Energy harvesting from constrained buckling of piezoelectric beams

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First Claim
1. An energy harvesting system for harvesting energy in response to application of an external force, said energy harvesting system comprising:
 a first beam member having a central member and at least one piezoelectric layer joined to the central member for deflection therewith, said first beam member having opposing first and second ends, said first beam member being elastically deformable in response to application of the external force;
a first mount coupling said first end of said first beam member, said first mount being generally stationary, the first mount comprises a lower beam;
a second mount coupling said second end of said first beam member, said second mount and said second end of said first beam member being generally moveable in response to application of the external force between a first position and a second position, thereby outputting energy from said at least one piezoelectric layer in response to said movement from said first position to said second position, the second mount comprises an upper beam;
a stop member physically opposing movement of said second mount and said second end of said first beam member in said second position; and
a plurality of oblique beams, a first pair of the plurality of oblique beams interconnecting the first mount to the first beam member and a second pair of the plurality of oblique beams interconnecting the second mount to the first beam member, the plurality of oblique beams transmitting a compressive bending force to the first beam member in response to the external force.
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Abstract
An energy harvesting system for harvesting energy in response to application of an external force. The energy harvesting system comprises a beam member having a central member and at least one piezoelectric layer joined to the central member for deflection therewith. The beam member includes opposing first and second ends and is elastically deformable in response to application of the external force. A first mount couples the first end of the beam member and is generally stationary. A second mount couples the second end of the beam member. The second mount and the second end of the beam member are generally moveable in response to application of the external force between a first position and a second position, thereby outputting energy from the at least one piezoelectric layer in response to the movement from the first position to the second position.
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Current Assignee
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9 Claims
 1. An energy harvesting system for harvesting energy in response to application of an external force, said energy harvesting system comprising:
a first beam member having a central member and at least one piezoelectric layer joined to the central member for deflection therewith, said first beam member having opposing first and second ends, said first beam member being elastically deformable in response to application of the external force; a first mount coupling said first end of said first beam member, said first mount being generally stationary, the first mount comprises a lower beam; a second mount coupling said second end of said first beam member, said second mount and said second end of said first beam member being generally moveable in response to application of the external force between a first position and a second position, thereby outputting energy from said at least one piezoelectric layer in response to said movement from said first position to said second position, the second mount comprises an upper beam; a stop member physically opposing movement of said second mount and said second end of said first beam member in said second position; and a plurality of oblique beams, a first pair of the plurality of oblique beams interconnecting the first mount to the first beam member and a second pair of the plurality of oblique beams interconnecting the second mount to the first beam member, the plurality of oblique beams transmitting a compressive bending force to the first beam member in response to the external force.  View Dependent Claims (2)
 3. An energy harvesting system for harvesting energy in response to application of an external force, said energy harvesting system comprising:
an upper beam member; a lower beam member being moveable relative to the upper beam member in response to application of the external force; a middle beam member having a central member and at least one piezoelectric layer joined to the central member for deflection therewith, said middle beam member having opposing first and second ends, said middle beam member being elastically deformable in response to application of the external force; a first pair of oblique beams interconnecting the upper beam member to the first and second ends of the middle beam member; a second pair of oblique beams interconnecting the lower beam member to the first and second ends of the middle beam member; wherein application of the external force causes application of a compressive force on the middle beam member thereby outputting electrical energy from the at least one piezoelectric layer.  View Dependent Claims (4, 5, 6, 7, 8, 9)
1 Specification
This application claims the benefit of U.S. Provisional Application No. 62/204,695, filed on Aug. 13, 2015. The entire disclosure of the above application is incorporated herein by reference.
The present disclosure relates to energy harvesting from constrained buckling of piezoelectric beams.
This section provides background information related to the present disclosure which is not necessarily prior art. This section provides a general summary of the disclosure, and is not a comprehensive disclosure of its full scope or all of its features.
According to the present teachings, a piezoelectric vibration energy harvester, having a piezoelectric beam, is provided to generate electricity under the weight of passing cars or crowds. The piezoelectric beam buckles to a controlled extent when the device is stepped on or otherwise depressed. The energy harvester can have horizontal or vertical configuration. Horizontal configuration is easier to install and can be tuned to low passing loads. The moving element of the device is one or more piezoelectric beams, such as a uniform bimorph beam or a segmented beam. The ends of the segmented beam are simple spring steel or brass while the central span is a piezoelectric bimorph.
