Biomedical Engineering Reference
In-Depth Information
Deflection time response (command = 0.105)
0.16
0.14
0.12
0.01
0.08
0.06
0.04
0.02
0
-0.02
0
2
4
Time (sec)
6
8
10
FIGURE 7.26
Beam tip step response.
v(x,t)
2
M
EI
∂
∂
v
2 2/3
M
moment
(v) = -
(1 +(
) )
∂
x
2
∂
x
v
f
(x)
Deflection
time
response
Final
deflection
vector
Large-angle
deflection equation
Open-loop
transfer
2
s
2
+ 2
ζω
s +
ω
ω
v
f
(0)
v(0, t)
v(x
1
, t)
v(x
2
, t)
2
v
f
(x
1
)
v
f
(x
2
)
2
ω
s
2
+ 2
v
f
(L)
2
ζω
s +
ω
v(L t)
FIGURE 7.27
Beam modeling process.
Once the shape of the open-loop transfer function of the tip was determined, a
rather significant assumption was made. For the purposes of model development, it
was assumed that each point on the beam responded like the tip; that is, the beam
was actually a set of second-order transfer functions responding independently to
their own final deflection commands. Given that assumption, the process to simulate
the deflection of the entire beam dynamically is detailed graphically in figure 7.27.
In this graphic, the modeling input is the moment value (assumed constant for
this portion of the effort). The moment is used in the solution of the nonlinear large-
angle deflection equation; the final deflection position is the solution to that equation
and is in the form of a single vector (with individual elements for each portion of
the curved beam). Each of the final deflection vector elements is used as a step
