Slide 2b.1 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Lecture 2b Optimal design of Regular Structures for Stiffness and Flexibility Principal features of optimal design of stiff structures and compliant mechanisms via analytical solutions for simple structural forms.

Slide 2b.2 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Contents Stiffest bar for a given volume of material –Equilibrium equations in strong form –And in weak form –Some insights Lightest beam for given stiffness –Interchanging the objective and an integral constraint Lightest beam for given deflection –What is right and what is wrong? –How to fix the formulation? Compliant mechanism: optimal juxtaposition of flexibility and stiffness –Proper way to formulate a compliant mechanism problem

Slide 2b.3 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Stiffest bar for a given volume of material C/s shape does not matter for axial load = load = cross-section profile Measure of stiffness = mean compliance = = axial deflection At equilibrium, mean compliance = 2 * strain energy Stiffest structure  minimum mean compliance  least strain energy

Slide 2b.4 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Optimal design problem for the stiffest bar Static equilibrium equation Volume constraint (Essential boundary conditions will also be there.) Lagrangian

Slide 2b.5 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Solution for the stiffest bar problem Recall the equilibrium equation: Self-adjointness  Strain energy density is constant across the bar Insight: Optimal use of material makes every point work equally hard. Necessary conditions:

Slide 2b.6 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Cross-section profile for the stiffest bar Substitute the above result into  Solve for given loading,. Use the natural boundary condition (i.e., the internal axial force is zero at the free end) to find. Example: = constant. Then, Volume constraint gives. Now, we know why Egyptian pyramids and temple towers linearly taper! This tells us why volume constraint must be active. Recall complementarity condition!

Slide 2b.7 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Highlights and some observations Self-adjointness Uniform strain energy density Lagrange multiplier has a physical meaning (as it usually does) Optimal shape for stiffest structures does not depend on… –Material property –Actual magnitude of loading but depends only on the profile of the loading (here, it was constant)

Slide 2b.8 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Weak form of equilibrium conditions in the stiffest bar problem Weak form of the equilibrium equation Volume constraint (Essential boundary conditions will also be there.) Lagrangian A scalar-function Lagrange multiplier is now replaced with a scalar-variable multiplier. An additional state variable ( ) arose but that can be easily dealt with.

Slide 2b.9 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Solution with the weak form Necessary conditions Same result as before In numerical optimization procedures, it is easier to handle integral (global) constraints. Hence, writing equilibrium equations in the weak form is advantageous. We can see that advantage even in manipulations involved in analytical solutions such as this.

Slide 2b.10 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh The lightest beam for given stiffness = load = width profile = transverse deflection In general, for other cross-sections. Mean compliance = Equilibrium equation Weak Strong (And, essential boundary conditions)

Slide 2b.11 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh The lightest beam problem statement Necessary conditions Lagrangian: Again the same story… the strain energy density is the same. Other viewpoint: uniformly stressed beam. This viewpoint leads to the “optimality criteria” method.

Slide 2b.12 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh The lightest beam for given deflection at a point: (towards flexibility) = load Consider a simply-supported (statically determinate) boundary condition. is to determined using the deflection constraint. = bending moment due to = bending moment due to unit dummy load applied at A A

Slide 2b.13 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh A problem and its fix What if is negative/zero somewhere in the span of the beam? Let us try by imposing a lower limit on the area of c/s… will be zero wherever (Recall complementarity) in other part(s) of the beam.

Slide 2b.14 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Is the problem really fixed? Observe that if area goes below the lower limit at some point, it has the effect of decreasing, which in turn, decreases and hence further. Finally, the entire span of the beam will reach the lower limit! Conclusion: Design for given deflection needs infinitesimally small volume. Therefore, including only flexibility (i.e., deflection) requirement is not a well posed problem.

Slide 2b.15 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Is this conclusion a surprise? Conclusion: Including only flexibility (i.e., deflection) requirement is not a well posed problem. A stiffness requirement, albeit only at one point. Making area as small as possible makes the objective and the constraint very happy! They have the same monotonicities, which makes it improper. Ill-posed Well-posed The objective and the constraint have opposite monotonicities with respect to the area—an indication of a properly posed optimization problem. A flexibility requirement at one point.

Slide 2b.16 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh A good fix: add stiffness requirement Use the two constraints to solve for and Now, can be prevented from becoming small by choosing as small as needed. Why?

Slide 2b.17 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Well posed flexibility (compliance) problem Adding a stiffness requirement prevents an overly flexible (and hence undesirable and indeed inappropriate, as we just saw) design solution. Insight: a flexible structure (and a compliant mechanism) should be: –As flexible as needed, but… –It should also be reasonably stiff. Practical viewpoint: some stiffness is needed to withstand the applied loads –Overly flexible structure has excessive motion but not the ability to effectively support or transfer the loads. Let us recall: –Structures support and transmit loads. –Mechanisms transfer/transform motion AND support and transmit loads.