Vibration and isolation

1. Vibration and isolation

2. Seismic isolation • Use very low spring constant (soft) springs • Tripod arrangement used for mounting early AFM’s • Had long period (rigid body) vibration but isolated above resonance • Large supported mass also lowers resonance • Measure with accelerometers, one for each direction

3. Optical table legs • Same basic idea – low spring constant support • Table will tilt as a whole but not transmit floor (seismic) vibration • Table will couple through electric wires, etc. • Equipment on table sensitive to acoustic vib. • Suspension becomes unstable if CG too high • Must use three active legs and one slave – otherwise unstable

4. Coupled vibration • Good and bad – electric cables bad • Coupling interferometer to mirror under test is good • Both vibrate together (in phase) so fringes still • Nice animation in references of coupled masses • Gives idea of multiple modes in solids • This is a one D model but can see effect in other D • Simulate in lab with sample of different springs and masses

5. Acoustic vibration • Sound (acoustic vibration) is air pressure waves • Transmitters (vibrating plates - speaker cones) are also good receivers • Decrease coupling by putting holes in plates • Making plates of damped materials (lead sheet) • Acoustic coupling often not recognized for what it is • Need a sound pressure sensor (microphone not accelerometer)

6. Resonances and stiffness • Cantilever wire (rod) off speaker cone • Note fundamental and higher modes • Note how sharp the resonance is, high Q • Use resonance, length and diameter to find E • Also, clamp wire as cantilever, add small weight measure deflection and calculate E • Do two methods give about the same value for E

7. Torsional rigidity For solid of uniform circular cross-section, the torsion relations are: T/J = GΦ/l where: Φ is the angle of twist in radians. T is the torque (N·m or ft·lbf). l is the length of the object the torque is being applied to or over. G is the shear modulus or more commonly the modulus of rigidity and is usually given in gigapascals (GPa), lbf/in2 (psi), or lbf/ft2. J is the torsion constant for the section . the product GJ is called the torsional rigidity. Applying small torque difficult so measure frequency and find G

8. Kinematic stackups • Problem illustrated by convex test plate • Measure frequency of vibration (rocking) • From geometry of part is this reasonable? • Would frequency be higher or lower if longer radius? • Nothing is flat so need to make flat to flat connections kinematic • One thing that helps is acoustic damping; thin air film

9. Measurement Schema • Measurement tools 10x more accurate (or sensitive) than tolerance implies confidence in measurement • In optics, measuring accuracy about same as tolerance – need error separation methods • Quantum optics – act of measuring changes result

10. Symmetry and reversal Assume f(x) over -1 to 1 To find fe(x) = 1/2[f(x) + f(-x)] fo(x) = 1/2[f(x) – f(-x)] Works for surfaces too fee = ¼[f(x,y) + f(-x,y) + f(x,-y) + f(-x,-y), etc

11. Centroid and remove rotationally symmetric error Relative pseudo aberration value 9.51 9.38 1.56

12. Pseudo astigmatism and coma .75 1.02 .71 .57