[mechanisms and machine science] vibration engineering and technology of machinery volume 23 || heat...

Heat Transfer and ThermoelasticDynamics of a Rotating Flexible Discin a Hard Disc Drive

Yong-Chen Pei and Huajiang Ouyang

Abstract Multi-field dynamic coupling takes place in computer hard disc drives,involving air flow, heat transfer and thermoelastic vibration of a rotating flexibledisc in an enclosure filled with air. Air velocities and pressure induced by discrotation in the enclosure are obtained by using penalty finite element method.Temperature increments in the rotating disc, driving shaft, enclosure and air floware determined interactively by including the external heat sources from the shaftdriving motor and enclosure circuit board, the internal heat source from aerody-namic heating due to viscous fluid dissipation, the heat convection in air flow, andthe free convection heat loss at enclosure’s outside surfaces. Natural frequencies ofthe rotating disc under the thermal stresses induced by the disc’s temperatureincrement and centrifugal force are determined. Effects of air flow, heat convectionand aerodynamic heating induced by disc rotation on heat balance in the enclosureand natural frequencies of the rotating flexible disc are investigated. This investi-gation is useful to hard drive design.

Keywords Rotating disc � Multi-physical interaction � Heat transfer � Air flow �Thermoelastic dynamics

1 Introduction

In computer hard disc drives, a high speed rotating disc works in the surroundingenvironment with air flow and heat. The air flow and heat transfer induced by discrotation and their influences on the dynamics of rotating discs are the fundamental

Y.-C. PeiInstitute of Mechanical Science and Engineering, Jilin University, Nanling Campus,Changchun 130025, People’s Republic of Chinae-mail: [email protected]

H. Ouyang (&)School of Engineering, University of Liverpool, The Quadrangle, Liverpool L69 3GH, UKe-mail: [email protected]

© Springer International Publishing Switzerland 2015J.K. Sinha (ed.), Vibration Engineering and Technology of Machinery,Mechanisms and Machine Science 23, DOI 10.1007/978-3-319-09918-7_53

599

and crucial issues to improve the performance of rotating discs [1]. Firstly, con-siderable vibration and even flutter instability of a rotating disc can be caused by theair flow induced by disc rotation. In view of acoustic and structural interactions,Jana and Raman [2] solved the wave equation governing the propagation ofinfinitesimal disturbances of air flow to analyse the vibration of a rotating flexibledisc, and they found that coalescence between the acoustic and structural modescould lead to disc flutter instability at supercritical speeds. By treating the couplingdynamics of the air flow and rotating disc as fluid and structure interaction prob-lems, Gad and Rhim [3] investigated dynamics of a flexible disc coupled to thin airfilm and rotating close to a rigid rotating wall. Cheng et al. [4] designed severalenclosure covers of an optical disc drive to improve the flow-induced vibration ofrotating discs. Secondly, heat transfer in the surrounding environment of rotatingdiscs is affected considerably by the air flow [1]. Effects of disc rotation and heattransfer on the viscous fluid flow and temperature distribution between heatedcontracting rotating discs were analyzed by Nazir and Mahmood [5]. Jiji andGanatos [6] analyzed the microscale steady axisymmetric flow and heat transfer intwo infinite parallel discs separated by a gas-filled micro-gap.

There are three limitations in the above investigations on the air flow and heattransfer in rotating disc systems: (1) Some research has been done on the air flowinduced vibration and flutter instability of a rotating disc [2–4], but there is no studyon the problem of heat transfer and thermoelastic dynamics coupling with the fluidflow. (2) Aerodynamic heating effect due to fluid’s viscous dissipation wasneglected [5], whereas the corresponding results cannot be confirmed by theexperimental observation by Kameya et al. [7] that the disc temperature increasesaround the disc’s outer edge. (3) Heat transfer of the rotating disc was ignored [5,6], although the dynamic characteristics of the rotating disc can be affected sig-nificantly by the thermal stresses induced by the disc temperature variation.

