Transcript f1.ppt

Shape Memory Alloy Cantilever Beam
Mike Hilldoerfer
Numerical Analysis for Engineers
April 10, 2001
Shape Memory Alloy Cantilever Beam
Background on SMAs
• Metals that possess the ability to ‘remember’ their
original size or shape
• Contain a characteristic phase transformation
temperature dependent upon metallurgical content
• Elastic Modulus different for the 2 phases
Shape Memory Alloy Cantilever Beam
Problem Description
• Cantilever beam composed of SMA subjected to
temperature gradient - both phases present
F
Beam loaded at its free end
x
L
Temperature gradient crosses phase
transformation temperature
T
Ms
Mf
a
Results in beam that is
1/2 Martensite, 1/2 Austenite --Modulus varies between phases
l
Ea
Em
a
l
Shape Memory Alloy Cantilever Beam
Mathematical Formulation
Problem separated into two functions
Function 1
Function 2
F
F
M1
L1
L2
d2y
EI 2  M  F ( x  L1 )  M 1
dx
dy
x2
EI
 V  F (  L1 x)  M 1 x
dx
2
1   x 3 L1 x 2  M 1 x 2 
 
Y1 
 F  

EI   6
2 
2 
where  M 1  F * L2
d2y
EI 2  M  F ( x  L2 )
dx
dy
x2
EI
 V  F (  L2 x )
dx
2
1   x 3 L2 x 2  

Y2 
 F  
EI   6
2 
Slope & Deflection of Function 1 used as origin for Function 2
a
Ymax
1   x 3 L1 x 2  M 1 x 2 
1
 

 F  
 
EI   6
2 
2  0 EI
L2
  x 3 L2 x 2 
dy L 2
 
x0
 F  
6
2
dx
 0
 
Shape Memory Alloy Cantilever Beam
Numerical Approaches & Results
• Simpson’s Method and Adaptive Quadrature Used
– Baseline…isothermal, constant modulus
– SMA with temperature gradient, varied modulus
SMA with temp
x
0.0
2.5
5.0
7.5
10.0
x
10.0
Analytical
dy/dx
y
0
-0.00175000
-0.00300000
-0.00495652
-0.00560870
Simpson’s
0
-0.00229167
-0.00833333
-0.01855072
-0.03202899
Adaptive
Quadrature
-0.03202354 -0.03202354
Shape Memory Alloy Cantilever Beam
Error Analysis & Conclusions
SMA Exposed to Temp Gradient
x
y
Adaptive
Quadrature
y
10.0
.00000545
.00000545
Simpson’s
All numerical solutions accurate within 0.1%
Both numerical method’s results the same
FEA validated solutions
• Solution makes sense
• Problem can be expanded to include plastic
deformation and strain recovery once heated above
transition temperature
• Beam can be developed into an actuator system