Recall that the finite element system of equations has the form
![{\displaystyle \mathbf {M} ~{\ddot {\mathbf {u} }}=\mathbf {f} _{\text{ext}}-\mathbf {f} _{\text{int}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b04a7adc97b64cee0822806d4f78e914bbfc6ae6)
We could also have written this equation as
![{\displaystyle \mathbf {M} ~{\ddot {\mathbf {u} }}+\mathbf {K} ~\mathbf {u} =\mathbf {f} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b6d42a22067418cc67fc1543a7bb391f9f835c)
For natural vibrations, the forces and the displacements are assumed to be
periodic in time, i.e.,
![{\displaystyle \mathbf {u} =\mathbf {u} ^{0}~\exp(i\omega t)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52cba09327bed499aa93134e8dc2175e6872b303)
and
![{\displaystyle \mathbf {f} =\mathbf {f} ^{0}~\exp(i\omega t)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1300b20f4d8e797c2175e76f14de7fdbb12738d)
Then, the accelerations take the form
![{\displaystyle {\ddot {\mathbf {u} }}=(i\omega )^{2}~\mathbf {u} ^{0}~\exp(i\omega t)=-\omega ^{2}~\mathbf {u} ^{0}~\exp(i\omega t)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2564e164584c0fbdf3bc3e0018ad9dd8a6997ce)
Plugging these into the FE system of equations, we get
![{\displaystyle [-\omega ^{2}~\exp(i\omega t)]\mathbf {M} ~\mathbf {u} ^{0}+\exp(i\omega t)~\mathbf {K} ~\mathbf {u} ^{0}=\exp(i\omega t)~\mathbf {f} ^{0}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/467160440f6bdce69a6c1b82a28429b6c693e6a3)
After simplification, we get
![{\displaystyle {\left(-\omega ^{2}\mathbf {M} +\mathbf {K} \right)~\mathbf {u} ^{0}=\mathbf {f} ^{0}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b07fe3c8a0c37d9aacc648e6b3aa8880fe1d54d0)
If there is no forcing, the right hand side is zero, and we get
the finite element system of equations for free vibrations
![{\displaystyle {\left(-\omega ^{2}\mathbf {M} +\mathbf {K} \right)~\mathbf {u} ^{0}=0~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af74bbe3eb4923e478bad859ae15c683ed20278d)
The above equation is similar to the eigenvalue problem of the form
![{\displaystyle \mathbf {A} ~\mathbf {x} =\lambda ~\mathbf {x} \qquad \equiv \qquad \left(\mathbf {A} -\lambda \mathbf {I} \right)\mathbf {x} =0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38804ffeaa872b436ed3e518141b26c16e09f44c)
Since the right hand side is zero, the finite element system of equations
has a solution only if
![{\displaystyle {\det(\mathbf {K} -\omega ^{2}\mathbf {M} )=0~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cae5d675252dd3cd36feef9ffa49ee572818146)
For a two noded element,
![{\displaystyle \mathbf {K} ={\begin{bmatrix}K_{11}&K_{12}\\K_{21}&K_{22}\end{bmatrix}}~{\text{and}}~\mathbf {M} ={\begin{bmatrix}M_{11}&M_{12}\\M_{21}&M_{22}\end{bmatrix}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/245f7d42a6aca9b16fabd1e9f88dcfb9e35fb60f)
Therefore,
![{\displaystyle \mathbf {K} -\omega ^{2}~\mathbf {M} ={\begin{bmatrix}K_{11}-\omega ^{2}~M_{11}&K_{12}-\omega ^{2}~M_{12}\\K_{21}-\omega ^{2}~M_{21}&K_{22}-\omega ^{2}~M_{22}\end{bmatrix}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/722f40479e6b7df45717f60b1dc4c4941a1cdd3e)
The determinant is
![{\displaystyle \det(\mathbf {K} -\omega ^{2}~\mathbf {M} )=(K_{11}-\omega ^{2}~M_{11})(K_{22}-\omega ^{2}~M_{22})-(K_{12}-\omega ^{2}~M_{12})(K_{21}-\omega ^{2}~M_{21})~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e86cd7a8402fd5a4881be884abfda4bea5103bd5)
This gives us a quadratic equation in
which can be solved
to find the natural frequencies of the element.