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Alexander, Michelle, The New Jim Crow: Mass Incarceration in
the Age of Colorblindness
Chapter 5: The New Jim Crow
It was no ordinary Sunday morning when presidential
candidate Barack Obama stepped to the podium at the Apostolic
Church of God in Chicago. It was Father’s Day. Hundreds of
enthusiastic congregants packed the pews at the overwhelmingly
black church eager to hear what the first black Democratic
nominee for president of the United States had to say.
The message was a familiar one: black men should better
fathers. Too many are absent from their homes. For those in
the audience, Obama’s speech was an old tune sung by an
exciting new performer. His message of personal
responsibility, particularly as it relates to fatherhood, was
anything but new; it had been delivered countless times by
black ministers in churches across America. The message had
also been delivered on a national stage by celebrities such as
Bill Crosby and Sidney Poitier. And the message had been
delivered with great passion by Louis Farrakhan, who more than
a decade earlier summoned one million black men to
Washington, D. C. for a day of “atonement” and recommitment
to their families and communities.
The mainstream media, however, treated the event as big
news, and many pundits seemed surprised that the black
congregants actually applauded the message. For them, it was
remarkable that black people nodded in approval when Barack
Obama said “If we are honest with ourselves, we’ll admit that
too many fathers are missing-missing from too many lives and
too man y homes. Too many fathers are MIA. Too many fathers
are AWOL. They have abandoned their responsibilities.
They’re acting like boys instead of men. And the foundations
of our families are weaker because of it. You and I know this is
true everywhere, but nowhere is this more true than in the
African American community.”
The media did not ask—and Obama did not tell—where the
missing facts might be found.
The following day, social critical and sociologist Michael
Eric Dyson published a critique of Obama’s speech in Time
magazine. He pointed out that the stereotype of black men
being poor fathers may well be false. Research by Boston
College social psychologist Rebekah Levine Coley found that
black fathers not living at home are more likely to keep in
contact with their children than fathers of any other ethnic or
racial group. Dyson chided Obama for evoking a black
stereotype for political gain, pointing out that “Obama’s words
may have been spoken to
1
black folk, but they were aimed at those whites still on the
fence about home to send to the White House.” Dyson’s critique
was a fair one, but like other media commentators he remained
silent about where all the absent black fathers could be found.
He identified numerous social problems plaguing black families
such as high levels of unemployment, discriminatory mortgage
practices, and the gutting of early-childhood learning programs.
Not a word was said about the prisons.
The public discourse regarding “missing black fathers”
closely parallels he debate about the lack of eligible black men
for marriage. The majority of black women are unmarried
today, including 70 percentage of professional black women.
Where have all the black men gone? Is a common refrain heard
among black women frustrated in their efforts to find life
partners.
The sense that black men have disappeared is rooted in
reality. The U. S. Census Bureau reported in 2002 that there are
nearly 3 million more black adult women than men in black
communities across the United States, a gender a gap for 26
percent. In many urban areas, the gap is far worse, rising to
more than 37 percent in places like New York City. The
comparable disparity for whites in the United States is 8
percent. Although a million black men can be found in prisons
and jails, public acknowledgement of the role of the criminal
justice system in “disappearing” black men is surprisingly rare.
Even in the black media—which is generally more willing o
raise and tackle issues related to criminal justice—an eerie
silence can often be found.
The Journal of Blacks in Higher Education reports the fact
that black women hold a large lead over black men in almost
every facet of higher education. Black women currently earn
about two thirds of all African-American bachelor's degree
awards, 70 percent of all master's degrees, and more than 60
percent of all doctorates. Black women also hold a majority of
all African-American enrollments in law, medical, and dental
schools. Looking exclusively to undergraduate higher education,
the latest Department of Education figures show that black
women account for 63.6 percent of all African-American
enrollment.
Ebony magazine for example ran an article in December 2006
entitled “Where Have the Black Men Gone?” The author posed
the popular question but never answered it. He suggested we
will find our black men when we rediscover God, family and
self-respect. The fact that Barack Obama can give a speech on
Father’s Day dedicated to the subject of fathers who are AWOL
without ever acknowledging the majority of young black men in
many large urban areas are currently under the control of the
criminal justice system is disturbing to say the least. What is
more problematic though, is that hardly anyone in the
mainstream media noticed the oversight. Hundreds of thousands
of black men are unable to be good fathers because they are
warehoused in prisons, locked in cages. They did not walk out
on their families voluntarily; they were taken away in
handcuffs, often due to a massive federal program known as the
War on Drugs.
More African American adults are under correctional control
today—in prison or jail, on probation or parole—than were
enclosed in 1850, a decade before the Civil War began. The
2
mass incarceration of people of color is a big part of the reason
that a black child born today is less likely to be raised by both
parents than a black children born during slavery. The absence
of black fathers from families across American is not simply a
function of laziness, immaturity, or too much time watching
Sports Center. Thousands of black men have disappeared into
prisons and jails, locked away for drug crimes that are largely
ignored when committed by whites.
Black people only represent 13.3 percent of the U.S.
population while white people make up about 77 percent. But
there are more black men in state prisons across the nation than
there are white men. The latest Federal Bureau of
Prisons statistics showed the white population was 58.7 percent
in August, compared to the 37.8 percent of blacks being housed
at federal institutions. At the state level, however, the
department said that black male prisoners represent 38 percent
of the population while white males account for 35 percent and
21 percent are Hispanic males. Overall, blacks are 5.1 more
times likely to be incarcerated than whites, and blacks represent
more than half of the prison population in 11 states.
The clock has been turned back on racial progress in
America, though scarcely anyone seems to notice. All eyes are
fixed on people like Barack Obama, Oprah Winfrey, and Robert
L. Johnson who have defied the odds and risen to power, family
and a great future. For those left behind, especially those
within prison walls, the celebration of racial triumph in
American must seem a tad premature. More black men are
imprisoned today than any other moment in our nation’s history.
More are disenfranchised today than in 1870, the year the
Fifteenth Amendment was ratified to laws that explicitly deny
the right to vote on the basis of race. Young black men today
may be just as likely to suffer discrimination employment,
housing, public benefits, and jury service as a black man in the
Jim Crow era—discrimination that is perfectly legal, because it
is based on one’s criminal record. This is the new normal, the
new racial equilibrium.
The launching of the War on Drugs and the initial
construction for the new system required the expenditure of
tremendous political initiative and resources. Media campaigns
were waged; politicians blasted soft judges and enacted harsh
sentencing laws; poor people of color were vilified. The system
now, however, requires very little maintenance or justification.
In fact if you are white and middle class, you might not even
realize the drug war is still going on. Most high school and
college students today have no recollection of the political and
media frenzy surround the drug war in the early years. They
were young children when the war was declared, or not even
born yet. Crack is out; terrorism is in.
Today, the political fanfare and the vehement, racialized
rhetoric regarding crime and drugs are no longer necessary.
Mass incarceration has been normalized, an all of the racial
stereotypes and assumptions that give rise to the system are now
embraced (or at least internalized) by people of all colors, from
all walks of life, and in every major political party. We may
wonder aloud “where have the black men gone?” but deep down
we already know. It is simply taken for granted that, in cities
like Baltimore and Chicago, the vast majority of young black
men are currently under the control of the criminal justice
system or branded criminals for life. This extraordinary
circumstance—unheard of in the rest of the world—is treated
here in America as a basic fact of life, as normal as separate
water fountains were just a half century ago.
