bài 4 - longitudinal motion (stick fixed) (2008)

Cơ học bay 2 – Longitudinal Motion (Stick Fixed)

Ngô Khánh Hiếu

1


Ngô Khánh Hiếu

2

Introduction (1/2)

- Lanchester had discovered that all flight vehicles possess certain natural frequencies or motions when disturbed from their equilibrium flight.

- The basic flight dynamics modes of an aircraft are: the Phugoid mode, the Short-period mode, the Dutch roll mode, and the Spiral divergence mode.

- The longitudinal motion consists of two distinct oscillations: the long-period oscillation (Phugoid mode), and the short-period oscillation (Short-period mode).

- Oscillating motions can be described by two parameters: the period of time required for one complete oscillation; the time required to damp to half-amplitude or time to double the amplitude.

- The phugoid or long-period is the one in which there is a large-amplitude variation of airspeed, pitch and flight path angle, and altitude but almost no angle-of-attack variation.

It is really a slow interchange of kinetic energy (velocity) and potential energy (altitude) about some equilibrium energy level. Typically, the period is 20-60 seconds.

The period is so long that the pilot usually correct for this motion without being aware that the oscillation even exists.


Ngô Khánh Hiếu

3

Introduction (2/2)

- The short-period mode is very fast, usually heavily damped, oscillation with a period of a few seconds. The motion is a rapid pitching of the aircraft about the center of gravity.

The oscillation is essentially an angle of attack variation. The time to damp the amplitude to one-half of its value is usually on the order of 1 second.

In this chapter, we examine the longitudinal motion of an airplane disturbed from its equilibrium state.


Ngô Khánh Hiếu

4

Second-order differential equations (1/7)

- Many physical systems can be modeled by second-order differential equations (ex., control servomotors, special cases of aircraft dynamics, etc.).

- Examine the motion of a mechanical system composed of a mass, a spring, and a damping device: the spring provides a linear restoring force that is proportional to the extension of the spring and the damping device provides a damping force that is proportional to the velocity of the mass.

2

2

d x c dx k 1x F(t)

m dt m mdt (1)

wherem: the mass,

c: the damping coefficient,

k: the spring constant,

F(t): the forcing function.

(1) is a nonhomogeneous, second-order differential equation with constant coefficients.

The general solution of (1) is the sum of the homogeneous (xcf(t)) and particular solution (xp(t)) of (1).


Ngô Khánh Hiếu

5


- The homogeneous solution of (1) is the solution of the equation (1) when the right-hand side of this equation is zero.

02

2

d x c dx kx

m dt mdt (2)

(3) is the characteristic equation of the equation (2). The roots of this equation are called the characteristic roots or eigenvalues of the system.

(2) is called a second-order homogeneous linear differential equation with constant coefficients.

- The equation (2) is solved by first letting “x = A.et”, and then finding two linearly independent solution (say x1(t) and x2(t)) such that x = C1x1(t) + C2x2(t) for constants C1 and C2.

tt

2 t

x A ex Ae

x A e

2 0c k

m m (3)

2

1,2c c k

2m 2m m


Ngô Khánh Hiếu

6


- This type of motion is referred to as an overdamped motion (it means that the motion will die out exponentially with time).

Case 1:

- This type of motion is referred to as a damped sinusoid having a natural frequency :

2

1,2c c k

2m 2m m

c k the roots are 2 distinct negative real roots2m m

1 2λ t λ tcf 1 2x (t) C .e C .e

Case 2: c k the roots are 2 distinct complex roots2m m

2k c

ω m 2m

2

1,2c k c

λ i 2m m 2m

c- t2m

cf 1 2x (t) e C .cos ωt C .sin ωt

where


Ngô Khánh Hiếu

7

Proofs:


Ngô Khánh Hiếu

8


- This particular motion is referred to as the critically damped motion. It represents the boundary between the overdamped exponential motion and the damped sinusoidal motion.

