Evaluating natural frequencies and mode shapes.pptx

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1. Evaluating natural frequencies and mode shapes

2. Multi degrees of freedom:

3. Multi degrees of freedom: We have already deployed this for 2DOF systems. – Eigenvalues -> Natural Frequencies of modes – Eigenvectors -> Initial Conditions X 1 ,X 2 It can be applied to higher order systems – And becomes increasingly tedious (More EoM , Bigger Matrices….) – Needs computational assistance Influence coefficients: -Used in some matrix method calculations - Stiffness Influence Coefficient – The stiffness matrix rewritten as the relation between displacement at point and the forces at other points in a system   Flexibility Influence Coefficient – ‘flexibility’ or inverse of stiffness

4. Dunkerly method: Useful for structures undergoing vibration testing. Attach an eccentric mass exciter to a structure and excite with frequency w 22 Note the frequencies  of maximum amplitude w 1 Calculate w 11

5. Dunkerly method example: An aircraft rudder tab shows a resonant frequency of 30Hz when vibrated by an eccentric mass shaker mass 1.5kg. By attaching a further 1.5kg to the shaker the frequency is lowered to 24Hz What is the true natural frequency of the tab?

6. Holzer method: • A tabular trial-and-error scheme to find natural frequencies and mode shapes of oscillating systems • A trial frequency is first assumed, and a solution is found when the constraints are satisfied. • Requires several trials • Gives node (zero displacement) positions • Used on semi-definite systems (needs a ‘free’ end.)

7. Torsional Systems:

8. Torsional Systems: When the sum of the Torques = 0 Graph of ST v w will give the modal frequencies Where the curve passes the w axis q values for each station will give the mode shapes

9. Holzer’s Method for Spring mass systems: