The Captive Trajectory System and Captive Airfoil Experiments

Chapter II: Experimental Setup and Data Processing Methodology

2.4 The Captive Trajectory System and Captive Airfoil Experiments

For the remainder of the experiments performed in support of this thesis, the Captive Trajectory (CTS) was used to provide the mounting system for the airfoil. A photograph of the airfoil mounted to the CTS downstream of the circular cylinder is provided in Figure 2.4.

The CTS is able to precisely drive a test object through a prescribed trajectory in the three translational axes, as well as pitch. It can also be integrated with

Captive Trajectory System

Cylinder

Airfoil

Force Sensor

Figure 2.4: Photograph of the airfoil (foreground) mounted to the Captive Trajectory System (CTS) downstream of a circular cylinder (background). Motion of the CTS is actuated by stepping motors, which individually allow for translation of the system in three translational axes and pitch. The airfoil is mounted to the CTS through its quarter-chord location, and a force sensor is used to measure the forces acting on it.

additional hardware to provide feedback for real-time control of a test object’s motion, referred to as acaptive trajectory; more details regarding CTS performance were reported by Shamai (2021). Both prescribed and captive trajectories were created for experiments in support of this thesis, as described in the following sections. Forces and torques on the airfoil were measured using the same force sensor as in the MFRS Experiments, the performance of which is discussed in Section 2.2.1; however airfoil position, velocity, and force data were available only as digital outputs from the CTS at a rate limited to 200 Hz, a significant reduction from the MFRS Experiments. This reduction in sample rate, a constraint imposed by the CTS itself, limits the frequency range and resolution possible for analysis of measured signals; however it is still several orders of magnitude larger than the expected dynamics in our flow.

For all experiments conducted using the CTS, the airfoil’s angle of attack was fixed

such that it remained stationary throughout the duration of each run, regardless of the pitching torque acting on the foil. This behaviour is different from that in the MFRS Experiments described earlier, as well as the pioneering experiments by Beal et al. (2006) in which they allowed for changes in pitch angle due to flow forcing.

As the observed pitching motions during the MFRS Experiments (which permitted free pitching, subject to the action of friction) were small, the geometric angle of attack was fixed to simplify the experimental setup in further Driven and Passive Captive Airfoil Experiments.

2.4.1 Motion through a Prescribed Trajectory

For all Driven Airfoil Experiments, discussed in detail in Chapter 3, the airfoil was mounted to the CTS and actively driven through a transverse sinusoidal trajectory with a known frequency and phase relative to oncoming vorticity, of the form

𝑦(𝑡) = 𝐴_𝑝sin(2𝜋 𝑓 𝑡 +𝜓), (2.3) where 𝑦(𝑡) is the transverse position as a function of time. The parameters 𝐴_𝑝, 𝑓 and 𝜓 were fixed based on data taken during previous MFRS experiments, which represent a truly passive mounting system (with no motors or other actuators to drive airfoil behaviour). Though the motion of the MFRS is unideal, in that it is not itself purely sinusoidal, choosing the parameters which determine the driven airfoil trajectories studied in this thesis based on characteristics of the MFRS allows us to directly connect these driven motions with the behaviour of a truly passive system.

To determine the appropriate driving frequency 𝑓, the transverse force and position of the airfoil from previous MFRS experiments were analyzed. For vortex shedding at a Reynolds Number of 40,000, the value of the Strouhal number,

𝑆𝑡 = 𝑓 𝐷 𝑈_∞

, (2.4)

has been shown to remain fixed around a value of 0.2 (for example Lefebvre and Jones, 2019, and references therein). This gives an expected dimensional shedding frequency of approximately 0.6 Hz. To confirm that this frequency was dominant in the airfoil dynamics, measured transverse force and position for all MFRS Experi- ments were windowed into temporal bins varying in length from approximately 5.5 s to 20.0 s (3.4𝑇to 12.5𝑇), and linear de-trending was applied to the position signal.

Then, the frequency content of each bin was computed using an FFT. In all cases, the windowed signals (with bins of any length) exhibited a strong primary peak in the region of 0.6 Hz; however the position signal sometimes exhibited additional

strong peaks at lower frequencies, perhaps due to long-term meander or dynamics induced by the mounting system. To remove this as a source of error in determining dominant oscillatory behaviour due to vortex shedding, only the transverse force signals were used to compute the oscillation frequency for the driven experiments.

The peak locations in all transverse force signal bins were averaged resulting in the selection of 0.6226 Hz, presented with fewer significant digits throughout the remainder of this thesis as 0.62 Hz, which gives a Strouhal number of 0.22.

