Mechanical Vibration Forced Undamped

7/31/2019 Mechanical Vibration Forced Undamped

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BFF3103 Mechanical Vibration

Forced Vibration Undamped


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Lesson-03 Objectives

Students will be able to:

Obtain the harmonic response of systemshaving a SDOF

Obtain the transfer function from the EOM

Determine the total response frequency, peakresponse and bandwidth

Analyze the displacement and transmitted force

of the systems having base excitation

Solve the problem related to undamped anddamped harmonic response SDOF vibration


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Todays Objectives

Students will be able to: Identify the harmonic response of SDOF

Obtained the response of an undamped

system under harmonic force

Solve the problem related to damped SDOF

force vibration


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Forced vibrations occurs when external

energy is supplied to the system during

vibration

The external force can be supplied

through either an applied force or an

imposed displacement excitation, which

may be harmonic, nonharmonic but

periodic, nonperiodic, or random in

nature.


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Harmonic response results when thesystem responses to a harmonicexcitation

Transient response is defined as theresponse of a dynamic system tosuddenly applied nonperiodic

excitations


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There are 4 categories of F(t):

(1) Harmonic (sin, cos) (3) Transient

(2) Periodic (4) Random


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Equation of Motion

The equation of Motion Using Newtons Second

Law of Motion:

)(tFkxxcxm

The homogeneous

solution of the equation:

0 kxxcxm


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Response of an Undamped System Under

Harmonic Force

tFtF cos)( 0

tFkxxm cos0

tnDtnCthx sincos)(

The homogeneous solution is

where is the natural frequency.2/1)/( mkn


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where denotes the static deflection

where X is the maximum amplitude ofxp(t)

tXtxp cos)(

220

1

n

st

mkFX

kFst

/0

Thus, t

mk

FtnDtnCtx

cos

2

0sincos)(

Because the exciting force and particular solution is

harmonic and has same frequency, we can assume a

solution in the form:


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Using initial conditions

Hence,

00

)0(and)0( xtxxtx

n

xD

mk

FxC

0,2

00

tmk

F

tx

tmk

Fxtx

n

n

n

cos

sincos)(

2

0

0

2

0

0

The max amplitude: 2

1

1

n

st

X


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where the quantity is

called the magnificationfactor, amplification factor, or

amplitude ratio. The

response of the system can

be identified to be of three

types from the figure.

stX /


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Case 1. When 0 < < 1, the denominator is positive

and the response is given without change. The

harmonic response of the system is in phase with

external force.

n/

2

1

1

n

st

X


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where the amplitude,

Case 2. When > 1, the denominator is negative

and the steady-state solution can be expressed asn/

tXtxp cos)(

1

2

n

stX


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Case 3. When = 1, the amplitude X given

becomes infinite. This condition, for which the forcing

frequency is equal to the natural frequency of the

system, is called resonance. Hence, the total responseif the system at resonance becomes

tt

tx

txtx

n

nst

n

n

n

sin2

sincos)( 00

n/


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Total Response

1for;cos

1

)cos()( 2

n

n

stn ttAtx


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1for;cos

1

)cos()(2

n

n

stn ttAtx

and


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Beating Phenomenon

If the forcing frequency is close to, but not exactly equal

to, the natural frequency of the system, beating may

occur. The phenomenon of beating can be expressed as:

ttmF

tx

sinsin

2

/)( 0


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The time between the points of zero amplitude or the

points of maximum amplitude is called theperiod of

beating and is given by

n

b

2

2

2

The frequency of beating defined as

nb 2

Mechanical Vibration Forced Undamped

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