2.72 Elements of Mechanical Design

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1 MIT OpenCourseWare Elements of Mechanical Design Spring 2009 For information about citing these materials or our Terms of Use, visit:

2 2.72 Elements of Mechanical Design Lecture 6: Dynamics and damping

3 Schedule and reading assignment Quiz None Topics Vibration physics Connection to real world Activity Reading assignment Skim last gear reading assignment (gear selection) Martin Culpepper, All rights reserved 2

4 Resonance Basic Physics Exchange potential-kinetic energy Energy transfer with loss Modeling 2nd order system model Spring mass damper Differential equations, Laplace Transforms c k PE m x Loss KE F(t) Why Do We Care Critical to understanding motion of structures- desired and undesired Generally not steady state Location error Large forces, high fatigue Gain ω n ω n = Martin Culpepper, All rights reserved 3 ω k m

5 Vibrations - Input Oscillation of System Why Categorize Different causes Different solutions A Forced Input Form Forced Steady State Command Signal Electrical (60 Hz) Free Transient Impacts A Free t t Martin Culpepper, All rights reserved 4

6 Vibrations - Source Source Disturbance Driven Device Motors Rotating Components Undriven Electrical (60 Hz) People (2 Hz) Cars (0 Hz) Nearby Equipment Command Step Response A 0 9 Undriven Disturbance Spectrum Driven Disturbance System t Vibration Command Response Martin Culpepper, All rights reserved 5

, Design of High-Speed, Meso-Scale Nanopositioners Driven by

7 Vibrations: Command vs. Disturbance Underdamped response Noise Free vibration Golda, D. S., Design of High-Speed, Meso-Scale Nanopositioners Driven by Electromagnetic Actuators, Ph.D. Thesis, Massachusetts Institute of Technology, Martin Culpepper, All rights reserved 6

8 Vibrations - Example Example Chinook Identify: Mode Form Source Response A Forced A t Free t 0 9 Undriven Disturbance 0 3 Spectrum Driven Disturbance A System t Response Martin Culpepper, All rights reserved 7

9 Attenuating Vibrations Change System Mass, stiffness, damping Adjust mode shapes Change Inputs Command: Input Modulation, Feedback Disturbance, reduce: undriven vibrations, e.g. optical table driven vibrations alter device structure (damping on motors, etc.) A t F ms 2 +bs+k Z Martin Culpepper, All rights reserved 8

11 Behavior Regimes () Low Frequency (ω<ω n ) (2) Resonance (ω ω n ) (3) High Frequency (ω>ω n ) Example Spring/Mass demo Amplification factor (output/input) 0 ζ = 0.05 ζ = 0.2 ζ = ζ = 2.0 ζ = 0.3 ζ = 0.4 ζ = 0.5 ζ = 0.6 ζ = 0.7 st order system ζ = 0.8 ζ = 0.9 ζ = Figure by MIT OpenCourseWare. ω/ω n 0 x = = F ms 2 + bs + k k(ω) Martin Culpepper, All rights reserved 0

12 Behavior Low Frequency Frequency Response Regimes () Low Frequency (ω<ω n ) (2) Resonance (ω ω n ) (3) High Frequency (ω>ω n ) Example Spring/Mass demo Low Frequency (ω<ω n ) System tracks commands Ideal operating range High disturbance rejection x F Resonance Frequency (ω ω n ) System Response >> command k eff k Disturbances will cause very large response Quality factor = magnitude of peak Figure by MIT OpenCourseWare. x Damping = Q High Frequency (ω ω n ) System Response << command High disturbance rejection Amplification factor (output/input) ζ = 0.05 ζ = 0.2 ω/ω n F = 2 ms ζ = ζ = 2.0 ζ = 0.3 ζ = 0.4 ζ = 0.5 ζ = 0.6 ζ = bs + k st order system ζ = 0.8 ζ = 0.9 ζ =.0 0 Martin Culpepper, All rights reserved

13 Behavior - Resonance Frequency Response Regimes () Low Frequency (ω<ω n ) (2) Resonance (ω ω n ) (3) High Frequency (ω>ω n ) Example Spring/Mass demo Low Frequency (ω<ω n ) System tracks commands Ideal operating range High disturbance rejection x Q Resonance Frequency (ω ω n ) < k F k System Response >> command k eff Disturbances will cause very large response Quality factor = magnitude of peak x Damping = Q = 2 F ms + bs + k High Frequency (ω ω n ) System Response << command High disturbance rejection Amplification factor (output/input) Figure by MIT OpenCourseWare. ζ = 0.05 ζ = 0.2 ζ = ζ = 2.0 ω/ω n ζ = 0.3 ζ = 0.4 ζ = 0.5 ζ = 0.6 ζ = 0.7 st order system ζ = 0.8 ζ = 0.9 ζ =.0 0 Martin Culpepper, All rights reserved 2

14 Behavior High Frequency Frequency Response Regimes () Low Frequency (ω<ω n ) (2) Resonance (ω ω n ) (3) High Frequency (ω>ω n ) Example Spring/Mass demo Low Frequency (ω<ω n ) System tracks commands Ideal operating range High disturbance rejection Resonance Frequency (ω ω n ) System Response >> command k eff Disturbances will cause very large response Quality factor = magnitude of peak x Damping = Q x High Frequency (ω ω n ) System Response << command F 2 mω Poor disturbance rejection Amplification factor (output/input) ζ = 0.05 ζ = 0.2 Figure by MIT OpenCourseWare. ζ = ζ = 2.0 ω/ω n F = 2 ms ζ = 0.3 ζ = 0.4 ζ = 0.5 ζ = 0.6 ζ = bs + k st order system ζ = 0.8 ζ = 0.9 ζ =.0 0 Martin Culpepper, All rights reserved 3

16 Application of Theory Relate Variables to Actual Parameters Vibrational Mode Mass Stiffness c k m x F(t) Damping Transfer between Model and Reality Iterative Start simple ( mass, spring) Add complexity Limits Example building (video) Iteration Mode? k? m? c? Martin Culpepper, All rights reserved 5

17 Strategies for damping Material Grain boundary Internal lattice Viscoelastic (elastomers/goo) Viscous Air Fluid Electromagnetic Friction Active Combinations Sponge Pros and cons of each Martin Culpepper, All rights reserved 6

19 Exercise (see next page too) Perform a frequency analysis of the part Develop & prove (FEA) how to increase nat. freq. via geometry change Any constraints you might have? Geometry changes can t be unbounded Explain effect of your change on vibration amplitude (relative to outer base) at given ω, via sketches & plots Xtra credit, assume: Flexure is contained between two parallel plates (on top and bottom) Viscous air damping in the gaps on both sides micron gap between the flexure sides and plates Elaborate on how well flexure is damped (don t just use intuition) Useful equations (c = damping coefficient, k = stiffness, m = mass) Damping ratio Frequency at peak with max gain Gain at peak amplitude c ζ = G = ω p = ω n 2ζ 2 p 2 km 4ζ 2 4ζ 4 Martin Culpepper, All rights reserved 8

Flexure Flexure design top Constrained on 4 sides Bottom See this diagram for

2.72 Elements of Mechanical Design

MIT OpenCourseWare http://ocw.mit.edu 2.72 Elements of Mechanical Design Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 2.72 Elements of