Ramos 1
Modeling and Analysis
Adela Ramos
Simulink #1
1. (1) A clothes iron has a sole plate weighing 1.75 kg with an exposed area of 0.05 m2. The
sole plate is made of steel, which has a heat capacity of 450 J/kgoC, and the heat transfer
coefficient for
Chapter 4
Problem 1
Using the definition of the Laplace transform, derive the transforms F(s) of the following
functions
a) f (t ) t
b) f (t ) e a t where a is a constant.
c) f (t ) cos t where is a constant.
d) f (t ) e a t cos t where a and are constant
TEST 5 Summer, 2016
Problem I (20 points)
Using the Laplace transform method obtain the analytical equation of
2
. d
5Q+5+30x=m0+42 w1thx(0)=0.612m;x =01 and fA(t)=30u(t) N,
(If dt d be 8
Problem 2 (20 points)
Using the method of y
Modeling and Analysis of Engineering Systems SIMULINK Exercise 1 (Due 10/16)
You simply need to submit (hardcopy) what is asked for in bold below. You may
work with others but you must submit your own work under your own name.
(1) A clothes iron has a sol
TEST 2
Problem 1
Obtain the solution of
y ' '8 y '15 y 30 60 u (t ) with y (0) 2 and y ' (0) 0
Before solving the example, note that the forcing function could have been written as
f (t ) 30 60 u (t ) as well.
Obtaining the Laplace of each term yields,
s
TEST 1 Spring 2009
Problem 1
The location of the roots of the characteristic equation denes the qualitative behavior of
systems. State the location of the roots for
a) system that will be continuously oscillatory forever
b) system that will be oscillatory
PROBLEM SET MODULE 1
Problem 1
Problem 1.1a
Problem 2
Problem 1.1b
Problem 3
Consider the tank of Section 11 and its steady state condition of w1 200 kg min ,
w2 100 kg min and h 3.24 m . Suppose that at time equal to 10 min w1 changes to
400 kg min and
TEST 2 Problem 2 Fall 2014
2
d 2T
dT
8
16 T 0
2
dt
dt
with T (0) 20 oC and
dT
dt
0
0 C
t 0
min
.
Taking the Laplace transform of the differential equation, and solving for the Laplace of the
dependent variable,
2 s 2T ( s ) 2 s (20) 2(0) 8sT ( s ) 8(20)
TEST 1 Fall, 2015
Problem 1 (20 points)
Find the complete solution to
2
d12w+2d—T+5T=t with T(0)=0and—6£1z =2
dt d t d t [:0
Problem 2 (20 points)
Suppose that the following equation describes the concentration of a component in a
fermentation reaction.
Ruth Fabian
Mini Project II
Modeling Analysis
November 2, 2015
Given:
W1= 200 kg/min
W2= 100 kg/min
h=3.24 m
146
dh
+166.7 h=w1 ( t ) + w2 (t)
dt
(1)
146
dhl
+ 46.34 hl +w 1 ( t ) +w 2 ( t )150.14
dt
(2)
The program below for the differential equation is
Chapter 3
Problem 31
a)
3
Using initial condition,
Thus,
0
3 ;
and
1
0
b)
2
0
0
2 ;
and
Using initial condition,
Thus,
1
1
c)
1
Using initial condition,
Thus,
4 1
4 0
0 ;
1 and
4
4
1
4
1
3
4
4
1
d)
0
;
and
1
1
Using ini
Chapter 4
Problem 41
a)
Integrating by parts
1

1
0
1
0
1

1
b)
where a is a constant
1
1
1

c)
cos
1
2
1
1
2
1
2
1
1

2
2
2
1

2
d)
cos
cos
2
1
2
1
1
2
1
2
1

1

1
2
2
2
Prob
Chapter 2
Problem 21
Going Up
+
y
y=0
Assuming no air resistance:
ground
2
1 1
2 2
2
Substituting (2) into (1) and rearranging,
0
3 with
Maximum height is when
35 /
0
From (3)
=>


=> 0
3.57
We can also obtain an expression for
CH
HAPTER 5
Problem 51
m
1 1
or
and
2
,
2 2
3 3
3
3
3
d
e
r
taining the an
lution.
We could reduce the number of equations for ease in obt
nalytical sol
Substitut
ting (3) and (2) into (1) and rearranging,
(
a
0
CHAPTER 9
Problem 91
R 4
+