3
Those cycling in and out of Illinois prisons today are
members of America’s new racial undercaste. The United
States has almost always had a racial undercaste –a group
defined wholly or largely by race that is permanently locked out
of mainstream, white society by law, custom, and practice. The
reasons and justifications change over time, as each new caste
system reflects and adapts to changes in the social, political,
and economic context. There are important differences between
mass incarceration and Jim Crow and a profound sense of déjà
vu. There is a familiar stigma and shame. There is an elaborate
system of control, complete with political disenfranchisement
and legalized discrimination in every major realm of economic
and social life.
Parallels Between Jim Crow and Mass Incarceration:
Listed below are several of the most obvious similarities
between Jim Crow and mass incarceration.
Historical parallels:
Jim Crow and mass incarceration have similar political
origins. Both the caste systems were born in part due to a
desire among white elites to exploit the resentments,
vulnerabilities, and racial biases of poor and working class
whites for political or economic gain. Segregation laws were
proposed as part of a deliberate and strategic effort to deflect
anger and hostility that had been brewing against the white elite
away from them and toward African Americans. The birth of
mass incarceration can be traced to a similar political dynamic.
Conservatives in the 1970s and 1980s sought to appeal to the
racial biases and economic vulnerabilities of poor and working
class whites through racially coded rhetoric on crime and
welfare. In the early years of Jim Crow, conservative white
elites opted with each other by passing ever more stringent and
oppressive Jim Crow legislation. A century later, politicians in
the early years of the drug war competed with each other to
prove who could be tougher on crime by passing ever harsher
drug laws—a thinly veiled effort to appeal to poor and working
class whites who, once again, proved they were willing to
forego economic and structural reform in exchange for an
apparent effort to put blacks back in their place.
Legalized discrimination:
The most obvious parallel between Jim Crow and mass
incarceration is legalized discrimination. During Black History
Month, Americans congratulation themselves for having put an
end to discrimination against African Americans in
employment, housing, public benefits, and public
accommodations, School children wonder out loud how
discrimination could ever have been legal in this great land of
ours. Rarely are they told that it is still legal. Many of the
forms of discrimination that relegated African Americans to an
inferior caste during Jim Crow continue to apply to huge
segments of the black population today—provided they are first
labeled felons. If they are branded felons by the time they reach
the age of twenty-one, they are subject to legalized
discrimination for their entire adult life. Large majorities of
black men in cities across the United States are once again
subject to legalized discrimination effectively barring them
from full integration into mainstream white society. Mass
incarceration has nullified many of the gains of the Civil Rights
Movement, putting millions of black men back in a position
reminiscent of Jim Crow.
4
Political Disenfranchisement:
During the Jim Crow era, African Americans were denied
the right to vote through poll taxes, literacy tests, grandfather
clauses, and felon disenfranchisement laws, even though the
Fifteenth Amendment to the U. S. Constitution specifically
provides that the right of citizens of the United States to vote
shall not be denied. Formally race-neutral devices were
adopted to achieve the goal of all white electorate without
violating the terms of the Fifteenth Amendment. The devices
worked quite well. Because African Americans were poor, they
frequently could not pay poll takes. And because they had been
denied access to education, they could not pass literacy tests.
Grandfather clauses allowed whites to vote even if they couldn’t
meet the requirements, as long as their ancestors had been able
to vote. Finally, because blacks were disproportionately
charged with felonies—in fact some crimes were specifically
defined as
felonies with the goal of eliminating blacks from the
electorate—felony disenfranchisement laws effectively
suppressed the black vote as well. The failure of our legal
system to eradicate all the tactics adopted during the Jim Crow
era to suppress the black vote has major implications today.
Felon disenfranchisement laws have been more effective in
eliminating black voters in the age of mass incarceration than
they were during Jim Crow. Less than two decades after the War
on Drugs began, one in seven black men nationally had lost the
right to vote and as many as one in four in those states with the
highest African American disenfranchisement rate. There are
millions of ex-felons who cannot vote in states that require ex-
felons to pay fines or fees before their voting rights can be
restored—the new poll tax. As legal scholar Pamela Karlan has
observed, “felony disenfranchisement has decimated the
potential black electorate.
Exclusion from Juries:
Another clear parallel between mass incarceration and Jim
Crow is the systematic exclusion of blacks from juries. One
hallmark of the Jim Crow era was all white juries trying black
defendants to the South. Although the exclusion of jurors on
the basis of race has been illegal since 1880, as a practical
matter, the removal of prospective black jurors through race-
based peremptory strikes was sanctioned by the Supreme Court
until 1985, when the court ruled in Batson v. Kentucky that
racially biased strikes violate the equal protection clause of the
Fourteenth Amendment. Today, defendants face a situation
highly similar to the one they faced a century ago. A large
percentage of black men are automatically excluded from jury
service because they have been labeled felons. The combined
effort of race-based peremptory strikes and the automatic
exclusion of felons from juries has put black defendants in a
familiar place—a courtroom in shackles, facing an all-white
jury.
Closing the courtroom doors:
The parallels between mass incarceration and Jim Crow
extend all of the way to the U. S. Supreme Court. In Dred Scott
v. Sanford, the Supreme Court immunized the institution of
slavery from legal challenge on the grounds that African
Americans were not citizens and in Plessy v. Ferguson, the
Court established the doctrine of “separate but equal.”
Currently, McCleskey v. Kemp serve much the same function as
Dred Scott and Plessy. In McCleskey, the Supreme Court
demonstrated that it is once again in protection mode—firmly
committed to the prevailing system of control. The Court has
closed the courthouse doors to claims of racial bias
5
at every stage of the criminal justice process, from stops and
searches to plea bargaining and sentencing. Mass incarceration
is off limits to challenges on the grounds of racial bias much as
its predecessors were in their time.
Racial segregation:
There are other less obvious, similarities between mass
incarceration and Jim crow. Jim Crow laws mandated
residential segregation and blacks were relegated to the worst
parts of town. Roads literally stopped at the border of many
black neighborhoods, shifting from pavement to dirt. Water
sewer systems and other public services that supported the
white areas of town frequently did not extend to the black areas.
Racial segregation rendered black experience largely inevitable
to whites, making it easier for whites to maintain racial
stereotypes about black values and culture. It also made it easier
to deny or ignore their suffering.
Mass incarceration achieves racial segregation by
segregating prisoners-the majority of whom are black and
brown-from mainstream society. In a sense, incarceration is far
more extreme form of physical and residential segregation than
Jim Crow segregation. Rather than merely shunting black
people to the other side of towns or corralling them in ghettos,
mass incarceration locks them in cages a form of apartheid
unlike any the world has ever seen. Bars and walls keep
hundreds of thousands of black and brown people away from
mainstream society. Segregation is also created and perpetuated
by the flood of prisoners who return to ghetto communities each
year. According to one study, during a twelve-year period, the
number of prisoners returning home to core counties-those
counties that contain the inner city of metropolitan area-tripled.
Mass incarceration thus perpetuates and deepens pe-existing
patterns of racial segregation and isolation, not just by
removing people of color from society and putting them in
prisons, but by dumping them back into ghettos, upon their
release. Youth of color who might have escaped their ghetto
communities—or helped to transform them—if they had been
given a fair shot in life and not been labeled felons, instead find
themselves trapped in a closed circuit of perpetual marginality,
circulating between ghetto and prison.