The piezoelectric beam is horizontal in the horizontal configuration. In this configuration, the vertical vehicle load is transferred to a horizontal buckling force through a scissorslike mechanism. The mechanism consists of seven rigid links. In the vertical configuration, the piezoelectric beam is vertical and directly sustains the vehicle load. The vertical (direct) configuration is designed to buckle under the load of the vehicle. This results in significant deformation of the piezoelectric and generates substantial amounts of power. If the beam'"'"'s buckling is not controlled, it will result in the fracture of the beam. The buckling deformation of the beam is constrained by limiting its axial deformation.
To better control the axial deformation of the buckled beam, the deformation has to be noticeable. This requires the length of the beam to be long. As the length of the beam increases its axial deformation becomes more significant. To reduce the cost of the present energy harvester, a segmented beam is preferred over a uniform beam for the buckling component. This choice allows having large axial deformation without more usage of costly piezoelectric.
In the present disclosure, the energy harvester is analytically modeled. The electromechanical coupling and the geometric nonlinearities have been included in the model for the piezoelectric beam. The design criteria for the device are discussed. It is demonstrated that the device can be realized with commonly used piezoelectric patches and can generate hundreds of milliwatts of power. The effect of the design parameters on the generated power and required tolerances are illustrated.
The present device could be implemented in the sidewalks producing energy from the weight of people passing over it. Other possible applications are portable smart phones chargers and shoe heel energy harvesting. The dance floor of a club is another applicable example for using this harvester. The vertical device could be implemented in roads, using the weight of the passing cars for generating electricity. The device is not prone to resonance and generates notable amounts of power from passing of each tire. It therefore can be used as a selfsufficient sensor for traffic control.
Further areas of applicability will become apparent from the description provided herein. The description and specific examples in this summary are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure.
The drawings described herein are for illustrative purposes only of selected embodiments and not all possible implementations, and are not intended to limit the scope of the present disclosure.
Corresponding reference numerals indicate corresponding parts throughout the several views of the drawings.
Example embodiments will now be described more fully with reference to the accompanying drawings.
Example embodiments are provided so that this disclosure will be thorough, and will fully convey the scope to those who are skilled in the art. Numerous specific details are set forth such as examples of specific components, devices, and methods, to provide a thorough understanding of embodiments of the present disclosure. It will be apparent to those skilled in the art that specific details need not be employed, that example embodiments may be embodied in many different forms and that neither should be construed to limit the scope of the disclosure. In some example embodiments, wellknown processes, wellknown device structures, and wellknown technologies are not described in detail.
The terminology used herein is for the purpose of describing particular example embodiments only and is not intended to be limiting. As used herein, the singular forms “a,” “an,” and “the” may be intended to include the plural forms as well, unless the context clearly indicates otherwise. The terms “comprises,” “comprising,” “including,” and “having,” are inclusive and therefore specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. The method steps, processes, and operations described herein are not to be construed as necessarily requiring their performance in the particular order discussed or illustrated, unless specifically identified as an order of performance. It is also to be understood that additional or alternative steps may be employed.
When an element or layer is referred to as being “on,” “engaged to,” “connected to,” or “coupled to” another element or layer, it may be directly on, engaged, connected or coupled to the other element or layer, or intervening elements or layers may be present. In contrast, when an element is referred to as being “directly on,” “directly engaged to,” “directly connected to,” or “directly coupled to” another element or layer, there may be no intervening elements or layers present. Other words used to describe the relationship between elements should be interpreted in a like fashion (e.g., “between” versus “directly between,” “adjacent” versus “directly adjacent,” etc.). As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items.
Although the terms first, second, third, etc. may be used herein to describe various elements, components, regions, layers and/or sections, these elements, components, regions, layers and/or sections should not be limited by these terms. These terms may be only used to distinguish one element, component, region, layer or section from another region, layer or section. Terms such as “first,” “second,” and other numerical terms when used herein do not imply a sequence or order unless clearly indicated by the context. Thus, a first element, component, region, layer or section discussed below could be termed a second element, component, region, layer or section without departing from the teachings of the example embodiments.
