Page 297 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
Equation (7.126a) is subject to either:
• a Dirichlet boundary condition where δθ = 0 , i.e., the bending angle
is fixed at either end of the segment;
• a Neumann boundary condition with an applied moment so that the
angular gradient of bending at either end of the segment becomes
⁄
⁄
∂θ ∂ξ = M ( EI) for an applied concentrated moment M .
Similarly, (7.126b) is subject to either of:
• a Dirichlet boundary condition where δw = 0 , i.e., the deflection is
fixed at either end of the segment;

• a Neumann boundary condition with an applied moment so that the
net deflection gradient at either end of the segment becomes
⁄
∂w ∂ξ – θ = Qk ⁄ for a concentrated force Q .
T
When the beam cross section is uniform, then the geometrical and mate-
rial quantities , , I , k and ρ become independent of the coordi-
EI
m T m
ξ
θ
nate along the beam. Eliminating the bending angle from (7.126a)
w
and (7.126b) in favour of (by taking the derivative of the first equation
ξ
w.r.t. , using the second to define , and assuming that k ≈ 1 ), we
θ
T
obtain the classical Timoshenko beam expression
4
2
∂ w ξ t,( ) ∂ w ξ t,( )
EI---------------------- + ρ----------------------
2
4
∂ξ ∂t
(7.127)
4
4
 EIρ m ∂ w ξ t,( ) I ρ ∂ w ξ t,( )
m m
– I + ------------- ---------------------- + ---------------------------------- = 0
 m k  2 2 k 4
T ∂ξ ∂t T ∂t
We take a harmonic displacement assumption
(
,
(
w ξ t) = w exp [ ikξωt)] which will be correct for the vibrational
–
0
modes. The geometrical mode factor w 0 remains undetermined for the
following calculation. In general we can then write that
(
,
∂ ( a + b) w ξ t) ( a + b) a b
(
,
-------------------------------- = i k ω w ξ t) (7.128)
a
b
∂ξ ∂t
294 Semiconductors for Micro and Nanosystem Technology