Energy harvesting from moving harmonic and moving continuous mass traversing on a simply supported beam

Measurement 150 (2020) 107080

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Energy harvesting from moving harmonic and moving continuous mass traversing on a simply supported beam I. Dehghan Hamani, R. Tikani ⇑, H. Assadi, S. Ziaei-Rad Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

a r t i c l e

i n f o

Article history: Received 19 January 2018 Received in revised form 10 June 2019 Accepted 19 September 2019 Available online 23 September 2019 Keywords: Energy harvesting Structural health monitoring Piezoelectric Moving harmonic mass Moving distributed mass

a b s t r a c t In this paper, energy harvesting from a simply supported beam using piezoelectric materials has been performed. Two different models for the moving objects were considered. First, the passing object was modeled as a concentrated mass with some amount of unbalances. The coupled governing equations of the beam, the piezoelectric patch, and the moving mass were obtained and discretized using assumed mode method. The beam response and the output voltage were calculated for different values of moving mass and unbalances. In the second model, the passing object was studied as a distributed mass. The governing equations of the system were extracted at three stages, namely entrance, main, and exit of the object from the beam. To validate the numerical findings, two test setups were erected, and the beam midpoint deflection and the piezoelectric power were measured experimentally. A good degree of correlation was found between numerical and measured values. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Vibration-based energy harvesting methods have been studied widely during the last decade. Piezoelectric materials are used to convert vibrations to electrical voltage. By attaching piezoelectric patches to vibrant structures, electrical energy can be generated. The energy harvested from structural fluctuations can be used in micro-scale electronics [1] or to supply energy for self-powered systems such as wireless sensor nodes [2]. Different kinds of loads pass along bridges and railways, which can cause these structures to shake. By using tolerable assumptions, a bridge span can be modeled as a simply supported beam traversed by a moving mass. Mainly two types of masses pass on bridges; the first type is concentrated masses which their length is negligible compared to the beam length. The second type is distributed loads which have a considerable length concerning the beam length such as trains. Many papers studied the effect of concentrated moving masses on the bridge with different methods. Foda and Abduljabbar [3], Lin and Trethewey [4], and Ting et al. [5] studied the problem of moving concentrated dynamic load from different theoretical points of view. Assadi et al. [6] presented a combination of theoretical analysis with experimental evaluation of the moving concentrated mass. Paul Cahill et al. [7] was investigated the vehicular ⇑ Corresponding author. E-mail address: [email protected] (R. Tikani). https://doi.org/10.1016/j.measurement.2019.107080 0263-2241/Ó 2019 Elsevier Ltd. All rights reserved.

parameters on the harvested power for a damaged bridge and an undamaged bridge using a cantilever piezoelectric beam. The sensitivity of harvested power to the positions and the intensity of the crack were studied. Moreover, concentrated loads passing a bridge can have harmonic behavior. Investigations on the response of beams under harmonic loads have been initiated by Timoshenko [8] and moved forward in details by Kolousek [9], Fryba [10], and Inglis [11]. Furthermore, studying the effect of different parameters in harmonic moving load problem such as moving speed, load frequency [12], and beam structure [13,14] has been performed theoretically. The moving distributed mass problem has also been investigated by many authors. Esmailzadeh and Ghorashi [15] studied the moving distributed mass and moving continuous force simultaneously and compared the results. Yang et al. [16] studied the vibration of simple beams traversed by high-speed trains by using moving load assumptions. Rieker and Tretheway [17] studied the moving distributed mass train problem using finite element method. Fryba [10] studied the theory of moving a continuous load on a simply supported beam. There are two approaches available for harvesting energy from the bridges under moving loads. The first approach installs an energy harvester on the bridge; this harvester, which is a cantilever beam with piezoelectric materials on it performs optimally when the bridge natural frequency is high. The second approach is to attach a piezoelectric patch directly to the bridge for energy

