1
2
Modelling and analysis of notebook computer chassis structure for optimization of
component mounting
3
4
5
Fukun Lai∗, J.Q. Mou, I. B. L. See, W. Z. Lin
Data Storage Institute, 5 Engineering Drive 1, Singapore 117608
Abstract
As one of the most important components in notebook personal computers, the chassis is critical for the performance
reliability and robustness of the computer as well as the hard disk drive (HDD). An appropriate chassis structure
could eectively reduce the position error signal (PES) of HDD induced by the built-in vibration sources, including
the speaker, cooling fan and CD/DVD driver, transmitted to the HDD via the chassis. Therefore, it is important to
study the vibration transmission from the sources to HDD in order to isolate the external disturbance on HDD through
optimization of chassis structure. This paper presents a theoretical model to analyze the vibration transmission from
the speaker, cooling fan and CD/DVD driver to the HDD, where the governing equations describing the vibration
transmission are formulated and their solutions are developed. The transmission characteristics are demonstrated
in the transfer function forms, which provide a straight approach to the analysis of the chassis of notebook. The
transmission routes in the model are then investigated in detail to provide a basic understanding and general design
rule for the notebook and chassis regarding vibration transmissibility. Furthermore, the results of the theoretical model
are veried by the frequency analysis of the notebook computers, and an investigation of the vibration transmission from
the vibration sources to the HDD is performed in detail. In addition, the solution to reduce the external disturbance on
HDD through optimization of chassis is also proposed. This study provides a simple and eective approach for chassis
design to reduce the vibration transmission to HDD in notebook computers.
6
Keywords:
notebook chassis design, theoretical model, vibration isolation, hard disk drives (HDDs), notebook
7 computer
8
1. Introduction
9
As the hard disk drives (HDDs) industry and academic community keep pushing the envelop of HDDs' capacity
10 and areal density, the track misregistration (TMR) budget decreases dramatically. Therefore, accurate head and track
11 positioning are required to ensure proper HDDs performance. However, the head position error signal (PES) of HDDs in
∗
Corresponding author, Tel: (+65)6874 5221
lai_fukun@dsi.a-star.edu.sg (Fukun Lai)
Email address:
Preprint submitted to Elsevier
November 20, 2012
1 notebook computer is very sensitive to the vibrations induced by the built-in vibration sources in a notebook computer,
2 including the speaker, cooling fan, CD/DVD driver. The vibration from the build-in source could be transmitted to
3 the HDD through two paths, i.e., through the air by the acoustic pressure, and through the chassis by structural
4 vibration. However, a previous study had shown that, the transmission from the build-in vibration source to HDD is
5 mainly through the chassis structure instead of the acoustic pressure in the air[1]. A proper chassis design is therefore
6 critical for reduction of the vibration of HDDs induced by built-in vibration sources in the notebook. In order to
7 better understand the vibration transmission through the notebook chassis and propose vibration reduction guideline
8 for chassis designers, a simple theoretical model is required.
9
Presently, a large volume of studies have been published on the vibration of HDDs and other components in personal
10 computers. For example, Yap et al. analyzed the noise and vibration for HDDs with spindle motor rotating at high
11 speeds by an analytical model [2]. Tandon et al. studied the vibration and noise of the HDDs for dierent spindle
12 motor speeds using the theoretical model, FEM and experimental methods [3]. Yasuhiro et al. devised a exible support
13 mechanism for HDDs to isolate external vibration [4]. Lim et al. analyzed dynamic characteristics and shock response
14 of the HDDs in notebook [5]. Many other researchers have also done a variety of investigations of the vibration of HDDs
15 [6, 7, 8]. However, very few publications have been found on the vibration transmission in notebook computer through
16 chassis. Among them, Hu et al. systematically investigated the built-in speaker induced structural-acoustic vibration
17 on HDDs by the developed FEM and experiments [1], and concluded that the acoustically transmitted vibration of
18 speaker can be ignored compared to the structurally transmitted vibration. Mou et al. analyzed notebook computer
19 chassis design for hard disk drive and speaker mounting [9]. Furthermore, the commercial notebook industry rarely
20 realized that the chassis design could substantially aect the performance of the HDD, especially, some notebook simply
21 place the vibration sources, e.g., speaker, very near to the HDD.
22
In this paper, a theoretical model to analyze the vibration transmission from the speaker, cooling fan, and CD/DVD
23 driver to the HDD is presented. In the model, the vibration source, chassis and HDD are simplied as the lumped
24 mass with spring and damper supported. Based on the simplication, the ordinary dierential equations governing the
25 dynamics of the chassis, HDDs and vibrations sources are developed. The model is composed of four degree of freedoms
26 (DOFs), in which the three components are modeled as lumped masses with mounting structures as the springs and
27 damping elements. The solution of the model is then developed to give the transmission characteristics in the transfer
28 function forms, which illustrate the vibration transmitted from the source to HDD via chassis in an eective form.
29 In order to determine the parameters in the theoretical model, such as mass, stiness and damping coecient, the
30 virtual experiments are performed through a static nite element analysis on the components. The theoretical model
2
1 is then applied to investigate the vibration transmission from the speaker, which is the most critical vibration sources
2 in notebook computer [1], to the HDD. The results are compared with the frequency analysis from a detailed full nite
3 element model simulation, which validates the present theoretical model.
4
5
2. A model of notebook structure
An accurate theoretical model for notebook structure should include the key factors while keep it as simple as
6 possible, as well as give a general understanding of the vibration transmission in the notebook. In this section, a
7 simplied model of notebook structure is developed, where the transmission from the vibration sources, such as speaker,
8 cooling fan and CD/DVD drives, to HDD in notebook could be characterized into four routes as shown in Figure 1(a).
9 The vibration from the source is rst transmitted to the chassis through the mount, shown as route AB in Figure
10 1(a). The vibration then propagates to the other side of the chassis, which is supporting the HDD, through route BC.
11 Since the chassis is usually mounted on a case with large area, part of the vibration energy is dissipated during the
12 transmission, and part of it is transmitted to ground through route BE and CE. Finally, the vibration is transmitted to
13 the HDD through HDD mounting structure. In addition to the key components, such as vibration sources, chassis, and
14 HDD, the present model also includes the mounting structures of the whole notebook and the vibration propagation
15 through the case of the notebook computer.
(a)
(b)
Figure 1: A notebook structural model for vibration transmission from sources to HDD (a), a four DOFs theoretical model of a notebook
computer(b).
16
3
1
From the above vibration route transmission discussion, a four DOFs vibration model of the notebook structure is
2 developed as shown in Figure 1(b). The motion of the vibration source can be accounted for by one DOF as a rigid
3 mass mvs , where the stiness and damping of the mount could be simplied to a spring with stiness kvs and a damper
4 with damping coecient cvs . A force fvs is applied on the source to excite the vibration. Similarly, the motion of the
5 HDD could also be modeled as a DOF as a mass mh , because this paper only considers the inuence of the external
6 disturbance on the HDD, and thus the vibration self-induced by the HDD is not been considered. The stiness and
7 damping of the mounting structure of HDD is modeled as a spring with stiness kh and a damper with coecient ch .
8 The vibration transmission through route BC is in one body, i.e., the chassis, however, the vibration energy is dissipated
9 through this route during transmission, and the dissipation depends largely on the relative distance between the two
10 mounting points, B and C. For example, if the distance of the route BC is larger, the stiness between the two points, B
11 and C, may be lower and the damping may be higher. Therefore, it is necessary to model the energy dissipation during
12 the vibration transmission route BC, as the dissipation could reduce the vibration transmitted to HDDs. However, the
13 dissipation in the chassis cannot be simulated properly with only one DOF. In order to accurately model the energy
14 transmission through the chassis without complicating the model, the chassis can be accounted for by two DOFs, by
15 decomposing into two parts, namely, chassis p1 and chassis p2 connected by a spring and damper. Hence, the two
16 DOFs system is not only able to model the vibration transmission through the chassis, but also capable of capturing
17 the vibration properties of the chassis. The two DOFs, with mass mc1 and mc2 , supports the vibration sources and
18 HDDs, respectively. The two DOFs are then connected by a spring with stiness kc and damper with coecient cc ,
19 while the spring and damper are shown in two parts in Figure 1(b). Finally, the chassis is supported by the notebook
20 mount on the ground, and the chassis mount is modeled as a spring with stiness kn and damper with coecient cn .
21 By assuming that only one vibration source is considered, these four DOFs system is sucient enough to simulate the
22 vibration transmission from the vibration sources to HDDs in a notebook computer in an eective way without losing
23 the key properties of the notebook computer.
24
25
3. Governing equations of the model
Based on the mass-spring-damper system shown in Figure 1(b) and the discussion above, the dynamic motion of
26 the notebook structure can be developed and expressed as,
MZ̈ + CŻ + KZ = F
4
(1)
1 where





M=



2

−cvs
cvs


 −cvs

C=




mvs
cc + cvs + cn
−cc

z̈vs


żvs


zv






















mc1

 z̈c1 
 żc1 
 zc1 
 Z̈ = 
 Ż = 
 Z=








mc2

 z̈c2 
 żc2 
 zc2 







mh
z̈h
żh
zh


k
−kvs

 vs



 −kvs kc + kvs + kn
−cc
−kc


 K=


cc + cn + ch −ch 
−kc
kc + kn + kh −kh



−ch
ch
−kh
kh


fvs







 0 



 F=





 0 



0
3 and zvs , zc1 , zc2 , and zh are the displacements of the vibration source, chassis p1, chassis p2 and HDD. As this model
4 only assumes the vibration from one source, i.e., either speaker, cooling fan or CD/DVD drives, only one force fvs is
5 applied to the system.
6
7
4. Vibration transmission routes
The characteristics of the vibration transmission routes can be expressed in a simple and eective way by transfer
8 functions. Through Laplace transformation, the above equation (1) can be solved in the frequency domain, and gives
9 the transfer functions from the vibration source to chassis and HDD. Let Zi (s) = ℘(zi (t)) be the Laplace transfer of
10 zi (t), the displacement transmissibility from vibration source to the HDD is therefore
HAD =
Zh
bh bvs bc kvs
=
δvs
Df
11 where
avs = s2 mvs + scvs + kvs
bvs = scvs + kvs
ac1 = s2 mc1 + s(cn + cvs + cc ) + kn + kc + kvs
bc = scc + kc
12
ac2 = s2 mc2 + s(cn + ch + cc ) + kn + kh + kc
ah = s2 mh + sch + kh
bh = sch + kh
Df = ac1 ac2 ah avs − ah avs b2c − ac1 avs b2h − ac2 ah b2vs + b2h b2vs
δvs = Fvc /kvs
5

