MICE TARGET WORKSHOP – SHEFFIELD 2 nd December Target vibration and general design issues 2. Simple Vibration - causes and forces. 3. Not so simple vibration. 4. What are the sensitive bits? 5.What we can chop off. 6.Concluding remarks. Eddie McCarron 1 Dec

Dynamic vibration analysis is difficult so I will present a simplified ‘static’ load case that hopefully avoids the challenge. Consider 1. Euler buckling. A slender column will buckle if the end load, P, is greater than (pi 2. EI)/L 2 where E = Young’s modulus, material property I = (pi (R 4 - r 4 )/4, second moment of area L = length of column 2. Driving force F = M. A where M= whole moving mass, a = acceleration. 3. Driving force F is generated by coil current Ic X Magnet field. The current, Ic, required is proportional to ‘air’ gap, t (and M & a). F Desired driving force from coils. d buckling will cause displacement d’ Secondary displacement caused by d. P = m. a Reaction to F. m = mass of tip (small), a = 80g Displacement curve is also acceleration curve. Peak acceleration at each end of curve (when ‘L’ is maximum). L A similar force diagram Could be drawn for this end. Usual discussion follows……… 1. Reduce L, M, m, a, t and F if possible. 2. Increase E, I and R if possible. 3. Stay away from natural frequency.. ….it would help if we knew the natural frequency of the system. Simple consideration of vibration. Un-damped vibration – energy is stored and released as strain in the shaft. 2

Not so simple consideration of vibration Consider the transverse field of the coils – it increases with radius! Even if the shaft were fairly straight statically and the coils perfectly concentric the displacement, d’, will cause a sideways movement of the magnet. The coil field will accentuate this movement. From ref[1] p282 it seems we have a ‘self-excited’ vibration. These have negative damping. p284 ‘A damping force, +ve or –ve, lowers the natural frequency….(particularly).. when the damping force is large in comparison with the spring or inertia forces… The next 50 pages describe examples of disasters like wing flutter on aircraft and Vortex shedding on chimney stacks (and Galloping Gerty). p378 moves to variable elasticity and a problem of a steel block on a stick between two magnets i.e. a simple model of the Target shaft. And ends the book with….. ….’it is evident that such analysis will be extremely difficult.’ d’ From simple vibration Ref 1. Mechanical Vibrations J.P. Den Hartog Dover ISBN d’’ From coils The analysis is difficult so vibration measurement of real targets is vital to predicting the performance of the target assembly and life. Physical vibration measurement will form a part of the QA and acceptance test of all future target assemblies. A. A knowledge of the vibration envelope of the target assembly is essential if the dip depth or frequency of dip is to be varied. Otherwise we won’t know whether we are moving towards or away from danger. B. Fast photography and velocity meters on the test target will provide essential data. C. Contacting a vibration consultant is also vital

Some suggestions Reduce L, M, m, a, t and F if possible. L = shaft length, M and m = masses, a = acceleration of shaft, t = magnet ‘air’ gap, F = drive force (= M. a, of course) 2. Increase E, I and R if possible. E = Young’s modulus, I = second moment of area, R = shaft radius 3. Stay away from natural frequency.. and increase it! F n = natural frequency Which are the sensitive mechanical bits? a. The magnet is most of the mass. The shaft mass is proportional to L. b. The deflection of a ‘built in’ cantilever (the ends bits) is given by δ = (P. L 3 )/3EI but.. The deflection of a ‘built in’ beam (the magnet bit) is given by δ = (P. L 3 )/192EI. c. Vibration frequency, f, is proportional to 1/L. Reducing L increases the natural frequency. d.Second moment of area, I, is proportional to shaft radius R 4. It is clear that we must try to increase R and decrease L. The magnet and coils efficiency is clearly important. The flux link is very sensitive to ‘air’ gap. Reducing ‘air’ gap would reduce current (and heating (I 2 )) and the driving voltage.

5 What can we easily chop off?

6 Summary of gains from the changes in length. Magnet section shaft- length reduced by 26%. Target end of shaft- length reduced by 39%. Optic end of shaft- length reduced by 32%. Concluding remarks. We are using a very hard coating for the bearings (DLC) applied by a competent company (Tecvac). It is clear that the bearing coating fails but we still have not found what causes the failure. If there were minimal side load on the bearing I doubt that the force and friction would cause a bearing failure. Significant side load can only be cause by vibration (the shaft assembly is now constructed very concentric). The cause of the vibration is longitudinal shaft curvature and magnetic misalignment. The latter being a negative damper. The high side load will cause higher friction and the temperature increase a differential expansion of the DLC and substrate. The driving frequency of the shaft is 33Hz but we don’t know the driving frequency of the magnetic misalignment. This varies from target to target with construction tolerances. A very concentric coil assembly will last a ‘long’ time (the one currently in synchrotron) a less concentric one will fail early. Whether the failures are caused by force or friction they are both exacerbated by vibration. Actions 1. A vibration test should be part of the QA build tests. 2. A vibration expert should be consulted. 3.The present target vibrations should be observed (fast photos) and measured (velocity meters). 4.A new target should be built from the findings of 1,2, & 3.