【EEEN30150分析】UNIVERSITY COLLEGE DUBLIN
SCHOOL OF ELECTRICAL, ELECTRONIC & COMMUNICATION ENG.
EEEN30150 MODELLING AND SIMULATION
Experiment MS1
SOLUTION OF EQUATIONS BY ITERATION
- Objective:
To employ the Regula-Falsi or the Newton-Raphson algorithm as appropriate in the solution of
equations. - Background Information:
See module notes and handouts for MS0_1 and MS0_2. In general you may solve problems using
just the command line, using loops, using inline functions, using scripts or using MATLAB function
m-files, save where the latter are explicitly called for, which it should be noted is common. All of
your MATLAB function M-files must include help files, version information and appropriate error
detection/correction. You will also, in any event, have to avoid bad practice such as choosing
meaningless names for variables and/or functions, embedding universal numbers into the code,
writing excessively inefficient code or failing to include comments. In printing your report you must
acknowledge that MATLAB employs colour coding, accordingly presentation of MATLAB
function m-files will need to be in colour. Samples of acceptable MATLAB function m-files are
presented in lectures and in Sample Report available from module Brightspace page.
The possible workload for this laboratory comprises: (i) two hour preparation such as downloading
and reading this laboratory description, revising or reviewing relevant module notes, textbooks
(such as Erwin Kreyzig, Advanced Engineering Mathematics) or websites, preliminary thoughts on
choice of second problem which you will solve, gaining familiarity with tools required to complete
report to satisfactory standard, (ii) two hours of this laboratory session where you should be aiming
at the very least to complete the first problem, (iii) eight hours after this laboratory session where
you will need to solve, or complete the solution of the second, more difficult problem and (iv) two
hours preparing a report on your work for submission.
The Teaching Assistants will grade this laboratory. Each TA will need to grade approximately 40
reports, which they should be able to do reasonably quickly. Accordingly, you should receive
feedback for this laboratory. As they will not even be due until very near the end of or after the
teaching trimester, you will not receive feedback for the team minor project MP2 (or the alternative
laboratory MS3). They are treated as examinations and you will learn your grade with the normal
issuing of the examination results. The module coordinator will grade the first solo minor project.
No grades are released until all grades are available.
Some comments on the style and substance of the report are called for. Mostly these comments
apply to all of the reporting for this module and are not specific to this laboratory, so take note. A
similar section regarding reporting will not appear in later handouts. This laboratory requires that
you solve two problems. Your report should therefore consist of two sections, one section for each
problem. The sample report issued would amount to one of those sections, although I have chosen to
divide it into subsections. Your report does not need to and therefore should not contain any
additional sections. Accordingly, it is neither required nor desired that an abstract, introduction or
conclusion be included, very much in contradiction of what you have been previously taught. I do
not repeat sections of the laboratory description. Rather this subsection essentially comprises the
“analysis”, that is to say the conversion of the problem to specific equations (Equations (1) and (2))
which need to be solved. Each problem can be solved by deciding upon or developing an algorithm
and implementing this using a number of commands. Accordingly, one of the most important
components in each section will be a record of the commands employed with good comments
explaining the objective and effect of each command. Remember that this module is primarily about
numerical algorithms not one random piece of software, so these comments are very important. Of
course a second important component in each section will be the presentation of the results. In
general in this module you will have choices as to how you present your results. If you are after the
highest grades which indicate excellence, then you must indeed excel and presentation is one area in
which you can do this. The report must be typed. Scanned hand-written material is unacceptable and
this year, from the outset, will not be graded. This comment is your one and only warning, so take
note. Electronic submission of your report must be as pdf attachment to e-mail sent to
paul.curran@ucd.ie for the purposes of identifying plagiarism and record. Production quality is your
responsibility and so you must take responsibility. Of course you must include your name and
student number to clearly identify yourself and it would be more secure to include this information
on each page either as a header or as a footer. Your report may well include equations. You should
employ a proper equation editor and you should employ proper mathematical notation. Visualisation
and presentation are important parts of this module and this extends to the reports.
It is not the job of either the module coordinator or the teaching assistants to edit your report.
