Chapter V: Experimental Study of Flextensional Flow-Energy Harvester
5.2 Flextensional Flow-Energy Harvester Design
5.2.2 Flexure Dynamics
Table 5.3: Table of structural material properties [93, 94].
Variable Aluminum Steel PZT Density [kg/m3] 2700 8000 7500 Young’s modulus [GPa] 68.9 193 64.5
Poisson’s ratio [ND] 0.33 0.29 0.31 Modal Finite-Element Analysis Results
Results from the modal finite element analysis consist of mode shapes and their cor- responding eigenvalues, stated as dimensional frequencies here. The fundamental mode of the system is at f1 = 169[Hz] and its shape is shown in figure 5.7. The amplitude of the modes shapes are arbitrary, and plotted for opposite phases in the figure. This is followed by the second mode at f2= 328[Hz], shown in figure 5.8.
There is also the first, second, and third flexure bending modes at f3 = 1067[Hz], f4 = 1335[Hz], and f5 = 1415[Hz]. The first beam torsional mode at f6 = 1962 [Hz], and a second transverse mode occurs at at f7= 2053[Hz].
(a) (b)
Figure 5.7: Results for fundamental mode of flexure at f1 = 169 [Hz]. Figures show snapshots of (a) up- and (b) down- wards movement of the mode shape. The red dashed-lines represent that static shape and contour levels are representative of stress concentration.
As seen in figure 5.7, the fundamental mode consists mostly of transverse move- ment of the flexure. This is the “flextensional” mode and couples structural motion strongest between the flexure and piezoelectric stacks, as evident by the lighter coloring representative of higher stresses around the flexure support to the piezo- elements. The mode also corresponds to a (mostly) rigid-body motion of the beam:
though the flexure moves up and down considerably, the cantilever itself remains straight through the cycle of motion. This becomes evident when comparing to the original static shape represented by the red dashed lines.
(a) (b)
Figure 5.8: Results for second mode of flexure at f2 = 328 [Hz]. Figures show snapshots of up- and down- wards movement of the mode shape. The red dashed lines represent that static shape and contour levels are representative of total dis- placement.
The modal analysis shows that the second and 7th modes are transverse bending modes. To understand whether only the beam is excited at those frequencies, we can compared the eigenvalues to those from classical Euler-Bernoulli beam theory.
We have defined the theory’s eigenfunctions in equation 2.79 and eigenvalues as the solutions to the characteristic equation 2.80, listed on table 2.2. Hence, we can calculate the theoretical clamped-free beam frequencies as
fi = (βiL)2 2πL2
s E I
ρsbhb. (5.10)
Iis the square cross-section moment of inertia for the beam in three dimensions,
I = h3bb
12 . (5.11)
Table 5.4 lists the FEA modal analysis and Euler-Bernoulli beam results, along with the description of each mode. Property parameters and values used are in tables 5.1 and 5.3. We see that the first two transverse modes frequencies are slightly over- predicted by≈5.5%in classical theory.
Table 5.4: Table of frequency predictions by finite element modal analysis (FEA) and Euler-Bernoulli (E-B) beam theory.
# FEA [Hz] E-B [Hz] Description
f1 169 - flextensional transverse mode
f2 328 346 fundamental cantilever transverse mode
f3 1067 - flexure spanwise bending mode
f4 1335 - flexure transverse bending mode f5 1415 - 2nd flexure transverse bending mode f6 1962 - fundamental cantilever torsional mode f7 2053 2167 2nd cantilever transverse mode
As results in flowing experiments will ascertain in section 5.4, the flow excites pri- marily the first two modes of the flexure. Hence, it is important that we ensure they can be captured as part of the model. Furthermore, the flow-energy harvester excites the flexure primarily through a transverse forceon the beam, where the pres- sure difference between the top and bottom flow paths apply the net force onto the structure. Since classical theory captures f2, and the mode shape in f1 is a rigid body motion of the cantilever, we conclude that it is tractable to represent the f1 mode shape as the elastic-translating leading edge boundary condition. Moreover, the simple harmonic oscillator discussed in section 2.3.3 has a physical analogue:
the flexure stiffness, damping, and mass that characterize the dynamics of the flex- tensional mode at f1.
In summary, the modal analysis provides the fundamental mode shapes and justifies the rigid body motion that characterizes the elastic-translating boundary condition.
It also suggests f1 as its representative frequency. Next we estimate the flexure static stiffness and obtain estimates for the simple harmonic oscillator boundary condition model.
Static Structural Finite Element Analysis
Figure 5.6 shows the static model details, including mesh, boundary conditions, and the position of the load Fa. To measure the static stiffness of the flexure, we obtain a relationship between applied force at the beam base versus the resulting displacement. Considering steady verson of equation 2.71, wherea¯represents the boundary condition amplitude, we can solve for the boundary condition stiffness as
k0= b fr
¯ a = Fa
¯
a . (5.12)
The applied force is thenFa = b fr, where fr is defined as the force per unit span and bis the span of the beam. Figure 5.9 shows a representative displacement contour forFa= 10[N].
Figure 5.9: Representative displacement at Fa = 10 [N] for static structural FEA.
Red dashed-lines represent the initial position beforeFais applied.
A series of 7 simulations were carried out with varyingFa. Figure 5.10 shows the results as a function of the beam base displacement. The slope of the line is the estimated stiffness k0FEA = 58,366[N/m]. The modal analysis has no mechanical damping in its solution, therefore f1= 169[Hz] is the solution of equation 2.71 for c0FEA =0. We can now estimate the effective mass component from f1as
m0FEA = k0
4π2f12 =0.052[kg]. (5.13) The valuesm0FEA = 0.052[kg] andk0FEA = 58,366[N/m] are approximations for those we should expect from the experimental measurements in section 5.3. They represent an idealization of the boundary conditions (i.e. no stack pre-stress) and
no damping. Though these conditions do not hold for the actual device, they help our understanding of the structure and provide a check on device fabrication and experimental set up.
0 0.2 0.4 0.6 0.8 1
7a [m] #10!3 0
10 20 30 40 50 60
Fa[N]
Figure 5.10: Fa as a function of displacement for 7 static structural FEA simula- tions. The slope represents a stiffness value ofk0FEA =58,366[N/m].