Case 3:

- The damping constant for this case is called the critically damping constant:

c k single real root2m m

λtcf 1 2x (t) C C t .e 1,2

cλ λ

2m

crc 2 km

- The damping constant for oscillatory motion can be specified in terms of the critically damping:

crc c where : the damping ratio

- For the undamped oscillation (c = 0), the natural frequency is called the undamped natural frequency (n):

nk

ω m


Ngô Khánh Hiếu

9


- The particular solution (xp(t)) is a solution that when substituted into the left-hand side of the differential equation yields the nonhomogeneous or right-hand side of the differential equation.

- The method for finding a particular integral is rather crude (it involves trial and error and educated guesswork). Hence, we try solutions which are of the same general form as the F(t) on the right-hand side of the differential equation.

- The general solution of a second order linear nonhomogeneous equation is the sum of it particular integral and the complementary function:

p cfx(t) x (t) x (t)

- When solving the ODE, if the nonhomogeneous term F(t) appears in the complementary function use as a trial particular integral t times what would otherwise be used.


Ngô Khánh Hiếu

10


- Since both the damping ratio and the undamped natural frequency are specified as functions of the system physical constants (ex., k, c, m), we can rewrite the differential equation in terms of the damping ratio and the undamped natural frequency:

22

n n2

d x dx2 ω ω x (t)

dtdtf (4)

(4) is the standard form of a second-order differential equation with constant coefficients.

- The equation (4) could be developed using any one of an almost limitless number of physical systems. For example, a torsional spring-mass-damper equation is given by:

n2

2n n cr2

cr

kω Id θ dθ

2 ω ω θ (t) where c 2 kIdtdt

cc

(c, k, and I are the torsional damping coefficient, torsional spring constant, and moment of inertia)

f


Ngô Khánh Hiếu

11



Ngô Khánh Hiếu

12

Pure pitching motion (1/7)

- It is the case in which the airplane’s center of gravity is constrained to move in a straight line at a constant speed, but the aircraft is free to pitch about its center of gravity.

- The equation governing this motion is obtained from Newton’s second law. And for this restricted motion, the variables are the angle of attack, pitch angle, the time rate of change of these variables, and the elevator angle. But,

e

θ α M fn Δα, Δα, q, δ

θ q

- The Taylor series expansion:

ee

M M M MM α α q δ

α q δα

eq α δ eα

α M M α M α M δ

(5)

eq y y α y δ yα e

M M M Mwhere M I , M I , M I , M I

q α δα


Ngô Khánh Hiếu

13


- Eq. (5) is a non-homogeneous second-order differential equation, having constant coefficients.

2q α

αλ M M λ M 0

(6)

- Eq. (5) is similar to a torsional spring-mass-damper system with a forcing function: the static stability of the airplane can be thought of as the equivalent of an aerodynamic spring, while the aerodynamic damping terms are similar to a torsional damping device.

- Eq. (6) is the characteristic equation of Eq. (5). This equation can be compared with the standard equation of a second-order system:

n α

2 2n n q

α

α

ω M

λ 2 ω λ ω 0 where M M

2 M

(7)

- Eq. (7) is the expression of Eq. (6) in terms of the standard equation of a second-order system with is the damping ratio and n is the undamped natural frequency.


Ngô Khánh Hiếu

14


- The roots of Eq. (7) are: 1,2

1,2

2n n

2n n

λ ω ω 1

λ ω iω 1

(8)

1 2λ t λ t

2 2trim

α t 1 11 1 e 1 e

α 2 21 1

Case 1: > 1 1,2 are two distinct roots: overdamped motion

Case 2: 0 < < 1 1,2 are two distinct complex roots: damped sinusoidal motion

(9) nω t

2 -1 2n2

trim

α t e1 sin 1 ω t where tan 1

α 1

Case 3: = 1 1,2 are identical roots: critically damped motion

(10) nω t

ntrim

α t1 1 ω t e

α


Ngô Khánh Hiếu

15


trim

α

α

nω t


Ngô Khánh Hiếu

16

Proofs (1/2):


Ngô Khánh Hiếu

17

Proofs (2/2):


Ngô Khánh Hiếu

18


- The roots of the characteristic equation tell us what type of response our airplane will have.