Symbolic production of race:
The parallel between mass incarceration and Jim Crow is
that both have serve to define the meaning and significance of
race in America. African Americans are not significantly more
likely to use or sell prohibited drugs than whites, but they are
made criminals at drastically higher rates for precisely the same
conduct. In Fact, studies suggest that white professional may
be the most likely of any group to have engaged in illegal drug
activity in their lifetime, yet they are the least likely to be made
criminals. Mass incarceration has produced racial stigma. Racial
stigma is produced by defining negatively what it means to be
black. The stigma of race was more than shame of the slave,
then it was the shame of the second class citizen; today the
stigma of race is the shame of the criminal. The critical point
for black men, the stigma of being a criminal in the area of
mass incarceration is fundamentally a racial stigma. Mass
incarceration like Jim Crow is a “race-making institution. ”It
serves to define the meaning and significance of race in
America.
6
Differences between Jim Crow and Mass Incarceration:
Jim Crow was explicitly race-based, whereas mass
incarceration is not. This statement initially pears self-evident,
but it is partially mistaken. Although it is common to think of
Jim Crow as an explicitly race-based system, in fact a number
of the key policies were officially colorblind. Poll workers had
the discretion to charge a poll tax or administer a literacy test,
or not, and they exercised their discretion in a racially
discriminatory manner. Laws that said nothing about race
operated to discriminate because those charged with
enforcement were granted tremendous discretion, and they
exercised that discretion.
Laws prohibiting the use and sale of drugs are facially race
neutral, but they are enforced in a highly discriminatory
fashion. The decision to wage the drug war primarily in black
and brown communities rather than white ones and to target
African Americans but not whites on freeways and train stations
has had precisely the same effect as the literacy and poll taxes
of an earlier era. A facially race-neutral system of laws has
operated to create a racial caste system.
Other differences are the absence of overt racial hostility
among politicians who support harsh drug laws and the law
enforcement officials charged with enforcing them. Mass
incarceration, like Jim Crow, was born of racial opportunism—
an effort by white elites to exploit the racial hostilities,
resentments, and insecurities of poor and working-class whites.
Another difference is the racial stigma during the Jim Crow
era contributed to racial solidarity in the African American
community. Racial stigma today, however—that is, the stigma
of black criminals—has turned the black community against
itself, destroyed networks of mutual support, and created a
silence about the new caste system among many of the people
most affected by it. The black communities debate about the
underlying causes of mass incarceration. Some argue that it is
attributable primarily to racial bias and discrimination, others
maintain that it is due to poor education, unraveling morals, and
a lack of thrift and perseverance among the urban poor.
One more difference is the white victims of racial caste as
mass incarceration harms far more whites than the Jim Crow
era. Black and brown people are the principal targets in the
Drug War; white people are collateral damage. The War on
drugs was declared as part of a political ploy to capitalize on
white racial resentment against African Americans, and the
Reagan administration used the emergency of crack and its
related violence as an opportunity to build a racialized public
consensus in support of an all-out war—a consensus that almost
certainly would not have been formed if the primary users and
dealers of crack had been white.
The racial caste system does not carry out the positive
economic mission of recruitment and disciplining of the
workforce. Instead it serves only to warehouse poor black and
brown people for increasingly lengthy periods of time, often
until old age. The new system does not seek primarily to benefit
unfairly from black labor, as earlier caste systems have, but
instead views African Americans as largely irrelevant and
unnecessary to the newly structured economy—an economy that
is no longer driven by unskilled labor. It is fair that we have
witnessed an evolution in the United States from a racial caste
system based entirely on exploitation (slavery), to one based
largely on subordination (Jim Crow), to one defined by
marginalization (mass incarceration).
7
Reflection Essay on The New Jim Crow by Michelle Alexander
Class Assignment that will be due on Thursday, June 1, 2017.
Please type the “Reflective Essay,” double spaced, New Times
Roman 12 inch front and no cover sheet is required.
Please include your name, title of the class, date and reference
the title of the chapter.
You are to type a reflective essay by summarizing the chapter in
two or more paragraphs, writing the similarities and differences
of the Jim crow era and Mass Incarceration. You may want to
choose three or more themes from the list to elaborate on in
your essay. This essay will include your reflections, thoughts
and apply it to our class and text discussions and other
articles/topics referenced in class. A conclusion must be
included in the essay. The essay should be at least
two complete or more pages.
There are guidelines for the reflection essay that should be an
original work on two or more pages written in paragraph format,
double spaced, 12 inch font in New Times Roman, and MLA
format. You will:
* Demonstrate your understanding of the text’s content, purpose
and reflect on its significance.
*
Essay should include the title, introduction with the thesis
statement ( the sentence that states the main idea of a writing
assignment), detailed body of paragraphs with main
ideas/themes of the similarities and differences, paragraphs of
your reflections (thoughts, analysis, etc.); and conclusion.
This essay will demonstrate the experiential reflection question
of What? So What? And Now What? Summarizing the text.
(What?) Discussing the significance of the content (So What?)
and the actual or hypothetical application of the comments of
the text to your understanding and application today’s African
American community and American society is important. (Now
What?) What are the implications? A paper might discuss the
main ideas/themes, and new information that you encountered
and then discuss the use and value of specific content from this
text.
Questions to ask yourself as you are taking notes on the text and
to remember as you reflect on the text:
1. Did the author bring out any controversial, relevant, or
otherwise noteworthy information?
2. How do the issues or ideas brought up in this text align with
or contact with past readings, teachings, or your experiences?
3. How does history repeat itself?
4. Has the text changed or challenged your way of thinking
about the topic/current issues?
ESSAY EVALUATION:
The essay should be at least two complete pagesof text,
(consider the heading), typed,12 inch font, Times New Roman,
double spaced, include the title of the essay, an introduction,
body of paragraphs, reflection paragraphs, and conclusion.
Please cite the resource (s) used. Include a heading with your
name, date, class, and Professor’s name.
Essay Requirements:
Demonstrates complete and thorough reflective thoughts.
Shows understanding of the core content of the video and
discusses pertinent information, and provide application by
comparing and contrasting the information presented.
Expresses relationships among ideas, organization enhances
effectiveness of communication
Shows connection of video to class discussion or current events.
The assigned length and all content contribute to the directions
of the essay.
Essay is revised for correct spelling, mechanics and correct
paragraph format.
Essay must cite the resources used.
Most points will be included in the contents of the essay.
There will be deductions if all requirements are not met.
This essay is worth 25 points and due on June 1, 2017. LATE
ESSAYS WILL NOT BE ACCEPTED THE FOLLOWING
WEEK
Midterm 1 &
Solution
s.pdf
(1)Fourier-expandthefollowingperiodicinputforce:
2π
ω
π
ω
3π
ω
4π
ω
0−
π
ω
A
−A
f (t)
t
f (t) =
a0
2
+ a1 cos(ωt) + a2 cos(2ωt) + ...
+ b1 sin(ωt) + b2 sin(2ωt) + ...
