A mathematical model and computational scheme to evaluate a cantilevered piezo- electric energy harvester under dynamic bending have been developed. The energy
103 104 105 106 107 Resistance ( ) 10-1 100 101 Voltage (V per g)
Voltage Amplitude vs Resistance
Erturk-Inman Model Present Model (FEM) Present Model (IGA) Present Model (Analytical)
8 8.5 9 9.5 10 105 5.4 5.45 5.5 5.55 5.6 (a) 103 104 105 106 107 Resistance ( ) 100 101 102 103 Power ( W per g 2)
Power Amplitude vs Resistance
Erturk-Inman Model Present Model (FEM) Present Model (IGA) Present Model (Analytical)
2.5 3 3.5 4 4.5 5 104 215 220 225 230 235 (b)
Figure 3.22: Bimorph 1 with h0/h = 104 - Variation of the (a) voltage amplitude and
(b) power amplitude with the resistance load
harvesting investigations by means of the hybrid analytical/ computational scheme have been discussed in this chapter. The capabilities and robustness of the scheme are shown by comparison with results from the literature.
The results obtained via the present hybrid scheme are well agreed with those obtained from analytical, numerical and experimental methods. It is shown in some details that the present hybrid scheme provided a faster computational time compared to an electro-mechanically coupled finite element method. Moreover, the hybrid scheme provided ease to utilise commercial software with minimum addition of computational coding for energy harvesting evaluation. By means of the hybrid scheme, the reverse piezoelectric effect was obtained from the dummy load structural simulation via a commercial software. Thus, full electromechanical coupling on the finite elements were not required. Hence, it can be used as an alternative tool during an early design stage to evaluate the harvester potential.
In relation to the aim of the present research to evaluate the energy harvesting potential of a large aircraft wing, the hybrid scheme is further implemented to inves- tigate a jet aircraft wingbox in Chapter 4. The investigation presented in Chapter 4 will serve to give some insights on the power harvested and concerns on the addition of piezoelectric material concerning the aircraft structure.
Chapter 4
Implementation of The Hybrid
Scheme on A Jet Aircraft Wingbox
In this chapter, the implementation of the novel hybrid scheme on a notional jet aircraft wingbox is presented. A typical long-ranged civil jet transport aircraft type is chosen. It is expected that with this type of wing structure, the order of power is much larger compared to those obtained from small lifting structure in the literature. A jet aircraft wingbox with 14.5 m half span is used for the energy harvesting evaluation. A piezo- electric material replaces the original upper skin0s material of the wingbox. A dynamic cruise load is applied. The results pointed out that the electrical power generated can be as much as 39.13 kW.
Some works and results presented in this chapter are parts of the author’s published works in Composite Structures, Volume 153, 2016.
4.1
Wingbox FEM analysis
A test case for a notional civil jet aircraft wingbox is simulated in the present work. The structural dynamic response is performed via FEM. A common practical case in the operational flight is considered. The dynamic excitation forces is equal with the cruise load. The excitation frequencies observed are around the 1st bending mode
natural frequency.
An aircraft wingbox model [84] is taken as the reference for the present simulation. However due to some details of the wingbox are not given, the geometry is simplified based on the available data.
Figure 4.1 shows the wingbox vertical stiffness distribution. Due to the detail of the ribs cross section is not available from [84], it is assumed that the ribs are plates with rectangular cross section. Figure 4.2 shows the wingbox layout from topside view. The available data from [84] are the span length of the spar (570 in or 14.48 m), the
0 100 200 300 400 500 600 0 0.5 1 1.5 2 2.5 10 11
Figure 4.1: Wingbox vertical stiffness distribution
Figure 4.2: Wingbox topside view layout
distance between the front spar and the rear spar at the wing root (90 in or 2.29 m) and at the wing tip (35 in or 0.88 m). For simplification, the front spar length is assumed perpendicular to the ribs at the root and tip. The rear spar is assumed to be straight connecting the trailing edge of the root and tip ribs. The ribs spacing are assumed uniform. Hence, there are 20 ribs with 30 inches spacing in the simplified model.
Other simplifications made for the present simulation are
1. The skins, ribs and spars are assumed as rectangular plates with uniform thick- ness. The thickness for the skins is 0.24 inches and for the ribs and the spars is 0.29 inches. These are maximum thickness values of the original model [84]. 2. The stringers and spar caps are not modelled for the present simulation.
3. The skins, ribs and spars are made of uniform plates with isotropic material, Al- 2219. Later on, for energy harvesting purpose, the upper skin material is replaced by a piezoceramic material, PZT-5A.
