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Consider a uniform cross section of bar of Length L made up of a material whose youngs modulus and density are given by E and S.

Consider a uniform cross section of bar of Length L made up of a material whose youngs modulus and density are given by E and S. Estimate the natural frequency of axial vibration of the bar using both consistent and lumped mass matrices.

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Using one element for entire rod i.e. he = l.

Using lumped mass matrix:

AEhe[1111]{u1u2}=ω2ζAhe2[1001]{u1u2}

Since he = l

AEl[1111]{u1u2}=ω2ζAl[120012]{u1u2}

Applying boundary condition, u1 = 0

AEl[1111]{0u2}=ω2ζAl[120012]{0u2}

AElu2=ω2ζAl2u2ω2=2Eζl2w=2Eζl2=1.414lEζ

Using consistent mass matrix

AEhe[1111]{u1u2}=w2ζAhe6[2112]{u1u2}

But, he = l

AEl[1111]{u1u2}=w2ζAl6[2112]{u1u2}

Applying boundary condition, u1 = 0

AEl[1111]{0u2}=ω2ζAl6[2112]{0u2}

AElu2=ω2ζAl62u2El=ω2ζl3ω2=3Eζl2ω=3Eζl2=1.732lEζ

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