1. Slide 3.1 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Lecture 3 Design as an Inverse Problem and its Pitfalls “What is right to ask”, an important thing in computational and optimal design, illustrated with the “design for desired mode shapes” problem.

2. Slide 3.2 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Contents Design for desired mode shapes –What is wrong with the optimal synthesis formulation? –Direct synthesis technique of a bar of a beam –Analytical solutions and insights –Solution using discretized models Stiff structure and compliant mechanism design problem formulations—a summary –Continuous model –Discretized model

3. Slide 3.3 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Why design for mode shapes? Resonant MEMS –Capacitive resonant sensors –Micro rate gyroscope AFM (atomic force microscope) cantilevers See: Pedersen,N., “Design of Cantilever Probes for Atomic Force Microscopy (AFM),” Engineering Optimization, Vol. 32, No. 3, 2000, 373-392. Swimming and flying mechanisms Acoustics, …

4. Slide 3.4 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Resonant-mode micromachined pressure sensor Pressure Top view Side view Resonant beam Capacitance is measured in this gap The mode shape of the beam influences the sensitivity of the sensor.

5. Slide 3.5 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh The cantilever in atomic force microscopy (AFM) (in the resonant mode) Laser Detector When AFM operates in the resonant mode, it helps to shape the cantilever to have a mode shape that has larger slope towards the tip. Pedersen,N., “Design of Cantilever Probes for Atomic Force Microscopy (AFM),” Engineering Optimization, Vol. 32, No. 3, 2000, 373-392.

6. Slide 3.6 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Rate gyroscope with a micromachined vibrating polysilicon ring Two degenerate mode shapes M. Putty and K. Najafi, “A Micromachined Vibrating Ring Gyroscope,” Tech. Digest of the 1994 Solid State Sensors and Actuators workshop, Hilton Head Island, SC, pp. 213-220.

7. Slide 3.7 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Principle of the rate gyroscope Foucault pendulum All of the above have degenerate pairs of mode shapes. When one mode shape is excited, the rotation of the base causes energy-transfer to the other mode due to Coriolis force. Wine glassRing Plane of oscillation rotates

8. Slide 3.8 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Design the spokes for improved mode shapes (and better sensitivity) (Lai and Ananthasuresh, 1999)

9. Slide 3.9 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Design for a desired mode shape of a bar Axially deforming bar Analysis: Given: Find: Mode shape Natural frequency Given: Synthesis: Find: Area of c/s

10. Slide 3.10 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Direct and optimal synthesis techniques error Designed Desired Finding the area profile to minimize the integrated error is the optimal synthesis technique. Solving this “inverse” differential equation is the direct synthesis technique

11. Slide 3.11 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Direct synthesis solution decides what should be! Furthermore, boundary conditions decide what mode shapes are possible; so, we cannot ask whatever we wish. Solution for area of c/s Can be specified also?

12. Slide 3.12 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Some examples Desired mode shape Frequency must be… Area of c/s (to cancel off the denominator in )

13. Slide 3.13 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Desired mode shape for a beam Assume that as before. Inverse eigenproblem for the beam: Solution? What are the conditions on the frequency to make a mode shape valid?

14. Slide 3.14 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Discretized model Using finite-difference derivatives…for a cantilever beam: is a diagonal matrix with

15. Slide 3.15 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Re-arrange the variables…

16. Slide 3.16 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Solution and conditions on frequency and mode shape From the last row of the previous system of equations: And then, solve for the areas: For details, see: Lai, E. and Ananthasuresh, G.K., “On the Design of Bars and Beams for Desired Mode Shapes,” Journal of Sound and Vibration, Vol. 254, No. 2, 2002, pp. 393-406.

17. Slide 3.17 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Return to the differential equation… For a cantilever, at the free end, i.e., at : (assuming is not zero) A condition to ensure positive : Another condition due to Gladwell: The number of sign changes in the mode shape and its first derivatives must be the same. See: Inverse Vibration Problems, G. M. L. Gladwell, 1986.

18. Slide 3.18 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh An example: valid mode shapes Explore which 6 th degree polynomials are valid mode shapes for a cantilever: With essential and natural boundary conditions imposed: Two other conditions: The number of sign changes in the mode shape and its first derivatives must be the same. Valid 1 st mode shapes Valid 1 st and 2 nd mode shapes

19. Slide 3.19 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Some examples of mode shapes and area profiles a 6 = 0 a 6 = 1 a 6 = 2 a 6 = 4 a 6 = 3 For details, see: Lai, E. and Ananthasuresh, G.K., “On the Design of Bars and Beams for Desired Mode Shapes,” Journal of Sound and Vibration, Vol. 254, No. 2, 2002, pp. 393-406.

20. Slide 3.20 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Now, we are ready for optimal synthesis… error Designed Desired Now, given a mode shape, we check if it is valid. If it is not, we can give the closest valid polynomial (or other) mode shape and get a solution.

21. Slide 3.21 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh An example Given (invalid) mode shape Rectified polynomial mode shape First derivative of the mode shape Area profile

22. Slide 3.22 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Return to stiff structure design Volume constraint Equilibrium equation + boundary conditions (displacements and tractions) ? Force or Strain-displacement relationship Design variables are in. Stress-strain relationship

23. Slide 3.23 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh or Stiff structure design Design variables

24. Slide 3.24 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh With the discretized model… or Stiffness matrix = Strain energy = Displacement vector = Equilibrium equation Volume constraint Strain energy Mean compliance Design variables are in.

25. Slide 3.25 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Return to compliant mechanism design Flexibility (deflection) constraint Stiffness (strain energy) constraint Equilibrium equations + boundary conditions (displacements and tractions) Design variables are in. ? Force Unit dummy load

26. Slide 3.26 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Alternatively… Since nonlinear constraints are more difficult to deal with, and multiple constraints make optimization harder, the problem is reformulated as: or Linear combination or ratio of two conflicting objectives.

27. Slide 3.27 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh With the discretized model… or Mutual strain energy = Strain energy = Geometric advantage Mechanical efficiency Output spring constant

28. Slide 3.28 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Modeling the work-piece in the compliant mechanism design problem ? Force Output spring to model the work-piece ?

29. Slide 3.29 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Main points An optimal design problem, as posed, should make sense. Design for desired mode shapes problem –Restrictions on “desired” mode shapes and frequencies Stiff structure design problem statement revisited Compliance design problem statement revisited –Flexibility and stiffness requirements should be optimally balanced –Work-piece can be modeled as an output spring

30. Slide 3.30 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Let’s make up some specifications… a)For a stiff structure b)For a compliant mechanism … so that we can compare designs given by the optimization program (PennSyn) and designs conceived by You!