1. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk *

2. Some Counterintuitive Problems in Vibration Hugh Hunt Cambridge University Engineering Department

3. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk Something is counter-intuitive if: it requires advanced/specialist knowledge it is obscure or difficult to observe it doesnt fit with our experience weve never noticed it before we believed what our teachers said

4. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk 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

5. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk design process concept iteration product vibration problems Vibration Consultant

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

7. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk Helmholtz Resonator Neck plug of mass m Contained air of stiffness k m k V1V1 Smaller volume of air: stiffness increased V2V2 Walls made flexible: stiffness decreased V2V2 V2V2 Water recreates rigid enclosure: stiffness increased

8. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk

9. The tips of the tuning fork move on the arcs 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, then the 880Hz component of tip force F is about 10% of the weight of the fork. F P Tuning Fork: P is a nodal point, so why do we get more sound when P is put on a table?

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

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

12. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk 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

13. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk Axisymmetric bodies Turbocharger blade vibration Questions: 1.Do the blades fatigue less rapidly if they are perfectly tuned, or is it better to mistune them? 2.Can vibration measurements made on a rotor be used to estimate its fatigue life?

14. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk Constrained-layer damping 1.Works by introducing damping material in places where shear strain is large 2.Material selection is important (i) not too rubbery (ii) not too glassy - just right! 3.Temperature dependent 4.Effective over wide range of frequencies 5.Compromises strength frequency amplitude

15. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk Tuned absorber 1.Works by attaching a resonant element, with just the right amount of damping 2.Works at one frequency only 3.Material selection again is important owing to temperature dependence of damping frequency amplitude

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

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

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

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

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

21. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk From aL the frequencies of free vibration are found using a j = 22.37, 61.67, 120.90, 199.86,... or a j The corresponding mode shapes are obtained by substituting j 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

22. Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk 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