- The period of the oscillation is related to the imaginary part of the root as follows:

2n

2πPeriod = where ω ω 1

ω

The rate of growth or decay of the oscillation is determined by the sign of the real part of the complex root. A negative real part produces decaying oscillation, whereas a positive real part causes the motion to grow.

- The expression for the time for doubling or halving of the amplitude is:

double halven

0.693t or t =

ω

and the number of cycles for doubling or halving the amplitude is:

double or halven

ωN(cycles) = 0.110

ω


Ngô Khánh Hiếu

19

n x [Period]

nω t2 -1 2

n2trim

α t e1 sin 1 ω t where tan 1

α 1

2n

n halve halve2n

sin 1 ω t 0.693ln(0.5) 0.693 ω t ln t

ω1

2n

2

sin 1 ω tln

1

Proofs:

= 0.1

= 0.115

= 0.9


Ngô Khánh Hiếu

20


Example problem 1:

A flat plate lifting surface is mounted on a hollow slender rod as illustrated in the figure. The slender rod is supported in the wind tunnel by a transverse rod. A low friction bearing is used so that the slender rod-flat plate system can rotate freely in pitch.

To have the center of gravity located at the pivot point ballast is placed inside the slender tube forward of the pivot.

a. Establish the equation of motion of the rod-flat plate system in terms of pitch angle ().

b. Estimate the damping ratio (), the undamped natural frequency (n), and the damped natural frequency () of the tube-flat plate assembly.

1 slug = 14.5939 kg; 1 Ibs = 0.4536 kg; 1 ft = 0.3048 m;

o = 1.225 kg/m3; uo = 25 ft/s

The following assumptions are made in the analysis:- Neglect the mass of the slender rod.- Neglect the contribution of pitching moment

contribution due to the slender rod.- Neglect the mechanical friction of the bearings.


Ngô Khánh Hiếu

21

Solution (1/3):

a...

yM = I θ

M MM = α q

α q

q y

q α

α y

MM I

qθ M θ M θ = 0 where

MM I

α

- The contribution due to is not included because this effect is due primarily to the

interaction of the wing wake on an aft surface.

- Because the center of gravity is constrained the angle of attack, , and the pitch angle, , are the same; the pitching rate, q, is the same as

.

This equation can be expressed in terms of the system damping ratio, and the system’s undamped natural frequency as follows:

n α2

n n q

α

ω M

θ + 2 ω θ + ω θ = 0 where M

2 M

The damped natural frequency of the system:

2q2

n α

Mω = ω 1 M

2


Ngô Khánh Hiếu

22

Solution (2/3):

b. The derivative M can be estimated from the following expression:

Lαα Lα α y

y

M lC αQSM = l. Lift l.C .α.Q.S M = I Iα

The derivative Mq can be estimated from the following equation:

q Lα q y Lα yoo

ql M lM = l.C . .Q.S M = I lC QS I

q uu

lα

Lαlα

CC 4.71 (/rad)

1+C AR

2y y_plate y_ballastI I I 0.01899 (kg.m )

2 2o

1Q u 35.5644 (N/m )

2

2α

q

M 38.3708 (/s )

M 1.4121 (/s)

n

halve

halve

0.114

ω 6.1944 (rad/s)

ω 6.1541 (rad/s)

Period = 1.021 (s)

t 0.98 (s)

N 0.96 (cycle)


Ngô Khánh Hiếu

23

Solution (3/3):

b.

o

θ(t)

θ

t (s)


Ngô Khánh Hiếu

24


Example problem 2:

