SUMMER RESEARCH 2012

Goal 1: 3D Flapping Wing Simulation

Goal 2: Build nonlinear differential equations in Simulink for a vehicle to mimic flapping wing motion

Goal 3: Form a-bk from the MS thesis and find the eigen values (it should be stable) with a closed loop (add a step input-a step graph should be obtained)

6/6/12

% Analysis of the dynamics of a crank arm connected to a

% mass−spring−damping system.

% Last modified on July 2, 2009

% Author: Stan Baek

lambda = .1; % ratio of crank arm length to connecting arm length

zeta_m = 0.1; % damping ratio of the motor

zeta = 0.1; % damping ration of the slider

theta = linspace(-pi,pi,30); % angular position of the crank arm

omega = linspace(0, 2, 30); % nondimensional angular velocity

[Theta Omega] = meshgrid(theta,omega);

beta = cos(Theta)...

+ lambda*cos(Theta).*sin(Theta)./sqrt(1-lambda^2.*cos(Theta).^2);

u = 2*zeta_m*Omega + beta.*sin(Theta)...

+ Omega.^2.*beta.*(2*zeta*sign(cos(Theta))*cos(Theta).^2 - sin(Theta));

mesh(Theta, Omega, u);

mesh(Theta, Omega, u);

set(gcf,'Position', [474 399 426 250]);

set(gca, 'XTick', [-pi -pi/2 0 pi/2 pi]);

set(gca, 'XTickLabel', {'−pi' '−pi/2' '0' 'pi/2' 'pi'});

set(gca, 'YTick', [0 0.5 1 1.5 2]);

set(gca, 'YTickLabel', {'0' '0.5' '1' '1.5' '2'});

xlabel('theta');

ylabel('Omega');

zlabel({'T'});

axis([-pi pi 0 2 -1 3.2])

omega = [0.8 0.9 1 1.1 1.2];

num_plot = length(omega);

cmap = colormap('jet');

colorOrder = cmap(floor(length(cmap)/num_plot): ...

    floor(length(cmap)/num_plot):length(cmap),:);

% Effect of lambda

omega = 0.9;

zeta_m = 0.1;

zeta_s = 0.1;

Lambda = [0.4 0.3 .2 0.1 0.01];

Theta = linspace(-pi,pi,50);

figure

set(gcf,'Position', [474 399 426 250]);

for i = 1:length(Lambda)

Omega=omega;

lambda = Lambda(i);

mu = lambda*cos(Theta)./sqrt(1-lambda^2*sin(Theta).^2);

gamma = (1+mu).*sin(Theta);

dmu = -lambda*sin(Theta)./sqrt(1-lambda^2*sin(Theta).^2)+...

    lambda^3*cos(Theta).^2.*sin(Theta)./(1-lambda^2*sin(Theta).^2).^(3/2);

T = Omega.*(2*zeta_m + 2*zeta_s*gamma.^2)+...

    Omega.^2.*gamma.*(cos(Theta) + mu.*cos(Theta) + dmu.*sin(Theta))...

        -gamma.*(cos(Theta) + cos(Theta)./mu - sqrt(1/lambda^2 -1) );

    plot(Theta, T, 'Color',colorOrder(i,:));

    hold on;

end

legend( strcat('\lambda = ', num2str(Lambda')));

set(gca, 'XTick', [-pi -pi/2 0 pi/2 pi]);

set(gca, 'XTickLabel', {'−pi' '−pi/2' '0' 'pi/2' 'pi'});

xlabel('theta');

ylabel('T');

Code for Mechanical Power:

% Analysis of the dynamics of a crank arm connected to a

% mass−spring−damping system.

% This script shows the output power of a geared motor with

% respect to constant driving angular speed.

% When the system is excited at the resonant frequency,

% the required output power is going to be minimum..

Theta = linspace(-pi,pi,300);

zeta = [0.1, 0.2, 0.3, 0.4];

zeta_m = 0.0;

lambda = 1/10;

OMEGA = linspace(0.001, 2, 200);

beta = cos(Theta) + ...

lambda*cos(Theta).*sin(Theta)./sqrt(1-lambda^2.*cos(Theta).^2);

Power = zeros(length(zeta), length(OMEGA));

for j = 1:length(zeta)

for i = 1:length(OMEGA)

Omega = OMEGA(i);

T = Omega*2*zeta_m + ...

