Modal analysis, beam

Results.

Example	Reference	Records	Largest deviation
Cantilever beam, bending modes	Closed form	3	< 0.001%
Simply supported beam, bending modes	Closed form	2	< 0.001%
Cantilever bar, axial vibration	Closed form	1	0.0064%

Problem. The first three bending frequencies of a cantilever, the first two of a simply supported beam and the axial frequency of a cantilever bar are tested.

Reference document. Classical closed forms for beam vibration are used.

The bending frequencies of a cantilever are computed from this relation. The first three roots of the frequency equation are given beside it.

f = \dfrac{(\beta L)^2}{2\pi L^2}\sqrt{\dfrac{EI}{\rho A}},\qquad \beta L = 1.8751;\ 4.6941;\ 7.8548

The bending frequencies of a simply supported beam and the axial vibration frequencies of a cantilever bar are given by these relations.

f_n = \dfrac{(n\pi)^2}{2\pi L^2}\sqrt{\dfrac{EI}{\rho A}},\qquad f_n = \dfrac{2n-1}{4L}\sqrt{\dfrac{E}{\rho}}

Numerical model. The beam is modelled with twenty-four elements and the axial bar with forty, with a consistent mass matrix. The supports are three-dimensional, so the spectrum also contains lateral bending, torsional and axial modes. The verified mode is found by its motion, and its position in the spectrum is given in the table.

Cantilever beam, bending modes

Closed form. The bending frequencies of a cantilever beam

f = \dfrac{(\beta L)^2}{2\pi L^2}\sqrt{\dfrac{EI}{\rho A}},\qquad \beta L = 1.8751;\ 4.6941;\ 7.8548

Quantity	Condition	Reference	Computed	Deviation
1st vertical bending mode frequency [Hz]	L = 6 · element count = 24 · mode = 2	6.79484	6.79484	0%
2nd vertical bending mode frequency [Hz]	L = 6 · element count = 24 · mode = 5	42.5825	42.5826	< 0.001%
3rd vertical bending mode frequency [Hz]	L = 6 · element count = 24 · mode = 7	119.232	119.233	< 0.001%

Simply supported beam, bending modes

Closed form. The bending frequencies of a simply supported beam

f_n = \dfrac{(n\pi)^2}{2\pi L^2}\sqrt{\dfrac{EI}{\rho A}}

Quantity	Condition	Reference	Computed	Deviation
1st vertical bending mode frequency [Hz]	L = 6 · element count = 24 · mode = 2	19.0734	19.0734	0%
2nd vertical bending mode frequency [Hz]	L = 6 · element count = 24 · mode = 5	76.2936	76.2939	< 0.001%

Cantilever bar, axial vibration

Parameters. L = 6 · element count = 40 · mode = 11

Closed form. The axial vibration frequencies of a cantilever bar

f_n = \dfrac{2n-1}{4L}\sqrt{\dfrac{E}{\rho}}

Quantity	Reference	Computed	Deviation
1st axial mode frequency [Hz]	210.314	210.328	0.0064%

Cantilever beam, bending modes#

Simply supported beam, bending modes#

Cantilever bar, axial vibration#

Cantilever beam, bending modes

Simply supported beam, bending modes

Cantilever bar, axial vibration