Spatially relative terms, such as “inner,” “outer,” “beneath,” “below,” “lower,” “above,” “upper,” and the like, may be used herein for ease of description to describe one element or feature'"'"'s relationship to another element(s) or feature(s) as illustrated in the figures. Spatially relative terms may be intended to encompass different orientations of the device in use or operation in addition to the orientation depicted in the figures. For example, if the device in the figures is turned over, elements described as “below” or “beneath” other elements or features would then be oriented “above” the other elements or features. Thus, the example term “below” can encompass both an orientation of above and below. The device may be otherwise oriented (rotated 90 degrees or at other orientations) and the spatially relative descriptors used herein interpreted accordingly.
In recent years, energy harvesting devices have attracted much interest in industrial and manufacturing sectors. The unique ability of converting ambient energy into electrical energy has led to several academic and commercial groups analyze and develop energy harvesting technology. Commonly, the generated electricity is stored in capacitors or batteries. R. Anton et al. have done a review in the field of energy harvesting and its current position for creating selfpowered devices. Vibration is one of the common sources in energy harvesting. David P. Arnold has studied compact magnetic power generators using different sources of energy, including vibration. Also, Zhongjie Li modeled and analyzed electromagnetic energy harvesters for applications in vibration energy harvesters. Another commonly used method of converting vibration into the electricity is by using the piezoelectric materials.
Using vibration and piezoelectric effect has been investigated by numerous groups. The piezoelectric effect converts mechanical strain into electric current or voltage. A piezoelectric material generates a small voltage whenever it is mechanically deformed. In CookChennault and his team'"'"'s attempt, energy harvesting from piezoelectric materials is studied. Piezoelectric harvesters are enabling a variety of applications: portable chargers, power sources for battlefield equipment, selfwinding wristwatches, etc. Beeby and his team studied the energy harvesting of the wireless selfpowered microsystems, including piezoelectric materials. In research done by Roundy and Wright a vibration based piezoelectric generator for wireless sensors is studied. In their article, the design of piezoelectric vibration to electricity converter is investigated and the purpose of their device is to be used as power source for wireless electronics.
Piezoelectric beams make harvesting energy from vibrating structures possible. One vibration source for energy harvesting is the vibrations of a bridge caused by the passing cars. In this method, the energy originates from the passing vehicle, translates to the bridge and causes vibrations in the bridge. This vibration is converted into electrical energy by a piezoelectric energy harvester device. When the energy translates from the vehicle to the bridge, the energy distributes over a very large structure. The energy absorbed by the vibrational energy harvester is therefore a very small fraction of the energy wasted by the vehicle. This is the first shortcoming of the traditional approach in bridge vibrational energy harvesting. The second disadvantage of conventional devices is that they will be useful only when installed on bridges. The vibrations of the regular portions of the roads are not sufficiently large to result in notable electrical energy. The third drawback of the vibrational energy harvesters is their limited frequency bandwidth. This issue, which has received significant attention in recent years, is still being discussed.
To avoid this phenomenon, we can directly use the force of the passing cars or crowds to generate electricity. According to some embodiments of the present teachings, the harvesting of energy from vibration of passing cars over piezoelectric patch for a uniform beam is provided.
According to the present principles, a vertical configuration and a horizontal configuration are disclosed. In some embodiments, as illustrated in
The present device is scalable and can be designed to fit in a shoe heel. This will make it possible to generate electricity with each step of a person. The power can be used for charging smart phones, players, or motion sensors. The sensors could operate without battery and their size will be reduced. In some embodiments, the harvester can be used as a suspension mechanism in a bicycle and using the vibration of the bicycle for generating electricity.
In the present disclosure, first the two design configurations (the vertical configuration and horizontal configuration) are explained. Both horizontal and vertical configurations are modeled and an electromechanical model for vibration and energy harvesting characteristics of the device is presented. The models are numerically solved to predict the performance of the device. At the end, the effect of the length and thickness of the active layer on the generated power is also discussed.
Device Configuration
First, we study the vertical configuration of an energy harvester device 10 which is a more trivial case. Energy harvester device 10 can comprise a vertical beam 12 mounted between a first anchor point 14 and a second anchor point 16. The first anchor point 14 can be configured to be stationary (e.g. ground) or can be configured to be movable in a direction to cause deflection of vertical beam 12. The second anchor point 16 can be coupled to a moveable member 18, such as a plate formed to be driven upon, which results in vertical deflection of the second anchor point 16 and corresponding bending movement or deflection of vertical beam 12. In some embodiments, vertical beam 12 can comprise a centrally disposed steel or brass layer 18 having two piezoelectric layers 20, 22 mounted thereto for movement therewith. However, it should be understood that only one piezoelectric layer 20 can be used. The bending movement or deflection of vertical beam 12, specifically steel or brass layer 18 and piezoelectric layers 20, 22, results in generation of electricity as a result of the deformation of piezoelectric layers 20, 22. The vertical configuration of the device (
As illustrated in
In some embodiment, a horizontal configuration of the energy harvester device 40 is provided. As illustrated in
Writing the static governing equation:
P=F cot(θ)
Δ=δ cot(θ) (1)
Where F is the force on top of the device, P is the force on two ends of the active beam, and θ is the angle between the top beam and the oblique beams (L_{o}). Δ is the gap distance and δ is the shortening in the segmented beam 46 after the beam 46 buckles. As it is seen in Eq. (1) if the angle is less than 45 degrees, P will be larger than F. It means that the mechanism magnifies the vertical force. In this case, the horizontal configuration results in a large force on the active beam with just a very small force over the harvester (F). This feature makes this harvester ideal for portable devices and for applications in which there is no access to large forces. The special design of the harvester 40 makes its placement easy in sidewalks or under a treadmill where the force over the device is not large. Another advantage of having the angle less than 45 degrees is the increase in the gap distance (Δ). As Eq. (1) shows, with the same amount of δ as the angle decreases the gap distance increases, which makes the building of the device much easier and more practical. If the angle is more than 45 degrees, P will be smaller than F, which makes this configuration suitable for installing on the roads and under large forces. By increasing the angle, we make sure that the applied force on the segmented beam will not exceed the allowable amount and will not result in breaking the beam.
Modeling and Governing Equations
The partial differential equation governing the vibrations of the piezoelectric beam is derived by combining those of a buckled beam and a piezoelectric energy harvesting bimorph. The mechanical governing equation is:
Where m is the total mass per unit length of the beam, w(x,t) is deflection along the yaxis
The coupling term α for the parallel connection can be written as:
Where e_{13 }is a piezoelectric coefficient. If the force is more than the critical buckling force, the beams buckle. In this disclosure, we assume that the force is larger than the first critical load and is less than the critical load of the higher modes. So, we only need to consider the first buckling mode shape in our calculations. For both buckled and unbuckled cases, the free vibration mode shapes are considered as the mode shapes. The equation for free vibration of the beam can be expressed as:
The solution for the free vibration can be shown as a linear combination of all natural motions of the beam (Section 11):
w(x,t)=Σ_{j=1}^{∞}ϕ_{j}(x)T_{j}(t) (6)
Where ϕ_{j }is jth natural mode shape and T_{j }is the temporal function. Substituting Eq. (6) in Eq. (5), the general solution for a segmented beam can be found in the form of:
Where i is the ith beam section and j shows the jth mode shape, a_{i1}, a_{i2}, a_{i3}, a_{i4 }and β_{j }are calculated from the boundary conditions, and C_{i }is:
In case of the uniform simply supported beam Eq. (7) reduces to:
Since the vibration mode shapes of a segmented beam does not stay the same as a uniform beam, the general form of a free vibration of a beam (Eq. (7)) is considered for each part. The beam has three segments; therefore, the deflection shapes are different for each part. So, there are in total 12 unknown coefficients (a_{11}, a_{12}, . . . , a_{34}). There are four boundary conditions at the two ends. The deflection (ϕ) and bending moment (EIϕ″) are zero.