Students can at times make de facto editing demands of their assessors by adopting the policy of
including everything on the off chance that it might be important. Appropriate concision is a very
important skill to develop. You will impress almost nobody by being unnecessarily long-winded and
you will be comprehensible to almost nobody by being unacceptably brief. Word counts and page
counts are deemed irrelevant here. You should have as many or as few words and pages as it takes
to effectively communicate all of the necessary information, but you should have no more and no
less. It is clear that in order to write a report to conform to these rather vague guidelines you will
need to determine what in fact is the necessary information. That is exactly as it should be and is in
fact the primary purpose in making these demands upon your editorial skills. It is noted elsewhere
that the most egregious example of poor editing in this module, which results in significant loss of
grades, has traditionally been omission. Many students undertake an analysis but do not report that
analysis or do so only in the form of code. Note yet again: present all of the important equations,
assumptions and arguments of your analysis. Moreover do so in the main body of the report, do
not push them into appendices (of which you should generally have none) or code (which amounts
to reporting through code, an entirely unacceptable practice). Need it even be said that another part
of the editor’s job is to ensure that all spelling is correct? With the widespread availability of spell
checkers incorrect spelling is now both unacceptable and inexcusable. American and English
spelling are both accepted. Extremely long lists of numbers running over many pages, which some
students appear to consider reasonable material and presentation, are indicative in fact of a missing
table. Tabulation or graphing are the appropriate means of presenting such data.
The issue of what code to present and what to withhold is fraught. Unfortunately it amounts, at least
initially to a case of guessing what the module coordinator requires. It is to be hoped however that
the majority can exercise a degree of common sense. So for example if you plot a graph then just
include the plot, do not bother to include the code whose sole purpose was to generate that plot, with
the exception discussed below. If you develop a MATLAB function m-file then the entire file
should be presented in the report (in colour). This file is employed in much the same way as one
would employ a graph. It amounts to your evidence. You would not present a graph without any text
commenting upon it and observing for the reader some of the more important information contained
within it. The same is true of the MATLAB function m-files which you present. You must actually
write a report. Just presenting code indicates your belief that you may report through code. You may
not. In spite of this clear stipulation, a large minority report through code nonetheless, probably
because they do not understand the phrase “report through code”. Accordingly, let us state it again
in a different manner. If all or almost all of your report is comprised of code then you can expect to
earn almost no grades.
Two additional points regarding code should be made: Firstly it is important in code development
that when there is a standard you write to that standard. Whereas in open source code such as
Python there is no universally accepted standard, it being to some degree a matter of taste, in
MATLAB code there is a standard since the code is owned by a single entity, Mathworks.
Accordingly, the standard for how a MATLAB function m-file should be written is the standard
used by Mathworks. Secondly a method for writing millions of lines of bug-free code is to not write
millions of lines of code. One way to achieve this is to modularise the code. Individual modules,
which essentially equate to MATLAB function m-files in MATLAB, should be simple but flexible.
Simplicity usually means that individual modules of code do not contain many lines of code and can
be fully debugged. Simplicity generally requires that a module of code should do just one job. An
error is to write MATLAB function m-files which do too many jobs, where in fact you should be
writing several MATLAB function m-files. Flexibility means that whereas the module does just one
job it can be used, usually through changing input arguments, to solve a variety of related problems.
When done correctly your core modules of code will be executed many times during the solution of
a problem. I refer to this as “reusing not rewriting”, although the more common phrase in Computer
Science is DRY (don’t repeat yourself). When a MATLAB function m-file is being used perhaps
hundreds of times in your solution of a problem, that m-file is code that I will wish to see. This is
the exception mentioned above. If a core module is repeatedly used to produce a series of plots or a
video, then I will wish to see that code even though I will also wish to see the main outcome of that
code, namely the plots or the video. Emphasis is on the word repeatedly.
Another example of inappropriate reporting occurs if you execute an identical line of code forty
times and subsequently print it forty times. All you need do is to print it one time and to report that
it was executed forty times. A “thought experiment” which you can attempt to perform to stop
yourself from doing something rather silly is “exaggeration”. Suppose that the amount of data which
you had was a thousand times what you do in fact have. Would you list thousands of pages of
numbers or print thousands of repetitions of an identical piece of code? At what point would it occur
to you to call a halt and realise that this has become entirely self-defeating and that some other
means of presentation ought to be sought? The purpose of the exaggeration thought-experiment is to
realise the need for this before much work on inappropriate presentation has been undertaken.
