Analysis of a Clamped Beam under Differential Support Displacements, Slides of Stochastic Processes

An in-depth analysis of a clamped beam under differential support displacements. It covers the concepts of displacement, slope, bending moment, and shear stress. The document also discusses the time-dependent nature of the boundary-condition solutions (bcs) and introduces new dependent variables and selects appropriate equations. Useful for students and researchers in the field of mechanical engineering, particularly those focusing on structural analysis and dynamics.

Typology: Slides

2012/2013

Uploaded on 04/24/2013

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Random vibration analysis of MDOF systems-4
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Download Analysis of a Clamped Beam under Differential Support Displacements and more Slides Stochastic Processes in PDF only on Docsity!

Random vibration analysis of MDOF systems-

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2

VIBRATION ANALYSIS OF CONTINUOUS SYSTEMS

^ 

2 2 ^ ^ ^ 

3 2 2 ( )^ ( )^ ( )^20

( , ) ICS:^ ( )^ ( , 0)

( )^ ( , 0)^

&^ BCS as appropriate. y^ yEI x x^

m x y^ c x y^ f

x t x^ x^

x^ t y x y x y^ x^ y x ^  x EI^ x  

 ^ ^

 ^ ^

 ^

 ^ ^

^  ^

   

^    (^12)   0 0

( , )^ ( )^

n^ n n n n^ n L L n k

n^ k

y x t^ a^ t^

x

EI^ m^

x

EI^ dx^

n^ k^ m^

dx^ n^ k

^    (^)      (^)   ^ ^

^ 

 

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4

2 (^

n^ n^ n^ n^ n

n^ n n n n L n

a^ a^ n^ L n

a^ p^ t x f x t dx

p^ t^

n x m x dx

 ^  ^    ^ ^   ^ 

^ 

^   

^

  ^

  

^  

   1

)()(]sin 0 cos)[exp() (),( n

t nndnndn nnn n^

dpth tBt Atx txy

  

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5 ^ ^  

^  ^ ^  

^    ^ ^

 ^ ^  ^ ^ ^ ^ 1  ^ ^  1

(^11)

Displacement:^

Slope:^ , Bending moment:

,^

Shear force:^

,^

Bending stressShear

N n^ n n N n n n N

n^ n n N n n n

y x t^ a^

t^ x

y^ x t^ a^

t^ x EI x y^ x t^ a

t^ EI x^ x

EI^ x^ y^ x t^

a^ t^ EI x^

x

 

         

^

^

 ^

^

^

^ 

^

^

^ 

^

   

Other quantities^ stressPrincipal stresses

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7

^ ^ ^ ^ 

^  ^ ^ ^   ^ ^ ^ ^ 

^  ^ ^ ^  

  ^ ^ ^ ^

^  ^ ^ ^ ^

^ ^  ^ ^ ^     ^ ^ ^  ^ ^ ^ ^  

 ^  ^  ^

  (^1)   (^21 21 21 21 21 ) Introduce a new dependent variable,^ , 0,^ 0,^

0 0 Select^ 0,^ 0;^

0 1; &^0 0 0, 0, 0 0

0 Select^ 0,^ 0;^

0 1; &^0 0 , , y x t^ w x t^ h Select^ ,^ 0;

x u t^ h^ x v t y^ t^ w^ t^ h

u t^ h^ v t

u t w^ t^ h^

h y^ t^ w^ t^

h^ u t^ h^ v t w t h^ h y l t^ w l t^ h^

l u t^ h^ l v t^

v t ^ ^ w l t^ h

 ^ ^

^     ^ ^

^   ^

 ^ ^

   ^ ^

^       ^ ^ ^ ^  

0; &^11 2  ^  ^   (^1 2)       1 2 ,^ ,^

0 Select^ ,^ 0;^

l^ h^ l 1; &^0 y^ l t^ w^ l t^ h

^      l u t^ h^ l v t   ^ w l t h l h^ l  ^ ^

  

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8 ^ ^

^  ^  ^ ^  ^

^ ^   (^1 2)    