Because of continuity and equilibrium conditions, we have four conditions at each of the interfaces. In other words, at these points, the two sides should have the same deflection, slope, shear force, and bending moment. Four boundary conditions at the two ends and eight equations at the two interfaces form the following 12 equations:
Where ϕ_{j1 }is the deflection of the first part, ϕ_{j2 }is the deflection of the middle part of the beam, ϕ_{j3 }is the deflection of the third part of the beam. E and I are the elastic modulus and moment of inertia. ϕ′_{ji }shows the slope in each part, E_{i}I_{i}ϕ″_{ji }is the bending moment of the beam, and E_{i}I_{i}ϕ_{ji}^{(3) }is the shear force of the beam. One way to find these coefficients is to get the 12 equations and write them in the matrix form:
BA=0 (11)
Where B is a 12×12 matrix and A is defined as:
A=[a_{11}a_{12}a_{13}a_{14}a_{21}a_{22}a_{23}a_{24}a_{31}a_{32}a_{33}a_{34}]^{T} (12)
Nontrivial solutions only exist if the determinant of matrix B is zero. In order to find the β that makes the determinant zero, we plot the logarithm of the determinant of matrix B in terms of different βs. The first drop in the plot of
If the determinant of B matrix is zero, there is not a unique answer for the set of the equations. So in order to solve them, one equation should be eliminated. Next, a value for one of the components of the vector A is assumed and other 11 remaining coefficients are found based on that component. We need to check to see if the choice of the arbitrary component has been proper. In some cases that assumed component in the coefficient vector is zero. It means that our first assumption is not correct and we need to take another element as the assumed component. In order to check to see if the assumption is correct, we remove one row and one column of the matrix B. The column is the column associated with that assumed coefficient. After removing one row and that column, we check the rank of the remaining square matrix. If the rank is n−1, 11 in this case, it means our assumption was correct and we can proceed. But if the rank is less than n−1 it means we need to remove another row instead of the one which was removed and check the ranking again. If all these matrices formed after removing one row fail to have the rank of n−1, it means that our first assumption has not been proper and we need to take another element as the assumed component.
The second way to find the coefficient is to reduce the 12×12 matrix using transformation matrices. In this paper, we name it as transformation matrices method wherever we want to refer to using this way of reducing the size of the initial matrix. By using this method we should calculate the determinant of a 4×4 matrix instead of a 12×12 matrix. This method would show a significant difference in calculation time when we have large matrices. Although in our case the first matrix (12×12) is not very big, due to the high number of numerical calculations for the optimization of the device, we use the transformation matrices method. This method is very useful and time saving when it comes to the optimization part. In order to find the optimized thickness or length of the beam, we need new numerical calculations for every configuration. Considering the number of different configurations, using this method reduces the overall time of the calculations significantly.
As it was mentioned earlier we have two boundary conditions at each end of the beam, totaling four conditions. Considering these boundary conditions at the two ends of the beam we have:
Where B_{C0 }and B_{Ce }are written based on the Eq. (10):
x* is l_{1}+l_{2}+l_{3}. Also, there are four equations relating the first part to the second and there are four more equations relating the second part of the beam to the third part (deformation of the beam is continuous).
Where
In which
x* is l_{1 }or l_{1}+l_{2 }depending on if we write the continuity and equilibrium equations at the intersection of the first segment with the second segment or if we write them at the intersection of second segment with the third segment. Finally we have:
And:
If we combine Eq. (19) and Eq. (20) we end up with:
The matrix B accordingly is defined as:
We next find the value of β that makes the determinant of matrix B equal zero. Comparing the number of equations in Eq. (10) and the size of the matrix in Eq. (22) shows that the size of the initial coefficient matrix (12×12) is reduced to a 4×4 matrix. As it mentioned before, the main advantage of using transformation matrices method is reducing the matrix size which decreases the numerical calculation significantly. When the size of the initial matrix increases, the decrease in overall time of the calculations is considerable. Also, another advantage of this formulation is avoiding extremely large values of determinant which might cause numerical errors.