Within the rather wide parameters set by these guidelines you may make your own decisions
concerning the report. Some will wish, as other modules will have demanded of them, to have
introductory and conclusory sections, i.e. sections both titled “Introduction” and “Conclusion” and
containing the kind of material normally present in such sections. As noted previously their
inclusion will be accepted, although the preference is that they be excluded since the majority of
conclusions are pointless and irrelevant and the majority of introductions amount to wholesale
repetition of the laboratory description and as such border upon plagiarism.
You must include with the report (at the front) a (digitally) signed cover sheet declaring that you
have read and understood the university policy on plagiarism. Failure to do so will result in an
automatic deduction of one grade step. Actually many of you will not have read this policy and
many more will not have understood it. The principle infringements likely in this module are
copying (mostly from other students) without citing. Downloading of large sections of code written
by others, copying large pieces of text from websites or from each other, these things do not
comprise plagiarism. Plagiarism only arises when you do these things without clearly stating in the
report that you have done them and giving sources for the material so that an assessor can determine
for themselves how much of your work is original and how much is just copied from others.
Obviously if almost all of your work is copied then it is not in fact your work and you have not
earned a good grade. It is a widely adopted practice to present copied text in italic and in quotes for
example, since indeed a quote is nothing more than copying, although it is not plagiarising.
Very many of you will effectively work in groups. This is acceptable and indeed encouraged to
some extent, although obviously not to excess. For those who struggle with the material a
reasonable level of help from a colleague will be most beneficial. For those who take to the material
well the exercise of attempting to explain it to others will greatly enhance both your understanding
and your reporting. When it comes to writing the report however the work should become your own.
Best practice is to say farewell to those with whom you have collaborated up to this point and
undertake the last task of reporting entirely on your own. In the event of obvious copying wholesale
of sections of report all students involved will suffer a loss of grades. No attempt will be made to
identify the content-creator. Allowing your report to be copied is equal to copying and suffers equal
penalty. If copying reaches extreme levels then the matter will be referred to the plagiarism
committee of the School of Electrical and Electronic Engineering for their adjudication and, where
appropriate, determination of penalty.
To finish on a positive note I reiterate point made in introductory slides. If you have questions
please ask. To those who are making an effort I promise that I will gladly support you in that effort.
Laboratories will be primarily run by the Teaching Assistants this year. Part of the reason for this is
to free up significant time on my part to engage via e-mail or small group zoom sessions with any
questions that you may have. - Experiment:
3.1. Required problem:
Problem: Clapeyron’s ideal gas law, as the name suggests, holds for ideal gases only where
there is essentially no interaction between atoms and molecules. The Redlich-Kwong equation
comprises a more accurate relationship for real gases. It is commonly written in one of two
equivalent forms (and it must be said that notation is not consistent across various accounts). We
have either:
and where the largest root of the cubic equation is taken. The parameters Tc and Pc are the
temperature and pressure at the critical point. Values are widely available for a variety of
substances. Find the critical temperature and pressure for Carbon Dioxide gas (CO2). In these
equations R is the ideal gas constant, P is the pressure, T is the temperature and v is the molar
volume, i.e. the volume occupied by 1 mole of the gas at the given pressure and temperature. Show
that the two forms of the equation given are indeed equivalent. By implementing either the RegulaFalsi
or the Newton-Raphson algorithms develop a MATLAB function m-file to find the molar
volume (in L/mol) of Carbon Dioxide (CO2) at a temperature of 293 K for pressures of 1 atm, 1.5
atm, 2 atm, 2.5 atm, 3 atm, 5 atm, 10 atm, 15, atm, 25 atm, 50 atm and 100 atm. Compare these
predictions with those of the ideal gas law:
Pv = RT
. You may find a numerical value for the ideal
gas constant R on the internet, but be careful with units. You should make this comparison using
good visualisation. For example plot the ideal molar volume and the calculated more realistic molar
volume on the same axes using the hold on command. You should also undertake to avoid the
inclusion of obvious machine artefact. Consulting online sources, for the given temperature
determine for what range of pressures is the Redlich-Kwong equation of state deemed to be
accurate.