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harvesting purposes. This method is more suitable for bridges with low natural frequencies [18]. Piezoelectric energy harvesting from ambient vibrations has also been studied in numerous cases. Sodano et al. [19] developed an energy harvesting setup for providing an additional power source for electronic devices. They estimated the potential output electric charge from their harvester and performed some experimental validation. Energy harvesting from structures under dynamic loads has been performed by several researchers. Ali et al. [20] analyzed energy harvesters for highway bridges and consequently estimated the output power for a single vehicle. Ertruk [21] studied the energy harvesting from civil infrastructures with a focus on moving load excitation and surface strain fluctuation. Shu et al. [22] illustrated that harvested power depends on many parameters such as the natural frequency, input vibration characteristics (acceleration and frequency), and the mechanical damping ratio of the system. Wang et al. [23] by modeling the railway system showed that patch-type piezoelectric could harvest the available energy to supply energy for a wireless sensor. Bai et al. [24] optimized the harvested energy from moving mass and showed that harvested energy depends on load resistance and travel time of the moving mass. Peigney et al. [25] used a prototype of a cantilever piezoelectric harvester, and by modeling, it showed that mean power of 0.03 mW can be produced, and they provided the design rule for piezoelectric harvesters for using on bridges. Cahill et al. [26] used a piezoelectric energy harvester device in the real bridge to harvest energy from passing train and finally for the purposes of structural health monitoring. There are several differences between a real bridge and a simply supported beam. The structure and support of a real bridge are more complicated than a simple beam; however, it can be a simple simulation for understanding the bridge response to a vehicle passing. In this article, the beam response to the harmonic and distributed loads have been shown. Also, the energy harvesting from the beam response has been illustrated by attaching a piezoelectric patch. We believe that these results show us a good view of the real bridge response to passing a vehicle and the energy harvesting from it. It is noted that attaching piezoelectrics on the real bridges needs novel and innovative methods to harvest more energy. For instance, direct attaching piezo patches on the bridge will not be commercial, and if we use piezo patches on the extra device and use bridge vibrations as a stimulator source of that device, more energy can be harvested. For example, we can use bridge vibration to induce piezo-device natural frequency and receive the highest piezoelectrics stimulation and energy. However, this is not our article purpose and needs more investigation. In this work, a theoretical study of a beam response under moving continuous mass and moving concentrated unbalance mass has been presented. Afterward, the amount of electrical energy harvested by attaching a piezoelectric patch on the beam has been calculated. Next, an experimental test rig capable of producing different types of moving masses passing on the beam was fabricated. Finally, the theoretical results of the beam response and output electrical energy from the beam under continuous mass and concentrated unbalance mass have been validated by using a set of experimental examinations. Comparing the results show good agreement between theoretical and experimental results.

2. Theory 2.1. Moving concentrated unbalance mass The schematic of the moving concentrated unbalance mass problem is shown in Fig. 1 schematically. The geometry used for creating the moving object is an unbalanced cylindrical object.

Fig. 1. The schematic of the moving concentrated unbalance mass problem.

The piezoelectric patch is attached to the midpoint of the beam where the vibration mode shape of the beam has its maximum value to have the maximum energy harvesting [27]. In this schematic model, L is the beam length, and a piezoelectric patch is attached to the beam on LP1 < x < LP2 region for energy harvesting purposes. The governing differential equation of the moving harmonic load problem is presented in Eq. (1). For simplifying the equation, yðx; tÞ represented by y.

EI

@4y @5y @y @2y þ c s I 4 þ ca þm 2 4 @x @x @t @t @t " # 2 @2y @ y @2y 2 þ QsinðXtÞ dðx  v tÞ ¼M g 2 v  2v @x2 @x@t @t

ð1Þ

In Eq. (1), the term of @@t2y shows the beam accelator, v 2 @@x2y indicates centrifugal acceleration due to the movement on the curve of 2

2

@ y the beam, and 2v @x@t demonstrates coriolis acceleration. The harmonic unbalance frequency ðXÞ is adjusted with the load speed ðv Þ and the diameter of the cylindrical object ðDÞ traversing the beam. 2

X¼

2v D

ð2Þ

The definition of other variables used in Eq. (1) can be seen in Table 1. Beam response is assumed to be in the form of Eq. (3); where /r ðxÞ presents the rth vibration mode shape of the system and gr ðtÞ is the modal coordinate system [15]:

yðx; t Þ ¼

1 X

/r ðxÞgr ðtÞ

ð3Þ

r¼1

By taking the first vibrational mode in to account, i.e. qffiffiffiffiffi 2 sinðpLxÞ, and substituting the assumed beam response /1 ðxÞ ¼ mL in to Eq. (1), and then by multiplying both sides of the equation by /1 ðxÞ, and integrating from 0 to L, Eq. (4) can be obtained.

Table 1 Definition of variables used in Eq. (1). Variable

Definition

yðx; tÞ E I cs ca m M Q

Beam response Beam Young’s modulus Beam cross section moment of inertia Equivalent coefficient of strain rate damping The viscous air damping coefficient Beam mass per length Concentrated unbalance mass Harmonic load amplitude due to unbalance mass Harmonic load frequency Dirac delta function

X d

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I. Dehghan Hamani et al. / Measurement 150 (2020) 107080 2

d g1 ðtÞ dg ðtÞ þ C e 1 þ K e g1 ðtÞ dt dt2 rffiffiffiffiffiffiffi 2 pv t sinð ¼ ½Mg þ QsinðXtÞ Þ mL L

Me

ð4Þ

where,

 2 M e ¼ 1 þ 2M sin pLv t mL p v    Ix1 þ cma þ 2M sin 2 pLv t C e ¼ cs EI mL L pv 2 2 pv t sin ð L Þ K e ¼ x21  2M mL L and x1 ¼ p2

qffiffiffiffiffiffi EI mL4

ð5Þ

is the fundamental frequency of a simply sup-

ported beam. By solving Eq. (4), the beam response under moving concentrated unbalance mass can be calculated. Having the beam displacement of the beam as a function of time, one can find the piezoelectric output voltage through Eq. (6) [20]:

Cp

dV ðt Þ V ðt Þ þ ¼ e31 hpc bp dt R

Z

Lp2

Lp 1

@ 3 yðx; tÞ dx @x2 @t

ð6Þ

In Eq. (6), bp is the width of piezoceramic, hpc is its distance with the neutral axis, C p is the piezoceramic patch capacitance, and e31 is plane-stress piezoelectric stress constant. Replacing the beam’s response into Eq. (6) gives,

dV ðt Þ V ðt Þ dg ðt Þ þ ¼ w1 1 dt RC P dt

Fig. 2. Schematic of the moving continuous mass problem.

where m and mf are mass per unit length of the beam and the moving mass, respectively. Parameter g is the acceleration due to gravity and, v is the constant velocity of the moving mass. Here, H is the Heaviside step function. Eqs. (12) and (13) show the governing equation for the main     L LþL step vf  t  vL and departure step vL  t  v f , respectively [28].

EI

ð7Þ

@4y @5y @y @2y þm 2 þ cs I 4 þ ca 4 @x @x @t @t @t " # 2   @2y @ y @2y 2 ðH v t  Lf  Hðv tÞÞ ¼ mf g  2  v  2v @x2 @x@t @t

ð12Þ

where

w1 ¼ 

e31 hpc bp Cp

Z

Lh

Lh

1

2

2

d /1 ðxÞ dx dx2

ð8Þ

By transferring the system of Eqs. (5) and (8) into a system of first-order differential equations, the numerical solution of system state variables can be obtained in the time domain. The state variables for the first vibration mode are considered as

x1 ðt Þ ¼ g1 ðtÞ;

x2 ð t Þ ¼

dg1 ðtÞ ; dt

x3 ðtÞ ¼ V ðt Þ

ð9Þ

where x1 ðtÞ and x2 ðtÞ are modal displacement and modal velocity, and x3 ðtÞ is the induced voltage in the piezoelectric patch. The set of Eqs. (5) and (8), in the form of first-order differential equations can be expressed as

9 8 9 8 x2 > > > qffiffiffiffiffi   > < x_ 1 > = > < = ½MgþQ sinðXt Þ C x K x 2 sin pLv t x_ 2 ¼  Me e2  Me e1 þ mL Me > > :_ > ; > > > : ; x3  x 3 þ w x2

EI

@4y @5y @y @2y þm 2 þ cs I 4 þ ca 4 @x @x @t @t @t " # 2   @2y @ y @2y 2 ¼ mf g  2  v  2v ðH v t  Lf  HðLÞÞ @x2 @x@t @t

Similar to the previous section, by assuming the beam response in the form of Eq. (4); and by considering the first vibrational mode qffiffiffiffiffi   2 sin pLx and substituting the assumed beam i.e. /1 ðxÞ ¼ mL response in to Eqs. (11)–(14), we get: 2

Me

d g dg þ Ce þ Keg ¼ Fe dt dt 2

ð10Þ

d g dg þ Cm þ Kmg ¼ Fm dt dt 2

Eq. (10) has been solved using the Runge-Kutta method in MATLAB. The initial conditions have been assumed to be zero for all three equations.