Steady state air flow, heat transfer and thermoelastic dynamics in the multi-fieldcoupling system of a flexible disc rotating in an enclosure are studied in this paperfor computer hard disc drives. The heat generated from the driving motor andenclosure circuit board is involved as the external heat source, and that from theaerodynamic heating of air flow is involved as the internal one. Temperature dis-tribution in the rotating disc, driving shaft, enclosure and air is determined, and thenatural frequencies of the rotating disc are computed under the thermal stressesinduced by disc temperature distribution and centrifugal force.

2 System Modeling and Governing Equations

As shown in Fig. 1, a flexible annular disc rotates at a constant angular speed Ω in acylindrical enclosure. The disc thickness h is assumed to be small compared with itsouter radius a, and the driving shaft with radius b is assumed to be rigid to clampand drive the disc. Due to electromagnetic induction and frictions in the drivingmotor, a heat flux qS is generated and transferred to the driving shaft.

600 Y.-C. Pei and H. Ouyang

The enclosure’s base is always heated by circuit board below in hard disc drives,and a heat flux qB is generated and transferred to the enclosure’s base. In the case ofhard disc drives, the read/write heads and suspension system are ignored in thecurrent work to reduce the huge computational workload and to highlight heatbalance and thermoelastic dynamics of the rotating flexible disc. Similarly, the samesimplification also can be found in many references [2–4].

This paper studies the rotating flexible disc system in its normal steady workingstate. The axisymmetric steady-state air flow and temperature distribution should bemuch more prominent [6] than the transient and asymmetrical ones, and thus thelatter can be neglected. However, as reported by Arafat et al. [10] and Pei et al. [11],the rotating flexible disc’s transverse deflection due to non-axisymmetric modes (m,n ≠ 0) is influenced significantly by disc’s axisymmetric temperature distributionand thermal stresses. Therefore, non-axisymmetric vibration of the flexible disc ismodeled and analyzed in the current work.

The axisymmetric Navier-Stokes and continuity equations [8, 9] for incom-pressible laminar flow can be written as

qa urouror

þ uzouroz

� u2hr

� �þ op

or¼la

o2uror2

þ ourror

þ o2uroz2

� urr2

� �

qa urouhor

þ uzouhoz

þ uruhr

� �¼la

o2uhor2

þ ouhror

þ o2uhoz2

� uhr2

� �

qa urouzor

þ uzouzoz

� �þ op

oz¼la

o2uzor2

þ ouzror

þ o2uzoz2

� �ð1Þ

our=or þ ur=r þ ouz=oz ¼ 0 ð2Þ

where p(r, z) is air pressure relative to the ambient atmosphere; ur(r, z), uθ(r, z),uz(r, z) are air velocity components in the radial r, circumferential θ and transversalz directions, respectively. The air velocity component uθ at the disc and shaftsurfaces equals to the local linear peripheral speed Ωr, while other velocity com-ponents on these surfaces are zeros. All velocity components vanish at the enclo-sure’s inside surfaces. Within the driving shaft gap, the relative air pressurep vanishes and our=or þ ur=r þ ouz=oz is zero in light of mass conservation [3].

Fig. 1 Rotating disc and its driving shaft in cylindrical enclosure

Heat Transfer and Thermoelastic Dynamics of a Rotating… 601

The steady state pressure difference P(r) between the upper and lower disc surfaces(b ≤ r ≤ a, z = ±h/2) can be expressed as P rð Þ ¼ p r; zð Þ z¼�h=2

�� p r; zð Þ z¼h=2

�� .Heat transfer takes place in the fluid medium of inside air flow, and between it

and the solid media of shaft, disc and enclosure. The energy equation [9] for thefluid and solid can be expressed separately as

qaca uroCor

þ uzoCoz

� �¼ ka

o2Cor2

þ oCror

þ o2Coz2

� �þ laUðr; zÞ ð3Þ

ksðo2C=or2 þ oC=or=r þ o2C=oz2Þ ¼ 0 ð4Þ

where ca is the specific heat of air, ka (air) and ks (shaft, disc and enclosure) is thethermal conductivity; Γ(r, z) is the temperature increment relative to the atmo-spheric temperature Ta far away from the modelled domain; q(r, z) = μaΦ representsthe aerodynamic heating flux due to viscous air dissipation, Φ is the dissipationfunction [9]. At the enclosure’s cover and walls, free convection boundary condi-tion is considered with a convective heat transfer coefficient ha. It is assumed thatthe heat generation power of the shaft base and enclosure base are PS and PB, andthen the averaged heat flux qS = PS/(πb

2) and qB = PB/[πR2 − π(b + Lg)

2]. Heatfluxes qS and qB are assigned at the shaft base and enclosure base. The adiabaticcondition is assumed at the driving shaft gap.