To determine the appropriate phase offset between the measured force and the airfoil motion (a proxy for the phase between the vortex shedding and the airfoil motion), the difference in phase angle computed from the FFT at the dominant forcing frequency in each bin between the force and motion was averaged. Although this phase difference varied widely especially for shorter window lengths, the mean value was determined to be quite close to𝜋/2. This value confirms a visual trend identified in the passive airfoil data, as well as enforcing the condition that the airfoil’s velocity is in-phase with the forcing. This is intuitively satisfying as it enforces that the rate of work done on the airfoil by the surrounding flow,

𝑃(𝑡) =𝐹_𝑦(𝑡) ¤𝑦(𝑡), (2.5) is always positive. Since there is no power input available for the airfoil to do mean work on the flow, this is a physically meaningful constraint; thus, for the Driven Airfoil Experiments presented in this thesis, the phase𝜓 for the motion is set such that velocity and transverse force are (approximately) in-phase. This choice however represents a significant simplification (and idealization, in terms of power extraction potential) relative to the true behaviour of the passive foil, for which friction-induced dynamics obscure the phase shift corresponding to an ideal free-airfoil system.

Finally, the appropriate amplitude for the driven sinusoidal motion was determined through analogy with the time-mean kinetic energy ¯𝐸 of the airfoil in the MFRS Experiments. Considering a pure sinusoid,

¯ 𝐸 = 𝐸¯_𝑘

1 2𝜌_∞

= lim

𝑇→∞

1 𝑇

∫ ^𝑇

0

[𝐴_𝑣sin(𝜔𝑡)]²𝑑 𝑡 = 𝐴²

𝑣

2 . (2.6)

Then, the mean kinetic energy of a signal that is not perfectly periodic can be

matched to an equivalent value for a pure sinusoid. We consider 𝐸¯ = 𝐴²

𝑣

2 , (2.7)

𝐴_𝑣 =√︁

2 ¯𝐸 =𝜔 𝐴_𝑝, (2.8)

𝐴_𝑝 =

√ 2 ¯𝐸 𝜔

, (2.9)

where 𝐴_𝑣 is the amplitude of the velocity, and 𝐴_𝑝 is the amplitude of the position for a pure sinusoid. By calculating ¯𝐸 as the mean kinetic energy over all MFRS trials (i.e., by summing square velocity at each recorded time step for all experi- mental runs then dividing by total time steps), then applying Equation 2.9, we can estimate the appropriate amplitude for our driven position signal as𝐴_𝑝= 5.1 mm, or approximately 5% of the airfoil chord. This is a relatively small amplitude of oscil- lation compared to previous studies (discussed further in Chapter 1), which usually consider flapping foil motions on the order of 0.5-1.0𝐷; however, the amplitude of oscillation observed by Beal et al. (2006) for a passively flapping foil exhibiting both thrust production and power extraction was much smaller, approximately 0.2𝐷. In addition, such a small amplitude of oscillation is still seen to give rise to a host of dynamics of interest in this study.

In summary, for the Driven Airfoil Experiments presented in the thesis, the air- foil was driven through a prescribed sinusoidal trajectory given by the following equation:

𝑦(𝑡) =0.0051sin(2𝜋(0.6226)𝑡+𝜓), (2.10) where𝜓was fixed for each individual trial to ensure that the velocity was in-phase with the force acting on the airfoil due to vortex shedding, and significant digits for 𝑓 and 𝐴_𝑝 are preserved in accordance with CTS precision. This trajectory is accurately realized by the CTS for all Driven Airfoil Experiments.

2.4.2 Motion computed from Real-Time Force Feedback

For the Passive Captive Airfoil Experiments discussed in Chapters 4 through 6, the CTS is used to provide real-time control of the airfoil’s motion, based on programmed dynamics. CPFD apparatuses of this type allow the experimenter access to much more detailed control of the behaviour of a test object, as discussed further in Chapter 1. In the present case, moving from an all-mechanical mounting system (the MFRS, discussed in detail in Section 2.3) to a cyber-physical realization through the CTS allowed for the specification of an idealized, friction-free passive

mounting system, and enabled much more precise control over the ‘physical’ details of such a system than would be possible using real mechanical components.

A flow chart showing the high-level operation of the CTS is given in Figure 2.5.

Low-level behaviour of the CTS (trajectory tracking, motor control, data acquisition etc) corresponding to the Measure, Output, and Actuate steps in Figure 2.5 is controlled internally, and such behaviour was not changed from basic settings tuned in a recent study by Shamai (2021). The high-level behaviour, or the determination of a desired trajectory based on measured sensor inputs (the Modelstep in Figure 2.5), was changed for each set of Passive Captive Airfoil Experiments as necessary to produce the desired airfoil dynamics.

The desired trajectory for a test object, in this case the airfoil, is specified at each CTS time step (a rate of 200 Hz) based on an update programmed by the user. For all Passive Captive Airfoil Experiments in this thesis, the trajectory update at each time step was calculated based on an impulse method, following Mackowski and Williamson (2011). The first step of this method is to calculate the real Impulse 𝐼 applied to the airfoil over the previous time step. The real impulse can be approxi- mated as the measured force multiplied by the time step, including an additional term necessary to counteract the fictitious force in the measurement due to the sensor’s real mass𝑚_realand accelerating reference frame (discussed in Section 2.2.1):

𝐼_𝑛 =

∫ ^𝑡_𝑛+1

𝑡_𝑛

𝐹_meas(𝑡) +𝑚_real𝑦¥(𝑡) dt≈ 𝐹_𝑛Δ𝑡+𝑚_realΔ𝑦¤_𝑛+1. (2.11)

Measure Model Output

Actuate

▪ Update for POSITION and VELOCITY.