In our calculation we use the mass normalized mode shapes such that they satisfy the condition given by:
∫_{0}^{L}ϕ_{ij}(x)m_{i}ϕ_{ij}(x)dx=1 (23)
After finding the deflection of the beam we replace w(x,t) with w(x,t)=Σ_{j=1}^{∞}ϕ_{j}(x)T_{j}(t) in Eq. 2 and we premultiply the equation by the mode shape. If we integrate the expression from zero to the length of the beam:
Due to orthogonally of the mode shapes the equations get decoupled:
M_{k}{umlaut over (T)}+c{dot over (T)}+(K_{k}+p_{k})T+N_{k}T^{3}+B_{k}V(t)=0 (25)
Where M_{k }is the modal mass of the kth mode. As mentioned before, we consider mass normalized mode shapes; c is the damping ration, K stands for the linear stiffness, the reduction of the stiffness coefficient due to the axial force is p, N is the nonlinear coefficient, and the coupling coefficient is β. These integrals were calculated analytically for a uniform beam:
The integrals in Eq. (26) were calculated numerically. The critical buckling force of the beam is the amount of axial force, P, that makes p equal to K.
Where R_{l }is load resistance, and the capacitance of the piezoelectric is
in which n is number of piezoelectric layers (2 in our device). The governing equations before the stop is engaged are:
If the axial force is less than the critical load, zero deflection would be a stable equilibrium point. When the axial force surpasses the first critical load, the modal coordinate representing the nonlinear equilibrium point is:
The governing differential equations in Eq. (2) are valid before the stop is engaged. When the support is engaged, the tire force is applied to the support and Eq. (28) is not valid. In that situation the axial displacement of the beam is fixed at the designed recess (
The model manifested in Eq. (30) implies that the effect of an axial fixed displacement of Δ is equal to the effect of an axial force of K_{eq}Δ. Unfortunately this model does not seem to be correct. The first incorrect result of Eq. (3) is on the static axial deformation of the beam. In static situation the axial deformation of the beam must be equal to Δ. The axial deformation of the beam is given by:
For a uniform beam the axial deformation resulted from Eqs. (30) and (31) is
This contradicts the boundary conditions. We suggest that the correct equivalent axial force should be
where p_{c }is the critical load. This value is consistent with d=Δ. The difference between the two values becomes obvious in the following thought experiment. Assume that the distance between the hinges is a slight amount (ε) shorter than the original length. Since the shortening of the neutral axis is negligble the beam has to buckle to accommodate to the boundary conditions. The equivalent axial force should therefore be larger than the critical buckling load. This condition is satisfied by our corrected equivalent load but is contradicted with the expression in the prior art (see Ref. 34 in Exhibit A). So, we suggest that the correct formula is:
Thus, after the stop is engaged, instead of Eq. (28) the governing equations are:
The parameters definitions are similar to those in Eq. (28). The only difference is the reduction of the stiffness coefficient due to the axial force:
p_{e}=−(P_{stp}+P_{c})∫_{0}^{L}ϕ″(x)ϕ(x)dx (34)
In this equation P_{c }is the first critical load for buckling and P_{stp }is the reaction force from the beam in the point when the stop is hit. For the segmented configuration we have:
P_{stp}=K_{eq}d_{max} (35)
Where d_{max }is the shortening in the beam when the stop is hit and K_{eq }is the equivalent stiffness of the segmented beam.
As it is mentioned, the critical force for buckling is defined as the amount of force that makes a linear part of Eq. (28) zero (p=K). So, the critical force is:
For a uniform beam the value of p_{e }and p_{c }are equal to:
where n is the number of the buckling mode. For the vertical configuration, the maximum shortening of the beam (d_{max}) is equal to the gap between the device and the support (Δ). Based on Eq. (1), for the horizontal configuration the equation which relates these two values is:
Δ=d_{max }cot(θ) (39)
Design
When there is a force over the device, the beam buckles. This buckling causes a transient vibration, which generates electricity. The buckling continues until the stop is hit. Based on the equation for shortening of the beam (Eq. (31)), the amount of maximum shortening of the beam (d_{max}) is proportional to the second power of the maximum allowable strain in the beam (∈_{max}^{2}):
d_{max}∝∈_{max}^{2} (40)
The allowable strain is related to the allowable stress as:
σ_{yp}, σ_{ys }is the yield stress for steel and piezoelectric and E_{p}, E_{s }is the module of elasticity for the piezoelectric material and the substrate. For the segmented beam, we do not have the exact equation which relates d_{max }to ∈_{max}. In order to find d_{max}, first the shortening of the beam, d_{0}, and ∈_{0 }are calculated. ∈_{0 }is the maximum strain in the beam based on the deformation for the beam. It should be noted that this value is different from maximum allowable strain in the beam (∈_{max}) which is calculated from Eq. (41).
