User:Egm6322.s12.team2.steele.m2/Mtg16

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg16.djvu|Page 16-0]]
Plan: SC-L1-ODE-CC: p.15-5 cont’d Example: Roll control of rocket Time constant 2nd-order ODEs (cont’d from p.13-8) 1st exactness condition for N2-ODEs 2nd exactness condition for N2-ODEs Application Solution for exact N2-ODE

[[media:Pea1.f11.mtg16.djvu|Page 16-1]]
SC-L1-ODE-CC: [[media:Pea1.f11.mtg14.djvu|Eqn(1) p.14-4:]] $$\mathbf{\dot X}(t) = \mathbf A(t) \mathbf x(t) + \mathbf B(t) \mathbf u(t)$$ R*3.11: use (2)-(3) p.15-4 [[media:Pea1.f11.mtg15.djvu|Eqns(2)-(3) p.15-4]] together with [[media:Pea1.f11.mtg14.djvu|Eqn(1) p.14-4:]] to show [[media:Pea1.f11.mtg15.djvu|Eqn(3) p.15-3]].

=Example: Roll Control of Rocket= Prevent rolling by activating the rocket ailerons $$\delta$$ = aileron deflection (angle) $$\phi$$ = roll angle $$\omega$$ = roll angular velocity Q = aileron effectiveness proportional to inverse of rocket roll intertia $$\tau$$ = roll time constant. See image of rocket in vertical position with air flow, force, and aileron.

[[media:Pea1.f11.mtg16.djvu|Page 16-2]]
Bryson & Ho 1975 p.169 Note: Time constant $$\tau$$

The smaller the time constant $$\tau$$ is, the larger the ratio $$\frac{1}{\tau}$$ is, and the faster $$\omega$$ goes down to zero.

[[media:Pea1.f11.mtg16.djvu|Page 16-3]]
R*3.12: Put (1)-(3) p.16-2 in the form of SC-L1-ODE-CC as in [[media:Pea1.f11.mtg14.djvu|Eqn(1) p.14-4:]] with A, B being constant matrices.

=2nd-order ODEs (cont’d from p.13-8)=

Number (1) is exact if there exists a function $$\phi(y\prime, y, x) = k_1$$

Use the same definition as in [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-3:]]

P(x) := y’(x)

[[media:Pea1.f11.mtg16.djvu|Page 16-4]]
Use (1) and (2) p.7-3 to rewrite (2) p.16-3 as

Number (2) is the required particular form of exact N2-ODEs, and is the equivalent counterpart of the particular form of exact N1-ODEs in [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-6]].

[[media:Pea1.f11.mtg16.djvu|Page 16-5]]
1st exactness condition for N2-ODEs: Thus the necessary condition for an N2-ODE to be exact is that it has the particular form (2) p.16-4. In other words, if an N2-ODE cannot be put in the particular form (2) p.16-4, then it cannot be exact. Cf. p.8-6 2nd exactness condition for N2-ODEs

If both 1st and 2nd exactness conditions are satisfied, i.e., (2) p.16-4 and (1)-(2), then from (4) p.16-4, we have

[[media:Pea1.f11.mtg16.djvu|Page 16-6]]
Where h(x,y) is a function of integration (not necessarily a constant) to be determined using (3) p.16-4. Application: Consider the N2-ODE Comparing (1) to (2) p.16-4, it follows that

Thus (1) satisfies the 1st exactness condition. R*3.13: 2nd exactness condition 1.	Derive (2) p.16-5, i.e., the 2nd relation in the 2nd exactness condition, by differentiating the definition of g(x,y,y’) in (3) p.16-4 with respect to p := y’ defined in [[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-3]]. (See F09 p.11-2)

[[media:Pea1.f11.mtg16.djvu|Page 16-7]]
2.	Derive (1) p.16-5, i.e, the 1st relation in the 2nd exactness condition. (See F09 p.11-3, Mtg 17) 3.	Verify that (1) p.16-6 satisfies the 2nd exactness condition

Solution for (1) p.16-6 (exact N2-ODE)

1st integration constant

Number (3) p.16-6:

(2) p.16-6