=
a0
2
+ an cos(nωt) + bn sin(nωt)( )
n=1
∞
∑
Find the coefficients a0 , an, and bn
an =
ω
π
f (t) cos(nωt)dt
0
2π
ω∫ =
ω
π
Aω
π
t cos(nωt)dt
0
π
ω∫ +
ω
π
−A( )cos(nωt)dtπ
ω
2π
ω∫
=
Aω 2
π 2
t cos(nωt)dt
0
π
ω∫ −
Aω
π
cos(nωt)dtπ
ω
2π
ω∫
=
Aω 2
π 2
1
nω
t sin nωt( )+ 1
n2ω 2
cos(nωt)⎡
⎣ ⎢
⎤
⎦ ⎥ 0
π
ω
−
Aω
π
1
nω
sin nωt( )⎡
⎣ ⎢
⎤
⎦ ⎥ π
ω
2π
ω
=
Aω 2
π 2
0 +
1
n2ω 2
cos(nπ )⎛
⎝⎜
⎞
⎠⎟
− 0 +
1
n2ω 2
cos(0)⎛
⎝⎜
⎞
⎠⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
−
Aω
π
0 − 0[ ]
=
Aω 2
π 2
1
n2ω 2
(−1)n −
1
n2ω 2
⎛
⎝⎜
⎞
⎠⎟
=
A
n2π 2
−1( )n −1⎡ ⎣ ⎤ ⎦
=
−
2A
n2π 2
when n ∈ odd
0 when n ∈ even
⎧
⎨
⎪
⎩ ⎪
a0 =
ω
π
f (t)dt
0
2π
ω
∫
=
ω
π
Aω
π
tdt
0
π
ω
∫ +
ω
π
−A( )dt
π
ω
2π
ω
∫ =
Aω 2
π 2
tdt
0
π
ω
∫ −
Aω
π
1⋅ dt
π
ω
2π
ω
∫
=
Aω 2
π 2
1
2
t2⎛
⎝⎜
⎞
⎠⎟
0
π
ω
−
Aω
π
t( ) π
ω
2π
ω =
Aω 2
2π 2
π 2
ω 2
− 0
⎛
⎝⎜
⎞
⎠⎟
−
Aω
π
2π
ω
−
π
ω
⎛
⎝⎜
⎞
⎠⎟
=
A
2
− A
= −
A
2
Solu:on
bn =
ω
π
f (t)sin(nωt)dt
0
2π
ω∫ =
ω
π
Aω
π
t sin(nωt)dt
0
π
ω∫ +
ω
π
−A( )sin(nωt)dtπ
ω
2π
ω∫
=
Aω 2
π 2
t sin(nωt)dt
0
π
ω∫ −
Aω
π
sin(nωt)dtπ
ω
2π
ω∫
=
Aω 2
π 2
−
1
nω
t cos nωt( )+ 1
n2ω 2
sin(nωt)⎡
⎣ ⎢
⎤
⎦ ⎥ 0
π
ω
+
Aω
π
1
nω
cos nωt( )⎡
⎣ ⎢
⎤
⎦ ⎥ π
ω
2π
ω
=
Aω 2
π 2
−
π
nω 2
cos nπ( )+ 0⎛
⎝⎜
⎞
⎠⎟
− 0 + 0( )⎡
⎣ ⎢
⎤
⎦ ⎥
+
A
nπ
cos 2nπ( )− cos nπ( )⎡ ⎣ ⎤ ⎦
= −
A
nπ
−1( )n + A
nπ
1− −1( )n⎡ ⎣ ⎤ ⎦
=
A
nπ
1− 2(−1)n⎡ ⎣ ⎤ ⎦
=
3A
nπ
when n ∈ odd
−
A
nπ
when n ∈ even
⎧
⎨
⎪ ⎪
⎩
⎪
⎪
(2)Givenastandardmass-spring-dampermechanicalsystem:
!!x + 2 !x + 5x = f (t), 0 < ς < 1
f (t) is a periodic force with Fourier expansion:
f (t) =
2
π
1
kk=1
∞
∑ cos kωt( )
Calculate the steady state response of x, which is xp, the
particular soluiton
for !!x + 2 !x + 5x = f (t).
Hint : xp = xp,k
k=0
∞
∑ = αk cos(kωt) + βk sin(kωt)
k=0
∞
∑ ,
Calculate αk & βk
!!x + 2 !x + 5x = f (t) =
2
π
1
kn=0
∞
∑ cos kωt( )
xp,k = αk cos(kωt)+ βk sin(kωt)
⇒
!xp,k = −αkkω sin(kωt)+ βkkω cos(kωt)
!!xp,k = −αkk
2ω 2 cos(kωt)− βkk
2ω 2 sin(kωt)
⇒
−αkk
2ω 2 cos(kωt)− βkk
2ω 2 sin(kωt)+ 2 −αkkω sin(kωt)+ βkkω cos(kωt)[ ]+ 5 αk
cos(kωt)+ βk sin(kωt)[ ]
=
2
kπ
cos kωt( )
= −k2ω 2αk + 2βkkω + 5αk( )cos(kωt)+ −βkk2ω 2 − 2αkkω +
5βk( )sin(kωt)
= 5− k2ω 2( )αk + 2kω ⋅ βk⎡ ⎣ ⎤ ⎦ cos(kωt)+ −2kω ⋅ αk + 5−
k2ω 2( )⋅ βk⎡ ⎣ ⎤ ⎦ sin(kωt)
⇒
5− k2ω 2( )αk + 2kω ⋅ βk = 2kπ
−2kω ⋅ αk + 5− k
2ω 2( )⋅ βk = 0
⎧
⎨
⎪
⎩
⎪
⇒
αk
βk
⎡
⎣
⎢
⎤
⎦
⎥ =
5− k2ω 2 2kω
−2kω 5− k2ω 2
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
−1 2
kπ
0
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
1
5− k2ω 2( )2 + 2kω( )2
5− k2ω 2 −2kω
2kω 5− k2ω 2
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
2
kπ
0
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
1
5− k2ω 2( )2 + 2kω( )2
2
kπ
5− k2ω 2( )
4ω
π
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
Solu:on
⇒
xp =
1
5− k2ω 2( )2 + 2kω( )2
2
kπ
5− k2ω 2( )cos kωt( )+ 4ω
π
sin kωt( )⎡
⎣ ⎢
⎤
⎦ ⎥ k=0
∞
∑
=
2
kπ
⋅
1
ωn
2 − k2ω 2( )2 + 2kω( )2
5− k2ω 2( )cos kωt( )+ 2kω sin kωt( )⎡ ⎣ ⎤ ⎦
k=0
∞
∑
m
J
y = 2sin 5t( )
x
b
k
(3)
A block with mass m is suspended inside a box via a spring (k)
and a damper (b). A roller with roatry inertia J
and radius r is hinged to the box. The block and the roller are
coupled by a set of gear and gear rack, such that
x = rθ. The box is shaked up and down following a function y =
2 sin 5t( ).
r
θ
This system can be described by a standard mass-spring-
damper-force format:
M!!x + B!x + Kx = F
Calculate M, B, K, and F.
Hint:
You can use either of the following methods to solve this
problem:
(1) Calculate the equivalent mass with respect to displacement
x.
