Timothy Burkhard - Phil Barat - John Gyurics ME-114 Final Project Timothy Burkhard - Phil Barat - John Gyurics

Background: Timothy Burhard and John Gyurics have recently finished designing a UAV model aircraft for the AUVSI competition. The design phase will soon be followed by building the composite aircraft, however the design will first be validated performing simulation analysis and also by building a much cheaper balsa version of it. Phil Barat is a Kite-Boarding enthusiast, who spends every waking moment possible out at the lake being tossed around by the forces of the wind, and is also very interested in the aero-dynamics involved in flight. For our Vibration and Controls project we decided to merge our goals and interests, and joined forces to develop a study of the longitudinal stability of an aircraft.

Objective: Perform a longitudinal Dynamic stability study on an aircraft, by working with the transfer functions that relates speed to the elevator in particular. Employ Matlab as our guiding tool to derive the Step-Response, Impulse Response, Root Locus, Bode Diagram, and the Sisotool. In “Airplane Flight Dynamics and Automatic Flight Controls”, Jan Roskam did a superb job at deriving the transfer functions for different style aircraft, and also demonstrating the correlation between a mass-spring-dampener system and a general aircraft dynamics system. Some of our work will utilize the resources made available by Jan, and further investigate and validate the work.

Along The Way: We started out by first analyzing a simple circuit, and generating the Root Locus Diagram, Body Diagram, Step Response Diagram and Impulse Response Diagram. We later turned our attention to Simulink, and performed an overall aircraft control loop analysis. Finally we tackled the Longitudinal Dynamic Stability Study on an aircraft, by performing the Step-Response, Impulse Response, Root Locus, Bode Diagram, and the development of the Sisotool.

Electronic Circuit

Bond Graph

MATLAB OUTPUT From the circuit, we enter the parameters of the two inductors, the resistor, and the capacitor in the campgsym.m file These lines need to be unsuppressed as well, to look like what is shown below: I2 = 2 ; R4 = 1 ; C6 = 1 ; I7 = 3 ;

fprintf ('\n Output # 1 is e7' ) C(1,:) = [-1/C6,-1/I7*R4,1/I2*R4]; MATLAB OUTPUT Then, the e7 (the efforts link which represents voltage) bond graph link must be chosen to produce a transfer function, state-space form, and related figures. It will need to look like this: % e7=P2/I2*R4-P7/I7*R4-Q6/C6 fprintf ('\n Output # 1 is e7' ) C(1,:) = [-1/C6,-1/I7*R4,1/I2*R4]; D(1,:) = [0];

MATLAB OUTPUT State-space form: B MATRIX A MATRIX [0] [ 0 1/3 0 ] [ ] [ 0 1/3 0 ] [ ] [-1 -1/3 1/2 ] [ 0 1/3 -1/2] C MATRIX [-1 -1/3 1/2] B MATRIX [0] [ ] [1] D MATRIX

Characteristic Polynomial MATLAB OUTPUT Characteristic Polynomial 3 2 s + 5/6 s + 1/3 s + 1/6 Transfer Functions Matrix H ... 2 s 3 --------------------- 3 2 6 s + 5 s + 2 s + 1

BODE Diagram and Root Locus

Step Response

Impulse Response

Impulse Response We can also use the SISOTOOL feature in MATLAB to design the control for this circuit. The following can be entered in the MATLAB command screen: num =[1 0 0] num = 1 0 0 >> den=[6 5 2 1] den = 6 5 2 1 >> g=tf(num,den) Transfer function: s^2 ----------------------- 6 s^3 + 5 s^2 + 2 s + 1 >> sisotool(g)

Impulse Response By moving the cursor over the plot, and following the directions at the bottom dialogue box, you can adjust the loop gain to design a desired system

Final Design Front View

Final Design Side View

Final Design Top View

Final Design Isometric View

Explanation of Static Stability A statically unstable body will tend to accelerate away from its equilibrium position when disturbed. A neutrally stable body will not accelerate toward or away from any position when disturbed; i.e., there is no particular equilibrium position. A positively stable body will accelerate toward its equilibrium position when disturbed.

Explanation of Aircraft Axes Movement about the lateral axis drawn from the nose to the tail of the aircraft is called roll. Movement about the longitudinal axis drawn from wingtip to wingtip is called pitch. Movement about the directional axis extending vertically through the fuselage is called yaw.

Lateral Stability dihedral and wing placement. Lateral (roll) stability is determined by: dihedral and wing placement. Center of Lift 3° .75”

Longitudinal Stability Longitudinal (pitch) stability is determined by the longitudinal stability margin (LSM) Center of Lift Center of Gravity LSM = 1”

Aerodynamic Requirements: “Positive static stability about all three axes” Directional Stability Direction (yaw) stability is determined by the directional stability margin (DSM) Center of Lateral Area Center of Gravity DSM = 2”

Airframe Design: Wings The Concept of an Airfoil An airfoil profile is a 2-D shape that is designed to behave in a particular way when subjected to external flow. Airfoils are usually designed to provide a desired coefficient of lift (CL) over a range of angles of attack (AoA) and Reynold’s numbers (Re) while minimizing drag. The characteristic behavior of an airfoil is empirically derived from wind tunnel testing and is plotted in Polar Curves.

Airframe Design: Wings Airfoil Selection At a cruising speed of 50 mph the UAV must generate a CL of .202 in order to stay aloft. At a landing speed of 18 mph the UAV must generate a CL of 1.26 in order to stay aloft. The NACA 3412 airfoil provides these lift coefficients at low drag.

Airframe Design: Fuselage Drag Reduction Top and side projections of the fuselage are NACA low-drag bodies. A front projection of the fuselage is an ellipse.

General Stability Source: Airplane Flight Dynamics and Automated Flight Controls, Jan Roskam

Transfer Functions for all three axes Source: Airplane Flight Dynamics and Automated Flight Controls, Jan Roskam

Overall Aircraft Control Loop

Airspeed Control by Elevator Deflection PID Loop

Roll Control by Aileron Deflection PI Loop

Altitude Step Response Airspeed Control = 20 knots

Altitude Step Response Airspeed Control = 25 knots

Dynamic Stability Criteria A linear system is stable if and only if the real parts of the roots of the characteristic equation of the system are negative. A linear system is convergent (stable) if the roots of the characteristic equation of the system are real and negative. A linear system is divergent (unstable) if the roots of the characteristic equation of the system are real and positive. A linear system is oscillatory convergent (stable) if the real parts of the roots characteristic equation of the system are negative. A linear system is oscillatory divergent (unstable) if the real parts of the roots of the characteristic equation of the system are positive. A linear system is neutral stable if one of the roots of the characteristic equation of the system is zero or if the real parts of the roots of the characteristic equation of the system is zero.

Transfer Function Matrix

Speed to Elevator transfer function

Step Response

Impulse Response

Root Locus

Bode Diagram

SISO

State - Space

References 1. Airplane Flight Dynamics and Automated Flight Controls, Jan Roskam 2. Aerosim Flight Control Blockset Help Files

Questions?