U. Glaeser

The critical distance x ∗ is
Therefore, the acceleration and the maximum speed depend on the critical values x ∗ and T ∗ :
They can also be numerically determined from the linear segment of the seek time characteristic, using
arbitrary points (x 1, T 1) and (x 2, T 2):
The seek time characteristic follows from x(T):
T( x)
This function satisfies the following properties:
x ∗
v max
x T ∗ a T
( )
∗
( ) 2
v-------------maxT
4
∗
= = = --------------- =
2
a 4x ∗ / T ∗
( ) 2
= =
• The initial nonlinear segment, for x ≤ x ∗ , is a square root function.
• The second segment, for x > x ∗ , is a linear function.
, v max 2x ∗ /T ∗
• At the critical point x = x ∗ , the first derivative of the square root function equals the first derivative
of the linear function (dT/dx = 1/v max).
From T = T ∗ (1 + x/x ∗ )/2 we can easily see that both x ∗ and T ∗ can be determined from the linear
segment of the measured seek time characteristic using the following four steps, illustrated in Fig. 8.3:
1. Extend the linear segment of the measured seek time characteristic to the vertical axis. For x = 0
we get the point T ∗ /2 on the vertical axis.
2. Move vertically up to the point T ∗ (this value is twice the distance between the origin and the
intersection of the linear segment and the vertical axis).
FIGURE 8.3 Graphical method for determining x ∗ and T ∗ . (From Dujmovic, J., Tomasevich, D., and Au-Yeung,
M., Proc. CMG, 1999, Reno, NV. With permission.)
© 2002 by CRC Press LLC
T *
T * / 2
2
vmax -------a
x2 – x1 ( x2 – x1) ----------------, a
T2 – T1 2
= = ------------------------------------------------------
( T1x 2 – T2x 1)
( T2 – T1) 2 x
-- 2
a
xx∗
= -------- = T
vmax x x ∗
+ T
------------vmax
∗
⎧
⎪
⎪
= ⎨
⎪
x
⎪ = ----- + -------- =
⎩
2 vmax T
[2]
[1]
[3]
Nonlinear
segment
[4]
∗ x
x ∗ ---- , x x ∗
≤
T ∗
-----
2
x
x ∗
⎛---- + 1⎞
∗
, x≥x ⎝ ⎠
Linear
segment
0 x
x *
0