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Evaluating natural
frequencies and mode
shapes
Multi degrees of freedom:
Multi degrees of freedom:
We have already deployed this for 2DOF systems.
–Eigenvalues -> Natural Frequencies of modes
–Eigenvectors -> Initial Conditions X1,X2
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
Dunkerly method:
Useful for structures undergoing vibration
testing.
Attach an eccentric mass exciter to a structure
and excite with frequency w22
Note the frequencies of maximum amplitude w1
Calculate w11
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?
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.)
Torsional Systems:
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
Holzer’s Method for Spring
mass systems:

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Evaluating natural frequencies and mode shapes.pptx

  • 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 X1,X2 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 w22 Note the frequencies of maximum amplitude w1 Calculate w11
  • 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.)
  • 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: