# Comb resonator design (2)

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1 Lecture 6: Comb resonator design () -Intro Intro. to Mechanics of Materials School of Electrical l Engineering i and Computer Science, Seoul National University Nano/Micro Systems & Controls Laboratory URL:

2 Stress Normal Stress: force applied to surface σ F / A measured in N/m or Pa, compressive or tensile Shear Stress: force applied parallel to surface τ F / A measured in N/m or Pa Young s Modulus: E σ / ε Hooke s Law: K F/ l EA/ l This material is intended for students in class in the Spring of 008. Any other

3 Strain Strain: ratio of deformation to length γ l/ l Shear Modulus G τ / γ Relation among: G, E, and ν G E (1 +ν ) This material is intended for students in class in the Spring of 008. Any other

4 Poisson s Ratio Tensile stress in x direction results in compressive stress in y and z direction (object becomes longer and thinner) Poisson s Ratio: ν ε / ε ε / ε y x z x transverse t strain / longitudinal l strain Metals: ν 0. ν 0.5 Rubbers: Cork: ν 0 This material is intended for students in class in the Spring of 008. Any other 4

5 State of stress The combination of forces generated by the stresses must satisfy the conditions for equilibrium: Fx Fy Fz 0 M M M 0 x y z Consider the moments about the z axis: ( τ ) ( τ ) M 0 A a A a z xy yx τ τ, τ τ and τ τ xy yx yz zy yz zy Only 6 components of stress are required to define the complete state of stress. This material is intended for students in class in the Spring of 008. Any other 5

6 Stress and Strain Diagram Ductile Materials This material is intended for students in class in the Spring of 008. Any other 6

7 Stress and Strain Diagram (cont d) Brittle Materials Stress-strain diagram for a typical brittle material This material is intended for students in class in the Spring of 008. Any other 7

8 Deformations Under Axial Loading From Hooke s Law: σ P σ Eε ε E AE From the definition of strain: ε δ L Equating and solving for the deformation: PL δ AE With variations in loading, cross-section or material properties: PL i i δ A E i i i This material is intended for students in class in the Spring of 008. Any other 8

9 Stress Concentration: Fillet σ max K σ σ ave This material is intended for students in class in the Spring of 008. Any other 9

10 Symmetric Member in Pure Bending Pure Bending: Prismatic members subjected to equal and opposite couples acting in the same longitudinal plane Internal forces in any cross section are equivalent to a couple. The moment of the couple is the section bending moment F x y σ da x x 0 M zσ da 0 M y σ da M z x This material is intended for students in class in the Spring of 008. Any other 10

11 Strain Due to Bending Consider a beam segment of length L After deformation, the length of the neutral surface remains L. At other sections, ( ρ y) θ ( ) L δ L L ρ y θ ρθ yθ ε ε ε x m x δ yθ ρθ y L ρ (strain varies linearly) c c or ρ ρ εm y ε m c This material is intended for students in class in the Spring of 008. Any other 11

12 Stress Due to Bending For a linearly elastic material, y σ x Eεx Eεm c y σ m (stress varies linearly) c For static equilibrium, y Fx 0 σx da σm da c σ m 0 yda c First moment with respect to neutral plane is zero. Therefore, the neutral surface must pass through the section centroid. This material is intended for students in class in the Spring of 008. Any other 1

13 Stress Due to Bending (cont d) For static equilibrium, Bending Momentum M t M df( h) h ( σ ( h) da) h A t w t h h σ max σ max M ( σ max da) h h da I t t t w t w t h ( ) ( ) h ( ) Mt smax EI t where σ max : magnitude of stress I : moment of inertia of the cross-section h : height of a beam t : thickness of a beam s max : maximum longitudinal strain This material is intended for students in class in the Spring of 008. Any other 1

14 Deformations in a Transverse Cross Section Deformation due to bending moment M is quantified by the curvature of the neutral surface 1 εm σ m 1 Mc ρ c Ec Ec I M EI Although cross sectional planes remain planar when subjected to bending moments, in-plane deformations are nonzero, ν y ν y εy νε x ε z νε x ρ ρ This material is intended for students in class in the Spring of 008. Any other 14

