ME 125NT
Intro to Nanotechnology
Due: 4/29/2008
1
Solutions Set 4
Problem 5.7 in book
A silicon cantilever beam is 300 μm long, 100 μm wide and 6 μm thick.
Silicon’s modulus of
elasticity is 110 GPa; its density is 2330 kg/m
3
.
(a) Determine the spring constant,
k
, of this beam.
(b)
Determine the natural frequency in radians per second.
(c) Express your answer from part
(b) In hertz.
ANSWER
A)
( )( )( )
( )
=
×
×
×
×
=
=



3
6
3
6
9
9
3
3
m
10
300
4
m
10
6
m
10
100
Pa
10
110
4
L
wt
E
k
M
beam
0.022 N/m
B)
The mass of the beam is:
( )( )( )( )
kg
10
2
4
m
10
6
m
10
100
m
10
300
kg/m
330
2
10
6
6
6
3




×
=
×
×
×
=
=
.
V
m
ρ
Natural frequency,
ω
n
, using the effective mass of the beam,
m
eff
=0.24
m
, is:
( )
=
×
=
=

kg
10
2
4
24
0
N/m
022
0
24
0
10
.
.
.
m
.
k
n
ω
1.48 × 10
4
rad/s
C)
=
=
π
2
n
n
f
2400 Hz
Problem 5.13 in book
A cantilever beam has length,
l=100
μm;
width,
w=40
μm
;
thickness,
t=5
μm
;
modulus of
elasticity,
E
M
=100 GPa; density,
ρ=
2000 kg/m
3
;
and damping coefficient
, b=10

7
N*
s/m
.
(a) What is the beam’s damped natural frequency, f (Hz)? (Do not forget, for a
cantilever beam
m
eff
=0.24 m in the equation for damped natural frequency.)
(b) What is its quality factor?
(c)
A periodically varying driving force with
F
0
=10 nN acts on the beam. Plot the
amplitude, A, of the beam’s oscillation as a function of the drive frequency, F
drive
. Include
F
d
on the graph so that the resonance peak is visable.
(d) On the same graph, plot the amplitude if the damping coefficient quadruples.
Label this curve “extra damping.”
(e) On the same graph, plot the amplitude if, instead, the effective mass of the beam
increases by 500 femtograms.
Label this curve “extra mass.”
(f) Finally, on the same graph, plot the amplitude if, instead, the spring constant
increases by 5 N/m.
ANSWER