(2)
1
By using the mass of notebook mn , stiness of chassis mount kn , and its natural frequency $n =
p
2kn /mn as
2 reference, the other masses, stiness, damping coecients can be dened as
kvs = αvs kn
cvc
mvs = 12 βvs mn
√
√
= ζvs 2αvs βvs mn kn
kc = αc kn
kh = αh kn
mc1 = 12 βc1 mn
√
√
cc = ζc 2αc βc mn kn
mh = 12 βh mn
√
√
ch = ζh 2αh βh mn kn
mc2 = 12 βc2 mn
√
cn = 2ζn mn kn
(3)
3 where α and β are the stiness and mass ratios, and ζ is the damping ratio. With the above non-dimensional parameters,
4 by dening
√
= βvs s2 /$n2 + 2sζvs αvs βvs /$n + αvs
√
ηvs = bkvs
= 2sζvs αvs βvs /$n + αvs
n
√
√
µc1 = akc1
= βc1 s2 /$n2 + 2s ζn + ζvs αvs βvs + ζc αc βc /$n + 1 + αc + αvs
n
√
ηc = kbnc = 2sζc αc βc /$n + αc
√
√
µc2 = akc2
= βc2 s2 /$n2 + 2s ζn + ζh αh βh + ζc αc βc /$n + 1 + αc + αh
n
√
µh = kanh = βh s2 /$n2 + 2sζh αh βh /$n + αh
√
ηh = kbhn = 2sζh αh βh /$n + αh
avc
kn
µvs =
(4)
5 the transfer function HAD can be expressed as
HAD =
6
ηh ηvs ηc αvs
2 + η2 η2
µvs µc1 µc2 µh − µh µvs ηc2 − µc1 µvs ηh2 − µc2 µh ηvs
h vs
(5)
The equation 5 is complex and it is dicult to analyze the eects of the individual mounts. In order to investigate
7 the eects of the mount of the vibration sources, the transmission routes AB, BC and CD are analyzed respectively in
8 the following subsections.
9
10
4.1. Transmissibility through route AB
at route AB is focused rst, therefore, the transfer function from the vibration source to the displacement of the
11 chassis p1 is
HAB =
Zc1
=
δvs
βvs s2
2
$n
+
2sζvs
√
αvs βvs
$n
+ αvs
h
2sζvs
βc1 s2
2
$n
+
√
αvs βvs
$n
+ αvs αvs
√
2s ζn +ζvs αvs βvs
$n
i 2
√
+ 1 + αvs − 2sζvs $αnvs βvs + αvs
(6)
12 To consider the eect of the stiness of the source mount, the damping ratio can be neglected, thus, the above equation
13 6 could be further reduced to
HAB =
2
αvs
2
(βvs s2 /$n2 + αvs ) (βc1 s2 /$n2 + 1 + αvs ) − αvs
6
(7)
1 The transmissibility from the vibration sources to the chassis p1 is therefore
TAB = |HAB | = |
ω2
βvs $
2
n
ω2
βc1 $
2
n
1
|
ω2
2 + 1 − β ω2 − β
− 1 /αvs
c1 $ 2
vs $ 2 /αvs
n
(8)
n
2 where the mass ratio βvs and βc1 , and the resonant frequency of notebook $n could be assumed to be constant. Since
3 the mass of the vibration sources usually is far less than the chassis, i.e., βvs βc1 , the transmissibility can be simplied
4 to
5
1
TAB = (9)
ω2
ω2
| 1 − βc1 $2
1 − βvs $
/αvs |
2 /αvs
n
n
√
From equation (9), the system has a resonance at frequency around ω = $n / βc1 , regardless how the stiness ratio
6 αvs changes. In order to examine how the transmissibility changes with dierent mount stiness, taking the derivative
7 of TAB with respect to αvs leads to
∂TAB
=
∂αvs
ω2
αvs αvs − 2βvs $
2
n
2
ω2
ω2
1 − βc1 $2
1 − βvs $2 /αvs
n
(10)
n
8 Therefore, if αvs < βvs ω 2 /$n2 or αvs > 2βvs ω 2 /$n2 , TAB decreases with the decrease of αvs . However, if βvs ω 2 /$n2 <
9 αvs < 2βvs ω 2 /$n2 , TAB increases with the decrease of αvs . In other words, there are two characteristic curves αvs =
10 βvs ω 2 /$n2 and αvs = 2βvs ω 2 /$n2 in the αvs −
ω
$n
plane, which decompose the plane into three regions as shown in
11 Figure 2(a), where the mass ratios are set as βvs = 0.01, and βc1 = 0.25, as it is a general value for notebook computers.
12 It is seen from Figure 2(a) that at the frequency range out of these two curves, reducing the stiness of vibration source
13 mount is always helpful in weakening the transmissibility, i.e., αvs ↓⇒ TAB ↓, however, at the frequency between the
14 two curves, smaller source stiness augments the transmissibility.
15
16
√
To study the eect of damping ratio ζvs on the transmissibility TAB , denoting e = αvs −βvs ω 2 /$n2 , f = 2 αvs βvs ω/$n ,
17 g = 1 + αvs − βvs ω 2 /$n2 , the transmissibility TAB can be derived as
s
TAB = αvs
(eg −
2 + f 2ζ 2
αvs
vs
2 )ζ 2
+ f 2 (e + g − 2αvs
vs
2 )2
αvs
(11)
18 Taking derivatives with respect to ζvs , the following relation is obtained,
