Helicopter project TTK4115 Linear System Theory

Helicopter project

TTK4115 Linear System Theory

Andreas Hystad & Thomas Stenersen Group 27

September 11, 2012

Contents

1 Mathematical modelling 3

1.1 Part I . . . 3

2 Appendix 5

List of Figures

1 Helicopter model . . . 3

List of Tables

1 List of variables and constants . . . 5

1 Mathematical modelling

1.1 Part I

The first part of the assignment was to develop the equations describing the helicopters motion – i.e. the differential equations for the pitch anglep, the travel angleλand the elevation angle e.

F_s

m g_h

m g_w ^Helicopter

Counterweight

λ e

(a) Helicopter side

F^b

F^f

p

(b) Helicopter front

Figure 1: Helicopter model

We use fig. 1a, 1b and Newtons 2. law for rotation to obtain the differential equations.

Στ =J_pp¨= (F_f −F_b)l_h=K_fl_hV_d m

¨

p= Kflh

J_p V_d=K1V_d (1) Where

K₁= K_fl_h Jp

(2)

Στ =J_tλ¨=−K_fl_asin(p)−K_al_a|λ|˙ λ˙ m

λ¨=−Kfla

J_t sin(p)− Kala

J_t |λ|˙ λ˙ =−K₂sin(p)−k_l|λ|˙ λ˙ (3) Where

K2 = Kflh

J_p , k_l

Στ =Jee¨ = (Ff +Fb)lacos(p) =Kflacos(p)Vs

m

¨

e= K_fla

Je

cos(p)V_s=K₃cos(p)V_s (4) Where

K₃ = Kfla

J_e (5)

Vs=Vf+Vb (6)

V_d=V_f−V_b (7)

V_f = 1

2(V_s+V_d) (8)

V_b = 1

2(V_s−V_d) (9)

¨

p=K₁V_d (10)

λ¨=−K₂sin(p)−kl|λ|˙ λ˙ (11)

¨

e=K₃V_s−K₄ (12)

Where

K₁ = l_hK_f Jp

= K₂ = laKf

J_t = K₃ = l_aK_f

Je

= K4 = l_am_gg

J = 42

2 Part II – Mono-variable control

In this part of the assignment, we wanted to implement mono-variable control for the elevatione, pitchp, and travel rate λ.

2.1 Problem 1

We want to use a PD controller to control the pitch anglep.

V_d=K_pp(p_c−p)−K_pdp˙ (13) (14) We apply the Laplace transform to (??) and (1) (with zero initial conditions):

s²p K1

=K_pp(p_c−p)−sK_pdp m

p(s²+sK1K_pd+K1Kpp) =pcK1Kpp

m p

p_c(s) = K1Kpp

s²+sK₁K_pd+K₁K_pp (15) As a starting point we set the regulator parameters such that the regulator is as fast as possible without oscillations – i.e. we choose the parameters such thatζ = 1. We choose K_pp= 1 and findK_pd.

ζ =

√K₁K_pd 2p

K_pp m

1 =

√K₁K_pd 2√

1 m

Kpd= 2

√K1

lol

3 Appendix

p – Pitch angle λ – Travel angle e – Elevation angle Vf – Voltage, front motor V_b – Voltage, rear motor V_d – Voltage difference,V_f−V_b V_s – Voltage sum, V_f +V_b Kpp – Controller gain K_pd – Controller gain K_rp – Controller gain pc – Pitch reference λ˙_c – Travel rate reference e_c – Elevation reference

l_a – Distance from axis of elevation to helicopter body [m]

lh – Distance from pitch axis to motor [m]

K_f – Motor force constant [N/V]

J_e – Moment of inertia about elevation axis [kg m²] Jt – Moment of inertia about travel axis [kg m²] J_p – Moment of inertia about pitch axis [kg m²] m_h – Helicopter body mass [kg]

mw – Counterweight mass [kg]

mg – Net mass of helicopter and counterweight [kg]

K_p – Force needed to lift the helicopter body from the table top (g·m_g) [N]

Table 1: List of variables and constants