Microsoft PowerPoint - Chapter 4 Biomechanics-이정찬.ppt [호환 모드]

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1 Biomechanics Jung Chan LEE

2 Mechanics 4M Mechanics 고체역학 : Solid Mechanics Solid Mechanics : deformation and fracture Statics : F=0 and M=0 동역학 : Dynamics Kinematics ( 기구학 ) : 운동의형태 Kinetics ( 운동역학 ) : 힘 과 에너지 유체역학 Fl idi (Fl id M h i ) 유체역학 : Fluidics (Fluid Mechanics) 유체정역학 : 비점성, 비압축성 유체동역학 : 점성, 압축성 ( 기체 )

3 열역학 : Thermodynamics 열역학제 0 법칙 : 열적평형상태 열역학제1법칙 : 열시스템의내부에너지는시스템에가해진열에너지와시스템이외부에한 -( 일 ) 의합이다. 열역학제 2 법칙 : 시스템의총엔트로피는시간을따라증가하려는경향이있다. 열역학제3법칙 : 절대영도에다가갈수록엔트로피는최소값상수로수렴 엔트로피 ( 일로변환할수없는에너지, 무질서도 ) 열전달 ( 전도, 대류, 복사 ) 열기관 ( 냉동기 )

4 Vector Mechanics Vector Math. 2D Norm Angle

5 3D

7 Vector 연산 Summation Product Dot Product ( 내적 ): 힘, 변위, 일 Cross Product ( 외적 ) : 토크, 모멘트, 비틀림

8 Dot Product ( 내적 ) : 스칼라값 힘, 변위, 일 힘을 F, 변위를 s, 힘과변위사이의각을 θ라고할때, 일 W 는

9 Cross Product ( 외적 ) : 토크, 모멘트, 비틀림

10 >> %(a) First write the direction vector d that points along F >> % as a 1D array: >> d = [ ] d = >> % Now write the unit vector of F, giving its direction: >> unit_vector = d/norm (d) unit_vector = >> % F consists of the magnitude 10 kn times this unit vector >> F = 10*unit_vector F = >> % Or, more directly >> F = 10*(d/norm(d) (d) ) F = >> % (b) First write the vector r_xz that points in the xz plane: >> r_xz = [12 0 9] r_xz = >> % The dot product is given by the sum of all the term by term >> % multiplications of elements of vectors F and r_xz >> F_dot_r_xz = sum(f.*r_xz) >> % or simply, dot(f,r_xz) F_dot_r_xz = >> % (c) Cross F with a vector that points from the origin to F. >> % The cross product is given by the cross function >> r_xz_cross_f = cross(r_xz,f) r_xz_cross_f = >> % Note that the cross product is not commutative >> cross(f,r_xz) ans = >> % Vectors are added and subtracted in MATLAB using the + and - >> %operations, respectively.

11 Coordinate Transformation

12 Euler Angle

14 % eulerangles.m % %E Euler angles for y-x-z rotation ti sequence % using MATLAB symbolic math toolbox % % x, y and z are thetax, thetay and thetaz, respectively % First define them as symbolic variables syms x y z % Writing equations as a matrix A A = [ cos(y), 0, -sin(y); 0, 1, 0; sin(y), 0, cos(y)] % equations as matrix B B = [ 1, 0, 0; 0, cos(x), sin(x); 0, -sin(x), cos(x)] %andequations as matrix C C = [ cos(z), sin(z), 0; -sin(z), cos(z), 0; 0, 0, 1] % The matrix equation 4.30 is created by multiplying matrices C, B % and A D = C*B*A >> eulerangles D = [cos(z)*cos(y)+sin(z)*sin(x)*sin(y), sin(z)*cos(x), -cos(z)*sin(y)+sin(z)*sin(x)*cos(y)] [-sin(z)*cos(y)+cos(z)*sin(x)*sin(y), cos(z)*cos(x), sin(z)*sin(y)+cos(z)*sin(x)*cos(y)] [cos(x)*sin(y), -sin(x), cos(x)*cos(y)]

16 function y = cosd(x) %COSD(X) cosines of the elements of X measured in degrees. y = cos(pi*x/180); function y = sind(x) %SIND(X) sines of the elements of X measured in degrees. y = sin(pi*x/180); function D = eulangle (thetax, thetay, thetaz) %EULANGLE matrix of rotations by Euler s angles. % EULANGLE(thetax, thetay, thetaz) yields the matrix of % rotation of a system of coordinates by Euler s % angles thetax, thetay and thetaz, measured in degrees. % Now the first rotation is about the x axis, so we use eqs A = [ cosd(thetax) sind(thetax) 0 -sind(thetax) cosd(thetax) ]; % Next is the y axis rotation (Eqs ) 23) B = [ cosd(thetay) 0 -sind(thetay) sind(thetay) 0 cosd(thetay) ]; % Finally, the z axis rotation (Eqs ) C = [ cosd(thetaz) sind(thetaz) 0 -sind(thetaz) cosd(thetaz) ]; % Multiplying rotation matrices C, B and A as in Eq gives the solution: D=C*B*A; >> eulangle(30,20,10) ans =

17 Static Equilibrium 정지 된물체에작용하는힘과모멘트고려 힘의합 =0 모멘트의합=0

20 가정 풀리의직경은무시 줄에걸리는장력은줄전체로동일 (T=F 1 =F 2 =F 3 ) i, j 성분끼리

21 x components y components Sol.