(2) Assume there is a tangent force between the block and the
roller, layout two dynamics equations using
F∑ = m!!x and F ⋅ r∑ = J!!θ, and finally combine all the
equatoin into one using x = rθ.
m!!x = −k x − y( )− b !x − !y( )−T
J!!θ = Tr
x = rθ
⇒
T = J
!!θ
r
= J
!!x
r2
m!!x = −k x − y( )− b !x − !y( )− J
r2
!!x
m +
J
r2
⎛
⎝⎜
⎞
⎠⎟
!!x + b!x + kx = ky + b!y
y = 2sin 5t( )
!y =10cos 5t( )
⇒
m +
J
r2
⎛
⎝⎜
⎞
⎠⎟
!!x + b!x + kx = 2ksin 5t( )+10bcos 5t( )
Solu:on
m
J
r
T
T
b !x − !y( )
k x − y( )
Midterm 2 &
Alexander, Michelle, The New Jim Crow Mass Incarceration in the A.docx

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Alexander, Michelle, The New Jim Crow Mass Incarceration in the A.docx

  • 1. Alexander, Michelle, The New Jim Crow: Mass Incarceration in the Age of Colorblindness Chapter 5: The New Jim Crow It was no ordinary Sunday morning when presidential candidate Barack Obama stepped to the podium at the Apostolic Church of God in Chicago. It was Father’s Day. Hundreds of enthusiastic congregants packed the pews at the overwhelmingly black church eager to hear what the first black Democratic nominee for president of the United States had to say. The message was a familiar one: black men should better fathers. Too many are absent from their homes. For those in the audience, Obama’s speech was an old tune sung by an exciting new performer. His message of personal responsibility, particularly as it relates to fatherhood, was anything but new; it had been delivered countless times by black ministers in churches across America. The message had also been delivered on a national stage by celebrities such as Bill Crosby and Sidney Poitier. And the message had been delivered with great passion by Louis Farrakhan, who more than a decade earlier summoned one million black men to Washington, D. C. for a day of “atonement” and recommitment to their families and communities. The mainstream media, however, treated the event as big news, and many pundits seemed surprised that the black congregants actually applauded the message. For them, it was remarkable that black people nodded in approval when Barack Obama said “If we are honest with ourselves, we’ll admit that too many fathers are missing-missing from too many lives and too man y homes. Too many fathers are MIA. Too many fathers are AWOL. They have abandoned their responsibilities. They’re acting like boys instead of men. And the foundations of our families are weaker because of it. You and I know this is true everywhere, but nowhere is this more true than in the
  • 2. African American community.” The media did not ask—and Obama did not tell—where the missing facts might be found. The following day, social critical and sociologist Michael Eric Dyson published a critique of Obama’s speech in Time magazine. He pointed out that the stereotype of black men being poor fathers may well be false. Research by Boston College social psychologist Rebekah Levine Coley found that black fathers not living at home are more likely to keep in contact with their children than fathers of any other ethnic or racial group. Dyson chided Obama for evoking a black stereotype for political gain, pointing out that “Obama’s words may have been spoken to 1 black folk, but they were aimed at those whites still on the fence about home to send to the White House.” Dyson’s critique was a fair one, but like other media commentators he remained silent about where all the absent black fathers could be found. He identified numerous social problems plaguing black families such as high levels of unemployment, discriminatory mortgage practices, and the gutting of early-childhood learning programs. Not a word was said about the prisons. The public discourse regarding “missing black fathers” closely parallels he debate about the lack of eligible black men for marriage. The majority of black women are unmarried today, including 70 percentage of professional black women. Where have all the black men gone? Is a common refrain heard among black women frustrated in their efforts to find life partners. The sense that black men have disappeared is rooted in reality. The U. S. Census Bureau reported in 2002 that there are nearly 3 million more black adult women than men in black communities across the United States, a gender a gap for 26 percent. In many urban areas, the gap is far worse, rising to
  • 3. more than 37 percent in places like New York City. The comparable disparity for whites in the United States is 8 percent. Although a million black men can be found in prisons and jails, public acknowledgement of the role of the criminal justice system in “disappearing” black men is surprisingly rare. Even in the black media—which is generally more willing o raise and tackle issues related to criminal justice—an eerie silence can often be found. The Journal of Blacks in Higher Education reports the fact that black women hold a large lead over black men in almost every facet of higher education. Black women currently earn about two thirds of all African-American bachelor's degree awards, 70 percent of all master's degrees, and more than 60 percent of all doctorates. Black women also hold a majority of all African-American enrollments in law, medical, and dental schools. Looking exclusively to undergraduate higher education, the latest Department of Education figures show that black women account for 63.6 percent of all African-American enrollment. Ebony magazine for example ran an article in December 2006 entitled “Where Have the Black Men Gone?” The author posed the popular question but never answered it. He suggested we will find our black men when we rediscover God, family and self-respect. The fact that Barack Obama can give a speech on Father’s Day dedicated to the subject of fathers who are AWOL without ever acknowledging the majority of young black men in many large urban areas are currently under the control of the criminal justice system is disturbing to say the least. What is more problematic though, is that hardly anyone in the mainstream media noticed the oversight. Hundreds of thousands of black men are unable to be good fathers because they are warehoused in prisons, locked in cages. They did not walk out on their families voluntarily; they were taken away in handcuffs, often due to a massive federal program known as the War on Drugs. More African American adults are under correctional control
  • 4. today—in prison or jail, on probation or parole—than were enclosed in 1850, a decade before the Civil War began. The 2 mass incarceration of people of color is a big part of the reason that a black child born today is less likely to be raised by both parents than a black children born during slavery. The absence of black fathers from families across American is not simply a function of laziness, immaturity, or too much time watching Sports Center. Thousands of black men have disappeared into prisons and jails, locked away for drug crimes that are largely ignored when committed by whites. Black people only represent 13.3 percent of the U.S. population while white people make up about 77 percent. But there are more black men in state prisons across the nation than there are white men. The latest Federal Bureau of Prisons statistics showed the white population was 58.7 percent in August, compared to the 37.8 percent of blacks being housed at federal institutions. At the state level, however, the department said that black male prisoners represent 38 percent of the population while white males account for 35 percent and 21 percent are Hispanic males. Overall, blacks are 5.1 more times likely to be incarcerated than whites, and blacks represent more than half of the prison population in 11 states. The clock has been turned back on racial progress in America, though scarcely anyone seems to notice. All eyes are fixed on people like Barack Obama, Oprah Winfrey, and Robert L. Johnson who have defied the odds and risen to power, family and a great future. For those left behind, especially those within prison walls, the celebration of racial triumph in American must seem a tad premature. More black men are imprisoned today than any other moment in our nation’s history. More are disenfranchised today than in 1870, the year the Fifteenth Amendment was ratified to laws that explicitly deny the right to vote on the basis of race. Young black men today
  • 5. may be just as likely to suffer discrimination employment, housing, public benefits, and jury service as a black man in the Jim Crow era—discrimination that is perfectly legal, because it is based on one’s criminal record. This is the new normal, the new racial equilibrium. The launching of the War on Drugs and the initial construction for the new system required the expenditure of tremendous political initiative and resources. Media campaigns were waged; politicians blasted soft judges and enacted harsh sentencing laws; poor people of color were vilified. The system now, however, requires very little maintenance or justification. In fact if you are white and middle class, you might not even realize the drug war is still going on. Most high school and college students today have no recollection of the political and media frenzy surround the drug war in the early years. They were young children when the war was declared, or not even born yet. Crack is out; terrorism is in. Today, the political fanfare and the vehement, racialized rhetoric regarding crime and drugs are no longer necessary. Mass incarceration has been normalized, an all of the racial stereotypes and assumptions that give rise to the system are now embraced (or at least internalized) by people of all colors, from all walks of life, and in every major political party. We may wonder aloud “where have the black men gone?” but deep down we already know. It is simply taken for granted that, in cities like Baltimore and Chicago, the vast majority of young black men are currently under the control of the criminal justice system or branded criminals for life. This extraordinary circumstance—unheard of in the rest of the world—is treated here in America as a basic fact of life, as normal as separate water fountains were just a half century ago. 3 Those cycling in and out of Illinois prisons today are