15 Bending of Beams Reaction Forces and Moments - For equilibrium F 0 F F 0, therefore F F R R M0 0 MR + FL 0, therefore MR FL This material is intended for students in class in the Spring of 008. Any other 15

16 Bending of Beams (cont d) Shear Forces and Moments (at any point in the beam) At every point along the beam equilibrium requires that, F 0 and M 0 F 0 F + V( x) 0 V F M 0 M( x) + F( L x) 0 M( x) F( L x) L This material is intended for students in class in the Spring of 008. Any other 16

17 Bending of Beams differential element Equilibrium of a fully loaded differential element: For equilibrium, F 0 and M 0 ( V + dv ) V dv F 0 qdx + ( V + dv) V 0 q - dx dx dx M 0 ( M + dm ) M ( V + dv ) dx qdx 0 ( M + dm) M dm V (neglecting q( dx ) terms). dx dx This material is intended for students in class in the Spring of 008. Any other 17

18 Bending of Beams differential element Approximation for radius of curvature: An increment of beam length dx is related to ds via dx cos( θ), for small θ dx ds ds The slope of the beam at any point is given by dw dw tan( θ), for small θ θ dx dx For a given radius of curvature, ds is related to dθ via 1 ds d, so for small d θ ρ θ θ d w dx ρ dx This material is intended for students in class in the Spring of 008. Any other 18

19 Bending of Beams differential element Basic Differential Equations for Beam Bending: d w 1 For small θ dx ρ Now that we have a relationship between w( x) and ρ we can express the moment and shear forces as a function of w( x) d w dm Moments: M EI, now recall V dx dx dw dv Shear: V EI, now recall q dx dx 4 d w Uniform Load: q EI dx 4 This material is intended for students in class in the Spring of 008. Any other 19

20 Analysis of Cantilever Beam Cantilever Beam with Point Load: M( x) F( L x) d w M F dx EI EI ( L x) Integrating the above equation twice, we have FL F w( x) A + Bx + x x EI 6EI Boundary conditions: w(0) 0 dw 0 dx x 0 This material is intended for students in class in the Spring of 008. Any other 0

21 Analysis of Cantilever Beam (cont d) Cantilever Beam with Point Load (cont d): Using the boundary conditions, we obtain the beam deflection equation, FLx x w( x) (1 ) EI L Maximum deflection : w( x) FL EI Spring constant t : k EI L EWH 4L This material is intended for students in class in the Spring of 008. Any other 1

22 Stress Concentration: Fillet σ max K σ σ ave This material is intended for students in class in the Spring of 008. Any other

23 Simple Beam Equations Relation between Load and deflection (1)- concentrated load cantilever guided-end fixed-fixed (a) cantilever beam Elongation x F L x x x Ehw F L Ehw x FxL 4Ehw (b) guided-end beam Deflection y z 4 y FL Ehw 4F FL z Ewh y z FL y Ehw FL z Ewh y z 1 16 FL y Ehw 1 FL z Ewh 1 16 [notation] L: length of beam h: height of beam w: width of beam (c) fixed-fixed beam This material is intended for students in class in the Spring of 008. Any other

24 Simple Beam Equations (cont d) Relation between Load and deflection ()-Distributed load cantilever guided-end fixed-fixed (a) cantilever beam (b) guided-end end beam Elongation Deflection y z x f L x x x E f L y 4 Ehw 4 fl z Ewh y 1 f L E f L y 4 Ehw 4 1 fl z Ewh y x 1 f L x 4 E f L y 4 Ehw 4 1 fl z Ewh z z (c) fixed-fixed beam [notation] L: length of beam h: height of beam w: width of beam This material is intended for students in class in the Spring of 008. Any other 4

25 Stability of Structures In the design of columns, crosssectional area is selected such that - allowable stress is not exceeded P σ σ all A - deformation falls within specifications PL δ δspec AE After these design calculations, may discover that the column is unstable under loading and that it suddenly becomes sharply curved or buckles. This material is intended for students in class in the Spring of 008. Any other 5

