A Novel Design and Performance Analysis of Piezoelectric Energy Harvester with Application to a Vehicle Suspension System Moving on Uniform Bridges

The development of energy recovery systems in vehicles is a form of synergy between energy efficiency and renewable energy to overcome the energy crisis and global warming problems. This paper addresses such challenges by introducing a novel design of a piezoelectric energy harvester (PEH) for a vehicle suspension system. The PEH is designed to capture the linear vibration of the suspension system subjected to road surface roughness and move on uniform bridges. The vibration is amplified by a pressurized liquid cylinder-piston mechanism which deforms the piezoelectric bar to produce electric power. An additional novel mechanism in the form of a piezoelectric protector is also proposed to avoid physical damage due to excessive unidirectional compressive force as a result of an unpredictable uneven road. Given the key design parameters, the electric power can be harvested up to 67.5 W for a PEH with a span of 150 m of the bridge, a velocity of 40 m/s, and a vehicle mass of 2,691.6 kg.

Energy harvesting; Hydraulic mechanism; Piezoelectric; Vehicle suspension

Design Concept and System Modelling  

        The PEH is proposed to be universally used on a wide range of vehicle suspension types. However, the space available on the suspension section is limited. Also, the vibration level experienced by the suspension is different for every vehicle. It leads to variations of load exerted on the piezoelectric transducer. To harvest the vibration energy optimally, an effective mechanism of the PEH is designed to address those challenges.

2.1. Design Concept

        Figure 1a shows the proposed PEH design concept (Figure 1b) mounted on a vehicle suspension system. There are two main developed mechanisms: force magnification and piezoelectric protector mechanism (Figure 1c). The former is designed to amplify the excitation force through a linear-spring mounted on the suspension system. It works based on a pressurized liquid cylinder-piston mechanism known as a hydraulic system, consisting of the master cylinder and caliper cylinder components. Due to the vehicle suspension’s limited space, the PEH that can be mounted onto the suspension system is limited in volume. Thus, the master cylinder components with small volumes and easily attached must be chosen.

        However, some factors such as road surface roughness, traffic conditions, and vehicle speed dynamics may significantly affect excitation force. If the road surface is flat with congested traffic conditions, the vehicle suspension's excitation force may be weak. Thus, the force magnification mechanism must be set to effectively capture the suspension’s vibration while amplifying its magnitude simultaneously. Meanwhile, the piezoelectric protector is an additional mechanism to avoid piezoelectric damage due to excessive unidirectional compressive force as a result of an unpredictable uneven road. It uses springs as the main component for regulating the pressure that deforms the piezoelectric.

Figure 1 System: (a) suspension [Adapted from www.pakwheels.com (Ali, 2015)], (b) PEH, (c) diagram block of modeling

2.2. System Modelling

The model of the overall system in Figure 1a comprises three main subsystems: the dynamics of the bridge, the vehicle, and the PEH. In this subsection, each will be detailed, and the parameter values of the respective subsystems will be tabulated afterward.

2.2.1.  Modelling of Force Magnification Mechanism

Figure 2 shows the overall structure and working principle of the proposed force magnification mechanism, which is the main subsystem of the PEH. The spring force F generated by the vehicle suspension's vibration is transmitted to the large piston, which moves the master cylinder and produces force F1. The force is then transferred to the fluid in the closed channel. The fluid will move the small piston and produce F2 in the caliper. The force F2 will deform the piezoelectric bar and generate electrical energy. In short, this mechanism applies Pascal’s Law (Equations 1-2). By taking into account the cross-sectional area of both pistons, A1 and A2, the value of F2 can then be calculated,

The ratio of cross-sectional area is denoted by n_p = A2/A1 where n_p>1 so the excitation force in the master cylinder into the caliper can be magnified. This magnified force will deform the piezoelectric at the end of the piston in the caliper. By taking the effect of friction force between piston and cylinder and working fluid mass in the pressurized cylinder-piston system are not accounted for, the ratio of the big piston of master cylinder S1 and small piston of caliper S2 to the respective cross-sectional area A1 and A2 is,

Since the excitation force exerted by the vehicle suspension system on the PEH is random, impulsive impact force can occur anytime. It then becomes an important consideration for PEH design. A novel design of a piezoelectric protector mechanism is proposed to minimize physical damage. Figure 2 provides the design concept and its components. If the magnitude of force F approaches or exceeds the limit of compressive strength of piezoelectric material, the linear spring will be compressed accordingly. The push rod then sticks to the stopper and discharges to press the piezoelectric. Thus, the piezoelectric will always be within its strength limit and a safe mode.

2.2.2.   Modelling of PEH

Since the PEH is mounted onto the existing vehicle’s suspension, it can be considered as two main structures: super- and sub-structure. The former is a vehicle, while the latter is a hydraulic system-based PEH. The piezoelectricity cylinder-piston mechanism depicted in Figure 3a can be simplified as a translational lumped mass and is shown in Figure 3b. It consists of an equivalent mass m_eq, damping coefficient c_eq, and spring constant k_eq. Two external springs are from master cylinder k_t and piezoelectric protector k_s. If the piston thickness and its mass density, t_cp, and  are respectively defined, then the mass of the small piston under pressurized working fluid can be computed using Equation 3,

A big piston deforms the piezoelectric transducer in Figure 2c through piston extension where its spring constant is not considered. Spring stiffness k_pb of the big piston may be computed using Equation 4,

Considering the working fluid, the small piston can be modeled as a spring on a mass. The equivalent spring stiffness k_eq can be calculated using Equation 5.

Figure 2 Overall structure and working principle (a) proposed force magnification mechanism (b) Proposed piezoelectric protector mechanism (c) Piezoelectric bar

When the work done by the spring force is converted into electricity, there is an equivalent electric resistance. The electrical damping coefficient c_eq can be derived as follows (Aouali et al., 2021; Pasharavesh, Moheimani, and Dalir, 2020; Wu, Wang, and Xie, 2015; Xie and Wang, 2015),

Equation 6 contains three important variables: d₃₃ is the piezoelectric strain constant in the polling direction, c_a is the electrical capacity of the piezoelectric bar, and f is the natural vibration frequency of the system.

Once the equivalent mass m_eq, spring constant k_eq, and damping coefficient c_eq are obtained, the proposed PEH results in a damped single-degree-of-freedom system subjected to the road surface roughness. Considering the equivalent magnified force F_m(t) at the PEH as expressed in Equation 7, the electrical charge that can be stored by the piezoelectric bar is calculated using Equation 8,

Figure 3 PEH model (a) Cylinder-piston mechanism (b) Equivalent model of the proposed PEH

Equation 7 and Figure 3b contain some important parameters: y_v and y_wt are displacements of the vehicle body and wheel tire, respectively, in the area where the PEH is mounted, y_eq is the displacement of the PEH, and n is the magnification factor (cross-sectional area ratio). The electrical current resulted by the piezoelectric bar can be calculated by deriving the electrical charge in Equation 8 with respect to time,

The electrical voltage resulted by the piezoelectric bar can be found by dividing the electrical charge Q_ep in Equation 8 over electrical capacity of the piezoelectric material c_a,

Generated electrical power is then obtained by multiplying the electrical current in Equation 9 and electrical voltage in Equation 10 with the number of piezoelectric p,

Equations 8-11 reflect that the outputs of PEH are electrical parameters such as electrical charge, current, voltage, and power. In particular, electrical charge Q_ep is seen as a very instrumental parameter and must be produced optimally. The RMS of the electric power within a predefined time duration can be then determined by using Equation 12,

2.2.3.   Modelling of Bridge Dynamics Traversed by a Mobile Half-Car Planar Model

Quarter-car model is the common model employed. However, such a model type is considered inadequate for many realistic cases, particularly for analyzing the total vehicle dynamics. In this paper, the vehicle consisting of the driver and passenger is modeled as a mobile half-car planar model, as depicted in Figure 4a. The vehicle moves on a bridge above the uneven road surface. The case becomes transverse elastic deformation of the bridge traversed by a mobile half-car planar model carrying the PEHs. They are modeled in Figure 4b. In deriving the governing equations of motion, simplifications and assumptions are defined:

1.        The overall system is modeled in linear behavior.

2.        A mobile half-car planar model has six DoFs, which consist of a body, two PEHs, two-wheel tires, a driver, and a passenger. The vehicle body is constrained to have the vertical motion (bounce) and the angular motion (pitch), where every wheel-tire bounces in its respective coordinate. Also, the driver and the passenger are considered to have only their vertical oscillations.

3.        The PEH is mounted between the vehicle body and the wheel-tire in front or rear positions. 

4.        Passenger seats, suspension, and wheel-tire systems are modeled as a combination of linear springs and viscous dampers which are connected in parallel arrangements.

5.        The resilience and damping of the suspension and wheel-tire systems are expected to be sufficient so as to be more realistic models for simulation and analysis.

6.        The wheel-tire system is assumed to be in contact with the surface of the uneven road at all times.

To derive the motion equations of the overall system, the energy method is employed. By defining x as the axis along the length of the beam measured from the left to right end support, and t as the travel time, then y(x,t) can be characterized as the vertical deformation of the bridge all the way of the undeformed neural axis of the bridge as depicted by Figure 4b. The kinetic and potential energies of the overall system in Figure 4a using linear strains assumption are given by Equations 13-14, respectively. Both equations belong to the bridge, vehicle body (bounce and pitch), PEH, wheel-tire (front and rear), driver, and passenger. Parameter  in Equation 13 is the mass density per unit length of the uniform beam, EI in Equation 14 denotes the flexural rigidity of the beam, while H(x) represents the Heaviside function. In particular, Equation 14 contains variables  which point out the locations of the contact points of the front and rear tires with the bridge surface.

Rayleigh’s dissipation function and generalized force are expressed in Equations 16-18,

By considering the Galerkin approximation, y(x,t) is written in Equation 19, where  presents mode shapes and q_i(t) points out the generalized coordinates for the elastic deflection of the beam element. Orthogonality conditions are given by Equation 20, where the term  denotes the Kronecker delta for i,j=1,2,..,n,

Figure 4 Vehicle system model (a) the bridge traversed by a mobile half-car planar model (b) transverse elastic deformation of the bridge traversed by a mobile half-car planar model

Defining the eight state variables as in Equation 21, Lagrange’s equations for those variables can be expressed in Equations 22-23,

Finally, motion equations of the overall system can be derived in general forms,

1.    The vertical motion (bounce) for the driver is given by,

2.    The vertical motion (bounce) for the passenger is presented by,

3.    The vertical motion (bounce) for the vehicle is defined as,

4.    The vertical motion for the front PEH is,

5.    The angular motion (pitch) for the vehicle can be expressed as,

6.    The vertical motion for the rear PEH is,

7.    The vertical motion for the front wheel-tire can be written as,

8.    The vertical motion for the rear wheel-tire can be written as,

9.    The equation motion of the bridge is,

Parameter k_tot1 is equivalent spring stiffness between shock-breaker with external spring of master cylinder k_t, k_tot2 is equivalent spring stiffness between wheel-tires with external spring of piezoelectric protector k_s while two coefficients D₁ and D₂ denote the predefined interval of the motion of the vehicle. The roughness of road surface (Wei and Taghavifar, 2017),

G_q(n₀) indicates the roughness coefficient of the road surface, n₀ refers to a reference spatial frequency with a value of 0.1 m^-1, f₀ is a minimal boundary frequency with a value of 0.0628 Hz, v(t) denotes the vehicle velocity, and w(t) presents a zero-mean white noise.

2.2.4. Numerical Solver

Equations 24-32 form a system of nine second-order coupled differential equations. They could be written in the state-space model by converting all equations into first-order differential equation systems. Consequently, state variables become (2+(2·DOF)+n) as the following,

From Equation 34, the system of first-order differential equations can be arranged in Equation 35. Considering the computational time and accuracy, the time step  is selected as 0.001 s. Consequently, the displacements and velocities of the system at the time t_i+1 can be arranged as in Equation 36. By assuming that the system is in an equilibrium position at i=0 and t₀=0 s, the initial conditions of all are equal to zero.

The proposed PEH's performance is evaluated using important design factors: the ratio of piston cross-sectional area, vehicle velocity, and road roughness coefficient. Dynamic responses of displacement, velocity, and acceleration in all nodal coordinates of the half-car planar model, particularly at the point of the PEH mounted on the vehicle suspension, are used. For vehicle dynamics, the road roughness coefficient refers to ISO/TC108/SC2N67 (Wei and Taghavifar, 2017), in which only the classes B, E, and H are used, with roughness coefficient (G_q(n₀), in cm³) are 64; 4,096; and 65,536 respectively. Meanwhile, Tables 1 and 2 list the beam and vehicle parameters. Dimension and material properties of the PEH are listed in Table 3. The harvested electric power is expressed in RMS value.

3.1. Dynamic Responses Analysis

Results are obtained by setting the numerical variables in simulations such as: (1) a random Gaussian white noise with a zero mean shown in Figure 5a, (2) road surface roughness of the car on the road in the class of B for 10 seconds are depicted in Figure 5b, (3) Two types of velocity are used in terms of variable and constant velocities as, where the former is shown in Figure 6a, and the latter is set to be constant of 30 m/s, (4) parameters of the bridge, vehicle, and PEH are based on Tables 1-3. Dynamic displacements are displayed in Figures 6b. Each figure compares the displacement of respective subsystems under both velocities. However, it should be noted that the values are selected only to show the dynamic characteristics. Any arbitrary values can be chosen as long as the time trajectory of the vehicle moving on the bridge is sufficient. Figure 7 reveals that the developed motion equations with the numerical solver can capture the dynamics of the overall system. The dynamic displacement obtained from variable and constant velocities is different. An amplification factor is produced when the maximum value of each displacement is compared. The ratio is averagely found around 1.23. It can be a variation if the trajectory profile is varied accordingly. However, those figures are intended to demonstrate the effect of velocity on the generated electrical power of the proposed PEH model, as justified by Figure 8a.

