Lecture 8

• Continuation of last time

• Stress Tensor- applied to blood forces

• Test

Mechanical Models

Voigt solution1

Z

1

Rs C Z

1

C

s1

Laplace domain V s( ) I s( ) Z s( )I o

s

1

C 1

s 0( ) s1

taat eeaa

21

12

1 = bi-exponential decay

Time domainV t( ) I o R 1 e

t

E o

E1 e

t

Z

Y

X

xx

yx

zx

xy

yy

zy

xz

yz

zz

3-Dimensional stresses (stress tensor)

Stress components @ Equilibrium

0

0

0

3

33

2

32

1

31

3

13

2

22

1

21

3

13

2

12

1

11

xxx

xxx

xxx

Blood Forces on ECs

Y.C. Fung

x

y

z

Analysis of EC upper membrane

0

,,

zyyzxzzx

xzzxzyyzyxxy

xx

yx

zx

xy

yy

zy

xz

yz

zz

Symmetrical

(Fluid Mosaic)

x

y

z

0

,

yxxy

yxxy

On surface facingblood

On surface facingcytosol

x

y

z

Static Eq

h

xxx

zyzxzz

yzyxyy

xzxyxx

dyT

yxz

zxy

zyx

0

0

x

y

z

h

xxx

yxxy

yxxy

dyT0

0

,

On surface facing

blood

We need membrane tension as f()x

y

z

Define

LTdxx

T

x

T

dyT

ce

dyy

dyx

egrateanddymultbyzyx

x

Lxx

h

xxx

hxy

hxx

xzxyxx

0

0

00

0

sin

0

int0

(if Tx= 0 @ x=0)

x

y

z

Stress on cell from flow

h

L

h

T

so

hT

xxx

xxx

@ x = -L

For = 1 N/m2 , L= 10 m, h = 10 nm

2310m

Nxx

m

NTx

610

0 LFor L= 1 cm, xx= 106

Fluid Pressure is omnidirectional

>

A

dZ

dx

dy

P1

P2

P3

P4q

P5

Rotate by 90, and see also:

P4 = p5

P1 dy dz = P2 sin(q) dz dy/sin(q)

P1= P2

Fx = 0

P3 dx dz = P2 cos(q) dz dx/cos(q)

P2=P3

Fy = 0

Fz = 0

Hence P1=P2=P3=P4=P5 =P

Coding of Probability

1

))(1(i

i

t

t

dttnKT

Integral pulse frequency modulation

Probability Pulse frequency and width Modulation

Pulse Width Modulator

2

1

)(t

t

dttu

Leaky integrator

Thresholder

Pulsesout

Inputs

Reset

Control System, I.e. climate control

Sensor Plant-

--

-

Output

Error

Perturbation

Feedback

Set Point

Temperature Control

1/s 1/s+

-1

3

X2 X1

0

1

1

0

03

10

C

B

A

)()(

)()()(

tCXty

tBUtAXtX