Lecture 6: Comb Resonator Design () -Intro. to Mechanics of Materials Sh School of felectrical ti lengineering i and dcomputer Science, Si Seoul National University Nano/Micro Systems & Controls Laboratory Email: dicho@snu.ac.kr URL: http://nml.snu.ac.kr
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/ Δ l EA/ l usage and possession is in violation of copyright laws
Strain Strain: ratio of deformation to length γ Δll / l Shear Modulus G τ / γ Relation among: G, E, and ν G E (1 + +ν ) usage and possession is in violation of copyright laws
Poisson s Ratio Tensile stress in direction results in compressive stress in y and z direction (object becomes longer and thinner) Poisson s Ratio: ε ε ν y ε z ε transverse strain longitudinal strain Metals : ν 0. Rubbers : ν 05 0.5 Cork : ν 0 usage and possession is in violation of copyright laws 4
State of Stress The combination of forces generated by the stresses must satisfy the conditions for equilibrium: F F F y z y z 0 M M M Consider the moments about the z ais: 0 ( τ ) ( τ ) M 0 ΔA a ΔA a z y y τ τ, τ τ and τ τ y y yz zy yz zy Only 6 components of stress are required to define the complete state t of stress. usage and possession is in violation of copyright laws 5
Ductile Materials Stress and Strain Diagram usage and possession is in violation of copyright laws 6
Stress and Strain Diagram (cont d) Brittle Materials Stress-strain diagram for a typical brittle material usage and possession is in violation of copyright laws 7
Deformations Under Aial 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 usage and possession is in violation of copyright laws 8
Stress Concentration: Fillet σ ma K σ σ ave usage and possession is in violation of copyright laws 9
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 y σ da 0 M zσ da 0 M y σ da M z usage and possession is in violation of copyright laws 10
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 m δ yθ ρθ y L ρ (strain varies linearly) c c or ρ ρ εm y ε m c usage and possession is in violation of copyright laws 11
For a linearly elastic material, Stress Due to Bending y σ Eε Eεm c y σ m (stress varies linearly) c For static equilibrium, y F 0 σ da σ m 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. usage and possession is in violation of copyright laws 1
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 ma σ ma M ( σ ma da) h h da I t t t w t w t h ( ) ( ) h ( ) Mt sma EI t where σ ma : magnitude of stress I : moment of inertia of the cross-section h : height of a beam t : thickness of a beam s ma : maimum longitudinal strain usage and possession is in violation of copyright laws 1
Deformations in a Transverse Cross Section Deformation due to bending moment M is quantified by the curvature of the neutral surface 1 ε σ 1 ρ c Ec Ec I m m Mc M EI Although cross sectional planes remain planar when subjected to bending moments, in-plane deformations are nonzero, ν y ν y εy νε ε z νε ρ ρ usage and possession is in violation of copyright laws 14
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 usage and possession is in violation of copyright laws 15
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( ) 0 V F M 0 M( ) + F( L ) 0 M( ) F( L ) L usage and possession is in violation of copyright laws 16
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 qd + ( V + dv) V 0 q - d d d M 0 ( M + dm ) M ( V + dv ) d qd 0 ( M + dm) M dm V (neglecting q( d ) terms). d d usage and possession is in violation of copyright laws 17
Bending of Beams Differential Element (cont d) Approimation for radius of curvature: An increment of beam length d is related to ds via d cos( θ), for small θ d ds ds The slope of the beam at any point is given by dw d tan( θ), for small θ θ dw d 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 d ρ d usage and possession is in violation of copyright laws 18
Bending of Beams Differential Element (cont d) Basic Differential Equations for Beam Bending: d w 1 For small θ d ρ Now that we have a relationship between w( ) and ρ. We can epress the moment and shear forces as a function of w ( ). d w dm Moments: M EI, now recall V d d dw dv Shear: V EI, now recall q d d 4 d w Uniform Load: q EI d 4 usage and possession is in violation of copyright laws 19
Analysis of Cantilever Beam Cantilever Beam with Point Load: M ( ) F ( L ) d w M F ( L ) d EI EI Integrating the above equation twice, we have FL F w( ) A + B + EI 6EI Boundary conditions: w(0) 0 dw 0 d 0 usage and possession is in violation of copyright laws 0