(9 grade steps)
3.2. Modelling and solving equations: Applications
Solve one of the following problems or alternatively a problem of your own choosing (provided it
has been approved by the module co-ordinator). Note carefully that core code in any case must take
the form of a MATLAB function m-file. A correctly reported problem solution will generally have
three components: (i) mathematical analysis of problem (either setting up of equations or
confirming equations given); (ii) numerical analysis of problem (implementing algorithm as
MATLAB function m-file to find approximate numerical solutions to equations; (iii) presentation of
results (commonly in the form of graphs or tables). Please take note. This is a numerical methods
module. You are to solve problems numerically using methods introduced in module, not
analytically, not using algorithms found from other sources (cited or otherwise) and not using builtin
MATLAB functions.
(12 grade steps)
Problem 3.2.1:A horizontal handrail is securely fixed at both ends. Make your own decision as to
what might comprise a reasonable material for the construction and thus find suitable values for the
Young’s modulus of elasticity and the density. The rail has an annular cross section giving a second
moment of area about either axis within the plane of the cross section of ( )
where r1
and r2 are respectively the inner and outer radius. By considering some standard sizes of handrails
determine appropriates values for the length L (between fixed points), the inner radius r1 and the
outer radius r2 of the rail. Obviously there is a significant level of approximation here as typical rails
are fixed (or pinned) at multiple points (as shown in Figure 1) and treating a single section between
fixed points as if it was unconnected to the rest of the rail is inappropriate. Nevertheless that is what
you may do on this occasion. We will model the rail (or section of rail) using the Euler-Bernoulli
beam equation. As the rail is fairly light we will ignore loading. Accordingly the free vibrations of
the rail are very approximately described by the partial differential equation:
where y is the transverse deflection in the vertical direction (as shown in Figure 1), m is the mass per
unit length, E is the Young’s modulus of elasticity and I is the second moment of area about an axis
perpendicular to the direction of deflection, being the same for every such axis as above. The fact
that the rail is fixed at both ends, i.e. we are just modelling the section of the rail between adjacent
supports, the following boundary conditions are applied:
It is possible to solve this partial differential equation using the method of separation of variables.
This method offers the following oscillatory solution:
y(x,t) (c sin(? x) c cos(? x) c sinh(? x) c cosh(? x))(c sin(?t) c cos(?t)) = 1 + 2 + 3 + 4 5 + 6
where c1, … , c6 and ? are constants of integration which are determined by appropriate boundary
and initial conditions.
y(x,t)
x x = 0 x = L
Figure 1: handrail
By substituting this formula back into the partial differential equation confirm that it is indeed a
solution, provided the parameters ? and ? are related by:
m
EI 2 ? = ? .
Interpret the formula for y(x,t) by showing that the frequency of vibration of the corresponding
deflection of the beam is ? in rad/sec.
By using the boundary conditions derive the constraints on the constants of integration c1, c2 , c3 , c4
and ? ? These constraints comprise a system of four simultaneous linear equations in the four
unknowns c1, c2 , c3 , c4 . Clearly the would-be solution offered by the method of separation of
variables is only non-zero if at least one of these unknowns c1, c2 , c3 , c4 is non-zero. By employing
the theory of determinants show that this occurs if and only if
cos(? L)cosh(? L)?1= 0.
Implement an algorithm of your choosing (Regula Falsi or Newton-Raphson) to determine the
smaller solutions ? of this equation, presenting an account of the code used. Find the lowest nine
frequencies of oscillation (free vibration) of the handrail in Hz (cycles per sec).
You do not have to use it, but there is an obvious scaling which can be employed here.
