Page 296 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Inhomogeneities

1 L  ∂w ξ t)  2 L  ∂θ ξ t)  2 

(
,
(
,
(
,
ξ
ξ
ξ
ξ
T ∗ ξ t) = 2 ∫ ρ () -------------------- d + ∫ I () ------------------- d  (7.124)
--- 




m
m
∂t
∂t


 0 0 
The proportionality constants of these two squared velocity terms reflect
the resistance of the beam to movement:
ξ
• ρ () = ρA is the mass per unit length of the beam material and
m
represents the sluggishness or linear inertia of the beam material to
being accelerated. The beam has a cross sectional area and a volu-
A
ρ
metric mass density ;
ξ
• I () = ρAi 2 is the second moment of mass or mass moment of
m g
inertia of the beam material and represents the sluggishness or inertia
of the beam cross section to angular acceleration. The radius of gyra-
tion i represents that radius that would give the same moment of
g
inertia, were all the mass of the cross section concentrated along a cir-
cular line. Again, since the geometry enters this term in a squared
sense, we can circumvent the constraints of material through prudent
geometrical design.
From the Hamilton principle (Box 2.3) for the variation of the action
t 2
∫
d
δA = δ ( T ∗ – V) t = W (7.125)
t 1
we obtain the two partial differential equations of motion [7.5]
(
∂  ∂θ   ∂w ξ t) 
,
(
,
------ E ξ()I ξ()------ + k ξ() -------------------- – θξ t)
∂ξ  ∂ξ T  ∂ξ 
(7.126a)
,
(
∂θ ξ t)
ξ
– I ()------------------- = 0
m
∂t
2
(
,
∂ w ξ() ∂   ∂w ξ t) 
,
(
ξ
(
,
ρ ()----------------- – ------ k ξ() -------------------- – θξ t)  – q ξ t) = 0 (7.126b)
∂ξ 

m
T
2
∂ξ ∂ξ
Semiconductors for Micro and Nanosystem Technology 293