of a vehicle. We assumed the paradigm of converting the energy otherwise dissipated as heat into work. We firstly investigated how much power can be harvested from a semicircular speed bump, finding that most power is harvested whilst passing the bump rather than in the oscillations after it, and that the harvested amount depends on the relation between the damping coefficient to the speed and the length of the bump. We then proposed a generalised Gaussian stochastic model for roads: AR(1). Under the AR(1) assumption, we derived an analytical expression for the amount of power that can be extracted as a function of the parameters of the car and road model. For a special case of AR(1) and a standard car known as the golden car, the extractable power can be directly determined from the International Roughness Index of the road. Another special case of AR(1) allows us to derive a key assumption underlying an ISO standard for characterising road roughness, such that parameters from the ISO standard, together with a free parameter of AR(1), can be employed to analytically calculate the extractable power.

Our findings should help evaluate the potential for energy harvesting from vehicle suspension across a range of road conditions. The results also suggest that the standard characterisations of roads should be augmented to include both free parameters in the AR(1) model. Furthermore, preliminary results not described here suggest that convolutional neural networks can significantly enhance the harvesting by controlling the damping coefficient efficiently, something which deserves further investigation. In particular, dynamically varying damping parameters should be investigated. Furthermore, the assumption of linear systems could be relaxed. Our analytical methods should be extended to more complicated non-linear damping forces from the harvester. The analytical results here should be compared with experimental data. The investigation can also be viewed as contributing to the general program of understanding how much power can be extracted from pseudo-random power sources [31,32].