User:Egm6322.s12.team2.steele.m2/Mtg14

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

[[media:Pea1.f11.mtg14.djvu|Page 14-0]]
Plan: Application (engineering): Motion of a particle in the air (e.g., soccer ball, projectile) Application: Control engineering – Linear systems Example: Coupled pendulums Note: Problem numbering in reports: Use the same numbering as in lecture notes, i.e., R*m.n or Rm.n (put the star when there is a star to distinguish starred problems from non-starred problems). Link to the corresponding lecture notes in all problem statements All figures: Category, license End of reports: Assignment table, signatures, tasks done. See F11 Team 4R1. \, in latex code

[[media:Pea1.f11.mtg14.djvu|Page 14-1]]
=Application (Engineering): Motion of a Particle in the Air (e.g., soccer ball, projectile)= See graph on p. 14-1. $$v := \parallel \boldsymbol v \parallel$$ Vector	norm = magnitude $$k, n \in \mathbb{R}$$	constants M	mass of particle=m G	acceleration of gravity=g

[[media:Pea1.f11.mtg14.djvu|Page 14-2]]
R*3.5: 1.	Derive the equations of motion

(1)-(2) form a System of Coupled N1-ODEs = SC-N1-ODEs (need numerical methods) 2. Particular case k=0 : Verify that y(x) is parabola. 3. Consider the case $$k \ne 0$$ and $$v_{x0} = 0$$

[[media:Pea1.f11.mtg14.djvu|Page 14-3]]
3.1 Is (1) either exact or can be made exact using IFM? Find $$v_y(t)$$ and $$y(t)$$ for m constant. (Note graph on p. 14-3). 3.2 Find $$v_y(t)$$ and  for “John Bernoulli posed other trajectory problems for the English, his particular  noire being Newton. Since the English and the Continentals were already at odds, the challenges were marked by bitterness and hostility. John Bernoulli then solved the problem of finding the motion of a projectile in a medium whose resistance is proportional to any power of the velocity.” Kline 1972, Mathematical thoughts…, v.2, p.475

[[media:Pea1.f11.mtg14.djvu|Page 14-4]]
=Application: Control Engineering – Linear Systems= Linear time-variant systems: $$\mathbf{x} : [t_0, +\infty) \rightarrow \mathbb R^{n \times 1}$$ 	state of system $$t \ \mapsto \ \mathbf x(t)$$ $$\mathbf{A}:[t_0, +\infty) \rightarrow \mathbb R^{n \times n}$$ $$\mathbf{B} : [t_0, +\infty) \rightarrow \mathbb R^{n \times m}$$ $$\mathbf{u} : [t_0 , +\infty) \rightarrow \mathbb R^{m \times 1}$$ Linear time-invariant systems: $$\mathbf{A}, \mathbf{B}$$ constant matrices

[[media:Pea1.f11.mtg14.djvu|Page 14-5]]
Example: Coupled pendulums See image with spring and pendulums.

c.f Bryson & Ho 1975, Applied optimal control, p.164.

[[media:Pea1.f11.mtg14.djvu|Page 14-6]]
R*3.6: 1.	Derive the equations of motion (1)-(2) p.14-5. 2.	Write (1)-(2) p.14-5 in matrix form (1) p.14-4 with

Col matrix		T=transpose 		row matrix

Col matrix Find $$\mathbf{A} \in \mathbb{R}^{4 \times 4}, \mathbf{B} \in \mathbb{R}^{4 \times 2}$$