▪ CTS creates desired trajectory.

▪ Desired trajectory calculated.

▪ Trajectory encounters PHYSICS.

Experiment

▪ New POSITION + VELOCITY + FORCING.

Figure 2.5: Flow chart showing the high-level behaviour of the Captive Trajectory System (CTS). The CTS allows the user to specify a model which governs a physical system’s response to measured inputs, in this case, measured forces acting on an airfoil. It then actuates the physical system in accordance with the desired behaviour.

Then, impulses applied by the virtual mounting system (for example, the action of a virtual linear spring or damper, as considered in the transverse direction for all Passive Captive Airfoil Experiments in this thesis) are computed in a similar fashion,

𝐼_virt,𝑛 =

∫ ^𝑡_𝑛+1

𝑡_𝑛

𝐹_virt(𝑡)dt≈ 𝐹_virt,𝑛(Θ,Δ𝑦¤_𝑛+1, 𝑦_𝑛)Δ𝑡 , (2.12) where the specific parameters Θ giving rise to the virtual forces correspond to parameters in the dynamics under test, for example [𝑚, 𝑏, 𝑘] for the spring-mass- damper system. Using the above expressions, an implicit update for the quantity of interest,Δ𝑦¤_𝑛+1is made possible according to

𝑚_virtΔ𝑦¤_𝑛+1 =𝐼_𝑛+𝐼_virt,𝑛. (2.13) Position is then updated using an Implicit Euler method. Trigger conditions appro- priate to each set of experiments are also included, such that trajectory updates are initiated only after trigger conditions are met. Before implementation on the CTS, trajectory update code was tested on an emulated CTS system with no hardware in the loop and simulated forcing, and results were validated against a simulated dynamical system with the desired dynamics created in MATLAB’s Simulink envi- ronment. The following code skeleton provides a general outline for the format of trajectory updates used for Passive Captive Airfoil Experiments in this thesis:

IF(Trigger Condition Met) Compute implicit update:

Solve Equation 2.13 forΔ𝑦¤_𝑛+1based on measured and virtual forces.

Update trajectory:

¤

𝑦_𝑛+1 =𝑦¤_𝑛+Δ𝑦¤_𝑛+1

𝑦_𝑛+1 =𝑦_𝑛+Δ𝑡𝑦¤_𝑛+1 ELSE

Wait for Trigger Condition

In general, performance of the CTS allowed for the desired trajectories computed at each time step to be realized with a high degree of accuracy. This is particularly true for transverse-only motion of the airfoil discussed in Chapter 4. For the 2D experiments discussed in Chapter 5, slight deviations from the desired trajectory in the 𝑥-direction motion only were apparent. The strict cause of these deviations is unknown, but they appeared to be more prevalent at higher forward speeds. To address the issue, a detailed review of controller gains and other motor-control level parameters set internally in the CTS is necessary. As these deviations did not seem

to critically affect the observations made regarding streamwise airfoil motion, such a detailed investigation is left for future experimental campaigns.

Another source of error in the computed trajectories stems from uncertainty regard- ing the real mass of the airfoil (and associated mounting hardware) necessary for the computation of the real impulse,𝐼_𝑛. The value of𝑚_realfor all Passive Captive Airfoil Experiments in this thesis was set to a value of 0.8005 kg based on a preliminary investigation of fictitious forces induced in the sensor, which was later found to be in error. Correcting the error and using an independently measured mass for the foil itself, the correct mass was found to be 1.2 kg, constituting a mass error𝛿𝑚 =0.4 kg. Though this is a relatively large error in terms of the real mass value𝑚_real=1.2 kg, it is a much smaller error relative to thevirtual mass of the system, for which the smallest value considered in this study is 6.1 kg. Propagating this error in mass to determine the effect on the force used to compute a trajectory, we find based on the expected acceleration of the system that the error should not be larger than 0.06 N, which is smaller than the uncertainty in the measured force itself (from Section 2.2.1,𝛿 𝐹 ≈0.1 N). Thus, although every effort should be made to correct the erro- neous mass in future tests, the effect of this error on the observed behaviour of the system is limited. In addition, the real mass of the system is a known source of error for CPFD systems of this type: Mackowski and Williamson (2011) acknowledge the real mass of their similar setup as a source of error in realized trajectories (despite its accurate specification), and recommend ensuring that𝑚_real ≪ 𝑚_virtfor optimal performance. For the majority of Passive Captive Experiments in this thesis, 𝑚_𝑣 was 18.4 kg or larger.