Figure 4.3: Wingbox model for finite element analysis
Figure 4.3 shows the wingbox model used for the finite element analysis. The skins, ribs and spars are modelled as quadrilateral shell elements with the thickness as mentioned earlier. The translations and rotations (for 3 directions) are fixed at the root and free at the tip. The different material configurations used for the simulation are
1. For the first wingbox model, its skins, ribs and spars are all modelled by Al- 2219 with modulus of elasticity 73.1 GPa (10.66 lb-in2), poisson0s ratio 0.33 and
density 2840 kg/m3(0.1 lb/in3). This model is called model A, hereafter.
2. For the second model, its upper skin replaced with piezoceramic material, PZT- 5A with modulus of elasticity 60.9 GPa (8.8x106 lb-in2), poisson0s ratio 0.33 and density 7750 kg/m3(0.27 lb/in3). PZT-5A is chosen due to its high electrome- chanical coupling. This model is called model B, hereafter.
The data given in [84] shows that the original model is weighted 2742.5 lbs and the maximum tip displacement is around 30 inches during the ultimate load (2.5 g up gust and 50,000 lbs thrust load). The weight of the aircraft itself is 170,000 lbs and the 2.5 g up gust is equal with 425,000 lbs. The comparison of the original model [84] and model A is shown in Table 4.1.
Modal analyses are conducted to verify the structure of model B. The natural frequencies comparison between model A and B is shown in Table 4.2. The modification
Table 4.1: Weight and tip displacement, Ztip, original model vs model A
Weight (lbs) Ztip - ultimate (in) Ztip - cruise (in)
Original model 2472.5 30 -
Model A 2415 33.8 13.5
in model B resulted in a lower natural frequency than model A for the same mode shape. This characteristic mainly influenced by the weight increment from 2415 lbs (model A) to 3929 lbs (model B). Meanwhile the stiffness just slightly decreases from model A to model B as the PZT-5A only applied to the upper skin. The square root of the mass ratio between model B and A is found to be 1.27. This ratio is reasonably close to the frequency ratio (fA/fB) for the first 3 bending modes shapes shown in Table 4.2.
It is important to note, a weight increment may result in a much more fuel to burn in an aircraft flight; hence, it will become a major challenge on the implementation of piezoelectric material in an aircraft. The significance of the weight increment is further discussed in Chapter 6.
Table 4.2: Natural frequency comparison, model A vs model B Natural frequency (Hz)
Mode Shape Model A Model B fA/fB
1st Bending 2.16 1.61 1.34
2nd Bending 9.01 6.74 1.34
3rd Bending 21.70 16.19 1.34
Moreover, for the energy harvesting purpose, Model B is analyzed by applying the frequency-dependent forced excitation via a FEM module [79]. The force amplitude is equal to the steady cruise lift, half of the aircraft weight (85,000 lbs). However, the detail of the airfoil is not available in [84], and the lift distribution is unknown. Therefore, the lift is simplified as a concentrated load acting on the wingbox.
An evaluation of the lift coefficient (Cl) distribution via Lifting Line Theory [85] is conducted to obtain the concentrated load point. A typical cambered NACA 6-series is assumed as the airfoil of the wing. To be noted, several configuration parameters of the wing are assumed, adopted the wing illustration given in [84]. Hence, as most of the inputs are based on assumptions, this Cl distribution may not be applied as a real load and herein only serve as a sketch to estimate the concentrated load point. The Cl distribution is depicted in Figure 4.4, the concentrated load point is estimated at around 6 m from the root. The excitation frequencies varied from 0.80, 1.31 to 1.45 Hz. These range of frequencies is close to the natural frequency of the 1st mode shape. It is important to note that the assumption of harmonically oscillating cruise load
0 5 10 15 Location in semispan [m] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Lift coefficient Lift distribution
Figure 4.4: Sketch of the lift coefficient distribution on the wing
may provide unrealistic loading condition in comparison to a typical cruise flight. One of the main consequences that may need to be noted is the structural responses may be overestimated. This is due to the vibration amplitude directly applies the load equal to the aircraft weight and the excitation frequencies near the first bending mode. Moreover, a continuous vibration, i.e., harmonic oscillation, may not occur on a normal flight as the aeroelastic damping is sufficient to reduce the oscillation and stabilise the response exerted by external disturbance, i.e., gust wing. Thus, the following evaluation in Section 4.2 may only be interpreted as an extreme and rare case, serves as the upper benchmark on the energy harvesting estimation.