Md

d g dg þ Cd þ Kdg ¼ Fd dt dt 2

1

2

2.2. Moving continuous mass The schematic model for the continuous mass traversing on a simply supported beam can be seen in Fig. 2. The governing partial differential equation of this problem can be written in three steps. Eq. (11) shows the governing equation for the entrance step, in which the moving load has not been fully   L located on the beam 0  t  vf [28]:

EI

@4y @5y @y @2y þm 2 þ cs I 4 þ ca 4 @x @x @t @t @t " # 2 @2y @ y @2y 2 ðHð0Þ  Hðv tÞÞ ¼ mf g  2  v  2v @x2 @x@t @t

  Lf 0t

ð14Þ

v

 Lf

2

Mm

RC P

ð13Þ

v



L

v

t

t

L

 ð15-1Þ

v

L þ Lf

v

 ð15-2Þ

where the definition of different coefficients used in these equations can be seen in Table 2. In this table Lf stands for moving mass qffiffiffiffiffi 2 and c ¼ pLv . length. Also A ¼ mL Finally, the beam response under moving continuous mass and induced voltage in the piezoelectric patch can be obtained by solving a set of equations in Eqs. (15) and (8). Here, a fourth order Runge–Kutta scheme was employed to solve these equations using the ode45 algorithm with 1e-6 absolute and 1e-3 relative error tolerance in MATLAB. 3. Experimental set-up

ð11Þ

Experimental investigation can help to validate theoretical results. In this work, an experimental setup has been used for measuring midpoint displacement of beam and piezoelectric output

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I. Dehghan Hamani et al. / Measurement 150 (2020) 107080 Table 2 Definition of coefficients used in Eq. (15). Variable

Definition

Value

Me

Entrance step mass coefficient

Mm

Main step mass coefficient

Md

Departure step mass coefficient

1 þ 12 mf A2 ðv t  2Lp sinð2ctÞÞ      2pL 1 þ 12 mf A2 Lf þ 2Lp sin 2ct  L f  sin ð2ct Þ    2pL 1 þ 12 mf A2 Lf þ L  v t þ 2Lp sin 2ct  L f

Ce

Entrance step damping coefficient

Cm

Main step damping coefficient

Cd

Departure step damping coefficient

Ke

Entrance step stiffness coefficient

Km

Step stiffness coefficient

Kd

Departure step stiffness coefficient

Fe

Entrance step loading

Fm

Main step loading

Fd

Departure step loading

voltage while moving harmonic load and moving continuous load traverses the beam. The experimental setup used for performing experiments has been previewed in Fig. 3. This setup can perform moving mass tests for various types of loads, including concentrated moving mass, concentrated unbalance moving mass, and continuous moving mass. The beam used for experimental tests is a 1000  50  3 millimeter planar steel

Fig. 3. (a) Schematic drawing of the experimental setup: (1) Ramp, (2) Light sensors, (3) Beam (4) Beam support, (5) Proximity sensor, (6) Microcontroller, (7) Concentrated unbalance moving mass (b) The experimental setup.

