The primary goal of the present dissertation is to improve the state of knowl- edge regarding the structural dynamic response of base-actuated (flapping) beams. In order to achieve this goal experimental, numerical, and analytical methods are used to characterize the time-dependent strain and displacement fields of flapping aluminum beams operating at both standard and reduced air pressures.
The remainder of the dissertation is organized in the following manner. First, in Chapter 2 a comprehensive description of the experimental setup is provided. The details of the experimental procedure are given and the as- sociated difficulties are addressed thoroughly. In Chapter 3, the nonlinear structural dynamics of the flapping beam is explored through experiment and numerical simulation. A brief outline of the experiments and a detailed de- scription of the computational model are provided. The experimental beam tip displacement and surface bending strain data are compared with those gathered from the numerical simulations in both time and frequency domains. These comparisons not only provide valuable insight towards the overall goal of the dissertation but also allow for various modeling assumptions to be tested. Additional numerical simulations are performed to analyze the beam response characteristics in terms of the bifurcations possible for the present problem.
One of the modeling assumptions which is made in Chapter 3 is that of a linear viscous damping model. In Chapter 4 this restriction is removed and the effects of nonlinear damping on the structural dynamics of flapping beams
are studied via experiment and numerical simulation. Experimental apparatus consisting of the flapping mechanism and the vacuum chamber is briefly sum- marized. A nonlinear inextensible beam model and the time-dependent bound- ary conditions used to approximate the experimental mechanism actuation are given. Then, the approximate solution of the problem in the spatial and time domains are presented along with the linear and nonlinear damping models. The utilized nonlinear damping models are of various simple functional forms which contain empirically determined constants. Such simple analytical mod- els for damping are used to compensate for the inability, or unwillingness, to solve the true (complex) fluid-structure interaction problem [60]. Such an approach is widely used in the literature [39, 84–87], and if the parameters are chosen correctly it yields an analysis framework which can accurately and efficiently predict large amplitude beam vibration response. The numerical solution consists of a 1-mode Galerkin method for spatial discretization and a high-order time-spectral method for temporal discretization. In addition, to explore the effect of damping on the stability of periodic solutions, Floquet theory is used in conjunction with the numerical solutions. The experimental setup consists of the flapping mechanism and a vacuum chamber as well.
While numerical simulation can provide detail and fidelity, oftentimes an- alytical solutions can provide insight into parameter dependence which is unattainable through simulation. In addition, approximate analytical solu- tions can be used to improve simulation efforts by uncovering possible scal- ing laws, thereby providing information to reduce the number of simulations needed. Furthermore, analytical solutions can provide guidance in how to improve numerical methods for solving the problem in question. As such, in Chapter 5, the nonlinear response of flapping beams to resonant excitations
under nonlinear damping is studied analytically. Using the method of mul- tiple time scales, modulation equations governing the steady-state amplitude phase evolution of the superharmonic and harmonic oscillations are obtained for the nonlinear ordinary differential equation which results from a 1-mode Galerkin spatial discretization of the inextensible beam theory. Frequency- response relationships and first-order approximate steady-state solutions at the superharmonic and primary resonances are determined. Approximate ex- pressions for the critical excitation amplitudes which lead to bistable solutions are calculated. The approximate results are determined to corroborate the experimental and numerical observations of the single-valued stable response amplitudes. The analytical results are compared with those obtained with numerical solutions based upon a time-spectral method in order to ascertain the validity of the approximate solutions.
Finally, summary of conclusions and suggestions for the future works are given in Chapter 6. In particular, a future work is given, and expanded upon in Appendix K, regarding the improved modeling of fluid damping.
CHAPTER 2
Experimentation
2.1 Scope of the Chapter
In this chapter, the details of the experimental setup constructed to simulate the flapping beam problem are presented. The highly dynamic and nonlinear nature of the problem requires a robust, reliable flapping mechanism and ap- propriate measurement procedures. As such, the flapping mechanism should be able to produce the commanded output (i.e., flapping frequency and flap- ping amplitude) as accurately as possible while operating under large dynamic forces. On the other hand the measurement hardware and installation meth- ods should be selected carefully for demanding cyclic response measurements. All these challenges and remedies are discussed in the present chapter.
The experimental setup consists of a flapping mechanism, a beam speci- men, response measurement equipment, and a vacuum chamber. The response measurement equipment include data acquisition peripherals for strain mea- surement, a high-speed camera, tungsten halogen lamps, and a speed controller for the electric motor which actuates the mechanism. A picture showing all major components of the experimental setup is given in Figure 2.1.