∂TAB
∝
∂ζvs
ω2
ω2
ω2
ω4
βc1 2 − 1 2αvs − 2αvs βc1 2 − (1 + 2αvs ) βvs 2 + βc1 βvs 4
$n
$n
$n
$n
7
(12)
(a)
(b)
Figure 2: The transmissibility trend through route AB for dierent mount stiness (a), and damping ratio(b).
1
Therefore, two characteristic curves exist in the αvs −
ω
$n
plane as shown in Figure 2(b), where the dashed curves
2 are the stiness characteristic curves corresponding to Figure 2(a). It is clearly illustrated in Figure 2(b) that, if
3
4
5
√
ω2
ω4
ω2
ω2
> 1/ βc1 or αvs > βvs $
/2 1 − βc1 $
, a larger ζvs is helpful to reduce the transmis2 − βc1 βvs $ 4
2 − βvs $ 2
n
n
n
n √
2
4
ω
ω
ω2
ω2
sibility. However if $ωn < 1/ βc1 and αvs < βvs $
/2 1 − βc1 $
, larger ζvs induces larger
2 − βc1 βvs $ 4
2 − βvs $ 2
n
n
n
n
√
transmissibility. It is noted that when αvs is large enough, the transmissibility at high frequency(i.e., $ωn > 1/ βc1 )
ω
$n
6 is proportional to the damping ratio.
7
8
4.2. Transmissibility through route BC
Transmission route BC is through the chassis itself as shown in Figure (1). In order to analyze solely the eect of
9 the chassis stiness, only two DOFs is considered. The transmissibility from the chassis p1 to p2 is therefore,
TBC = |
10 where $vs =
p
αc
|
2 ) (1 + α − α β ω 2 /β $ 2 ) − α2
(1 + αc − αs βc1 ω 2 /βvc $vc
c
s c2
vs vc
c
(13)
kvs /mvs is the reference frequency. When the chassis stiness is changed, both of kc and kn can be
11 assumed to have changed with approximately the same ratio, thus αc is kept approximately constant. Furthermore,
12 since αc is generally far less than 1, i.e., the chassis stiness is far less than that of the chassis mount, it is reasonable
13 to neglect αc2 in the above equation, which leads to
TBC ≈ |
αc
|
2 ) (1 + α − α β ω 2 /β $ 2 )
(1 + αc − αs βc1 ω 2 /βvs $vs
c
s c2
vs vs
8
(14)
1 To study the eect of the chassis stiness on TBC , only αvs is concerned in the above equation, hence, taking the
2 derivative of TBC with respect to αvs leads to
2
2
2
4
αc (1 + αc ) βc1 ω 2 /βvs $vs
+ βc2 ω 2 /βvs $vs
− 2αs βc2 βc1 ω 4 /βvs
$vs
∂TBC
=
2 )2 (1 + α − α β ω 2 /β $ 2 )2
∂αvs
(1 + αc − αs βc1 ω 2 /βvs $vs
c
s c2
vs vs
3 Therefore, there are three parallel characteristic curves in the αvs −
αvs =
ω
$vs
(15)
plane, namely,
(1 + αc ) βvs
(1 + αc ) βvs
(1 + αc ) (βc1 + βc2 ) βvs
2 , αvs =
2 , αvs =
2
βc1 $ωvs
βc2 $ωvs
2βc1 βc2 $ωvs
4 which divide the plane into four regions as shown in Figure 2(b), where βvs = 0.01, βc1 = 0.25, βc2 = 0.75 and
5 αc = 0.1 are set for a notebook computer. From the gure, it is observed that, in general, at high frequency, i.e.,
6
ω
$vs
7
p
>
p
(1 + αc ) βvs /βc2 αvs , the soft chassis could reduce the transmissibility, while at low frequency, i.e.,
ω
$vs
<
(1 + αc ) βvs /βc1 αvs , a hard chassis could reduce the transmissibility.
(a)
(b)
Figure 3: The transmissibility trend through route BC for dierent mount stiness (a), the transmissibility trend through route CD for
dierent mount stiness and damping ratio(b).
8
9
10
4.3. Transmissibility through route CD
The route CD is very critical in isolating the vibration from chassis, since it connects the chassis and HDD. Similar
11 to the above analysis, only the chassis p2 and the HDD are considered in investigating the eect of the HDD mount.
9
1 The transmissibility from the chassis p2 to HDD can be derived as
TCD = |
αh
|
(1 + αh − βc2 ω 2 /$n2 ) (αh − βh ω 2 /$n2 ) − αh2
(16)
2 Since the HDD mount is generally much softer than the chassis mount, it can be assumed that αh2 ≈ 0, which leads to,
TCD ≈ |
αh
|
(1 + αh − βc2 ω 2 /$n2 ) (αh − βh ω 2 /$n2 )
3 and
ω
$n
(17)
2 2
ω
βc2 $n − 1
βh
∂TCD
=
2
∂αh
[(1 + αh − βc2 ω 2 /$n2 ) (αh − βh ω 2 /$n2 ) − αh2 ]
4
Therefore, two characteristic curves exist in the αh −
ω
$n
(18)
plane as shown in Figure 3(b), by set βh = 0.08, and
5 βc2 = 0.25 for a notebook computer. There are two dierent regions for the relationship between transmissibility and