  • 6. members of America’s new racial undercaste. The United States has almost always had a racial undercaste –a group defined wholly or largely by race that is permanently locked out of mainstream, white society by law, custom, and practice. The reasons and justifications change over time, as each new caste system reflects and adapts to changes in the social, political, and economic context. There are important differences between mass incarceration and Jim Crow and a profound sense of déjà vu. There is a familiar stigma and shame. There is an elaborate system of control, complete with political disenfranchisement and legalized discrimination in every major realm of economic and social life. Parallels Between Jim Crow and Mass Incarceration: Listed below are several of the most obvious similarities between Jim Crow and mass incarceration. Historical parallels: Jim Crow and mass incarceration have similar political origins. Both the caste systems were born in part due to a desire among white elites to exploit the resentments, vulnerabilities, and racial biases of poor and working class whites for political or economic gain. Segregation laws were proposed as part of a deliberate and strategic effort to deflect anger and hostility that had been brewing against the white elite away from them and toward African Americans. The birth of mass incarceration can be traced to a similar political dynamic. Conservatives in the 1970s and 1980s sought to appeal to the racial biases and economic vulnerabilities of poor and working class whites through racially coded rhetoric on crime and welfare. In the early years of Jim Crow, conservative white elites opted with each other by passing ever more stringent and oppressive Jim Crow legislation. A century later, politicians in the early years of the drug war competed with each other to prove who could be tougher on crime by passing ever harsher drug laws—a thinly veiled effort to appeal to poor and working class whites who, once again, proved they were willing to
  • 7. forego economic and structural reform in exchange for an apparent effort to put blacks back in their place. Legalized discrimination: The most obvious parallel between Jim Crow and mass incarceration is legalized discrimination. During Black History Month, Americans congratulation themselves for having put an end to discrimination against African Americans in employment, housing, public benefits, and public accommodations, School children wonder out loud how discrimination could ever have been legal in this great land of ours. Rarely are they told that it is still legal. Many of the forms of discrimination that relegated African Americans to an inferior caste during Jim Crow continue to apply to huge segments of the black population today—provided they are first labeled felons. If they are branded felons by the time they reach the age of twenty-one, they are subject to legalized discrimination for their entire adult life. Large majorities of black men in cities across the United States are once again subject to legalized discrimination effectively barring them from full integration into mainstream white society. Mass incarceration has nullified many of the gains of the Civil Rights Movement, putting millions of black men back in a position reminiscent of Jim Crow. 4 Political Disenfranchisement: During the Jim Crow era, African Americans were denied the right to vote through poll taxes, literacy tests, grandfather clauses, and felon disenfranchisement laws, even though the Fifteenth Amendment to the U. S. Constitution specifically provides that the right of citizens of the United States to vote shall not be denied. Formally race-neutral devices were adopted to achieve the goal of all white electorate without violating the terms of the Fifteenth Amendment. The devices worked quite well. Because African Americans were poor, they
  • 8. frequently could not pay poll takes. And because they had been denied access to education, they could not pass literacy tests. Grandfather clauses allowed whites to vote even if they couldn’t meet the requirements, as long as their ancestors had been able to vote. Finally, because blacks were disproportionately charged with felonies—in fact some crimes were specifically defined as felonies with the goal of eliminating blacks from the electorate—felony disenfranchisement laws effectively suppressed the black vote as well. The failure of our legal system to eradicate all the tactics adopted during the Jim Crow era to suppress the black vote has major implications today. Felon disenfranchisement laws have been more effective in eliminating black voters in the age of mass incarceration than they were during Jim Crow. Less than two decades after the War on Drugs began, one in seven black men nationally had lost the right to vote and as many as one in four in those states with the highest African American disenfranchisement rate. There are millions of ex-felons who cannot vote in states that require ex- felons to pay fines or fees before their voting rights can be restored—the new poll tax. As legal scholar Pamela Karlan has observed, “felony disenfranchisement has decimated the potential black electorate. Exclusion from Juries: Another clear parallel between mass incarceration and Jim Crow is the systematic exclusion of blacks from juries. One hallmark of the Jim Crow era was all white juries trying black defendants to the South. Although the exclusion of jurors on the basis of race has been illegal since 1880, as a practical matter, the removal of prospective black jurors through race- based peremptory strikes was sanctioned by the Supreme Court until 1985, when the court ruled in Batson v. Kentucky that racially biased strikes violate the equal protection clause of the Fourteenth Amendment. Today, defendants face a situation
  • 9. highly similar to the one they faced a century ago. A large percentage of black men are automatically excluded from jury service because they have been labeled felons. The combined effort of race-based peremptory strikes and the automatic exclusion of felons from juries has put black defendants in a familiar place—a courtroom in shackles, facing an all-white jury. Closing the courtroom doors: The parallels between mass incarceration and Jim Crow extend all of the way to the U. S. Supreme Court. In Dred Scott v. Sanford, the Supreme Court immunized the institution of slavery from legal challenge on the grounds that African Americans were not citizens and in Plessy v. Ferguson, the Court established the doctrine of “separate but equal.” Currently, McCleskey v. Kemp serve much the same function as Dred Scott and Plessy. In McCleskey, the Supreme Court demonstrated that it is once again in protection mode—firmly committed to the prevailing system of control. The Court has closed the courthouse doors to claims of racial bias 5 at every stage of the criminal justice process, from stops and searches to plea bargaining and sentencing. Mass incarceration is off limits to challenges on the grounds of racial bias much as its predecessors were in their time. Racial segregation: There are other less obvious, similarities between mass incarceration and Jim crow. Jim Crow laws mandated residential segregation and blacks were relegated to the worst parts of town. Roads literally stopped at the border of many black neighborhoods, shifting from pavement to dirt. Water sewer systems and other public services that supported the white areas of town frequently did not extend to the black areas. Racial segregation rendered black experience largely inevitable to whites, making it easier for whites to maintain racial
  • 10. stereotypes about black values and culture. It also made it easier to deny or ignore their suffering. Mass incarceration achieves racial segregation by segregating prisoners-the majority of whom are black and brown-from mainstream society. In a sense, incarceration is far more extreme form of physical and residential segregation than Jim Crow segregation. Rather than merely shunting black people to the other side of towns or corralling them in ghettos, mass incarceration locks them in cages a form of apartheid unlike any the world has ever seen. Bars and walls keep hundreds of thousands of black and brown people away from mainstream society. Segregation is also created and perpetuated by the flood of prisoners who return to ghetto communities each year. According to one study, during a twelve-year period, the number of prisoners returning home to core counties-those counties that contain the inner city of metropolitan area-tripled. Mass incarceration thus perpetuates and deepens pe-existing patterns of racial segregation and isolation, not just by removing people of color from society and putting them in prisons, but by dumping them back into ghettos, upon their release. Youth of color who might have escaped their ghetto communities—or helped to transform them—if they had been given a fair shot in life and not been labeled felons, instead find themselves trapped in a closed circuit of perpetual marginality, circulating between ghetto and prison. Symbolic production of race: The parallel between mass incarceration and Jim Crow is that both have serve to define the meaning and significance of race in America. African Americans are not significantly more likely to use or sell prohibited drugs than whites, but they are made criminals at drastically higher rates for precisely the same conduct. In Fact, studies suggest that white professional may be the most likely of any group to have engaged in illegal drug
  • 11. activity in their lifetime, yet they are the least likely to be made criminals. Mass incarceration has produced racial stigma. Racial stigma is produced by defining negatively what it means to be black. The stigma of race was more than shame of the slave, then it was the shame of the second class citizen; today the stigma of race is the shame of the criminal. The critical point for black men, the stigma of being a criminal in the area of mass incarceration is fundamentally a racial stigma. Mass incarceration like Jim Crow is a “race-making institution. ”It serves to define the meaning and significance of race in America. 6 Differences between Jim Crow and Mass Incarceration: Jim Crow was explicitly race-based, whereas mass incarceration is not. This statement initially pears self-evident, but it is partially mistaken. Although it is common to think of Jim Crow as an explicitly race-based system, in fact a number of the key policies were officially colorblind. Poll workers had the discretion to charge a poll tax or administer a literacy test, or not, and they exercised their discretion in a racially discriminatory manner. Laws that said nothing about race operated to discriminate because those charged with enforcement were granted tremendous discretion, and they exercised that discretion. Laws prohibiting the use and sale of drugs are facially race neutral, but they are enforced in a highly discriminatory fashion. The decision to wage the drug war primarily in black and brown communities rather than white ones and to target African Americans but not whites on freeways and train stations has had precisely the same effect as the literacy and poll taxes of an earlier era. A facially race-neutral system of laws has operated to create a racial caste system.
