Diagnosis and Control of Nonlinear Oscillations of a Fluttering Plate

Abstract

This chapter focuses on both diagnosing and controlling the nonlinear dynamic responses of a fluttering plate excited by a high-velocity air flow. Six modes of the motion are considered for obtaining the numerical solutions of the system, and the modes are used to investigate the nonlinear dynamic responses of the fluttering. Due to the different characteristics of the diagnosing methods for nonlinear systems, Lyapunov Exponent method is employed to detect the system motion of each mode, while the Periodicity Ratio method is utilized to detect the behavior of entire system motion subjected to non-periodic excitations generated by the air flow. A newly developed control strategy, modified FSMC method, is applied to control the nonlinear oscillatory responses of the system. The approaches presented in this chapter have research and engineering application significances in the fields of aerodynamics, nonlinear dynamics, aircraft design, and design of space vehicles.

Key Symbols

$D$

Plate stiffness

$E$

Modulus of elasticity

$h$

Plate thickness

$K$

Spring constant

$L$

Panel length

M

Mach number

$m$

Mode number

${N_x}$

In-plane force

$N_x^{(a) }$

Applied in-plane force

$p-{p_{\infty }}$

Aerodynamic pressure

$\varDelta p$

Static pressure differential across the panel

$P$

${{{\varDelta p{l^4}}} \left/ {Dh } \right.}$

${R_x}$

${{{N_x^{(a) }{L^2}}} \left/ {D} \right.}$

$r$

Mode number

$s$

Mode number

$t$

Time

${U_{\infty }}$

Flow velocity

$W$

${w \left/ {h} \right.}$

$w$

Plate deflection

$\alpha$

Spring stiffness parameter

$\beta$

${{({M^2}-1)}^{{{1 \left/ {2} \right.}}}}$

$\lambda$

${{{2q{a^3}}} \left/ {{\beta D}} \right.}$

$\mu$

${{{\rho L}} \left/ {{{\rho_m}h}} \right.}$

$\nu$

Poisson’s ratio

$\rho$

Air density

${\rho_m}$

Plate density

$\tau$

$t{{({D / {{{\rho_m}h{l^4}}} })}^{{{1 \left/ {2} \right.}}}}$

${a_{s1 }}$

The displacement corresponding to the $s\mathrm{th}$ mode

${a_{s2 }}$

The velocity corresponding to the $s\mathrm{th}$ mode

${x_{s1 }}$

The displacement of the reference signal of the $s\mathrm{th}$ mode

${x_{s2 }}$

The velocity of the reference signal corresponding to the $s\mathrm{th}$ mode

$\mathrm{a}$

The column vector of the velocity and acceleration of the $s$ modes

${f_s}\left( {\mathrm{a},\tau } \right)$

The expression of the acceleration corresponding to the $s\mathrm{th}$ modes

${g_s}\left( {\mathrm{a},\tau } \right)$

The expression of the reference signal acceleration of the $s\mathrm{th}$ mode

${d_s}\left( {\mathrm{a},\tau } \right)$

The uncertain external disturbance corresponding to the $s\mathrm{th}$ mode

${u_s}$

The control input corresponding to the $s\mathrm{th}$ mode

$R$

Real number

$ue{q_s}$

The equivalent control input corresponding to the $s\mathrm{th}$ mode

${\eta_s}$

A positive real number

$k{f_s}$

The normalization factor of $\mathrm{a}$ corresponding to the $s\mathrm{th}$ mode

$n$

The number of points in Poincare map

$NPP$

The number of periodically overlapped points in Poincare map

$Q(.),P(.)$

The step functions

$\gamma$

The periodicity ratio

${f_s}(\mathrm{a},\tau )$

The expression of the acceleration corresponding to the $s\mathrm{th}$ modes

$J$

The Jacobian matrix

$LE(\hat{\lambda})$

Lyapunov Exponents