Yeah I agree with your analysis. The frequencies come in pairs which represent the bending in y and z direction. Since my design is symmetric, the two frequencies should be equal and when I add a spring it would break the symmetry. However, you can see that not all frequencies are in pairs (for example the 5th and the 10th mode) and I think they represent the frequencies of rotation.Note regarding my last post: I think Ansys solution is right and probably the problem is in the equation that I used for my analytical solution. n