22 z y R y F 2, F 3

23 Moment of Inertia 물체의회전력을지속하게끔하는질량분포에너지 ( 물체각부분의질량 )X( 회전축까지의거리 )^2 의합

24 Mechanics of Materials Stress( 응력 ) Strain( 변형율 ) Elastic Modulus

25 Poisson s Ratio : 축방향 strain 에대한횡방향 strain 의비

27 Stress-strain curve

28 Brittle( 취성 ) vs. Ductile( 연성 )

29 Spring-back hysteresis

30 Viscoelastic Model K B 스프링상수 : K F=Kx 스프링상수 : K F=Kx 댐핑상수 : B F=B*dx/dt=Bv

31 Simple Spring Model : F=Kx

32 Maxwell Model : dx/dt=1/k*df/dt+1/b*f

33 Voight Model : F=Kx+B*dx/dt

34 Kelvin Model : F+B/K*dF/dt=Kx+2B*dx/dt

35 Cardiovascular Dynamics Blood Rheology Solid vs. Fluid against Shear stress

36 점성과전단응력 압력, 온도, 밀도 : 시스템의특성을나타내는열역학적변수 점도 : 움직이는유체에서의국소응력과유체요소의변형률 (strain) 과의관계 치약, 케첩, 혈액 (?) 천천히흘려주면흐르지만, 강하게쏟으면고정 ( 전방농후 ) 천천히흘려주면약간흐르지만, 강하게쏟으면무너짐 ( 전방희박 ), 생크림, 페인트

37 % Power Law Fit of Blood Data % % Store shear strain rate and stress data in arrays alpha=[ ]; [1.5,2,3.2,6.5,11.5,16,25,50,100] T = [12.5,16,25.2,40,62,80.5,120,240,475] ; % Take natural logs of both x = log(alpha) ; y = log(t) ; % Use MATLAB s polyfit function to do linear curve fit coeff = polyfit (x,y,1) % Write curve fit coefficients as a new x-y function for plotting x1=[0;0.01;5] y1=polyval (coeff,x1) % Plot the original data as o points plot(x,y, o ) hold on % Overlay a plot of the curve-fit line plot(x1,y1) grid on title ( Power Law Function ) xlabel( ln Strain Rate [ln (1/s)] ) ylabel( ln Shear Stress [ln dyne/cm2] )

38 비압축성, 압축성유동 연속식을간단하게하기위해밀도변화를무시 : 비압축성 혈액유동을포함하는대부분의생체공학적인유동은비압축성유동으로해석 ex) 압축성유체공학 : 기체를주로다루는내연기 ex) 압축성유체공학 : 기체를주로다루는내연기관, 항공역학

39 레이놀즈수 (Reynolds number) 무차원변수 : 층류유동과난류유동을구분하는기준 점성력에대한관성력의비 Re 가매우작은경우 : 점성력이지배, 관성을무시 Re가매우큰경우 : 관성력이지배, 점성의효과를무시 유체밀도 ρ, 점성계수 μ, 특성길이 l, 속도 V

40 난류유동 (Turbulent flow) 유선이발견되지않음 유체는질량의소용돌이 속도분포 : 중앙에서최대, 벽부근에서중앙의절반정도 원형관에서 Re > 2000 일때발생

41 층류유동 (Laminar flow) 유체의입자가서로층을이루어, 뒤섞임없이질서있게흐르는상태 원형관내에서의층류유동 벽부근속도는 0, 관의중앙에서최대 관내에서속도분포는포물선 평균속도는중심속도의약 배 Re < 2000

42 Poiseuille flow 일정한단면을가진원형튜브내를흐르는정상상태의층류유동 튜브의반지름이 2 배늘어나면유량은 16 배로증튜브의반지름이 2 배늘어나면유량은 16 배로증가 ( 관저항이 1/16 로감소 ) 튜브의반지름이 ½ 로줄어들면유량은 1/16 으로감소 ( 관저항이 16 배로증가 )

43 에너지보존방정식 수직응력, 전단응력, 중력, 외부에서받는압력을고려

44 Navier-stokes Eq. 거의모든유동현상을풀수있다? 유체흐름의운동량미분방정식 가정 1) 뉴턴유체 가정 2) 비압축성유체 에너지보존식에연속방정식 ( 질량보존 ) 을적용 - CFD 프로그램의기본원리 Cylindrical Coord.

45 Hemodyamics( 혈역학 ) 혈액의압력과유량에대한측정과해석 CO : Cardiac Output 심박출량 MAP : Mean Artery Pressure 평균동맥압 BP(dia) : Diastolic Blood Pressure 이완기혈압 BP(sys) : Systolic Blood Pressure 수축기혈압 PVR : Pulmonary Vascular Resistance 폐혈관저항 SVR : Systemic Vascular Resistance 체순환혈관저항 = TPR(Total Peripheral Resistance) RAP : Mean Right Atrial Pressure 평균우심방압 CVP(Central Venous Pressure) 중심정맥압 HR : Heart Rate 심박수 D'arcy's Law Flow = Pressure/Resistance CO=(MAP-RAP)/TPR

4.1 힘의모멘트 스칼라공식 4.1 힘의모멘트 스칼라공식 모멘트크기 (resultant moment) 2

Engineering Mechanics 정역학 (Statics) 4장힘계의합력 1 GeoPave Lab. 4.1 힘의모멘트 스칼라공식 1 4.1 힘의모멘트 스칼라공식 4.1 힘의모멘트 스칼라공식 모멘트크기 (resultant moment) 2 4.1 힘의모멘트 The moment does not always cause r otation. The actual