  • 12. Other differences are the absence of overt racial hostility among politicians who support harsh drug laws and the law enforcement officials charged with enforcing them. Mass incarceration, like Jim Crow, was born of racial opportunism— an effort by white elites to exploit the racial hostilities, resentments, and insecurities of poor and working-class whites. Another difference is the racial stigma during the Jim Crow era contributed to racial solidarity in the African American community. Racial stigma today, however—that is, the stigma of black criminals—has turned the black community against itself, destroyed networks of mutual support, and created a silence about the new caste system among many of the people most affected by it. The black communities debate about the underlying causes of mass incarceration. Some argue that it is attributable primarily to racial bias and discrimination, others maintain that it is due to poor education, unraveling morals, and a lack of thrift and perseverance among the urban poor. One more difference is the white victims of racial caste as mass incarceration harms far more whites than the Jim Crow era. Black and brown people are the principal targets in the Drug War; white people are collateral damage. The War on drugs was declared as part of a political ploy to capitalize on white racial resentment against African Americans, and the Reagan administration used the emergency of crack and its related violence as an opportunity to build a racialized public consensus in support of an all-out war—a consensus that almost certainly would not have been formed if the primary users and dealers of crack had been white. The racial caste system does not carry out the positive economic mission of recruitment and disciplining of the workforce. Instead it serves only to warehouse poor black and brown people for increasingly lengthy periods of time, often until old age. The new system does not seek primarily to benefit
  • 13. unfairly from black labor, as earlier caste systems have, but instead views African Americans as largely irrelevant and unnecessary to the newly structured economy—an economy that is no longer driven by unskilled labor. It is fair that we have witnessed an evolution in the United States from a racial caste system based entirely on exploitation (slavery), to one based largely on subordination (Jim Crow), to one defined by marginalization (mass incarceration). 7 Reflection Essay on The New Jim Crow by Michelle Alexander Class Assignment that will be due on Thursday, June 1, 2017. Please type the “Reflective Essay,” double spaced, New Times Roman 12 inch front and no cover sheet is required. Please include your name, title of the class, date and reference the title of the chapter. You are to type a reflective essay by summarizing the chapter in two or more paragraphs, writing the similarities and differences of the Jim crow era and Mass Incarceration. You may want to choose three or more themes from the list to elaborate on in your essay. This essay will include your reflections, thoughts and apply it to our class and text discussions and other articles/topics referenced in class. A conclusion must be included in the essay. The essay should be at least two complete or more pages. There are guidelines for the reflection essay that should be an original work on two or more pages written in paragraph format, double spaced, 12 inch font in New Times Roman, and MLA format. You will: * Demonstrate your understanding of the text’s content, purpose and reflect on its significance. * Essay should include the title, introduction with the thesis statement ( the sentence that states the main idea of a writing assignment), detailed body of paragraphs with main ideas/themes of the similarities and differences, paragraphs of
  • 14. your reflections (thoughts, analysis, etc.); and conclusion. This essay will demonstrate the experiential reflection question of What? So What? And Now What? Summarizing the text. (What?) Discussing the significance of the content (So What?) and the actual or hypothetical application of the comments of the text to your understanding and application today’s African American community and American society is important. (Now What?) What are the implications? A paper might discuss the main ideas/themes, and new information that you encountered and then discuss the use and value of specific content from this text. Questions to ask yourself as you are taking notes on the text and to remember as you reflect on the text: 1. Did the author bring out any controversial, relevant, or otherwise noteworthy information? 2. How do the issues or ideas brought up in this text align with or contact with past readings, teachings, or your experiences? 3. How does history repeat itself? 4. Has the text changed or challenged your way of thinking about the topic/current issues? ESSAY EVALUATION: The essay should be at least two complete pagesof text, (consider the heading), typed,12 inch font, Times New Roman, double spaced, include the title of the essay, an introduction, body of paragraphs, reflection paragraphs, and conclusion. Please cite the resource (s) used. Include a heading with your name, date, class, and Professor’s name. Essay Requirements: Demonstrates complete and thorough reflective thoughts. Shows understanding of the core content of the video and discusses pertinent information, and provide application by comparing and contrasting the information presented. Expresses relationships among ideas, organization enhances
