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ajtgraves

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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:

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'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)

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.

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