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U. Glaeser

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v max<br />

0<br />

0<br />

v<br />

Disk Head Velocity<br />

Acceleration<br />

at<br />

vt ()<br />

Deceleration<br />

t1 t2 T *<br />

T *<br />

2<br />

T T<br />

*<br />

−<br />

2<br />

Deceleration<br />

FIGURE 8.2 A simplified disk head velocity diagram. (From Dujmovic, J., Tomasevich, D., and Au-Yeung, M., PrProc.<br />

CMG, 1999, Reno, NV. With permission.)<br />

heads linearly accelerate until they achieve the maximum speed vmax, then travel at this constant speed,<br />

and eventually decelerate linearly.<br />

The maximum time T ∗ that the disk mechanism can travel without being limited by the maximum<br />

velocity is called the critical seek time.<br />

T ∗ is the time to accelerate disk heads to the maximum speed and<br />

then to decelerate them to a standstill. In cases where disk heads travel at the constant maximum speed the<br />

critical seek time T ∗ denotes the acceleration plus deceleration time, i.e. the total seek time minus the<br />

time heads spend traveling at the constant maximum speed. Assuming constant acceleration/deceleration,<br />

the critical seek time is T ∗ = 2vmax/a. During the critical seek time interval the mechanism travels the<br />

distance x ∗ , called the critical distance . In the case of constant acceleration/deceleration, we have v = at,<br />

x(t) = at 2 /2, and x ∗ = 2x(T ∗ /2) =<br />

Let us now investigate a general symmetrical case where the function v(t) has the property that the<br />

acceleration time equals the deceleration time. We differentiate the small distance model (T ≤ T ∗ ) and<br />

the large distance model (T ≥ T ∗ 2<br />

vmax/a. ) as follows:<br />

∫<br />

T<br />

x( T)<br />

= v()dt t<br />

x ∗<br />

=<br />

0<br />

⎧<br />

⎪<br />

⎪<br />

⎪<br />

⎪<br />

= ⎨<br />

⎪<br />

⎪<br />

⎪<br />

⎪<br />

⎩<br />

∫<br />

T ∗ /2<br />

0<br />

∫<br />

T/2<br />

v()dt t<br />

+<br />

∫ v()dt t , T T ∗<br />

≤<br />

0<br />

T/2<br />

acceleration deceleration<br />

⎧<br />

⎪<br />

⎨<br />

⎪<br />

⎩<br />

∫<br />

T ∗ /2<br />

v()dt t<br />

T<br />

∫<br />

T− T ∗ /2<br />

The distance x ∗ is the total distance for acceleration from 0 to v max and deceleration from v max to 0. In<br />

the constant acceleration case (v(t) = at), this model yields<br />

Time<br />

+ vmax dt + v()dt t x ∗<br />

=<br />

0<br />

T ∗ /2<br />

T T ∗ – /2<br />

acceleration max velocity deceleration<br />

⎧<br />

⎪<br />

⎨<br />

⎪<br />

⎩<br />

v()dt t<br />

⎧<br />

⎪<br />

⎨<br />

⎪<br />

⎩<br />

© 2002 by CRC Press LLC<br />

+<br />

∫<br />

⎧<br />

⎪<br />

⎨<br />

⎪<br />

⎩<br />

⎧<br />

⎪<br />

⎪<br />

⎨<br />

⎪<br />

⎪<br />

⎩<br />

T<br />

T− T ∗ /2<br />

v()dt t<br />

⎧<br />

⎪<br />

⎪<br />

⎨<br />

⎪<br />

⎪<br />

⎩<br />

acceleration deceleration<br />

x( T)<br />

=<br />

∫<br />

T<br />

⎧<br />

⎪<br />

⎪<br />

⎨<br />

⎪<br />

⎪<br />

⎩<br />

aT 2<br />

vmax --------<br />

4 2T ∗ -------- T 2 , T T ∗<br />

= ≤ =<br />

vmax T v ⎧<br />

⎪<br />

⎪<br />

⎨<br />

⎪ ⎛ max<br />

– -------- ⎞<br />

⎪<br />

= v<br />

⎝ a ⎠ max T –<br />

⎩<br />

2vmax ----------a<br />

T<br />

Seek Time<br />

∗<br />

⎛ T<br />

----- ⎞ ∗<br />

, T≥T ⎝ 2 ⎠<br />

t<br />

vmax T T ∗<br />

( – ), T T ∗<br />

+<br />

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