6 HDD mount stiness. At high frequency, i.e., ω > $n αh /βh , reducing the HDD mount stiness could be eectively
p
7 in reducing the transmissibility, however, at low frequency, i.e., ω < $n αh /βh , the reverse will happen.
p
8
In considering the damping ratio of HDD mount, a similar relation also exists between TCD and ζh as
ω2
ω2
ω2
ω4
−
1
2α
−
2α
β
−
(1
+
2α
)
β
+
β
β
(19)
h
h c2 2
h
h 2
c2 h 4
$n2
$n
$n
$n
√
ω2
ω4
ω2
ω2
The two characteristic curves in the αh − $ωn plane are $ωn = 1/ βc2 and αh = βh $
−
β
β
/2
1
−
β
−
β
2
c2 h $ 4
c2 $ 2
h $2
∂TCD
∝
∂ζh
9
βc2
n
n
n
10 as shown in Figure 3(b). A larger ζh is helpful to reduce the vibration transmissibility, in the two regions that
√
ω2
ω4
ω2
ω2
> 1/ βc2 or αh > βh $
−
β
β
/2
1
−
β
−
β
2
c2 h $ 4
c2 $ 2
h $ 2 , however, in the region between the two curves,
n
n
n
n
√
2
4
ω
ω
ω2
ω2
12 i.e., when $ωn < 1/ βc2 and αh < βh $
/2 1 − βc2 $
, a larger ζh also induces larger trans2 − βc2 βh $ 4
2 − βh $ 2
n
n
n
n
√
13 missibility. It is noted that when αh is large enough, the transmissibility at high frequency(i.e., $ωn > 1/ βc2 ) is
11
ω
$n
14 proportional to the damping ratio.
15
5. Application to the analysis of vibration transmission in notebook
16
5.1. Parameter determination
17
The stiness and inertial parameters in the present theoretical model equation (1) can be extracted from both the
18 experiments and nite element analysis. For example, the stiness and mass of the components in the notebook can be
19 measured directly. On the other hand, a simple static nite element analysis (FEA) of the components can also give
20 accurate enough data, since the vibration is assumed linear in the present model. For the purpose of illustration, an
21 example is given to determine the stiness and mass parameters of speaker from a simple static FEA.
22
10
n
(a)
(b)
Figure 4: The speaker structure and its mount (a), a mass-spring-damper simplication of the speaker structure (b).
1
Figure 4(a) shows the structure of the speaker and its mounts. The aim is to simplify the speaker and its mounts
2 to a model of a mass, a spring and a damper as shown in Figure 4(b) and nd the parameters. Firstly, the potential
3 energy of the simplied mass in Figure 4(b) is,
V =
1 T s
dx ·K · dx
2
(20)
4 where Ks is the stiness of the speaker with its mounting, and it is a 3 × 3 symmetric matrix with six independent
s
s
s
s
s
s
5 elements, namely, Kxx
, Kyy
, Kzz
, Kxy
, Kyz
and Kxz
, and dx is the displacement of the mass, with the spring and
6 damper xing on the ground. The displacement dx represents the relative displacement between the mounting position
7 and the force applying position from the actual structure, i.e., dx = x1 − x2 =[dx, dy, dz]T , as shown in Figure 4(a).
8
In order to evaluate the Ks matrix, the six independent elements have to be obtained. Therefore, six equations
9 are required to be formulated. In order to formulate the six equations, the FEA is performed to obtain the potential
10 energy of the speaker and the mounts, by xing one point and applying displacement to another point. This method
11 to obtain the lumped parameters from FEA has successfully applied to the Micro electromechanical systems (MEMS)
12 analysis [10]. It is known that the potential energy from FEA can be computed from,
V =
1X
σi εi dΩi
2 i
(21)
13 where σi is the stress at a node, εi the corresponding strain, and dΩi the volume of the element. Six static simulations are
14 performed with the boundary conditions shown in Table 1. With the obtained potential energy V = [V1, V2, V3, V4, V5, V6 ]T ,
15 equation (20) can be rearranged as
V = A · Ks
11
(22)
Analysis
∆x(m)
∆y (m)
∆z (m)
1
2
3
4
5
6
1e-6
0
0
1e-6
1e-6
0
0
1e-6
0
1e-6
0
1e-6
0
0
1e-6
0
1e-6
1e-6
Table 1: Displacements imposed to the point in the FEA analysis.
1 where each row the matrix A is from Table 1 as,

