Some Counterintuitive Problems in Vibration

1. * Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

2. Some Counterintuitive Problems in Vibration Hugh Hunt Cambridge University Engineering Department Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

3. Something is counter-intuitive if: • it requires advanced/specialist knowledge • it is obscure or difficult to observe • it doesn’t fit with our experience • we’ve never noticed it before • we believed what our teachers said Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

4. VIBRATION “Common sense will carry one a long way but no ordinary mortal is endowed with an inborn instinct for vibrations”. “Vibrations are too rapid for our sense of sight … common sense applied to these phenomena is too common to be other than a source of danger”. Professor Charles Inglis, FRS from his “James Forrest” Lecture, Inst Civil Engineers, 1944 Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

5. VibrationConsultant vibration problems concept “design process” iteration product Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

6. Important concepts • Stiffness Frequency = • Mass • Nodal points • Vibration modes • Non-linearity • Damping The mkc model Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

7. V2 V1 V2 V2 Smallervolumeof air: stiffnessincreased m Water recreates rigid enclosure: stiffnessincreased Wallsmadeflexible: stiffnessdecreased k Helmholtz Resonator Neck plug of mass m Contained air of stiffness k Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

8. Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

9. The tips of the tuning fork move on thearcs of circles and centrifugal inertia forces are generated, twice per cycle. Suppose tip amplitude is 0.2mm, oscillating frequency is 440Hz, moving mass is 20% of the fork mass, thenthe 880Hz component of tip force Fis about 10% of the weight of the fork. F Tuning Fork: “P” is a nodal point, so why do we get more sound when “P” is put on a table? P Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

10. AXIAL VIBRATION mode 4 mode 3 mode 2 mode 1 Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

11. EULER BENDING VIBRATION mode 4 mode 3 mode 2 mode 1 Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

12. A vibrating beam marked out with the nodal points is very useful. The location of the nodal points are: Position of nodal points for a beam of L=1000mm (measured in mm from one end) mode 1: 224 776 mode 2: 132 500 868 mode 3: 94 356 644 906 mode 4: 73 277 500 723 927 mode 5: 60 226 409 591 774 940 mode 6: 51 192 346 500 654 808 949 mode 7: 44 166 300 433 567 700 834 956 mode 8: 39 147 265 382 500 618 735 853 961 See the Appendix for details of how to derive these Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

13. Axisymmetric bodies • Turbocharger blade vibration • Questions: • Do the blades fatigue lessrapidly if they are perfectly tuned, or is it better to mistune them? • Can vibration measurements made on a rotor be used to estimate its fatigue life? Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

14. Constrained-layer damping • Works by introducing damping material in places where shear strain is large • Material selection is important • (i) not too rubbery • (ii) not too glassy • - just right! • Temperature dependent • Effective over wide range of frequencies • Compromises strength amplitude frequency Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

15. amplitude frequency • Tuned absorber • Works by attaching a resonant element, with just the right amount of damping • Works at one frequency only • Material selection again is important owing to temperature dependence of damping Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

16. The mkc model has great virtues: - simple - huge range of application - “intuitive” … with a bit of thought Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

17. Appendix Nodes of a Vibrating Beam Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

18. Free vibration of a beam mass per unit length m flexural rigidity EI, length L y z Equation of motion: For vibration, assume y(x,t)=Y(x)cos(wt), so This has general solution Boundary condition for a fee end at z=0: Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

19. so i.e.C=A and D=B Boundary condition for a free end at z=L: so and or, in matrix form, Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

20. For a non-trivial solution, the determinant must be zero, so 1 0 Exact solutions for aL: 4.730 7.853 10.996 14.137 Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

21. From aL the frequencies of free vibration are found using aj= 22.37, 61.67, 120.90, 199.86, ... or aj The corresponding mode shapes are obtained by substituting aj into the matrix equation to find the ratio between A and B so that The location of nodal points is then found by looking for where Y(z)=0 Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

22. The location of the nodal points needs to be computed numerically, and the values are: Position of nodal points for a beam of L=1000mm (measured in mm from one end) mode 1: 224 776 mode 2: 132 500 868 mode 3: 94 356 644 906 mode 4: 73 277 500 723 927 mode 5: 60 226 409 591 774 940 mode 6: 51 192 346 500 654 808 949 mode 7: 44 166 300 433 567 700 834 956 mode 8: 39 147 265 382 500 618 735 853 961 Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk