Bending Strain vs Flapping Frequency Results

The effect of change in the surrounding air pressure on the response of flapping beams is observed to be similar at both tested flapping amplitudes, 15◦ _and

30◦_{. Figures 4.2 and 4.3 show the experimental bending strain data obtained}

at 15◦ _{in air (101.3 kPa) and 70% vacuum (30.3 kPa), respectively. Note}

that ω0 denotes the flapping frequency normalized by the first-mode bending

frequency of the cantilever beam; i.e., ω0 = ωf/ωN. Through comparison

of the two figures, it can be noted that the vibration amplitudes at second and third-order superharmonic resonance frequencies (i.e., at ω0 = 0.33 and

ω0 = 0.50) amplify considerably as the surrounding air pressure is decreased.

On the other hand, the vibration amplitudes at frequencies other than the resonant frequencies do not vary with a change in ambient pressure. Similar behavior can be noted by comparing Figures 4.4 and 4.5, which show the experimental bending strain data obtained at 30◦ _{in air and 70% vacuum,}

respectively. When compared to flapping at 15◦_{, the experiments conducted}

for flapping at 30◦_{have much broader secondary resonance peaks (at ω}

0 = 0.31

and ω0 = 0.43). These peaks do become slightly more pronounced as the

ambient pressure is reduced.

The first mode damping ratio of the cantilever beam is measured as 0.013 based on the small amplitude free vibration response in air. This experimentally- determined damping ratio, ξvis= 0.013, is used for the linear viscous damping

force, fd,vis, throughout the study. In Figure 4.2, the frequency response curve

obtained with the linear viscous model is compared against the experimental data obtained at a flapping amplitude of 15◦_{. Overall the viscous model es-}

timation is in good agreement with the experiments at flapping frequencies up to ω0 ≈ 0.83. The exception to this result is in regions of secondary res-

onances, ω0 ≈ 0.33 and ω0 ≈ 0.50. In regions of secondary resonance, the

bending strain amplitude is severely overestimated, e.g. for ω0 = 0.50 the

simulation overestimates the strain by an order of magnitude. In addition to the current simulation results, the result (labeled ATFEM in the figure) found using the time-marching, nonlinear finite element model solution discussed in Chapter 3, which contains the same linear viscous damping model used in the current work, is shown in Figure 4.2 (and Figure 4.4). While the nonlinear finite element model includes both in-plane and out-of-plane deformation, one can see that the current Galerkin 1-mode solution of the inextensible beam theory gives comparable results.

The response curves obtained with the displacement-2nd _{power damping}

(fd,disp) and velocity-3rd power damping (fd,vel) models are also included in

Figure 4.2. The values of the damping parameters ηdisp and ηvel are chosen

through an (approximate) minimization of the following error measure:

e = N X i=1  ǫexp_{i − ǫ}model i ǫexp_i 2 , (4.7.1)

where N is the number experimental data points for ω0 ∈ [0.3, 0.6] and ǫexp

and ǫmodel are the experimental and model (including fd,vis) values of bending

strain (standard deviation of dynamic bending strain signal). Note that Eq. (4.7.1) implies the sum of the squares of the normalized (by the experimental value) differences between the experimental and model values at the same flapping frequency [52,62]. For additional methods, in both frequency and time domains, for parameter estimation in linear and nonlinear structural dynamics

please see the review article by Kerschen [134].

For flapping at 15◦_{in air, the values found are 0.45 m}−1_{· s and 3600 m}−1_{· s}−1

for ηvel and ηdisp, respectively. The values for the error measure e, for various

values of the damping parameters, are given in Table 4.1 for flapping at 15◦

and 30◦ _{in both air and 70% vacuum.}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 ω₀ 0 1000 2000 3000 4000 5000

S.D. of SURFACE BENDING STRAIN [

µ -strain] experiment f_d,vel + f_d,vis f_d,disp + f_d,vis f_d,vis ATFEM

Figure 4.2: Experimental frequency response of the beam bending strain along with theoretical response curves obtained with different damping models (with, ¯ηvel = 0.30, ¯ηdisp = 3.61, ξvis = 0.013) for flapping at 15◦, in air. The

curve labeled ATFEM represents the solution obtained with time-marching, nonlinear finite element model presented in Chapter 3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 ω₀ 0 1000 2000 3000 4000 5000

S.D. of SURFACE BENDING STRAIN [

µ -strain] experiment f_d,vel + f_d,vis f_d,disp + f_d,vis f_d,vis

f_d,mat (Crawley & van Schoor [44])

Figure 4.3: Experimental frequency response of the beam bending strain along with theoretical response curves obtained with different damping models (with, ¯ηvel = 0.14, ¯ηdisp = 1.87, ξvis = 0.013) for flapping 15◦, in 70% vacuum

(21 inHg vacuum).

As shown in Figure 4.2, for flapping in air at 15◦_{, the results obtained with}

both nonlinear damping models show good agreement with the experiments for frequencies well beyond the range over which they are fitted. It is noted, however, that the displacement-based model, fd,disp, does not give any indica-

tion of superharmonic resonance peak at ω0 = 0.33 and, similar to the linear

viscous model, fails to provide a realistic damping force in the primary reso- nance region. Conversely, the model fd,vel yields better predictions for both

secondary resonance peaks and also for the primary resonance behavior of the beam.

The frequency response of the beam bending strain obtained at 15◦ _in

70% vacuum is shown in Figure 4.3. In addition to the response curves of linear and nonlinear damping models, the data obtained with a model which only includes the nonlinear stress-dependent material damping force fd,mat

is also given in the figure. The material damping model estimates follow a similar trend as the linear viscous model fd,vis with the secondary resonance

peaks greatly overestimated. Moreover, the HDHB numerical scheme does not converge in the resonance regions when the material damping model is employed alone. The experiments in the present study are not run under the same vacuum conditions (0.133 kPa) as the experiments conducted by Crawley and van Schoor [129] and the results presented in Figure 4.3 suggest that the contribution of the internal damping to the overall damping force is trivial even under 70% vacuum.

Table 4.1: List of nonlinear damping parameters ηvel [m−1· s] and ηdisp

[m−1 _{· s}−1_{] along with the associated error measure e [see Eq. (4.7.1)] used}

to determine the best damping parameter estimate for flapping at 15◦ _and

30◦_{, in both air and 70% vacuum (21 inHg vacuum).}

15◦

In air In 70% vacuum

ηvel e ηdisp e ηvel e ηdisp e

0.30 0.751 2800 0.815 0.45 0.525 5400 0.538 0.35 0.690 3000 0.792 0.60 0.450 5800 0.528 0.40 0.663 3300 0.773 0.65 0.440 6100 0.526 0.45 0.656 3500 0.768 0.70 0.436 6200 0.524 0.50 0.663 3600 0.767 0.75 0.437 6400 0.525 0.55 0.679 3800 0.769 0.80 0.440 6600 0.526 30◦ In air In 70% vacuum

ηvel e ηdisp e ηvel e ηdisp e

0.30 3.247 2000 5.601 0.90 3.502 6400 5.892 0.50 2.507 3000 4.307 1.50 2.634 7000 5.348 0.55 2.472 3600 4.184 1.80 2.577 9000 4.558 0.60 2.456 3800 4.166 1.90 2.575 10000 4.420 0.70 2.454 4000 4.153 2.00 2.577 12000 4.310 0.80 2.470 4600 4.120 2.10 2.583 14000 4.270

In the simulation, the reduced pressure condition is modeled through a change in the air density ρa, in Eqs. (4.5.4) and (4.5.3), which is assumed to

be directly proportional to the air pressure. As such, in order to simulate 70% vacuum conditions, ρa is set to 0.36 kg/m3 which corresponds to 30% of the

air density at 101.3 kPa. In Figure 4.3, the response curves for the nonlinear damping models fd,disp and fd,vel are obtained using parameter values ηdisp and

ηvel which are determined based upon the aforementioned error minimization

procedure. The values of the damping parameters ηdispand ηvel in 70% vacuum

reveals that the velocity-3rd power damping model provides a slightly bet- ter prediction of the strain response when compared to the displacement-2nd

power model. In particular, it is able to better predict the response near the secondary (ω0 ≈ 0.33) and primary (ω0 ≈ 0.93) resonance regions. It should

be noted that a posteriori measurement of the natural frequency of the beam after flapping in the primary resonance region (ω0=0.97, 1.03, 1.10, and 1.17),

for reduced air pressure, show some reduction in the natural frequency likely indicating some (unmodeled) yielding behavior.

Shown in Figures 4.4 and 4.5 are the experimental and theoretical frequency response data obtained for flapping at 30◦_{in air and 70% vacuum, respectively.}

The most notable result, which can be observed in both figures, is that the experimental data form a broad “hump” over the range of frequencies en- compassing the second and third-order superharmonic resonance frequencies. Close examination of the data collected in 70% vacuum (Figure 4.5) reveals that, in addition to being amplified in magnitude, the local peaks in this hump, which correspond to third and second-order superharmonics, occur at slightly higher values of the flapping frequency when flapping takes place at reduced air pressure.

As can be seen in Figure 4.4, once again simulation which includes only the linear viscous damping force fd,vis severely overestimates the strain amplitude

in the regions of secondary resonance. In addition, as the secondary resonance frequencies are approached the simulation fails to converge, likely indicating either a breakdown in the beam model assumptions or the absence of a pe- riodic solution. It also appears that the location of the secondary resonance peaks are overestimated by as much as 0.725 Hz (ω0 = 0.05). In addition

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ω₀ 0 1000 2000 3000 4000 5000

S.D. of SURFACE BENDING STRAIN [

µ -strain] experiment f_d,vel + f_d,vis f_d,disp + f_d,vis f_d,vis ATFEM

Figure 4.4: Experimental frequency response of the beam bending strain along with theoretical response curves obtained with different damping models (with, ¯ηvel = 0.46, ¯ηdisp = 4.02, ξvis = 0.013) for flapping at 30◦, in air. The

curve labeled ATFEM represents the solution obtained with time-marching, nonlinear finite element model presented in Chapter 3.

cate minimal contribution to the overall damping force, similar to what was observed for flapping at 15◦_.

Also shown in Figure 4.4 are the theoretical curves obtained with the non- linear external damping models. Damping parameters ηvel in air and in 70%

vacuum are determined, according to Eq. (4.7.1), as 0.70 and 1.90 m−1_{· s,}

respectively. However, a qualitative agreement between the experiment and model could not be established for the displacement-2nd _{power model by using}

the aforementioned error minimization scheme. The error values decrease as ηdisp is increased until the model curve “flattens out” and the secondary reso-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ω₀ 0 1000 2000 3000 4000 5000

S.D. of SURFACE BENDING STRAIN [

µ -strain] experiment f_d,vel + f_d,vis f_d,disp + f_d,vis f_d,vis

f_d,mat (Crawley & van Schoor [44])

Figure 4.5: Experimental frequency response of the beam bending strain along with theoretical response curves obtained with different damping models (with, ¯ηvel = 0.38, ¯ηdisp= 3.61, ξvis= 0.013) for 30◦, in 70% vacuum (21 inHg

vacuum).

the ηdispin air and in 70% vacuum are determined besed upon visual judgment

as 4000 and 12000 m−1_{· s}−1_{, respectively. Unlike flapping at 15}◦_{, for flapping}

at 30◦ _{the nonlinear damping models fail to match, either qualitatively or}

quantitatively, the experimental strain response characteristics in regions of secondary resonance. While a previous study performed on a clamped beam by Mei and Prasad [86] indicated the ability of the nonlinear displacement- 2nd _{power damping model to predict broadening of resonance peaks, here it is}

observed that neither this damping model nor the velocity-3rd _{power damp-}

ing model are able to predict significant broadening of the strain response in the superharmonic resonance region. Similar observations can be made for flapping at 30◦ _{in a 70% vacuum, for which results are shown in Figure 4.5.}

4.7.2 Discussion of Bending Strain vs. Flapping Frequency Results