Solving boundary conditions for vibrating beam

  • #1
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Hi there,

I'm solving the equation for the transverse vibrations of a Euler-Bernoulli beam fixed at both ends and subject to axial loading. It's a similar problem to that described by Rao on page 355 of his book "Vibration of Continuous Systems" (Google books link), except the example he uses is for a simply supported beam.

The general solution takes the form of ##y(x) = C_1cosh(αx) + C_2sinh(αx) + C_3cos(βx) + C_4sin(βx)## ,
where ##C_1##, ##C_2##, ##C_3## & ##C_4## are the constants I need to find. The BCs are standard:

  • ##y(0)=y(L) = 0## (zero displacement at ends)
  • ##y'(0)=y'(L) = 0## (zero gradient at ends)
When I substitute these in the ##y(0)## and ##y'(0)## conditions give ##C_1 + C_3 = 0## and ##αC_2 + βC_4 = 0##, respectively, while the ##y(L)## and ##y'(L)## conditions give:

1) ##C_1cosh(αL) + C_2sinh(αL) + C_3cos(βL) + C_4sin(βL) = 0##

2) ##αC_1sinh(αL) + αC_2cosh(αL) – βC_3sin(βL) + βC_4cos(βL) = 0##

Clearly the first 2 conditions can be used to reduce these last two equations into functions of ##C_1## and ##C_2##only:

3) ##C_1[cosh(αL) - cos(βL)] + C_2[sinh(αL) - (α/β)sin(βL)] = 0##

4) ##C_1[αC_1sinh(αL) + βsin(βL)] + C_2[βcosh(αL) - αcos(βL)] = 0##


We can now solve for ##C_1## (or ##C_2##) and use this to write all the terms of the original governing equation in terms of it alone. However, there are two possible expressions for ##C_1## (and ##C_2##), depending on which equation is used. 3) gives:

##C_2 = -C_1[cosh(αL) - cos(βL)] / [sinh(αL) - (α/β)sin(βL)]##

whereas 4) gives:

##C_2 = -C_1[αC_1sinh(αL) + βsin(βL)] / [βcosh(αL) - αcos(βL)]##


These are clearly different, but are they both correct? Which one should be used?


Many thanks in advance for your help, it would be much appreciated.
 
Last edited:

Answers and Replies

  • #2
SteamKing
Staff Emeritus
Science Advisor
Homework Helper
12,798
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Hi there,

I'm solving the equation for the transverse vibrations of a Euler-Bernoulli beam fixed at both ends and subject to axial loading. It's a similar problem to that described by Rao on page 355 of his book "Vibration of Continuous Systems" (Google books link), except the example he uses is for a simply supported beam.

The general solution takes the form of
##y(x) = C_1cosh(αx) + C_2sinh(αx) + C_3cos(βx) + C_4sin(βx)##,
where C1 , C2 , C3 & C4 are the constants I need to find. The BCs are standard:

  • ##y(0)=y(L) = 0## (zero displacement at ends)
  • ##y'(0)=y'(L) = 0## (zero gradient at ends)
When I substitute these in the ##y(0)## and ##y'(0)## conditions give ##C_1 + C_3 = 0## and ##αC_2 + βC_4 = 0##, respectively, while the ##y(L)## and ##y'(L)## conditions give:


1) ##C_1cosh(αL) + C_2sinh(αL) + C_3cos(βL) + C_4sin(βL) = 0##


2) ##αC_1sinh(αL) + αC_2cosh(αL) – βC_3sin(βL) + βC_4cos(βL) = 0##


Clearly the first 2 conditions can be used to reduce these last two equations into functions of ##C_1## and ##C_2##only:


3) ##C_1[cosh(αL) - cos(βL)] + C_2[sinh(αL) - (α/β)sin(βL)] = 0##


4) ##C_1[αC_1sinh(αL) + βsin(βL)] + C_2[βcosh(αL) - αcos(βL)] = 0##


We can now solve for ##C_1## (or ##C_2##) and use this to write all the terms of the original governing equation in terms of it alone. However, there are two possible expressions for ##C_1## (and ##C_2##), depending on which equation is used. 3) gives:

##C_2 = -C_1[cosh(αL) - cos(βL)] / [sinh(αL) - (α/β)sin(βL)]##

whereas 4) gives:


##C_2 = -C_1[αC_1sinh(αL) + βsin(βL)] / [βcosh(αL) - αcos(βL)]##



These are clearly different, but are they both correct? Which one should be used?


Many thanks in advance for your help, it would be much appreciated.

I've re-worked your Latex commands slightly to make your post more legible.
 

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