Problem 2 (20 points) A sailing ship of mass, m , is initially at rest, i.e. v(0) = 0 . At time t = 0, a strong wind arises of magnitude Vo = 10m/s
Vo v(t) m
Assume that the force of the wind on the sails in the direction of travel is given by Fw (t) = Bw
Table 1: Table of Laplace Transforms Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 f (t) (t) us (t) t tn eat teat
1 n1 eat (n1)! t
F (s) 1 1
s s2 sn+1 1 (s+a) 1 (s+a)2 1 (s+a)n a s(s+a) ba (s+a)(s+b) (ba)s (s+a)(s+b) a s2 +a2 s s2 +a2 s+a (s+a)2 +b2 b
Troubles at the Origin: Consistent Usage and
Properties of the Unilateral Laplace Transform
Kent H. Lundberg, Haynes R. Miller, and David L. Trumper Massachusetts Institute of Technology
The Laplace transform is a standard tool associated with the analys
2.003 Fall 2003
Complex Exponentials
Complex Numbers
Complex numbers have both real and imaginary components. A complex number r may be expressed in Cartesian or Polar forms: r = a + jb (cartesian) = re (polar)
The following relationships convert from
11/8/04 3:14 PM % % % %
/Users/davidtrumper/ActiveAl_./complicatedbode.m
1 of 1
define systems individually and then combine via
multiplications (probably should use state space for higher
accuracy in real computations
D L Trumper 11/8/04
= = = = tf(5
Standup and Stabilization of the Inverted Pendulum
by Andrew K. Stimac Submitted to the Department of Mechanical Engineering in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science at the Massachusetts Institute of Technology June
Bode plot of Underdamped Second Order System
w
2 n
( s 2 + 2 x wn s + w2 ) n
1/(2x)
Magnitude
1 2
wn
0
o
Approximation of phase plot as a staircase Actual phase plot
Phase
o
90
180
o
10
1
wn
10
2
10
3
Frequency (rad/sec)
Magnitude and Frequency have be
Problem
A  RLC
circuit
analysis
R vi
+ 
L
+
C
vo

1. Wr ite the transfer function, V out (ss) , for the circui t shown above. V in 2. Gi ven C = 1 m F , find the values of R and L such that x = 0 .707 and the undamped natural frequency is 5 kHz. ( Don'
Page 1
2.003 Spring 2003 Sample Problems for Quiz 2
Question 1: X 1 V 1 X 2 V 2 1000 N/m 1.25 Kg M 1 Neglect Friction Draw the PoleZero diagram and Bode Plot of the following transfer functions: X 1 (S) ; V 2 10 Kg M 2 Position Velocity
F
F
Question 2:
F
2.003 Spring 2003 Quiz 2  Sample problem Set 2 Solutions Problem A  RLC circuit analysis 1. Vo 1 = Vi LCs2 + RCs + 1 2. n L R L R = = = = 2 5000 = 31, 400 r/s = 1 = 0.001 H = 1 mH 2 n C 2n = 2 0.707 31, 400 44.4 1 LC
3. There are no zeros, poles at root
2.003 Spring 2003 Quiz 2: Sample Set1 Solutions 1. Mass Spring system with no damper. 10 s^2 + 1000 X1 / F = 12.5 s^4 + 11250 s^2 1000 V2 / F = 12.5 s^3 + 11250 s Use MATLAB to verify Bode plots. 2. Transfer function from Bode plot 1) f=2p/3 1 2) Trans
2.003 Fall 2002
Final  Sample problems
Problem 1 Match each of the following polezero diagrams to the corresponding Bode plot and step response from the following two pages. For example is you think that polezero diagram (1) corresponds to step respons
Problem 3 (20 points) A fender is mounted on a automobile though dampers (to absorb collision energy) and springs (so that the fender can recover after lowspeed collisions). During a crashtest, the automobile is moving at 2 m/s when its fender strikes a
2.003 Quiz 1 This quiz has three problems. The numerical weighting of each problem is identical. The quiz is closedbook, but you may reference one page of notes (both sides) that you have prepared. Problem 1 (20 points) This problem concerns the secondo
Bode Plot of First Order System
1 ts + 1
Asymptotes
1 0.707
Magnitude
1
1/t 0
0
Approximation of phase plot as a staircase
45
0
Phase
Actual phase plot
90
0
10
3
1/t
10
1
Frequency (rad/sec)
Magnitude and Frequency have been plotted on log scale; Pha
2.003 Fall 2002
Final  Sample problems  Answers
Problem 1 To solve this problem it is easiest if you use the initial and nal value theorems. Final Value theorem: lim lim f (t) = sF (s) t s0 Initial Value theorem: f (0+ ) = lim sF (s) s 1, c, j 2, h, o 3
Think of problem 2 as consisting of the spring, mass and dampers moving for a long time (inf<t<0) and then we stop the mass M as t=0+ (this is clear from the given velocity source pulling M.) For t<0 the velocity source can be achieved by, say, a force F