Ai =
2
∆x2
2
∆y 2
2
∆z 2
2
∆x∆y
∆y∆z







s
∆x∆z , K = 







s
Kxx
s
Kyy
s
Kzz
s
Kxy
s
Kyz
s
Kxz
















The stiness of the speaker and its mounts Ks is therefore solved by
Ks = A−1 · V
3
(23)
The inertial parameter of the mass model can be determined by the conservation of kinetic energy of the respective
4 components. Taking the speaker and its mounting as an example, the relative velocity of the the mass centroid and the
5 mounting points is dẋ = ẋg − ẋm = [dẋ, dẏ, dż]T . Therefore, the kinetic energy of the component is
T =
1
dẋ · M · dẋ
2
(24)
6 where M is the matrix of the mass, which has also six independent elements. Through the FEA, the kinetic energy of
7 the speaker and its mounts can be computed as
T =
1X
ρi viT · vi dΩi
2 i
(25)
8 where ρi is the density of the material at node i, vi = [vxi , vyi , vzi ]T is the velocity at node i. Similarly, six FEAs
9 have to be performed to obtain the six independent elements in the mass matrix M. However, from the static FEA,
10 no velocity can be extracted. Since this is a linear problem, the velocity distribution can be assumed to be linearly
12
Parameter
Value
Parameter
Value
Parameter
Value
Parameter
Value
ks
cs
ms
1.92 × 105
0.64
0.02
kh
ch
mh
3 × 105
10
0.12
kc
cc
mn
2.3 × 107
50
1.5
kn
cn
1.6 × 107
1.0
Table 2: The parameters for the theoretical model from the notebook, where all the units are in international standard.
1 proportional to the displacement distribution [10], namely,
(26)
vi = D · ui
2 where ui = [uxi , uyi , uzi ]T is the displacement of node i. Similar, the relative velocity of the mass centroid and the
3 mounting points dẋ is also assumed to be linearly proportional to their relative displacement du̇ with the same constant
4 D. Therefore, it is possible to compute the mass matrix from linearly static FEA simulation shown in Table 1 as
(27)
M = 2A−1 · Tv
5 where M is the vector of the mass matrix, and Tv is the kinetic energy from the displacement boundary condition in
6 Table 1 as