1 2

1 2 1 2 3 2 1 1 1

(^1 12 ) 1 2

(^32 ) 2 2 3

Select^000 1;^

0;^0 0;^

iv^ iv^ ivEI w h u^ h u^ m w^ iv iv^3 21 Similarly^3

h u^ h v^ c w^

h u^ h v h h h^ x^ ax^ bx^

cx^ d h^ h^ l^

h^ h^ l x x h^ x^ L^

L x x ^ h x^ L^ L

  ^ ^ ^
^ ^ ^
^    ^ ^ ^
^ ^ ^     ^      
^ ^ 

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10 ^ ^

^  (^1 2)   ^   

1 2

1 2 2 3 2

3 2 3 2

3 2 3 2

3 2 3 2

3

0,^ 0;^ 0,^

iv^ iv^ ivEI w h u^ h u^ iv , 0;

m w^ h u^ h v^

c w^ h u^ h v EIw^ mw^ cw x^ x^

x^ x m u^

v L L L^ L x x x^ x c u^

v^

f^ x t L^ L^ L

L

w^ t^ w^

t ^ w l t^ w

  ^ ^ ^
^ ^ ^
^   ^ ^  ^
^ ^
^ 
^ ^ ^
^ 
^
^ ^
^ 
^ ^
^ 
^
^
^ ^
^ 
^ ^ ^
^ ^
^
^ ^
^ 
^ ^
^ 
^
^   
^ ^ ^
^ 

^     ^ ^    ^ ^ ^ ^ ^ 

^  ^ ^ ^ ^ ,^0 l t , 0 (^0 01 2) , 0 0 0 ^ ^ ^ ^ ^  1 2 w x^ h^ x u

h^ x v w x^ h^ x u

      h^ x v    Eigenfunction expansion method can now be used.

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11 ^ ^ ^ ^ ,^ ,^ ^  ^ ^ ^  ^1 y x t^ w x t^ ^ h^ x u t^ h^ x v t

 ^ ^

 

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13

^ ^

^ ^ ^ 

^ ^ ^

^ ^ ^

^ 

2 2

3 2 2

2 ( )^0

( )^ ( )^ exp ICS:^ ( )^ ( , 0)

( )^ ( , 0) BCS:^ 0,^ 0;^

0,^ 0;^ ,^

0;^ ,^0 y^ yEI x EI^ x^

m x y^ c x y^

i^ t^ x

x^ x^

x^ t y^ x^ y x^ y

x^ y x y^ t^ EIy^

t^ y L t^

EIy^ L t

^ ^ 

^

 ^ ^

 ^ ^

^  ^

 ^ ^

^  ^

 ^  ^

 ^ ^

^    ^  lim ,? y x t    t 

Harmonically driven beam: Green’s functionsin frequency domain

exp^ i^ t^ ^ ^ 

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14 (^12)   0 0

( , )^ ( )^

n^ n n n n^ n L L n k

n^ k

y x t^ a^ t^

x

EI^ m^

x

EI^ dx^

n^ k^ m^

dx^ n^ k

^    (^)      (^)   ^ ^

^ 

 

 ^ ^ ^ ^ ^

 2 0 ^ ^ ^ 

L exp exp^ ; 1, 2,

n^ n^ n^ n^ n

n^

n n

a^ a^

a^ i^ t^

x^ x^ dx i t

 ^ ^  n

^    

^ ^

^

   

^ 

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16

^ ^ ^

 ,^ ,^ ,    

, , , is complex valued, , is the generalization of the FRF discussed earlier

G x^ G^

x

 ^ G xG x ^    

Note ^   

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17 0 100 200 300

400 500 600

700 800

-2 10 -4 10 -6 10 -8 10 -10 10 -12 10

frequency rad/s FRF

0.3 x L  0.3 L 

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19

0 100 200 300

400 500 600

700 800

-2 10 -4 10 -6 10 -8 10 -10 10 -12 10 -14 10

frequency rad/s FRF

0.9 x L  0.3 L 

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20

^ ^

^ 

^ ^ ^

^ ^ ^

^ 

2 2

3 2 2

2 ( )^0

( )^ ( )^ ( ) ICS:^ ( )^ ( , 0)

( )^ ( , 0) BCS:^ 0,^ 0;^

0,^ 0;^ ,^

0;^ ,^0 y^ yEI x EI^ x^

m x y^ c x y^ f t

x

x^ x^

x^ t y^ x^ y x^ y

x^ y x y^ t^ EIy^

t^ y L t^

EIy^ L t

^ 

^

 ^ ^

 ^ ^

^  ^

 ^ ^

^  ^

 ^  ^

 ^ ^

^    ^  , ,^ ,    ^ ^ 

Y^ x^ G x

F ^  ^ 

Response of beam to a general load

f(t)

Note: the Fourier transform of f(t) is taken to exist

f(t)

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