Subjected to the air pressure and thermal stresses and centrifugal force, thegoverning equation of the rotating flexible disc [10, 11] can be written as

qho2wot2

þ 2Xo2wotoh

þ X2 o2w

oh2

� �� qha2X2 1

roor

rrrowor

� �þ 1r2

ooh

rhowoh

� ��

þ Eh3

12ð1� v2Þr4wþ Eh2aT

1� vr2HQ ¼ PðrÞ þ o2w

or2oWror

þ o2W

r2oh2

� �

þ owror

þ o2w

r2oh2

� �o2Wor2

� 2o2wroroh

� owr2oh

� �o2Wroroh

� oWr2oh

� �

ð5Þ

r4W ¼ �EhaTr2HM ; HM ¼Zh=2

�h=2

Cðr; zÞdz=h; HQ ¼Zh=2

�h=2

Cðr; zÞzdz=h2 ð6Þ

where w(r, θ, t) is disc’s transverse deflection; ρ is mass per unit volume of the disc,E is Young’s modulus, ν is Poisson’s ratio, and αT is coefficient of linear thermalexpansion; σr and σθ are the radial and circumferential membrane stress resultantsdue to centrifugal force [12]. At the clamped edge r = b and free edge r = a, theboundary conditions of the transverse deflection w and thermal stress function Ψcan be found in Deng and Ouyang [12].


3 Solution Procedures for Air Flow and Temperature

With the penalty finite element method by Elman et al. [13] and Reddy [14], a smallpenalty parameter ε (ε → 0) is introduced into the continuity equation Eq. (2),

epþ our=or þ ur=r þ ouz=oz ¼ 0 ð7Þ

By using the Q2 − P−1 element [13] to discretise the solution domain in the r-z plane, finite element equations of the system can be yielded from Eqs. (1–2) andEqs. (3–4) respectively,

N Xð ÞX=2þ XBþ Dð ÞX þX XB=2þ Dð Þ ¼ 0 ð8Þ

x ¼ h Xð Þ � qBQB Xð Þ þ qSQS Xð Þ½ � ð9Þ

where the unknown variables for air velocities are collected as vector X, those fortemperature increment are collected as x; Matrix N(X) and vector h(X) are derivedfrom the nonlinear convection terms, B and D are coefficient matrices and vectors,QB(X) and QS(X) are related to the heat generation qB and qS. From Eq. (9), tem-perature increment Γ(r, z) can be divided into two parts induced separately by theinternal aerodynamic heating (q) denoted as ΓAir(r, z) with solution x = h(X), and bythe external shaft and enclosure base heat generations (qB, qS) denoted as ΓBase(r, z)with x = −[qBQB(X) + qSQS(X)].

Equation (8) is a nonlinear quadratic vector equations, the iterative algorithmpresented by Poloni [15] is used to solve X numerically.

The total heat generation power of aerodynamic heating can be determined by

PA ¼ 2pZZ

Va

laU rdrdz ð10Þ

where Va represents the solution domain of air media.

4 Solution Procedure for Disc’s Natural Frequency

The transverse vibration of a rotating flexible disc [11] can be assumed as

wðr; h; tÞ ¼X1n¼�1

einhX1m¼0

um;nðrÞwm;nðtÞ ð11Þ

where φm,n(r) is the shape function of stationary disc mode (m,n).Then an ordinary differential equation for the disc mode of n nodal diameters can

be derived from Eqs. (11) and (5) with Galerkin’s method [11],


€wn þ i2nX _wn þ ðSn �Hn þ X2Ln � n2X2IÞwn ¼ f n ð12Þ

where fn = […fm,n…]T, fm;0 ¼ 1qh

Rabum;0ðrÞ½PðrÞ � Eh2aT

1�v r2HQðrÞ�rdr and fm,n = 0

for n ≠ 0, since both P(r) and Eh2αT∇2ΘQ(r)/(1−v) are axisymmetric; wn = […wm,

n…]T for m = 0, 1, 2, …; matrices Sn and Ln can be found in Pei et al. [11], and thematrix induced by thermal stress can be written as