M=







7


T
i ρi ui
P
Mxx 


 P


T
Myy 

i ρi ui

 P


T
Mzz 

i ρi ui
 , Tv = 

 P
T

Mxy 
i ρi ui



 P

T
Mxz 


i ρi ui

 P
T
Myz
i ρi ui

· ui dΩi 

· ui dΩi 



· ui dΩi 


· ui dΩi 


· ui dΩi 


· ui dΩi
The damping coecient is dicult to be determined from FEA. In this analysis, a 2%~5% damping ratio is usually
8 assumed. The parameters determined from the method in the last section for the real notebook are listed in Table 2.
9
10
11
5.2. Characteristics of the Four DOFs
A transmission analysis is conducted and compared with the harmonic analysis of a nite element model of a
12 notebook as shown in Figure5(a). This notebook model has been validated with experiments by [1] in a previous study.
13 Figure 6(a) shows the vibration transmission from speaker to chassis and HDD, when the speaker plays music or sounds,
14 where the structural responses of the speaker, chassis and HDD are shown under the excitation on the diaphragm of
15 the speaker. As shown in the gure, there are four main peaks corresponding to the four dominant modes at resonant
13
1 frequencies of 275Hz, 486Hz, 653Hz and 1248Hz respectively in the system. These frequencies are the structural response
2 resonant modes corresponding to the speaker, chassis and HDD. Since the vibration energy is rstly transmitted to
3 the speaker from the diaphragm, its vibration spectrum thus has the highest magnitude. As the energy transmits to
4 chassis and HDD from speaker, the magnitude of the vibration decrease gradually, due to the energy dissipation by
5 the mounting and chassis structure. Therefore, the mounts and the chassis structure are critical for the reduction of
6 the vibration energy during the transmission. Figure 6(b) illustrates the corresponding FEM result by implementing a
7 harmonic analysis of the response of chassis and HDD, where the vibration source of the speaker (the dashed magenta
8 curve I in the gure) is from the experiment measurement subject to a sweep sinusoidal currenti = i0 sin(2πf t) input
9 to the speaker[1]. The frequencies of the four peaks agree well with the FEM results as shown in the gure, where
10 the oscillation of the diaphragm induces the whole speaker to vibrate, which subsequently causes the vibration of the
11 chassis and HDD.
(a)
(b)
Figure 5: Finite element model (a), of a real notebook computer(b).
12
13
14
15
5.3. Analysis of vibration sources mounting
In order to reduce the vibration of HDD transmitted from the vibration source, the inuence of the the speaker
16 mount and position on the vibration of HDD is investigated detailed, since the speaker is the most critical built-in
17 vibration source with a wide spectrum span [1]. As stated in the theoretical model, the speaker mount and structure
18 are modeled by the spring and damping elements with stiness ks and damping coecient cs . Once the speaker
14
(a)
(b)
Figure 6: Response of speaker, chassis and HDD subject to external speaker force excitation from, theoretical model (a), and nite element
analysis (b).
1 structure is xed, the mount determines these two parameters. From the theoretical model shown in Figure 1, varying
2 these two parameters directly aects both the local vibration transmission route AB and subsequently AD. Therefore,
3 speaker mount is the rst barrier to reduce the vibration transmission. A good mount is able to isolate the vibration
4 energy from speaker to chassis as much as possible. Figure 7(a) illustrates the eect of dierent speaker mounts on the
5 transmissibility of the vibration to the HDD, where the stiness ks and damping coecient cs are normalized by the
6 respective values listed in Table 2. As expected from the route analysis illustrated in Figure 2, a soft mount (the dashed
7 curve III in Figure 7(a)) with higher damping can eectively isolate the vibration above 440Hz and below 300Hz, while
8 a 30% reduction of stiness could generally result in a minimum 25% reduction of vibration transmissibility. However, a
9 hard mount (the solid curve I in Figure 7(a)) could have the reverse eect at the frequency ranging from 300 to 440Hz,
10 namely, a 30% increase of the stiness results in at least 33% increment of the vibration transmissibility, while at some
11 frequencies, it could reach more than 2 times increase.
12
13
14
5.4. Analysis of chassis structure
The chassis is not only functioned as an assembly for all the components of the notebook, but it is also an important
15 vibration transmission route from the vibration point of view. A good chassis should be less sensitive to the vibration
16 source, and more eective to dissipate vibration energy transmitted through it. Figure 8(a) shows the inuence of the
17 chassis mounting stiness and damping, kn and cn , and the chassis stiness and damping kc and cc , on the vibration
15
(a)
(b)
Figure 7: Eect of the dierent speaker mounting on HDD vibration from the theoretical model(a), illustration of dierent positions of
speaker(b).
1 transmission from the speaker to HDD, in the present theoretical model, where the stiness kc and damping coecient
2 cc are normalized by the respective values listed in Table 2. It is observed that with a softer chassis and mounts (the
3 dashed line in Figure 8(a)), the vibration transmission is augmented below 600Hz, while a harder chassis and mounts
4 (the solid curve I in Figure 8(a)) amplify the transmissibility above 625Hz. This eect can be validated from the nite
5 element analysis as shown in Figure 8(b), where the two frequencies are 596 and 625Hz respectively. This observation
6 proves the route analysis in section 4 as shown in Figure3(a). Therefore, by using lower stiness and higher damping,
7 the vibration of transmission at high frequencies above 625Hz can be signicantly reduced, especially at the peak
8 where the higher frequency moves to lower values. However, the vibration transmission at frequencies below 600Hz is
9 increased. As the HDD vibration at low frequency may be compensated by the servo controller, to reduce the vibration
10 transmission at high frequency, a softer chassis may be helpful. However, if the chassis stiness is too low, the HDD
11 vibration at low frequency will be increased substantially and out of range of the servo controller.
12
13
Apart from the mount, the relative distance between the vibration sources and HDD in chassis design is also one
14 of the key factors relevant to vibration transmission from the sources to HDD by route BC shown in Figure 1. In
15 general, if the relative distance between the speaker and HDD is larger, the stiness between them would be smaller
16 and damping coecient would be larger. This behavior is mimicked in the present model by the parameters kc and cc ,
17 namely, varying these two parameters can generally model the distance eect between the vibration sources and the
16
(a)
(b)
Figure 8: Eect of chassis stiness and damping on the vibration of HDD from theoretical model(a), and nite element analysis(b).
1 HDD. Figure 8(a) has demonstrated the eect of varying the kc and cc , which shows that a larger relative distance
2 between sources and HDD is able to dissipate more vibration energy and thus is better for the performance of HDD.
3 Therefore, it is recommended to arrange the vibration sources to be mounted far away from the HDD in chassis design,
4 such as position #4 and #5 as illustrated in Figure 7(b).
5
6
5.5. Analysis of HDD mounting
HDD mount is able to functional as an isolation from the built-in vibration sources in the notebook. It is therefore
7 necessary to have a good HDD mount. The present theoretical model, where kh and ch represent the characteristics of
8 the HDD mount, gives a guidance for the selection of the HDD mount. From the model, the inuence of the stiness
9 and damping coecient of the mount on the vibration of HDD is shown in Figure 9, where the stiness kh and damping