The dimensional longitudinal stability derivatives of De Havilland Canada airplane estimated from the formulas in chapter 3 are:

a. In case of “pure pitching motion”, establish the equation of motion of De Havilland Canada airplane in terms of the change in angle of attack ().

b. Estimate the damping ratio (), and the undamped natural frequency (n) of this airplane for “pure pitching motion”.

e

2α

α

q

δ

M 0.2805 (/s )

M 0.5412 (/s)

M 1.5817 (/s)

M 4.7515 (cycle)


Ngô Khánh Hiếu

25

a. For “pure pitching motion” the equation of De Havilland Canada airplane in terms of the change in angle of attack is:

eq α δ eα

α M M α M α M δ

eα 2.1229 α + 0.2805 α 4.7515 δ

b. The damping ratio:

q α

α

M M2.0042 overdamped motion

2 M

The undamped natural frequency:

n αω M 0.5296 (rad/s)

Solution (1/2):


Ngô Khánh Hiếu

26

Solution (2/2):

trim

α(t)

α

t (s)


Ngô Khánh Hiếu

27

Stick fixed longitudinal motion (1/7)

- The longitudinal motion of an airplane without control input (controls fixed) disturbed from its equilibrium flight conditions is characterized by two oscillatory modes of motion: one mode is lightly damped and has a long period (the long-period or phugoid mode); the second is heavily damped and has a very short period (the short-period mode).


Ngô Khánh Hiếu

28


- The linearized longitudinal equations developed in chapter 3 can be written as a set of first-order differential equations, called the state-space or state-variable equations and represented mathematically as:

x: the state vector

η: the control vectorx Ax + Bη where

A, B: matrices contain the aircraft's

dimensional stability derivatives

- The linearized longitudinal set of equations:

e Tu w o δ e δ Tu X u + X w g.cosθ θ + X δ X δ

e Tu w o o δ e δ Tw Z u + Z w + u q g.sinθ θ + Z δ Z δ

e δ T δe Tu u w w q o δ e δ T

w w w w wq M M Z u + M M Z w + M M u q + M M Z δ M M Z δ

θ q

(11)


Ngô Khánh Hiếu

29


- Rewriting the equation in the state-space form yields:

u

wx =

q

θ

e

T

δη =

δ

e T

e T

e e T T

δ δ

δ δ

δ δ δ δw w

X X

Z ZB =

M M Z M M Z

0 0

u w

u w o

u u w w q ow w w

X X 0 g

Z Z u 0A = M M Z M M Z M M u 0

0

0 1 0

Note that the force derivatives Zq and usually are neglected because they contribute very little to the aircraft response.

wZ


Ngô Khánh Hiếu

30


- The homogeneous solution to Eq. (11) can be obtained by assuming a solution of the form:

rλ trx = x e (12)

- Substituting Eq. (12) into Eq. (11) yields:

r rλ I A x 0 (13)

where I is the identity matrix

1 0 0 0

0 1 0 0I =

0 0 1 0

0 0 0 1

- The roots r of equation: rλ I A 0

are called the characteristics roots or eigenvalues.


Ngô Khánh Hiếu

31


- The eigenvectors for the system can be determined once the eigenvalues are known from the following equation:

(14)j ijλ I A P 0

where Pij is the eigenvector corresponding to the jth eigenvalue.

- The set of equations making up Eq. (14) is linearly dependent and homogeneous; therefore, the eigenvectors cannot be unique.

- One relatively straightforward technique for finding the eigenvectors is the following:


Ngô Khánh Hiếu

32


- Because the long-period mode is characterized by changes in pitch attitude , altitude, and velocity at a nearly constant angle of attack, an approximation to the long-period mode can be obtained by neglecting the pitching moment equation and assuming that the change in angle of attack is 0.

(15)o

wα α = 0 w = 0

u

bài 4 - longitudinal motion (stick fixed) (2008)

Documents

period of time

characteristic equation

homogeneous solution

equilibrium flight

spring constant

complete oscillation

spiral divergence mode

dutch roll mode