2fx1  12 mf A2 v ðcos ð2ctÞ  1Þ    2pL 2fx1  12 mf A2 v cos ð2ctÞ  cos 2ct  L f    2pL 2fx1  12 mf A2 v 1  cos 2ct  L f  2 2 p2  2 1 L x  2 mf A v L v t  2p sinð2ctÞ      2 2 p2 2 1 L x  2 mf A v L Lf þ 2p sin 2ct  2pLLf  sin ð2ctÞ      x2  12 mf A2 v 2 pL 2 Lf þ L  v t þ 2Lp sin 2ct  2pLLf mf Ag pL ð1  cosðctÞÞ     pL mf Ag pL cos ct  L f  cosðctÞ     pL mf Ag pL cos ct  L f þ 1

plate with two thin plastic strips attached on it which avoids the moving mass from falling off the beam. The three-point bending test has been performed on the beam after attaching plastic rails to take stiffness of strips into account. The speed of the moving mass on the bridge is being measured at both ends of the beam in order to verify its steadiness of the moving objects. This speed is measured by two sets of light sensors connected to a microcontroller. The entrance and departure speeds of the moving object are presented on the microcontroller board LCD. The midpoint displacement of the beam is measured with a proximity sensor, and the output voltage of this sensor is being viewed and saved by using an oscilloscope. The output voltage of the piezoelectric patch (MIDE QP20N model) is recorded by a data-logger capable of having 20 kHz sampling rate. In order to simulate the harmonic moving load on a bridge, an unbalanced cylindrical object (Fig. 4) is released from a specific height on the ramp to provide the desired speed for traversing of the object on the beam. The harmonic load frequency can be adjusted with the cylindrical object speed and its diameter (Eq. (3)). Two kinds of masses have been used in the experiments, the larger mass weight is about 1.09 N, and its unbalanced mass is 0.004 kg; the unbalanced mass central distance is 0.012 m. The smaller mass weight is 0.527 N and an unbalanced mass of 0.004 kg, which is located at 0.0077 m of its center. Fig. 4 shows the cylindrical geometries used for experimental validations. To create a continuous moving mass for the experiment, several steel spheres are connected together by a rectangular steel frame, as shown in Fig. 5(a). For carrying out tests by continuous moving mass, the experimental setup needs to be modified. This is because the continuous mass cannot easily release from the ramp and also the friction along its path on the beam cannot be neglected. In other words, the entrance and the departure speeds during the test are not equal, and thus, the assumption of steady speed is violated.

Fig. 4. Cylindrical Geometries used for creating harmonic load.

I. Dehghan Hamani et al. / Measurement 150 (2020) 107080

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Fig. 5. (a) Continuous mass used for performing experimental tests (b) DC motor control system used for moving the continuous mass.

Fig. 6. (a) Modified setup for performing moving continuous mass tests: (1) Table, (2) Light sensors, (3) Proximity sensor, (4) Beam support, (5) Beam, (6) Motor, (7) Continuous moving mass, (8) Microcontroller (b) The experimental stand.

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I. Dehghan Hamani et al. / Measurement 150 (2020) 107080

The experiments have been repeated five times for each speed in each part, and all the acquired data have been assessed. Finally, the average of recorded data for each speed of certain moving part has been reported.

Table 3 Value of parameters used in the theoretical approach. Parameter

Symbol

Numerical value

Beam Mass per length Length Modulus of elasticity Inertial cross section

m L E I

1:17 kg=m 1m 225 GPa

cs

7:9  105 Ns=m5

ca

0 Ns=m 6:5 Hz

Measured strain-rate damping coefficient Viscous air damping coefficient Measured fundamental frequency of the beam Piezoelectric patch Capacitance

x1

1:125  10

hpc bp LP1 LP2

2  107 F 11:58 C=m2 0:0015 m 0:0205 m 0:47 m 0:52 m

M mu du M mu du

0:054 kg 0:004 kg 0:0077 m 0:110 kg 0:004 kg 0:012 m

mf Lf

1:39 kg=m 0:1 m

Cp

Effective piezoelectric stress constant Thickness Width Piezoelectric Location Piezoelectric Location Moving harmonic mass Small cylinder Mass Unbalanced mass Central distance Large cylinder Mass Unbalanced mass Central distance Distributed mass Mass per length Length

4. Experimental and numerical studies 10

e31

4

m

Therefore, by using an adjustable DC motor connected to the moving mass, the continuous mass has been moved along the beam with a constant speed. Fig. 5 shows the DC motor control system and the continuous mass used for performing experimental tests. A view of the modified stand has been presented in Fig. 6. The value of different parameters used in the theoretical approach due to the experimental setup characteristics has been presented in Table 3.

As already mentioned and in order to obtain more accurate results, a three-point bending test was performed on the beam and rails. The measured bending stiffness of the beam was then used in beam dynamic response and tabulated in Table 3. Also, a modal analysis test was carried out on the beam to obtain its first natural frequency and also the damping coefficients of the understudy structure. The measured values for the first natural frequency and damping coefficient are also given in Table 3. 4.1. Moving concentrated unbalance mass The problem of moving a harmonic load on a beam has been studied in this manuscript in two separate cases. The first case is when the harmonic load frequency is the same as the beam first natural frequency, which will lead the beam to resonate. It is noteworthy that the resonance speeds for small and large masses are different. Having the fundamental frequency of the beam and the diameter of each cylinder, one can easily calculate the critical speed of each masses using Eq. (2). The critical speeds for small and large masses are found to be 0.53 m/s and 0.72 m/s, respectively. Fig. 7 shows the midpoint displacement of the beam under a resonance condition. The horizontal axis of the graphs is from zero up to the time that the moving mass leaves the beam. For the small mass with the speed of 0.53 m/s, it takes 1.88 s for the cylinder to pass the length of the beam while this time for the large mass is about 1.38 s. It is clear from Fig. 7 that the vibration amplitude of the beam midpoint starts from zero and increases steadily by time till the moving mass exit from the beam right support. The

Table 4 Comparison of theoretical and experimental expected output power for the harmonic mass. Mass weights ðNÞ

Unbalanced mass speed m

0.527 0.527 1.09 1.09

0.53 (Resonance) 1.3 (Non-resonance) 0.72 (Resonance) 1 (Non-resonance)

s

Errorð%Þ ¼ ExperimentCalculated  100. Experiment

Theoretical method ðlWÞ

Experiment ðlWÞ

Error (%)

0.2030 0.0163 0.2820 0.0648

0.2290 0.0164 0.2710 0.0669

11.3 0.6 4.1 3.1

Fig. 7. Displacement of beam midpoint for resonated condition (a): Small mass with v = 0.53 m/s (b): Large mass with v = 0.72 m/s.

I. Dehghan Hamani et al. / Measurement 150 (2020) 107080

measured values follow the analytical curves appropriately. The experimental amplitudes are lower than the analytical ones. One reason is due to the approximation of the damping coefficient of the structure. The second type of the performed tests is non-resonated beam experiments in which the harmonic load frequency is far from the beam natural frequency. Fig. 8 shows the displacement of beam mid-point in the non-resonating condition. The frequency of the harmonic load due to unbalance moving mass, in this case, is 16.5 Hz and 9 Hz for small and large mass respectively, which is

7

10 Hz and 2.5 Hz above the beam fundamental frequency. Using Eq. (2), the concentrated mass speeds are found to be 1.3 m/s and 1 m/s for small and large cylinders, respectively. Again, comparing the results for the beam response generally reveals that the number of peaks on theoretical and experimental results is equal, and the theory is following the trend of experimental results appropriately. The open circuit induced voltage of the piezoelectric patch has been shown in Figs. 9 and 10, respectively, for resonating condition and non-resonating condition.

Fig. 8. Displacement of beam midpoint for non-resonated condition (a): Small mass with v = 1.3 m/s (b): Large mass with v = 1 m/s.

Fig. 9. Piezoelectric output voltage for resonated condition (a): Small mass with v = 0.53 m/s (b): Large mass with v = 0.72 m/s.

Fig. 10. Piezoelectric output voltage for non-resonated condition (a): Small mass with v = 1.3 m/s (b): Large mass with v = 1 m/s.

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I. Dehghan Hamani et al. / Measurement 150 (2020) 107080

Fig. 11. Displacement of beam midpoint for moving continuous mass (a): v = 1.5 m/s (b): v = 2 m/s.

Fig. 12. Piezoelectric output voltage for moving continuous mass (a): v = 1.5 m/s (b): v = 2 m/s.