+
+
L =2H
vS 20 e 4t

t =0

The voltage supply and switch can be mathematically expressed as
20
KVL
0
1 1 . , 2
From R:
2 2 . , 3
3 3
From L:
. , 3
.
.
.
,
.
Substituting (2), (3) into (1):
0
With
0
0
TEST 4 Fall, 2015
If you want the possibility of partial credit, SHOW ALL YOUR WORK.
Problem 1 (20 points)
Consider the following circuit.
a) Obtain the initial currents through all the elements and the voltages at nodes A &
B
b) Obtain the model that d
TEST 4 Spring 2016
Problem 1
a) Obtain the models for the currents in both meshes as well as the voltage drop across the
capacitor in the circuit shown below. (10 points)
b) Obtain the transfer function relating the voltage drop across the capacitor to th
TEST 3 Summer, 2016
Problem 1 (30 points)
Consider the circuit shown below. a) Develop the model that describe the currents through all
the electrical components, b) describe in detail the steps to obtain the analytical solution.
R1=2g2 R2=4Q
Problem 2
CH
HAPTER 6
Problem
m 6.1
1 1
or
and
2
,
2 2
3 3
3
3
3
d reduce the number of equations
e
forr ease in obttaining the annalytical sollution.
We could
Substitutting (3) and (2)
( into (1) and
a rearranging,
0 4
n
TEST 1 Spring, 2016
Problem 1 (40 points)
The following differential equation describes the mechanical system shown below
dzx dx dzx dx
+P+kx= t or 10 +20+kx= t
at2 dt f1 at2 dt f1
m
a) It is desired to size the spring, that is obtain the value of k,
TEST 1 Summer, 2016
FOR PARTIAL CREDIT YOU MUST SHOW ALL YOUR WORK
Problem 1 (20 points)
The following mathematical model (equation) describes the
current in a RC circuit once a
voltage supply is applied at t = 0.
For R=1000E ; C=2.5x10F;a)= 600Hz
(Q stan
TEST 1 Fall, 2016
' l
Problem 1 (20 points) 7  '
Industrial engineers are sometime interested in studying population dynamics. A
common equation describing the population growth is the following differential equation,
dp p
_= 1_n
dt r[ ij
where r is call
Chapter 4
Problem 4.1
a)
Integrating by parts
1

1
0
1
0
1

1
b)
where a is a constant
1
1
1

c)
cos
1
2
1
1
2
1
2
1
1

2
2
2
1

2
d)
cos
cos
2
1
2
1
1
2
1
2
1

1

1
2
2
2
Prob
Chapter 1
Problem 11
a)
1
3
2
2
2
3
2
4
16
3
2
4
3
6
b)
4
3
16
3
2
16
14
14
3
12
Problem 12
a)
w1 (t ) 200 200 u (t 10)
b)
146
At the new steady state
dh
166.7 h w1 w2
dt
dh
0 , therefore,
dt
166.7 h w1 w2 400 100 500
h 9m
and
Problem
TEST 6 Spring 2016
Problem 1 (20 points)
. . . . d1 ' d .
Obtain the analytical solution usmg Laplace transforms of 5 d i +7~dx =j4 (t)
I I
. . . . . dx m _
Assume that the initial conditions are 3:00) = Om and we?" 2 0w , and that the applied force
I s
TEST 1 Spring 2017
Problem 1(30 points)
The following model describes the current in an electrical circuit,
d2i(t) dim _ _, . _ di
d +6 dt +9z(z)=27u(r)+3e2u(r) Withl(0):0A;E :OA/s
r:0
a) Will this electrical system be stable or not? Explain why.
b) Will
TEST 3 Fall, 2016
Problem 1 (30 points)
The model that describes the motion of a rotating device is 6 + 40 = 20 u(t) .
Assuming that the initial conditions that all initial conditions are zero, (a) using the
Laplace transform method obtain the analytical
TEST 5
Spring 2016
Problem 1 (20 points)
Using the characteristic equation and undetermined coefficients methods, obtain the analytical
solution (all initial conditions are zero) of,
0.47
d 2x
dx
6.87
114 x 0.0347 1.379 t
2
dt
dt
Problem 2 (20 points)
C