Hn ¼ HMðbÞHbn þH0

MðbÞHdn þ

X1m¼0

JmðHMÞHrm;n ð13Þ

From Eq. (13), dynamic property of the rotating disc is influenced by the meantemperature increment ΘM in three ways: ΘM(b), Θ’M(b) and Jm(ΘM). For thetemperature distributions separately induced by the internal aerodynamic heatingand by the external heating from the shaft and enclosure base, the three componentsΘM(b), Θ’M(b) and Jm(ΘM) of thermal membrane stress resultants are denoted asHAir

M ðbÞ, H0AirM ðbÞ, JmðHAir

M Þ, and HBaseM ðbÞ, H0Base

M ðbÞ, JmðHBaseM Þ.

Substituting the homogenous solution wn = e(λ − inΩ)tAn into the homogenousform of Eq. (12) yields an eigenvalue problem [λ2I + (Sn − Hn + Ω2Ln)]An = 0 aspresented in Pei et al. [11]. Then the dimensionless natural frequency of disc mode(m,n) is introduced as ωm,n = Im(λ)/Ω0,0, where Ω0,0 is the natural frequency of discmode (0,0) for a non-rotating disc [11] without any thermal stress.

5 Data Analysis and Results

Default geometric parameters [4] of the shaft-disc-enclosure system in Fig. 1 areselected as b = 15 mm, a = 60 mm, h = 1 mm, R = 62 mm, L1 = 6 mm, L2 = 4 mm,Lw = 2 mm, Lt = 1 mm, Lb = 1 mm and Lg = 1 mm. From Tan et al. [16] the heatgeneration powers of the shaft base and enclosure base are selected as PS = 3 W andPB = 2 W. In addition, some fundamental physical parameters of air, disc, shaft andenclosure are listed in Table 1.

Table 1 System fundamental physical parameters

Airmedia

Air fluid ρa = 1.205 kg/m3, μa = 1.8135 × 10−5 kg/m/s, νa = 1.505 × 10−5

m2/s, ca = 1.012 × 103 J/kg/K, ka = 0.024 W/m/K

Solidmedia

Rotor Disc ρ = 2,700 kg/m3, E = 70 × 109 Pa, v = 0.35, ks = 200 W/m/K,αT = 23 × 10−6 K−1

Shaft ks = 60 W/m/K

Enclosure Cover ks = 60 W/m/K, ha = 8 W/m2/K

Wall ks = 230 W/m/K, ha = 6 W/m2/K

Base ks = 230 W/m/K


With Ω increasing from 0 rpm to 8,000 rpm, the total heat generation power ofaerodynamic heating PA in Eq. (10) is illustrated in Fig. 2. It can be seen that PA

increases very fast, and can get up to several watts, which are in the same order asheat generation power of the shaft base and the enclosure base at 8,000 rpm. It canbe approximated by curve fitting to a polynomial in terms of Ω (krpm) asPA = −1.3041 × 10−2Ω + 1.8334 × 10−2Ω2 + 3.0987 × 10−3Ω3 with the maximumfitting error 6.5963 × 10−3 W, that is to say PA increases with Ω cubed. FromEq. (10), PA is the volume integral of aerodynamic heat flux q(r, z) = μaΦ, and Φ isrelated to the square of the gradients of air velocity components ur, uθ and uz [9].With the increase of Ω, velocity components ur, uθ and uz increase, and theirgradients also increase simultaneously.

