In this section, a general procedure for conducting experiments (at ambient and reduced pressures) with the flapping test-bed will be summarized:
i. All joints of the flapping test-bed are lubricated with white lithium grease to maintain a frictionless motion. The mechanism is brought to the neu- tral position (i.e., set to 0◦ flapping angle, see Figure 2.3b) and mov-
Figure 2.13: Vacuum chamber: (a) stationary cover end with electrical feedthrough for strain gage and motor cables, (b) flapping mechanism placed in the chamber, (c) access cover end, (d) aluminum cover with double O-rings. ing parts (main gear and rocker link) are immobilized with the aid of Scotch®
tape.
ii. The test-bed frame is supported with a dead weight and leveled with a “bull’s eye” spirit level. The beam to be tested is attached to the clamp- ing fixture and leveled. Strain gage instrument leadwires are connected to the bridge completion accessories.
iii. The strain measurement program is turned on and null correction (offset calibration) is performed to remove any offset value from the unstrained gage readings. First bending mode natural frequency of the beam is mea- sured by slightly deflecting the beam and setting it into free vibration. The tapes preventing the mechanism from motion are removed after the
free vibration test is performed.
iv. Strain data acquisition period (3 ∼ 6 s), sampling frequency (2000 Hz), and folders in which the raw data and analysis results to be saved are set on the strain measurement program.
v. The high-speed camera is focused on the beam edge and the filming parameters (e.g., frame rate, number of frames, etc.) are adjusted for the initial test. A low-speed camera (Canon, Inc.) is mounted on a tripod, focused on the experiment site and brought to stand-by.
vi. The motor controller is powered up and the settings related to desired velocity profile are configured. The motor speed data recorder is brought to motion-trigger mode. The trapezoidal velocity profiles consist of ac- celeration, constant speed (target speed), and deceleration regions. In all experiments the same duration (or velocity rate) is used for both acceleration and deceleration with values varying between 2 s and 5 s depending on the target speed (slower rates were used for higher target speeds).
vii. All lights are connected to the same switch to turn them on/off at the same time.
viii. With the experimental setup ready, the low-speed Canon camera is started first, lights are turned on and the motor is activated; strain data acquisition and high-speed camera are triggered once target speed is reached and stabilized.
the motor is set to decelerate and halt. The lights are turned off and low-speed camera is stopped.
Testing at other flapping frequencies, at the same flapping amplitude, are carried out starting from the third item in the above list. For the tests con- ducted in the vacuum chamber; the test-bed is placed in the chamber and items listed above are followed by excluding the recording with the high-speed camera and illumination with the halogen work-lights. As an exception, the chamber “door” is sealed following the third item and the vacuum pump is run until the desired level of reduced pressure is reached at which point the pump is stopped and remaining steps of the above-mentioned list are performed. Other flapping frequencies, at the same amplitude, are tested after pulling vacuum, if needed, to compensate for the loss. The vacuum pump is not run during the flapping experiments.
CHAPTER 3
Experimental & Numerical Characterization of the
Structural Dynamics of Flapping Beams
3.1 Scope of the Chapter
In this chapter,∗ the nonlinear structural dynamics of aluminum slender beams
is examined both experimentally and computationally. In the experiments the periodic flapping motion is imposed on the clamped edge of the cantilever beam using the 4-bar crank-and-rocker mechanism. Aluminum beams with nominal dimensions of 150 mm × 25 mm × 0.4 mm are tested in air over a range of flapping frequencies up to 1.3 times the linear first modal frequency at two different flapping amplitudes, 15◦ and 30◦. The response of the beam
is characterized experimentally through bending strain and tip displacement data obtained from a foil strain gage and high-speed camera, respectively. It was determined that for the particular combination of beam specimen (di- mensions, material properties) and forcing parameters investigated, all exper- imental responses were periodic. The frequency response curves based upon the experimental bending strain data reveal a secondary superharmonic peak in addition to the primary resonance peak. As the flapping frequency is in- creased, the response of the beam is observed to change from symmetric (with respect to equilibrium position) periodic vibrations with a period equal to the flapping period to asymmetric vibrations with higher harmonic content fea-
∗The material presented in this chapter was published in Journal of Sound and Vibration,
turing local oscillations in the time histories. Experimental tip displacement results show that the beam spends more time during stroke reversals when the flapping frequency is near the primary and secondary resonance regions. In addition to experiment, numerical simulations are performed using two-node, isoparametric degenerate-continuum based geometrically nonlinear beam ele- ments. The HHT-α version of the Newmark finite difference scheme is used to discretize the problem in time and a linear viscous damping model is assumed. Overall the numerical simulations agree well with the experiments and capture most of the nonlinear dynamical features of the beam response. It is, however, found that in resonance regions the simulations overpredict response magni- tudes, possibly due to the use of the linear damping model and linear elastic constitutive model. Additional numerical simulations of the beam tip response reveal dynamics which include periodic, asymmetric periodic, quasi-periodic, and aperiodic motions.
In Section 3.2, the computational model which is comprised of a beam finite element and finite difference time integration scheme is presented in detail. In Section 3.3, the experimental setup is briefly explained since more detailed presentation is given previously in Chapter 2. In Section 3.4, the results gath- ered from the experiments will be analyzed and compared with those obtained from the finite element simulations. In Section 3.5, additional numerical sim- ulations will be performed to explore the beam response characteristics with varying flapping frequency and flapping amplitude. Finally, conclusions are given in Section 3.6.