Problem 3.2.2:When water flows through a pipe there is friction between the water and the inner
surface of the pipe. One method of modelling the effect of this friction is to compute the resulting
head loss. The Darcy-Weisbach equation expresses the head loss due to friction:
where L is the pipe length, Dh is the hydraulic diameter, V is the average velocity of the fluid flow
being equal to the volumetric flow rate per unit cross-sectional wetted area, g is the local
acceleration due to gravity and fD is a dimensionless coefficient called the Darcy friction factor. One
must be careful not to confuse the Darcy friction factor with the Fanning friction factor f. The two
are related by the rather trivial relation:
f f D
= 4 .
Rather strangely the hydraulic diameter is defined to be four times (not twice) the hydraulic radius.
This is due to the curious nature of the definition of the hydraulic radius. The hydraulic radius is
defined as:
P
A
Rh =
where A is the cross-sectional area of the flow normal to the direction of the flow and P is the wetted
perimeter, i.e. the length of the perimeter in contact with the water. The cross-sectional wetted area
is the cross-sectional area of the pipe which is wetted, i.e. in contact with water. This is equal to A.
For a pipe of circular cross section of internal diameter d where the flow completely fills the pipe
the hydraulic radius is given by:
So the hydraulic radius comes out as one quarter of the internal diameter in this case. The hydraulic
diameter is defined so that it is equal to the actual internal diameter in this case, i.e. as four times the
hydraulic radius. In this respect it is the definition of the hydraulic radius which is strange, being
essentially half what it arguably could be.
If rather than head loss one requires the pressure loss due to friction then the standard formula
?p = ? ghf
may be employed, where ? is the density of the water.
To use the Darcy-Weisbach equation one must know the Darcy friction factor. There are two
methods for determining this. One is graphical. A Moody chart is an experimentally derived plot of
Darcy friction factor vs Reynolds’ number (Re) for a variety of relative roughness values (? /D) and
flow regimes assuming fully developed flow in a circular pipe. Alternatively for laminar flow (i.e.
very low Reynolds’ number) Poiseuille gives the formula:
Re - f D
=
whereas for turbulent flow (i.e. high Reynolds’ number) Colebrook gives the more difficult formula:
For relative roughness of 0.001, 0.002, 0.003, 0.004, 0.005, 0.007, 0.01, 0.02 and 0.05 employ the
Colebrook equation to determine the Darcy friction factor for a range of Reynolds’ numbers from - to 500,000,000, presenting your results in the manner of a Moody chart.
A circular pipe has a radius 1.2 m, length 40 m and a relative roughness 0.007. Water flows through
the pipe. The flow is turbulent, with a Reynolds’ number of 1,800,000, but it is uniform, meaning
that the depth of the water is constant along the pipe. The depth of the flow is 175 cm and the
volumetric flow rate is 12.5 m3
/sec. Use the standard value g = 9.81 m/s2
.
Calculate the Darcy friction factor for the given Reynolds’ number and relative roughness.
The depth of the flow is larger than the radius of the pipe. The situation is as depicted in Figure 2.
q r
y
h
Figure 2: water in circular pipe of depth y exceeding radius
Show that. Hence show that the wetted perimeter
is the cross sectional area of the water normal
to the flow. Hence determine the value of the hydraulic diameter and the average velocity of the
flow. You may now employ the Darcy-Weisbach equation to determine the head loss. What is the
associated loss of pressure?
Problem 3.2.3:An electrical wire is 1 m long and has a circular cross section with cross-sectional
area of 70 mm2
. The wire is made of copper which has a resistivity ? = 16.78 n?·m at 20oC and a
thermal conductivity kwire = 401 W/m·K. The wire is enclosed in an electrically insulating sheath
made of flexible PVC. Flexible PVC has a thermal conductivity of about kPVC = 0.15 W/m·K and
begins to decompose at about 140oC. The insulated wire is surrounded by air at an ambient
temperature of 20oC. The dominant mechanism of heat transfer in the air is convection with a heat
transfer coefficient of h = 25 W/m2K. Making the unrealistic assumption that all physical
parameters remain constant over the range of temperatures involved and assuming a DC current of
615A is flowing in the wire, what is the minimum acceptable thickness of the PVC sheath such that
the plastic does not begin to decompose?
Assuming steady-state has been achieved with the surface temperature of the wire at T1, ignoring
edge effects occurring at the ends of the wire and assuming the dominant mechanism of heat
transfer within the material to be conduction the temperature distribution may be described by the
steady-state heat equation with a heat generation term. In polar co-ordinates this is:
with the heat generation term: ? =2W/m3
where R is the resistance of the wire, l is the
length of the wire, A is the cross-sectional area of the wire and i is the current flowing through the
wire.
This equation may be solved giving the general solution: () = 1() ??(24) + 2 for
arbitrary constants of integration C1 and C2.
By substituting this formula into the ordinary differential equation show that it is indeed a solution.
We have one explicit boundary condition: () = 1 but we have a second implicit boundary
condition: (0) finite. Show that these boundary conditions lead to the solution:
The outward heat flow at the surface of the wire is: ? = ?
at = where Awire
is the surface area of the wire. This heat flow equals the heat flow into the PVC insulating sheath
through the inner surface. Show that:
? =
2
.
Again assuming steady-state has been achieved the surface temperature of the wire at T1 ,the outer
surface temperature of the PVC insulating sheath at T2, ignoring edge effects occurring at the ends
of the wire and assuming the dominant mechanism of heat transfer within the material to be
conduction the temperature distribution in the PVC may again be described by the steady-state heat
equation. In polar co-ordinates this is:
1
(
) = 0 , () = 1
To reflect the heat transfer across the inner boundary we also have a second boundary condition
? = ?
at = where ? is the heat flow into the PVC as previously calculated.
The general solution of this equation is: () = 1
() + 2
where C1 and C2 are constants of
integration.
By substituting this formula into the ordinary differential equation show that it is indeed a solution.
Show that the boundary conditions lead to the solution:
() = 1 ?
?
2
(
) .
Show that the heat flow out across the outer boundary of the cable as given by ?
at
= is equal to ? the heat flow in, where Acable is the outer surface area of the cable (i.e. the
wire plus insulating sheath).
Again assuming steady-state has been achieved with ,the outer surface temperature of the PVC
insulating sheath at T2 , ignoring edge effects occurring at the ends of the wire, assuming the
surrounding air at a suitable distance from the sheath to be at temperature ∞ = 20oC , this being
the ambient temperature and assuming the dominant mechanism of heat transfer within the air at the
surface of the sheath to be convection the heat transfer away from the sheath is given by Newton’s
law of cooling:
? = ?(2 ? ∞).
We seek the value of rcable such that T1 =140oC. Show that in this event the following equations
hold:
2 = 1 ?
?
2 ln (
)
? = ?2(2 ? ∞)
? =
2
Hence find rcable. The insulation thickness which is actually sought equals rcable – rwire .
Problem 3.2.4: The npn BJT has been modelled by Ebers and Moll as a pair of ideal diodes,
back-to-back.
n+ p nEmitter
Base Collector IC
IB
IE
Figure 3: npn bipolar junction transistor idealised structure
There are two pn junctions (base-emitter and base-collector). Under the conditions where the
voltages VBE and VCE are positive the base-emitter pn junction is forward-biased whereas the basecollector
pn junction is reverse-biased. Normally the current flowing through a reverse-biased pn
junction should be small. But normally the p-doped base region should have relatively few mobile
electrons. With the base-emitter pn junction forward-biased there can be a massive influx of mobile
electrons from the emitter (where they are in relatively great abundance). The base region, being ptype
notwithstanding, actually has a very large number of mobile electrons and the result is that the
current flowing through the reverse–biased base-collector pn junction is far from small (relatively
speaking). A minor extension of the analysis of Ebers and Moll leads to the following equations for
the collector and base currents:
Ebers and Moll in their original paper establish that
Let this quantity equal IS hence
where
Kirchoff’s voltage law yields an additional equation concerning the voltages:
VCE = ?VBC +VBE
In these equations k = 1.3806503 x 10-23 m2kg s-2 K-1
is Boltzmann’s constant, q = 1.60217646 x
10-19 C is the magnitude of the electron charge, T is the temperature in K. The quantity VT = kT/q is
called the thermal voltage at temperature T. As is reasonably common in electronics we will
perform the analysis for a temperature of 300 K. Model parameter IS is the reverse saturation
current. Model parameter VA is called the Early voltage. It models the Early effect whereby as the
collector-base voltage becomes more positive the depletion region of the base-collector pn junction
becomes wider. The result is that the base region effectively becomes narrower. As a result fewer of
the mobile electrons injected from the emitter recombine in the base region, so that there are more
of them to cross into the collector giving rise to an increase in the collector current. Model
parameters ?F and ?R are the forward and reverse common-emitter current gain respectively.