  • 15. effectiveness of communication Shows connection of video to class discussion or current events. The assigned length and all content contribute to the directions of the essay. Essay is revised for correct spelling, mechanics and correct paragraph format. Essay must cite the resources used. Most points will be included in the contents of the essay. There will be deductions if all requirements are not met. This essay is worth 25 points and due on June 1, 2017. LATE ESSAYS WILL NOT BE ACCEPTED THE FOLLOWING WEEK
  • 17. 0− π ω A −A f (t) t f (t) = a0 2 + a1 cos(ωt) + a2 cos(2ωt) + ... + b1 sin(ωt) + b2 sin(2ωt) + ... = a0 2 + an cos(nωt) + bn sin(nωt)( )
  • 18. n=1 ∞ ∑ Find the coefficients a0 , an, and bn an = ω π f (t) cos(nωt)dt 0 2π ω∫ = ω π Aω π
  • 19. t cos(nωt)dt 0 π ω∫ + ω π −A( )cos(nωt)dtπ ω 2π ω∫ = Aω 2 π 2 t cos(nωt)dt 0 π
  • 20. ω∫ − Aω π cos(nωt)dtπ ω 2π ω∫ = Aω 2 π 2 1 nω t sin nωt( )+ 1 n2ω 2 cos(nωt)⎡ ⎣ ⎢ ⎤
  • 21. ⎦ ⎥ 0 π ω − Aω π 1 nω sin nωt( )⎡ ⎣ ⎢ ⎤ ⎦ ⎥ π ω 2π ω = Aω 2 π 2
  • 22. 0 + 1 n2ω 2 cos(nπ )⎛ ⎝⎜ ⎞ ⎠⎟ − 0 + 1 n2ω 2 cos(0)⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣ ⎢ ⎤
  • 23. ⎦ ⎥ − Aω π 0 − 0[ ] = Aω 2 π 2 1 n2ω 2 (−1)n − 1 n2ω 2 ⎛ ⎝⎜ ⎞ ⎠⎟
  • 24. = A n2π 2 −1( )n −1⎡ ⎣ ⎤ ⎦ = − 2A n2π 2 when n ∈ odd 0 when n ∈ even ⎧ ⎨ ⎪ ⎩ ⎪ a0 = ω π
  • 26. −A( )dt π ω 2π ω ∫ = Aω 2 π 2 tdt 0 π ω ∫ − Aω π 1⋅ dt π
  • 28. t( ) π ω 2π ω = Aω 2 2π 2 π 2 ω 2 − 0 ⎛ ⎝⎜ ⎞ ⎠⎟ − Aω π 2π
  • 30. ω π f (t)sin(nωt)dt 0 2π ω∫ = ω π Aω π t sin(nωt)dt 0 π ω∫ + ω π −A( )sin(nωt)dtπ
  • 31. ω 2π ω∫ = Aω 2 π 2 t sin(nωt)dt 0 π ω∫ − Aω π sin(nωt)dtπ ω 2π ω∫
  • 32. = Aω 2 π 2 − 1 nω t cos nωt( )+ 1 n2ω 2 sin(nωt)⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 0 π ω + Aω π 1
  • 33. nω cos nωt( )⎡ ⎣ ⎢ ⎤ ⎦ ⎥ π ω 2π ω = Aω 2 π 2 − π nω 2 cos nπ( )+ 0⎛ ⎝⎜
  • 34. ⎞ ⎠⎟ − 0 + 0( )⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + A nπ cos 2nπ( )− cos nπ( )⎡ ⎣ ⎤ ⎦ = − A nπ −1( )n + A nπ 1− −1( )n⎡ ⎣ ⎤ ⎦ = A nπ
  • 35. 1− 2(−1)n⎡ ⎣ ⎤ ⎦ = 3A nπ when n ∈ odd − A nπ when n ∈ even ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪
  • 36. (2)Givenastandardmass-spring-dampermechanicalsystem: !!x + 2 !x + 5x = f (t), 0 < ς < 1 f (t) is a periodic force with Fourier expansion: f (t) = 2 π 1 kk=1 ∞ ∑ cos kωt( ) Calculate the steady state response of x, which is xp, the particular soluiton for !!x + 2 !x + 5x = f (t).
  • 37. Hint : xp = xp,k k=0 ∞ ∑ = αk cos(kωt) + βk sin(kωt) k=0 ∞ ∑ , Calculate αk & βk !!x + 2 !x + 5x = f (t) = 2 π 1 kn=0 ∞
  • 38. ∑ cos kωt( ) xp,k = αk cos(kωt)+ βk sin(kωt) ⇒ !xp,k = −αkkω sin(kωt)+ βkkω cos(kωt) !!xp,k = −αkk 2ω 2 cos(kωt)− βkk 2ω 2 sin(kωt) ⇒ −αkk 2ω 2 cos(kωt)− βkk 2ω 2 sin(kωt)+ 2 −αkkω sin(kωt)+ βkkω cos(kωt)[ ]+ 5 αk cos(kωt)+ βk sin(kωt)[ ] = 2 kπ cos kωt( ) = −k2ω 2αk + 2βkkω + 5αk( )cos(kωt)+ −βkk2ω 2 − 2αkkω + 5βk( )sin(kωt) = 5− k2ω 2( )αk + 2kω ⋅ βk⎡ ⎣ ⎤ ⎦ cos(kωt)+ −2kω ⋅ αk + 5−
  • 39. k2ω 2( )⋅ βk⎡ ⎣ ⎤ ⎦ sin(kωt) ⇒ 5− k2ω 2( )αk + 2kω ⋅ βk = 2kπ −2kω ⋅ αk + 5− k 2ω 2( )⋅ βk = 0 ⎧ ⎨ ⎪ ⎩ ⎪ ⇒ αk βk ⎡ ⎣ ⎢ ⎤
  • 40. ⎦ ⎥ = 5− k2ω 2 2kω −2kω 5− k2ω 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ −1 2 kπ 0 ⎡
  • 41. ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = 1 5− k2ω 2( )2 + 2kω( )2 5− k2ω 2 −2kω 2kω 5− k2ω 2 ⎡ ⎣ ⎢ ⎢
  • 43. 1 5− k2ω 2( )2 + 2kω( )2 2 kπ 5− k2ω 2( ) 4ω π ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥
  • 44. ⎥ ⎥ Solu:on ⇒ xp = 1 5− k2ω 2( )2 + 2kω( )2 2 kπ 5− k2ω 2( )cos kωt( )+ 4ω π sin kωt( )⎡ ⎣ ⎢ ⎤ ⎦ ⎥ k=0 ∞
  • 45. ∑ = 2 kπ ⋅ 1 ωn 2 − k2ω 2( )2 + 2kω( )2 5− k2ω 2( )cos kωt( )+ 2kω sin kωt( )⎡ ⎣ ⎤ ⎦ k=0 ∞ ∑ m J
  • 46. y = 2sin 5t( ) x b k (3) A block with mass m is suspended inside a box via a spring (k) and a damper (b). A roller with roatry inertia J and radius r is hinged to the box. The block and the roller are coupled by a set of gear and gear rack, such that x = rθ. The box is shaked up and down following a function y = 2 sin 5t( ). r θ This system can be described by a standard mass-spring- damper-force format: M!!x + B!x + Kx = F Calculate M, B, K, and F.
  • 47. Hint: You can use either of the following methods to solve this problem: (1) Calculate the equivalent mass with respect to displacement x. (2) Assume there is a tangent force between the block and the roller, layout two dynamics equations using F∑ = m!!x and F ⋅ r∑ = J!!θ, and finally combine all the equatoin into one using x = rθ. m!!x = −k x − y( )− b !x − !y( )−T J!!θ = Tr x = rθ ⇒ T = J !!θ r = J !!x
  • 48. r2 m!!x = −k x − y( )− b !x − !y( )− J r2 !!x m + J r2 ⎛ ⎝⎜ ⎞ ⎠⎟ !!x + b!x + kx = ky + b!y y = 2sin 5t( ) !y =10cos 5t( ) ⇒ m + J r2
  • 49. ⎛ ⎝⎜ ⎞ ⎠⎟ !!x + b!x + kx = 2ksin 5t( )+10bcos 5t( ) Solu:on m J r T T b !x − !y( ) k x − y( ) Midterm 2 &