Note that for the uniform beam the maximum strain occurs at the middle of the beam. In the next step, based on Eqs. (40) and (42) we can find the maximum allowable shortening in the beam (d_{max}). In order to find d_{max }we should calculate this value both for substrate and the piezoelectric layer:
Each of these values was less, should be considered as the maximum shortening of the beam. Using maximum shortening, we can find the value for Δ. The support location, identified by the gap Δ, should be adjusted to prevent fracture of the beam. The other thing we should notice in designing the device is that the system is designed to have low damping coefficient to enhance the power generation. As a result the beam might buckle more than the static buckling. So, extra caution should be taken in defining the gap value to avoid fracture in the piezoelectric beam.
The solution is divided into different parts: The first part is when the force is over the device and the beam moves down until it hits the support. In this part, Eq. (2) is the governing equation for the vibration of the piezoelectric beam. After the beam buckles the support is engaged, Eq. (32) is the governing equation of the system. When the force is removed, the piezoelectric beam tends to spring back and Eq. (2) gives the solution of the system. Since there is no force over the harvester in this situation, p equals to zero in the equation. Since both equations (2) and (32) are nonlinear, we therefore use numerical integrations to solve the model and predict the power generated by the energy harvesting device.
To design the device we need to make sure that the axial stress is less than the yield stress of piezoelectric and substrate parts of the beam. If the force exceeds this limit, the axial stress will break the harvester. The horizontal configuration could be implemented in treadmills or sidewalks, using loads of passing crowds over the harvester. So, the amount of force we expect on the upper beam is about human weight. Since this force would become larger in the case of jumping or running over the device, we consider a safety factor in our design. The same as horizontal design, for the vertical configuration, which could be implemented in the roadways generating electricity using the force of the passing cars, we consider a safety factor to make sure the device does not break under the weight of cars. The next step is to check if this force is more than the first critical load of the buckling of the beam. The critical buckling force is calculated based on Eq. (37). In order to calculate the first critical buckling force, the first mode shape needs to be replace as ϕ(x) in the Eq. (37) and for calculating the second critical buckling force the second mode shape. Similarly, for calculating the nth buckling critical force, ϕ(x) in Eq. (37) is replaced by the nth mode shape. Since there usually is charge cancellation associated with the higher modes, we only utilize the first buckling mode shape in our design. Thus, the force over the piezoelectric beam has to be more than the first critical load and less than the second critical load. First and second critical buckling forces are shown for different thicknesses and lengths of piezoelectric layers. For a uniform simply supported beam, the critical buckling force is calculated using Eq. (38). As the equation shows, the second critical buckling force in a uniform beam is four times the first critical buckling force. The interesting result about the segmented beam is that the ratio of the second to the first critical buckling force changes with the length and thickness of the middle part.
Results
In this section we study three different cases. The first one is a uniform beam in the vertical (direct) configuration. The second case is the vertical configuration with a segmented beam. In both cases the force over the device is considered to be weight of a car. The last case is a segmented beam in the horizontal (indirect) configuration. This device is designed to generate electricity using human weight. The output power and the gap size are calculated for all three cases and ultimately the effect of design parameters on the output power and gap size is studied.
As the first case, we investigate a case study for the direct configuration, using the uniform beam (
The instantaneous power across an 800Ω resistive load is plotted in
Next, we study both configurations using the segmented beam instead of the uniform beam (
As illustrated in the plot, most part of the electricity is generated after the force is removed. To make sure that the optimized resistance is used, the average power
for different resistances were calculated. It is shown in
Where C is the capacity of the piezeoelectric layer and w is the natural frequency of the beam. The duration of force being applied to the harvester is assumed 0.1 seconds. After the force is removed the beam springs back. The average power for generated electricity in
For the indirect configuration (
With this angle, we make sure that there is sufficient force for buckling of the beam. If we decrease the angle, the force needed for buckling the device would be less, but having a very small theta angle makes the building of the device impractical and there is always the building limitations that should be considered. The safety factor for the axial load is 13.6 and 2 for the substrate and piezoelectric layers (the yield stress is assumed 40 MPa and 250 MPa for these layers). Since the motion of the beam is oscillatory even after the stop is hit, another safety factor is considered in the last step of the design. The main advantage of this device over the vertical configuration is the increase in the gap between the top beam and support. A larger gap would make the building and implementing of the device much easier.