26 Stability of Structures (cont d) Consider model with two rods and torsional spring. After a small perturbation, ( θ ) K restoring moment L L P sin θ P θ destabilizing moment Column is stable (tends to return to aligned orientation) if P L ( θ ) θ < K P < P cr 4K L This material is intended for students in class in the Spring of 008. Any other 6

27 Stability of Structures (cont d) Assume that a load P is applied. After a perturbation, the system settles to a new equilibrium configuration at a finite deflection angle. P L ( θ ) sinθ K PL P θ 4K P sinθ cr Noting that sinθ < θ, the assumed configuration is only possible if P > Pcr. This material is intended for students in class in the Spring of 008. Any other 7

28 Euler s Formula for Pin-Ended Beams for Buckling Consider an axially loaded beam. After a small perturbation, the system reaches an equilibrium configuration such that dy M P dy P y + y 0 dx EI EI dx EI Solution with assumed configuration can only be obtained if π EI P > Pcr L E P π ( Ar ) π E σ > σcr A L A Lr where r I/A ( ) This material is intended for students in class in the Spring of 008. Any other 8

29 Extension of Euler s Formula A column with one fixed and one free end, will behave as the upper-half of a pin-connected column. The critical loading is calculated l from Euler s formula, P cr π EI L e π E σ cr L r L e ( ) e L equivalent length This material is intended for students in class in the Spring of 008. Any other 9

30 Net Torque Due to Internal Stresses Net of the internal shearing stresses is an internal torque, equal and opposite to the applied torque: ( ) T ρ df ρ τ da Unlike the normal stress due to axial loads, the distribution of shearing stresses due to torsional loads can not be assumed uniform. This material is intended for students in class in the Spring of 008. Any other 0

31 Shaft Deformations From observation, the angle of twist of the shaft is proportional p to the applied torque and to the shaft length: φ T φ L Cross-sections Cosssect o s of noncircular cu (non- axisymmetric) shafts are distorted when subjected to torsion. Cross-sections for hollow and solid circular shafts remain plain and undistorted because a circular shaft is axisymmetric. This material is intended for students in class in the Spring of 008. Any other 1

32 Shearing Strain Since the ends of the element remain planar, the shear strain is equal to angle of twist: ρφ L γ ρφ or γ L Shear strain is proportional to twist and radius cφ ρ γ max and γ γ L c max This material is intended for students in class in the Spring of 008. Any other

33 Stresses in Elastic Range Multiplying the previous equation by the shear modulus, ρ Gγ Gγ c max From Hooke s Law, τ G γ, so ρ τ τmax c J The shearing stress varies linearly with the radial position in the section. τ max max T da da J τ ρτ c ρ Tc τ J max and T ρ J τ c J 1 c 4 π 4 4 ( c ) 1 c 1 π This material is intended for students in class in the Spring of 008. Any other

34 Deformations Under Axial Loading The angle of twist and maximum shearing strain are related: cφ γ max L The shearing strain and shear are related by Hooke s Law, τmax Tc γ max G JG TL φ JG With variations in the torsional loading and shaft cross-section along the length: φ TL i i J G i i i This material is intended for students in class in the Spring of 008. Any other 4

35 Torsion of a rectangular bar torsional beam comb Assume that the torsional beam is isotropic material Torsional stiffness (when, t h >W h ) G 19 wh 1 1 th k thwh 1 π 5 π tanh n, n 1,, L h th n n wh [ref] S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, pp. 09 1, This material is intended for students in class in the Spring of 008. Any other 5

36 Reference F. P. Beer, E. R. Johnston, and Jr. J.T. DeWlof, Mechanics of Materials", McGraw-Hill, 00. J. M. Gere and S. P. Timoshenko, Mechanics of Materials, PWS Publishing Company, S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw- Hill, Chang Liu, Foundations of MEMS, Pearson, 006. Nicolae O. Lobontiu, Mechanical design of microresonators, McGraw- Hill, 006. This material is intended for students in class in the Spring of 008. Any other 6

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