Comparing theoretical and experimental induced voltages implies that the theoretical output voltage also follows suitably the trend of the experimental results. While the beam is in resonating mode, the output voltages are extremely high, but the beam cannot be used in this way as resonance can do major damage to the structure. The expected value of the output power can be defined as:

V2 P ¼ rms R

ð16Þ

The expected values of the output power for theoretical and experimental methods are compared in Table 4. 4.2. Moving continuous mass The results for the beam midpoint response under moving continuous mass with different speeds have been presented in Fig. 11. The experiments are carried out for two moving speeds, namely 1.5 and 2 m/s. The small discrepancy between measured and predicted displacements of the beam midpoint is due to the fact that the motor used in measured test rig cannot provide an entirely constant speed for the moving distributed mass. This error will also affect the experimental voltage and power output of the harvesting system. The induced voltage of the piezoelectric patch is presented in Fig. 12 for two speeds of continuous mass. The expected values of output power theoretical and experimental methods are compared in Table 5.

Table 5 Comparison of theoretical and experimental output power for the continuous mass. Continuous mass speed (m/s)

Theoretical method ðlWÞ

Experiment ðlWÞ

Error (%)

1.5 2

0.131 0.169

0.152 0.179

13.8 5.6

Errorð%Þ ¼ ExperimentCalculated  100. Experiment

For finding the effect of continuous load length on the harvested power while the overall moving mass is constant, we have defined a dimensionless parameter as: n ¼ Lf =L. The overall mass is considered to be 0:139 kg. This parameter shows the ratio of the moving mass length to the total beam length. When n is zero, the moving mass is a concentrated mass. Different values of n between zero

Table 6 Effect of continuous load length on harvested power. n

Harvested power (lW) while V ¼ 1 m=s

Harvested power (lW) while V ¼ 2 m=s

0 0.2 0.4 0.6 0.8

0.0808 0.0756 0.0542 0.0364 0.0237

0.1841 0.1399 0.1103 0.8421 0.6302

I. Dehghan Hamani et al. / Measurement 150 (2020) 107080

to 0.8 were selected for this parametric study. The harvested powers for two different moving speeds are tabulated in Table 6. This table shows an overall decrease in output power while the continuous load length increases. This means that a concentrated mass can produce more prominent vibration and consequently higher harvested power in compare to a distributed mass. The results also reveal that by decreasing the moving speed, the power harvested by the piezoelectric patch will decrease monotonically. 5. Conclusions This paper studies piezoelectric energy harvesting from a simply supported Euler-Bernoulli beam traversed by two different mass types. By selecting proper beam parameters, the EulerBernoulli beam model can be used to simply represent a bridge span. The inertial, centrifugal, and gyroscopic effects of the moving mass have been considered in derived equations which is necessary for heavy and high-speed masses passing on a beam. The moving mass models introduced in this manuscript are concentrated mass with unbalance and distributed mass. To certify the obtained results, two test rigs were fabricated and based on them some parameters such as beam midpoint deflection, the piezoelectric voltage and power were measured. For concentrated mass with unbalance case, the coupled governing equations of the system were extracted and discretized. The beam midpoint deflections were calculated for two different moving mass with unbalances. To validate the mechanical model of the system, the numerical results of the beam midpoint deflections were compared with the measured values. A good agreement was observed between the numerical and the measured values. Concentrated mass with unbalance can pass over the beam with different velocities. The passing vehicle velocity can excite the beam in resonate and non-resonate conditions. It was found that when the moving mass excites the beam in resonate condition, the beam midpoint amplitude of vibration steadily increases with time until the moving mass left the beam. For non-resonate case, the beam vibration is much lower than the resonate case. Due to the high amplitude of vibration in the resonate case, the harvested power is an order of magnitude more than the non-resonate case. However, in reality, resonate case may cause some permanent damage to the bridge structure and reduces its usage lifetime. In concentrated mass with unbalance, the findings reveal that the piezoelectric voltage and power increase by increasing the mass of the moving part and also by increasing the amount of unbalance. For the case of distributed mass, the voltage output of the piezoelectric patch has variations through time which are caused by the different stages of the mass on the beam, namely the entrance, main and exit steps. The results also showed that the maximum voltage occurs at the exit step of the distributed traveling mass. By increasing the distributed mass speed, the piezoelectric harvester voltage and power output increase monotonically. The results also indicate that by increasing the length of the moving mass over the bridge, the extracted power decreases. In other words, the maximum power achieves when all the moving mass is lumped at a point. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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