As mentioned before the most important advantage of using a segmented beam over uniform beam is the increase in the gap distance where the support is placed. This gap distance even increases more when using a horizontal configuration. The gap distance for the case study of the horizontal device is 0.8 mm, which is larger than vertical configuration (about 10 times) and much larger than uniform beam (more than 500 times).
In horizontal configuration of the device, by increasing in the theta angle the gap distance increases too. Building limitations should be considered, before decreasing the theta angle to have a larger gap size. The length of the lateral beam does not affect this distance significantly (
The effect of the length of the piezoelectric layer and its thickness in the horizontal device is also illustrated in
By increasing the thickness of the piezoelectric layer, at first the gap distance is increasing, but after a point it begins to decrease. The reason is because after a point the thickness of the piezoelectric is not the deciding factor in breaking the beam. For thin piezoelectric layers, substrate will break instead of the piezoelectric layer. So it is crucial to ensure the safety of both piezoelectric layer and the substrate. The deflection shape for a beam with a thick piezoelectric and a thin steel layer is illustrated in the
The effect of the increase in the thickness of the active layer (piezoelectric layer) and also the length of the middle part of the beam on the output power for the vertical device is illustrated in
Alternative Configurations
Testing of the Shoe Energy Harvester (L=11.00″, W=3″, H=1.87″) resulted in a maximum power output of 32.89 mW with the use of a shorter single spring steel beam, which was covered with four piezo patches. This output shows a substantial increase from the projects previous prototype (L=15.75″, W=3″, H=2″) which utilized a longer brass substrate. These results demonstrate that spring steel substrates are worth investigating as its power output exceeds that of any other prototype. This power is generated without using the stop mechanism and it is recommended to use spring steel in the double beam configuration which has a built in stop mechanism. The power output is predicted to be increased when using double beam instead of a single beam.
As described herein, the present invention, in its horizontal configuration, may include a singlebimorph beam device (see
In some embodiments, a stop member 70, as illustrated in
In this article, generation of electricity from uniform and segmented piezoelectric beams was studied. The mode shapes of the beam were calculated and the electromechanical equations were solved for vertical and the introduced horizontal configurations. The load over the harvester, results in buckling of the piezoelectric beam. This transition results in oscillations of the piezoelectric beam and causes power generation. After the force is removed (the person takes another step or the car passes over) the beam springs back to the unbuckled shape and more power is generated. To prevent the beam from fracture or damage, a mechanical stop is placed to limit the beam'"'"'s deformation. The vertical design can be implemented in roads, using the weight of the passing cars to generate electricity. By using the segmented beam instead of a uniform beam the gap distance increases, which makes this design more practical in terms of manufacturing. The other configuration of the device (horizontal) makes it easy to implement in treadmills, shopping centers floors, or dance floors and also makes it possible to use the human body weight as the source of applying force to the device and generating electricity. The gap distance is also larger for this new configuration. It was shown that this distance is increased around 500 times in comparison to vertical configuration using a uniform beam. Gap distances for different theta angles and different lateral beams were discussed and it was shown that theta angle is a critical parameter in designing the horizontal device. Governing equations of the system were derived for two cases (before engagement of the stop and after it). The transient vibrations of the beam have been analytically modeled. The geometric nonlinearities of the beam have been taken into account. The deflection and output power of the device were calculated and studied for two case studies. It was shown that there is an optimized resistance for each design that generates the maximum average power output. Finally, the effect of the length and thickness of the piezoelectric layers on the output power and also the gap distance were investigated.
The foregoing description of the embodiments has been provided for purposes of illustration and description. It is not intended to be exhaustive or to limit the disclosure. Individual elements or features of a particular embodiment are generally not limited to that particular embodiment, but, where applicable, are interchangeable and can be used in a selected embodiment, even if not specifically shown or described. The same may also be varied in many ways. Such variations are not to be regarded as a departure from the disclosure, and all such modifications are intended to be included within the scope of the disclosure.