Omega^2*beta.*(2*zeta(j)*sign(cos(Theta)).*cos(Theta).^2 - ...

sin(Theta)) + beta.*sin(Theta);

Power(j, i) = norm(T*Omega)/sqrt(length(T));

end

end

figure;

plot(OMEGA, Power(1,:), 'b', OMEGA, Power(2,:), 'r');

hold on;

plot(OMEGA, Power(3,:), 'g', OMEGA, Power(4,:), 'k');

xlabel('Omega');

ylabel('MechPower');

axis([0 1.5 0 1]);

6/3/12

5/29/12

/

5/27/12

5/16/12

Video 1: 3D Transient inlet and outlet block simulation

CFD References

[1] Bret K. Stanford and Philip S. Beran, U.S. Air Force Research Laboratory, "Analytical Sensitivity Analysis of an Unsteady Vortex-Lattice Method for Flapping-Wing Optimization", Journal of Aircraft, DOI: 10.2514/1.46259, (2010)

[2] F. Lesage, N. Hamel, X. Huang, W. Yuan, M. Khalid and P. Zdunich, DRDC Valcartier, Institute for Aerospace Research, NRC, Advanced Subsonics, Inc, "Initial investigation on the aerodynamic performance of flapping wings for nano air vehicles," Technical Memorandum, 2007-550 9, (2008)

[3] Daniel Norrison, RMIT University, "Nomad Flutter and Flow Simulation Acceleration for Elastic Wings", PhD, (2009)

[4] Gyung-Jin Park , Hanyang University , "Optimization of the Flapping Wing Systems for a Micro Air Vehicle", AOARD-09-4103, (2010)

[5] Daniel T. Prosser, Rochester Institute of Technology, "Flapping Wing Design for a Dragonfly-Like Micro Air Vehicle" MS, (2011)

[6] Zach Votaw, Zach Gaston, Aron Brezina, Asela Benthara, Haibo Dong, Wright State University, "Design and Analysis of a Bio-Inspired Warping Tail for MAV Applications", AIAA, (2010)

[7] Dominic D. J. Chandar and M. Damodaran, Nanyang Technological University, "Computation of Low Reynolds Number Aerodynamic Characteristics of a Flapping Wing in Free Flight", (2009)

[8] Ria Malhan, Vinod K. Lakshminarayan, James Baeder, Inderjit Chopra, Pierangelo Masarati, Marco Morandinik and Giuseppe Quaranta, University of Maryland, Stanford University, Politecnico di Milano, "CFD-CSD Coupled Aeroelastic Analysis of Flexible Flapping Wings for MAV Applications: Methodology Validation" AIAA 2012-1636, (2012)

[9] Jonathan A. Lichtwardt and David D. Marshall, California Polytechnic State University, "Investigation of the Unsteady Behavior of a Circulation Control Wing Using Computational Fluid Dynamics", AIAA 2011-1041, (2011)

Controls References

[1]Katherine Sarah Shigeoka, The University of Utah, "Velocity and Altitude Control of an Ornithopter Micro Aerial Vehicle", MS, (2007)

[2]Erick Durand, Massachusetts Institute of Technology, "Time Domain Modeling of Unsteady Aerodynamic Forces on a Flapping Airfoil", MS, (1998)

[3] Atilla Yilmaz, "Design and Development of a Flapping Wing Mico Air Vehicle",Design and Development of a Flapping Wing Micro Air Vehicle, MS, (2009)

[4] Ben Parslew, The University of Manchester, "Low Order Modelling of Flapping Wing Aerodynamics for Real-Time Model Based Animation of Flapping Flight", MS, (2005)

[5]Vladislav Gavrilets, Massachusetts Institute of Technology, Avionics Systems Development for Small Unmanned Aircraft", MS, (1998)

[6]John W. Roberts, Massachusetts Institute of Technology, "Motor Learning on a Heaving Plate via Improved-SNR Algorithms", MS, (2009)

[7] David L. Raney and Martin R. Waszak, NASA Langley Research Center, "Biologically Inspired Micro-Flight Research",  SAE 2003-01-3042, (2003)

[8] "Nuri Kundak and Bernard Mettler, University of Minnesota, "Experimental Framework for Evaluating Autonomous Guidance and Control Algorithms for Agile Aerial Vehicles, (2007)

[9] Matt Burkhalter, Aisha Grieme, Ben Harris, Matt Martin, Anid Monsur, Jake Pratt, Kaela Rasmussen and Chris Skaggs, Iowa State University, “FLAPFLAP-1-SP10”, technical report, (2010)

[10] Travis J. Higgs, Air Force Institute of Technology, Air University, "Modeling, Stability, and Control of a Rotatable Tail on a Micro Air Vehicle", MS, (2005)