The npn BJT is employed in the very basic amplifier circuit shown in Figure 4. The collector supply
voltage VCC = 15 V. The base bias resistor RB = 22 k?. The load resistor RL = 1.2 k?. The base
bias voltage E = 1.5 V. The input signal e is sinusoidal being Asin(?t). The amplitude of the input
signal is A V. The frequency of the input signal is ? rad/sec. Assume the input signal frequency to
be 1.2 kHz. Model the npn BJT by the modified Ebers-Moll equations above with the model
parameters IS = 0.85 pA, VA = 120 V, ?F = 250 and ?R = 8. Assume the amplitude, A, of input signal
e is 20 mV. Write down the equations which describe the circuit and hence by employing any
numerical algorithm from the module notes you choose (Regula Falsi or Newton-Raphson)
determine the output voltage Vo appearing across the load resistor RL (as indicated in Figure 4) for
several (at least three) cycles of the input signal. To facilitate comparison, plot the output signal Vo
and the input signal e(t) vs time t for several cycles on the same axes. Hence deduce whether
amplification has indeed occurred and determine the gain. For the purposes of this example we will
define the gain as:
Peak - to - peak variation of input signal
Peak - to - peak variation of output signal Gain = .
RB
VBE
IB IC VCE
RL
VCC
E e
Vo
Figure 4: basic amplifier circuit
Problem 3.2.5:A model of the space shuttle is shown in Figure 5. Thrust is provided by the two
solid rocket boosters and the three liquid-fuelled orbiter engines. The centre of mass of the fully
loaded craft (external tank, boosters and orbiter) is shown in Figure 5 and marked as G. At lift off
the orbiter engine thrust is directed at an angle q. NASA offers the following information:
External tank: 78,000 lb (empty), 143,000 US gallons of liquid oxygen, 383,000 US gallons of
liquid hydrogen. Find the total mass (in kg) of the external tank at lift off.
Solid rocket booster: 185,000 lb (empty), 1,100,000 lb fuel. Find the total mass (in kg) of the two
solid rocket boosters at lift off.
Orbiter: 165,000 lb (empty), payload 65,000 lb (including fuel). Find the total mass (in kg) of
the orbiter at lift off.
The combined thrust from the solid rocket boosters is
23.6 10 N
6
TB
= ?
. The combined thrust from
the three orbiter engines is
5.03 10 N
6
TS = ?
. Clearly any residual moment about the centre of mass
of the craft will cause it to rotate about G to catastrophic effect.
Set up an equation for the net moment about G at lift off as a function of q. What is the required
angle q at lift off such that the net moment about G is zero?
Research fuel usage and thereby determine the mass of the external tank when each of the solid
rocket boosters has used up 25% of its fuel. Making the incorrect assumption that the attitude of the
shuttle is unchanged (i.e. ignore roll and similar manoeuvres and assume a simple vertical flight
path) determine the new location of the centre of mass and the new required angle q such that the
net moment about this new centre of mass remains zero. You may assume that the loading and use
of fuel is such that that the centre of mass does not shift in the vertical (i.e. longitudinal) direction.
Solid rocket booster External tank Orbiter G
推荐阅读
- APS 105 Computer Fundamentals
- javascript|Vue.js全家桶仿哔哩哔哩动画 (移动端APP)
- CPSC 4050/6050 分析
- python|网络图片爬虫(几个简单步骤实现网页图片的爬取，详细步骤，超详细，简单易懂)
- Java|抽取幸运观众(JAVA基础案例教程第二章-课后作业)
- 案例|Java 登录注册系统 【java基础案例教程 案例2-6】
- java|Spring MVC +Spring 框架学习总结-全网最好用的学习笔记
- Linux笔记|Linux Nginx-Web
- python笔记|Python 面向对象（一）