Analysis of Cantilever Beam (cont d) Cantilever Beam with Point Load (cont d): Using the boundary conditions, we obtain the beam deflection equation, FL w( ) (1 ) EI L Maimum deflection : w( ) FL EI Spring constant : k EI L EWH 4L usage and possession is in violation of copyright laws 1
Stress Concentration: Fillet σ ma K σ σ ave usage and possession is in violation of copyright laws
Simple Beam Equations Relation between Load and deflection (1)- concentrated load Cantilever Guided-end Fied-fied (a) Cantilever beam Elongation y Deflection F L Ehw 4 y FL Ehw (b) Guided-end end beam z 4F FL z Ewh y z F L Ehw FL y Ehw FL z Ewh y z FL 4Ehw 1 16 FL y Ehw 1 FL z Ewh 1 16 L : length of beam h : height of beam w : width of beam (c) () Fied-fied beam usage and possession is in violation of copyright laws
Simple Beam Equations (cont d) Relation between Load and deflection ()-Distributed load Cantilever Guided-end Fied-fied (a) Cantilever beam Elongation f L E f L E f L 4 E (b) Guided-end beam y f L y 4 Ehw y 1 f L y 4 Ehw y 1 f L y 4 Ehw Deflection 4 fl z z Ewh z 4 1 fl z Ewh z 4 1 fl z Ewh (c) Fied-fied beam L : length of beam h : height of beam w : width of beam usage and possession is in violation of copyright laws 4
Stability of Structures In the design of columns, cross-sectional area is selected such that - Allowable stress is not eceeded P σ σ A all - Deformation falls within specifications PL δ δ AE spec After these design calculations, may discover that the column is unstable under loading and that it suddenly becomes sharply curved or buckles. usage and possession is in violation of copyright laws 5
Stability of Structures (cont d) Consider model with two rods and torsional spring. After a small perturbation, ( ΔθΔ θ ) K restoring moment L L P sin Δ θ P Δθ destabilizing moment Column is stable (tends to return to aligned orientation) if P L ( θ ) Δ θ < K Δ P < P cr 4K L usage and possession is in violation of copyright laws 6
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 > P cr. usage and possession is in violation of copyright laws 7
Euler s Formula for Pin-Ended Beams for Buckling Consider an aially loaded beam. After a small perturbation, the system reaches an equilibrium configuration such that dy M P dy P y + y 0 d EI EI d 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 ( ) usage and possession is in violation of copyright laws 8
Etension of Euler s Formula A column with one fied 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 usage and possession is in violation of copyright laws 9
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 aial loads, the distribution of shearing stresses due to torsional loads can not be assumed uniform. usage and possession is in violation of copyright laws 0
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 of noncircular (non- aisymmetric) shafts are distorted when subjected to torsion. Cross-sections for hollow and solid circular shafts remain plain and undistorted because a circular shaft is aisymmetric. usage and possession is in violation of copyright laws 1
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 ma and γ γ L c ma usage and possession is in violation of copyright laws
Stresses in Elastic Range Multiplying the previous equation by the shear modulus, ρ Gγ Gγ c ma From Hooke s Law, ρ τ τma c τ G γ, so The shearing stress varies linearly with the radial position in the section. J 1 4 π c τ ma ma T da da J τ ρτ c ρ Tc τ J ma and T ρ J τ c J 4 4 ( c ) 1 c 1 π usage and possession is in violation of copyright laws
Deformations Under Aial Loading The angle of twist and maimum shearing strain are related: cφ γ ma L The shearing strain and shear are related by Hooke s Law, τma Tc γ ma 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 usage and possession is in violation of copyright laws 4
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,,5... 5 L h th n n wh [Ref] S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill,,pp pp. 09 1, 1970. usage and possession is in violation of copyright laws 5
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, 1997. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, 1970. Chang Liu, Foundations of MEMS, Pearson, 006. Nicolae O. Lobontiu, Mechanical design of microresonators, McGraw-Hill, 006. usage and possession is in violation of copyright laws 6