In this section, the explicit functional forms for different damping models fd(v, ˙v) considered in the equation of motion, Eq. (4.3.1), are given and the
corresponding modal forces ¯fd are derived. Four different damping models
which aim to represent energy dissipation mechanisms of different origins are studied. These include linear viscous damping fd,vis, stress-dependent material
damping fd,mat, velocity-3rdpower damping fd,vel, and displacement-2nd power
damping fd,disp. In contrast to the linear viscous model, the latter three are
nonlinear damping models. It should be noted that the total damping force fd
for both the velocity-3rdpower damping and displacement-2nd power damping
includes fd,vis. As will be described below, the unknown parameter in both
fd,vel and fd,disp is computed through comparison of the simulation results
with experiment. These simulations include the linear damping term whose coefficient is known via small (free) vibration experiments.
Linear viscous damping is the most widely assumed form of dissipation operative in various nonconservative systems. This is due to its mathematical convenience and its fairly good agreement with physical observation. It simu- lates, in a simple linear manner, the impeding force acting on a body creeping
through a viscous fluid in a laminar flow regime as observed in a dashpot. The damping force is assumed to be proportional to the relative velocity between the body whose motion is hampered and the surrounding medium. That is:
fd,vis = cvis˙v. (4.5.1)
In the present study the viscous damping constant, cvis, is approximated ex-
perimentally based upon the linear (i.e., small amplitude) free vibration re- sponse of the cantilevered beam tested in air. In this regard, the source of the measured dissipation can attributed to both the beam material itself (mate- rial damping or internal damping) and the surrounding air medium (external damping).
The second type of damping which is investigated is internal (material) damping. Using a carefully-designed experimental setup, Crawley and van Schoor [129] proposed a material damping model based upon the empirical data obtained from the free-free vibration response of aluminum beam samples. They showed that at vibration frequencies below the so-called Zener relaxation frequency [130], the average material damping in the aluminum beam samples increases exponentially with increasing maximum stress level in the samples. According to the proposed stress-dependent nonlinear damping model, the functional dependence of internal damping on the stress is given by [129]:
ξmat = α exp[β σmax/σy], (4.5.2)
where ξmat is the damping ratio, σmax is the maximum bending stress and σy
are given in Reference [129] as α = 7.73 × 10−4 and β = 4.06.
In situations where a solid body is exposed to high relative velocities, the damping force can be expected to depend nonlinearly on the relative veloc- ity [85, 131, 132]. In such cases nonlinear damping models involving quadratic or higher powers of the relative velocity would be appropriate to model the damping force induced by the surrounding medium. As such, fluid damp- ing force experienced by a solid body is known to be contributed by normal and shear stresses (form and friction drag) and is traditionally modeled as quadratic velocity damping model. In the case of air damping acting on the flapping beam, the skin friction drag is expected to be negligible, whereas damping due to normal stresses and convected shed vortices is expected be significant. To account for the damping due to separated flow conditions and convected vortices, a nonlinear damping model other than the quadratic veloc- ity model would be more appropriate. Accordingly, we consider the velocity-3rd
power damping of the following form:
fd,vel = cvel˙v3 = (ρabηvel) ˙v3, (4.5.3)
where, cvel is the velocity-3rd power damping coefficient which is a function
of beam geometry and fluid properties. We assume cvel to be the product of
the air density ρa, beam width b and an empirically determined parameter ηvel
which has dimensions of time/length.
The final dissipation model which is investigated in this study is the dis- placement 2nd power damping. This type of dissipation model has been asso-
ciated with the nonlinear damping of aluminum plates subjected to different levels of sound pressure [86, 87]. Accordingly, the damping force density takes
the following form:
fd,disp = cdispv2˙v = (ρabηdisp)v2˙v, (4.5.4)
where cdisp is the displacement-2nd power damping coefficient which is again
assumed to be the product of ambient air density, beam width and an empir- ically determined parameter ηdisp which has dimensions of 1/(length×time).
The generalized damping forces associated with the above-mentioned mod- els should be derived so that Eq. (4.4.4) can be used. In order to determine the modal damping force ¯fd,viscorresponding to the linear viscous damping model,
Eq. (4.5.1), along with Eqs. (4.3.3) and (4.3.4), is substituted for fd(ζ, ˙ζ, g, ˙g)
in Eq. (4.3.5). Applying Galerkin’s method with a 1-mode approximation in Eq. (4.4.1), nondimensionalizing with Eqs. (4.4.3), and dividing through by ρAc, one obtains:
¨¯aI1+ 2ξvisω¯N(I1˙¯a + I31) + . . . = 0, (4.5.5)
where,
ξvis= cvis/(2ωNρAc),
ωN (ωN = 1.8752
q
EI
ρAcL4) is the first bending mode frequency of the cantilever
beam, and I31 is defined in Appendix E. In Eq. (4.5.5), the first term is
recognized as the linear inertial modal force, the ellipsis denote the terms which are not shown for the sake of brevity and the remaining terms represent the damping force ¯fd,vis, i.e.:
¯
The generalized damping force ¯fd,mat is obtained by replacing ξvis in Eq.
(4.5.5) with the expression for ξmat which is given in Eq. (4.5.2). However,
one needs to first determine the maximum bending stress, which varies during flapping, in terms of the transverse displacement v(s, t) [114]:
σmax = κEh = v′′/√ 1 − v′2Eh = aφ ′′+ g′′ p1 − a2φ′2− 2aφ′g′+ g′2Eh, (4.5.7)
where κ is the curvature and h represents half of the beam thickness. Sub- stituting Eq. (4.5.7) into Eq. (4.5.2), expanding the exponential function in a three term Taylor series and evaluating φ′ and φ′′ at the beam root (i.e.,
s = 0) where the maximum bending stress occurs results in (after nondimen- sionalization) the following expression for ξmat:
ξmat = α[1 + Ψ + (1/2)Ψ2+ (1/6)Ψ3], (4.5.8)
with the nondimensional parameter Ψ given as:
Ψ = β Eh Lσy
¯ φ(0)′′¯a
p1 − ¯g′2. (4.5.9)
Thus, the generalized damping force corresponding to the stress-dependent nonlinear damping model can be written as:
¯
fd,mat = 2ξmatω¯N(I1˙¯a + I31). (4.5.10)
which yields:
¯
fd,vel = ¯ηvel( ˙¯a3I32+ 3 ˙¯a2I33+ 3 ˙¯aI34+ I35), (4.5.11)
with,
¯
ηvel = ηvelρab(EI)1/2(ρAc)−3/2.
Similarly, one can determine ¯fd,disp as:
¯
fd,disp = ¯ηdisp( ˙¯a¯a2I32+ ¯a2I33+ 2 ˙¯a¯aI36+ ˙¯aI37+ 2¯aI38+ I39), (4.5.12)
with the dimensionless variable given by,
¯
ηdisp = ηdispρabL4(EIρAc)−1/2.
The integrals I32, I33, . . . , I39 in Eqs. (4.5.11) and (4.5.12) are defined in Ap-
pendix E.
We would like to reiterate that when either the displacement-2nd power
or velocity-3rd power damping models are used in the simulations, the linear
viscous damping force is also included such that the total damping force acting on the beam is ¯fd,vis+ ¯fd,vel (or ¯fd,vis+ ¯fd,disp).