With the increase of Ω and r/a, the mean temperature increments HBaseM ðrÞ,

HAirM ðrÞ, ΘM(r) and ΘQ(r) of the disc are illustrated in Fig. 3. Induced by qS and qB,

HBaseM ðrÞ decreases along r/a and with Ω due to the convection induced by disc

rotation as a “cooling fan”, as shown in Fig. 3a. However, HAirM ðrÞ increases con-

siderably with Ω and r/a due to the aerodynamic heating in Fig. 3b. As sum ofHBase

M ðrÞ andHAirM ðrÞ, the total ordinary mean temperature incrementΘM(r) decreases

first and then increases with Ω in Fig. 3c. ΘQ(r) increases with Ω, and increases firstand then decreases with r/a in Fig. 3d. It almost vanishes at both edges of the disc.Nevertheless, since the disc is very thin in the current investigation, Γ(r, z) variesslightly across the disc thickness and thus the magnitude of ΘQ is quite small com-pared with ΘM as shown in Fig. 3. Moreover, P(r) and the thermal momentEh2αT∇

2ΘQ(r)/(1−v) act as the steady state forces on the disc in Eq. (12), and they donot affect natural frequency ωm,n, to be seen in the following numerical results anddiscussions.

With the increase of Ω, natural frequency ωm,n is illustrated in Fig. 4a–f.Compared with the natural frequency without any thermal stress, disc natural fre-quencies ωm,n for the disc modes (0,0), (0,1), (1,0) and (1,1) are decreased by thethermal stress from Θ’M(b), whereas it is increased by the thermal stress fromΘM(b) and Jm(ΘM). However, the entirely opposite situation (ωm,n being decreasedor increased) takes place for the disc modes (0,2), (0,3). It can be concluded that theeffect of the thermal stress from Θ’M(b) on natural frequency is quite opposite tothat from ΘM(b) and Jm(ΘM). Natural frequency ωm,n with the thermal stress just

Fig. 2 Total heat generationpower of aerodynamic heatingvarying with disc speed


from Θ’M(b) and Jm(ΘM) increases consistently with Ω from 0 rpm to 8,000 rpm,whereas the natural frequency with the thermal stress just from ΘM(b) decreasesfirst and then increases for disc modes (0,0), (0,1), (1,0) and (1,1), it increases forthe disc mode (0,2), and it increases first and then decreases for disc modes (0,3).These can be explained by the effects of ΘM(b), Θ’M(b) and Jm(ΘM) varying with Ωin Fig. 3. As shown in Fig. 4a–f, the effect of the thermal stress from ΘM(b) on ωm,n

is very considerable, and it is more dominant than that from Θ’M(b) and Jm(ΘM).Thus it can be observed that the natural frequency with the thermal stress just fromΘM(b) is very close to the natural frequency with all thermal stresses.

Fig. 3 Mean temperature increments varying with disc speed and disc radial position. (a) Ordinarymean temperature increment HBase

M ðrÞ induced by heat generations in the shaft and enclosure’sbase; (b) Ordinary mean temperature increment HAir

M ðrÞ induced by the aerodynamic heating;(c) Total ordinary mean temperature increment HM rð Þ ¼ HBase

M ðrÞ þHAirM ðrÞ; (d) Total quadratic

mean temperature increment ΘQ(r)


6 Conclusions

(1) Total aerodynamic heating power of viscous air dissipation increases very fastwith disc speed, and can get up to several watts, which are in the same order asheat generation power of the shaft driving motor and the enclosure base circuitboard.

(2) With the increase of disc speed, temperature increment decreases first due tothe convection induced by disc rotation as a “cooling fan”, and then increasesdue to the considerable aerodynamic heating.

(3) Natural frequency of the rotating flexible disc is influenced by temperatureincrement in three ways: the mean temperature increment and its radialderivative at disc clamped edge, and the distribution characteristics of meantemperature increment along disc radial direction, where the first way is moredominant than the others. The positive or negative effect of these three wayson the natural frequency varies with disc mode, and that of the second way isalways quite opposite to the others.

Fig. 4 System natural frequency varying with disc speed. – –: Without any thermal stress; ∙∙∙∙:With thermal stress just from ΘM(b); —: With thermal stress just from Θ’M(b); -∙-: With thermalstress just from Jm(ΘM); —: With all thermal stresses. (a) Disc mode (0,0); (b) Disc mode (0,1);(c) Disc mode (0,2); (d) Disc mode (0,3); (e) Disc mode (1,0); (f) Disc mode (1,1)


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[mechanisms and machine science] vibration engineering and technology of machinery volume 23 || heat...

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