Figure 5 Road data (a) Gaussian white noise with zero mean value (b) road surface roughness

Figure 6 Vehicle and bridge system (a) Trajectory profile for vehicle (b) Bridge displacements

3.2. Parametric Study for PEH

Three key design parameters are chosen:  vehicle speed variations, road roughness coefficient, and ratio of the cross-sectional area of the piston. Each variation is tested with three white noise time series to check the robustness of the generated electric power. The white noise has a similar mean value with different intensities, shown in Figure 8b respectively. The first parametric study involves variations in vehicle velocity and its effect on the harvested electric power, while keeping the road roughness coefficient in the C class and maintaining a piston cross-sectional area ratio of 4. The piezoelectric bar and the cylinder-piston dimensions are based on Tables 2-3. The results show that the highest RMS of the generated power is 135 W.  From Figure 9a, it can be seen that the increase in vehicle velocity is followed by the increase of P_rms. The increment has a linear form. This finding owes to the fact that an increase in velocity leads to an increase in dynamic displacements of the bridge and suspension system, respectively. As a result, the dynamic displacement of the equivalent model of the cylinder-piston mechanism and piezoelectric bar (PEH) in Figure 3, either in the front or rear parts increases correspondingly. Those increments correspond to the increase in charge, voltage, current, and generated electric power of the PEH as indicated by Equations 9-12. Under three white noise time series, the difference slightly deviates within 6% for the generated electrical power. Thus, the proposed PEH model is less sensitive to noise variances.

        In the next case, we examine variations in the road roughness coefficient while keeping the vehicle velocity at 40 m/s and the cross-sectional area ratio at 4. Road-roughness coefficients are based on ISO/TC108/SC2N67 (Class A-H). Piezoelectric bar dimensions and piston-cylinder dimensions are still based on Table 3. The results are displayed in Figure 9b. Similar to the previous variation, an increase in the road roughness coefficient results in an increase in the RMS of the electric power generated by the proposed PEH. However, this increment follows a nonlinear form. The results show that the highest RMS of the generated power is 230 W. However, this value is only found on a very rough road. Such a road may be found as off-road, which is not common as a public road. City road is commonly found in the range of class A to class D so the road-roughness coefficients are considered in those ranges. In this variation, the proposed PEH model is also less sensitive to noise variations. The difference is within 7.5%.

Table 1 Beam parameters

Table 2 Vehicle parameters

Table 3 Property and dimension of a piezoelectric bar (Piezo, n.d.)]

The last case is variations of piston cross-sectional area ratio to the harvested electric power by keeping the vehicle velocity of 30 m/s, and road roughness coefficients in the class of C. The results are displayed in Figure 9c. A nonlinear relation between the RMS of the electric power generated by the proposed PEH and the piston cross-sectional area ratio is found. The results show that the highest RMS of the generated power is 93.5 W under piezoelectric bar and cylinder-piston dimensions in Table 2. This is to be expected since the increase of cross-sectional area ratio n_p decreases the deflection of the PEH. This can be referred to as magnified force F_m(t) in Equation 7.

Since the force increases with the ratio of n, then the P_rms increases up to a certain limit. When the ratio of n_p is at higher values, the equivalent damping coefficient c_eq becomes predominantly, and the P_rms decrease according to Equation 6. Hence, the optimum value of n_p is selected to be 4, which can still be accommodated in the design. For this variation, the proposed PEH model also seems less sensitive to the noise variations as in the previous case. The difference is found within 8.7% for the generated electrical power. This result suggests that the proposed PEH is robust due to the deviation being relatively low to the variations of white noise as an environmental factor.

Figure 7 Dynamic displacements of (a) vehicle body (b) front tire (c) rear tire (d) PEH

Figure 8 Harvested energy (a) Generated electric power (b) White noise time series

Figure 9 RMS of generated electric power under variation of (a) vehicle velocity (b) road class (c) piston cross-sectional area ratio

In this paper, a novel design of a PEH for a vehicle suspension system has been developed. The structural dynamics of the bridge, vehicle in terms of half-car model, and PEH system are characterized to calculate the RMS of generated electric power. The finding results show that the RMS of generated electric power increases with an increase in the velocity of vehicles, the ratio of the cross-sectional area of the piston cylinder, and the road roughness coefficient. The increment has a linear form for vehicle velocity variation and a nonlinear form for the last two variations. The electric power can be harvested up to 67.5 W for a PEH with a span of 150 m of the bridge, a velocity of 40 m/s, and a vehicle mass of 2,691.6 kg. Based on the findings, it is possible to harvest a higher power by increasing the number of PEHs, high-capacity type piezoelectric transducer, the size of the vehicle and speed, the number of passengers, and other parametric designs as long as the vehicle and PEH are safe. However, experimental work is envisaged for future work.

        Financial support from Electronics and Information Research Organization under contract: SK IPT no. 2/III/HK/2022 and facility from Research Centre for Smart Mechatronics, National Research and Innovation Agency (BRIN) are greatly appreciated.

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