10 coecient ch are normalized by the respective values listed in Table 2. As expected from the route analysis shown in
11 Figure 3(b), the isolation eect of the HDD mounting can be achieved at high frequency by the soft mount with lower
12 stiness and higher damping. For example, at 1000Hz, with the same damping, a 33% reduction of the stiness could
13 reduce 35% of the transmissibility (i.e., from the curve I to the blue dashed curve III in Figure 9). However, it is also
14 noted that the rst mode is shifted to a lower frequency and the transmission amplitude at the frequencies around 200
15 Hz is increased due to the softer mounting. The stiness therefore cannot be too low as shown in Figure 3(b), where
16 the rst mode of the dashed line is less than 200Hz, which is very easy to be set in resonance under external excitation.
17 The nite element analysis from the input of speaker is also analyzed, as shown in Figure 9(b), where the vibration
18 spectrum of the speaker is from the experimental data. The stiness and damping coecient are also converted to the
17
1 Young's modulus and damping ratio of the materials while performing the FEA. The FEA validates the conclusion
2 from the theoretical analysis, where a soft mount helps to reduce the vibration at high frequency above 200Hz ( the
3 dashed curve III in Figure9(b)), and a harder mount helps to reduce the vibration at low frequency below 200Hz ( the
4 solid curve I in Figure9(b)).
5
As well known, HDDs are generally actuated at high frequencies higher than 1000Hz, e.g., 1.4 kHz and 1.7 kHz
6 [11, 12], and the vibration at low frequencies can be compensated by the servo controller optimization relatively easy
7 [13][12]. Hence it is critical to reduce the vibration of HDDs at high frequencies through proper chassis and mounting
8 design. To achieve this target, the HDD mount can therefore be made softer from the above analysis. For example,
9 the HDD can be screwed to chassis by using foams and/or rubber grommet, which is helpful to reduce the stiness of
10 speaker mounting and increase damping for dissipation of vibration energy.
(a)
(b)
Figure 9: Eect of dierent HDD mounting on the vibration of HDD from the theoretical model (a), and nite element analysis (b).
11
12
13
6. Conclusion and discussion
A theoretical model has been developed for the analysis of the vibration transmission from the vibration sources to
14 HDD in the notebook computers. The key components of the notebook PC, including the vibration source, the chassis,
15 and the HDD, are modeled to a four DOFs with spring and damper connecting them. After the simplication, four
16 governing ODEs have been developed to generally characterize the dynamics and the vibration transmission inside a
17 notebook.
18
1
In order to have a fundamental understanding of the vibration transmission in notebook, a detail analysis of the
2 transmission routes from the sources to HDD via chassis is investigated individually. The investigation provides a
3 general guidance on the eect of dierent stiness and damping ratio of the mounts on the vibration transmissibility
4 at dierent frequencies. From the analysis, it could be concluded that, in general, lower stiness and higher damping
5 ratio of the mounts and chassis are helpful in reducing the transmissibility at high frequency, but increasing it at low
6 frequency.
7
The application of the model to the analysis of the notebook chassis design has also been demonstrated. In order to
8 determine the parameters in the model, including the stiness, mass, a method from static FEA of the key components
9 is presented for illustration purpose. The study on mounting of speaker shows that a soft mount is eective to dissipate
10 the energy transmitted from them to the chassis at the high frequency range, e.g, above 500 Hz. The dissipation can
11 reduce the vibration energy transmitted to the HDD, which helps the HDD operate in a more proper way. However,
12 at low frequency, the vibration transmission is actually increased even with a soft mount, while the HDD servo control
13 algorithm is relative easier to compensate the vibration increment at low frequency range. In order to provide useful
14 vibration isolation, it is also important to choose a soft mount on the HDD. The isolation eect of a softer mount is
15 enhanced at high frequency, but reduced at low frequency from the studies. However, it is still valuable to have a soft
16 isolation, since the low frequency vibration can be compensated by servo control in the HDD. Apart from the mounting
17 of the vibration sources and the HDD, the chassis between these two components are also signicant for vibration
18 transmission. These are related to three factors, namely, the structure and material of the chassis itself, the relative
19 distance between the HDD and vibration sources, and the mount between the chassis and the ground. Based on the
20 study, in order to alleviate the eect of vibration that would result in HDD performance degradation, the structure and
21 the material of the chassis should be less sti with higher damping, while the vibration sources should be placed as far
22 away as possible from the HDD.
23
Reference
24
[1] Hu, Y., Yoshida, S., Nakamura, S., Watanabe, K., Lin, W., Ong, E., et al. Analysis of built-in Speaker-
25
Induced Structural-Acoustic vibration of hard disk drives in notebook PCs. Magnetics, IEEE Transactions on
26
2009;45(11):49504955.
27
28
[2] Fah, Y., Feng, G., Ying, Y.. Vibro-acoustic analysis of hard disk drives. In: Magnetic Recording Conference,
2002. Digest of the Asia-Pacic. 2002, p. TUP1901TUP1902.
19
1
2
3
4
5
6
7
8
9
10
11
12
13
14
[3] Tandon, N., Rao, V., Agrawal, V.. Vibration and noise analysis of computer hard disk drives. Measurement
2006;39(1):1625.
[4] Matsuda, Y., Nakamura, S., Sega, M., Katou, Y.. Flexible support mechanism for hard disk drives to decrease
vibration disturbance. Magnetics, IEEE Transactions on 2009;45(11):51085111.
[5] Lim, S., Chang, Y., Park, N., Park, Y.. Optimal design of rubber mounts supporting notebook HDD for shock
and vibration isolation. In: Asia-Pacic Magnetic Recording Conference, 2006. 2006, p. 12.
[6] Semba, T., White, M., Huang, F.. Adaptive cancellation of Self-Induced vibration. Magnetics, IEEE Transactions
on 2011;47(7):19581963.
[7] Harmoko, H., Yap, F., Vahdati, N., Li, C.. Design and analysis of shock and random vibration isolation of
operating hard disk drive in harsh environment. Shock and Vibration 2009;16(2):143154.
[8] Otake, N., Watanabe, K., Shimizu, T., Tomida, K., Arisaka, T.. Controlling vibration of HDD actuator by
using dummy heads. Journal of Advanced Mechanical Design, Systems, and Manufacturing 2010;4(1):8287.
[9] Mou, J.Q., Lai, F., See, I.B.L., Lin, W.Z.. Analysis of notebook computer chassis design for hard disk drive
and speaker mounting. In: Proc. of the 2012 Int'l Conf. Computer Design. 2012, p. 4448.
15 [10] Braghin, F., Leo, E., Resta, F.. Modelling the stiness and inertial characteristics of MEMS' supporting beams.
16
In: Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electronics and Micro-Systems,
17
2006. EuroSime 2006. 7th International Conference on. 2006, p. 18. doi:10.1109/ESIME.2006.1644027.
18 [11] Li, Y., Guo, G., Wang, Y.. Reset control for midfrequency narrowband disturbance rejection with an ap19
plication in hard disk drives. Control Systems Technology, IEEE Transactions on 2011;19(6):1339 1348. doi:
20
10.1109/TCST.2010.2087759.
21 [12] Du, C., Xie, L., Zhang, J., Guo, G.. Disturbance rejection for a data storage system via sensitivity loop
22
shaping and adaptive nonlinear compensation. Mechatronics, IEEE/ASME Transactions on 2008;13(5):493 501.
23
doi:10.1109/TMECH.2008.2000637.
24 [13] Lee, H.S.. Controller optimization for minimum position error signals of hard disk drives. IEEE Transactions on
25
Industrial Electronics 2001;48(5):945950. doi:10.1109/41.954558.
20