J. Cent. South Univ. (2017) 24: 891-899

DOI: 10.1007/s11771-017-3491-z

Optimization of passive control performance for different hard disk drives subjected to shock excitation

Seyed Rashid Alavi^{1, 2}, Mehdi Rahmati^{1, 2}, Saeed Ziaei-Rad¹

1. Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran;

2. Isfahan Mathematics House, Isfahan 81698-51177, Iran

Central South University Press and Springer-Verlag Berlin Heidelberg 2017

Abstract: Laptop personal computers (LPCs) and their components are vulnerable devices in harsh mechanical environments. One of the most sensitive components of LPCs is hard disk drive (HDD) which needs to be protected against damages attributable to shock and vibration in order to have better magnetic read/write performance. In the present work, a LPC and its HDD are modeled as two degrees of freedom system and the nonlinear optimization method is employed to perform a passive control through minimizing peak of HDD absolute acceleration caused by a base shock excitation. The presented shock excitation is considered as half-sine pulse of acceleration. In addition, eleven inequality constraints are defined based on geometrical limitations and allowable intervals of lumped modal parameters. The target of the optimization is to reach optimum modal parameters of rubber mounts and rubber feet as design variables and subsequently propose new characteristics of rubber mounts and rubber feet to be manufactured for the HDD protection against shock excitation. The genetic algorithm and the modified constrained steepest descent algorithm are employed in order to solve the nonlinear optimization problem for three widely-used commercial cases of HDD. Finally, the results of both optimization methods are compared to make sure about their accuracy.

Key words: hard disk drive; passive control; shock excitation; sequential quadratic programming; genetic algorithm; rubber

1 Introduction

In general, laptop personal computers (LPCs) consist of light-weight devices and are used almost everywhere. Thus, one of the most important advantages of LPCs is their portability. Nevertheless, this property allows the LPC components to be vulnerable against harsh mechanical environments. Hard disk drive (HDD) is one of the most prominent components of LPCs employed to store plenty of data and retrieve them when required. In particular, the read/write performance of a HDD is very sensitive to shock and vibrational excitations. Shock excitations of HDD are derived from external excitation sources which come from transportation (like car, airplane, train, etc) or falling from hands onto ground. These excitations take place in a very short time but they have many side effects on the system, thus the shock analysis theory [1] should be considered in order to design mounting system of laptop HDD to minimize effects of shock excitations and subsequently reduce the shock mechanical damages and reach the optimum system.

There are a remarkable number of research efforts about influences of shock and vibration on the HDD performance in the literature. LIM [2] simulated flexural vibrations in HDD spindle systems by using finite element method (FEM). System components were analytically modeled by him and commercial HDD systems were selected as examples to prove the accuracy of formulated models through computation of natural modes. MURTHY et al [3] studied the shock performance of the head/disc interface in 2.5 inch and 3.5 inch HDDs by using numerical methods and experiment. The FEM was employed to model the HDD and simulate the shock response. TANDON et al [4] investigated vibration and noise generated by some computer HDDs. They computed natural frequencies of disk platters both theoretically and by FEM, and also obtained frequency range of noise generation. In continuation, FEM was employed in order to investigate the dynamic response of 1-in. HDD under external shock and vibration loads by MURTHY et al [5]. YAP et al [6, 7] simulated the mounted HDD just by single degree of freedom (SDOF) model. They explained the process of design and test for operating HDD subjected to shock and random vibration based on military standard 810E. Also, the vibration isolation system was designed and analyzed by them. HARMOKO et al [8] used state-space formulation to model structural components of the HDD and investigate effects of various HDD components on shock tolerance. KUWAJIMA et al [9] proposed a type of latch mechanism for suspending shock. Their latch mechanism is useful for installation in the small HDD of a mobile device. In addition, PARK et al [10] investigated shock and vibration analysis of the laptop HDD on rubber mounts. They developed a lumped parameter model for shock analysis and verified the obtained results by a linear drop test. They just used a specific kind of rubber to be mounted under laptop body. On the other hand, improving the global performance of engineering systems is a main goal for designers and manufactures; accordingly, a substantial number of studies have been conducted so as to optimize the technical performance of various engineering systems such as cooling towers, wind turbines and data storage systems through experiments or mathematical modeling [11-14].

In this investigation, the laptop body and HDD have been modeled as two degrees of freedom (2DOFs) system despite almost all published articles. Consequently, the effects of rubber feet mounting on the system performance are considered. The considered SDOF to 2DOFs expansion of system model can improve the performance of vibration compensation. Then, the genetic algorithm and modified constrained steepest descent algorithm (Pshenichny-Lim-Belegundu- Arora, PLBA) which was extended based on the sequential quadratic programming (SQP) are employed in order to solve the nonlinear optimization problem for three widely-used commercial cases of HDD. Furthermore, the results of both optimization methods are compared to make sure about the accuracy of results. The target of optimization process is to minimize the peak of HDD absolute acceleration and subsequently reach optimum modal parameters (and also spatial parameters) of rubber mounts and rubber feet as design variables. Thus, new spatial characteristics of rubbers are proposed to be manufactured for HDD protection against shock excitation. In addition, the presented shock excitation is considered as a half-sine pulse of acceleration.

2 Mathematical modeling

To the best of our knowledge, there are three typical and widely-used laptop HDDs known as 1.8, 2.5 and 3.5 inch. In this investigation, protection for all types is studied. HDD is rested on 16 rubber mounts, as shown in Fig. 1. Laptops generally have 4 to 6 rubber feet.Figure 2 shows a laptop with the five rubber feet.

As shown in Fig. 3, the 2DOFs model is utilized to study the HDD shock response caused by moving vehicle which laptop installed on it or falling from hands onto ground. The HDD and laptop body are modeled as two separate rigid bodies where their masses are respectively denoted by m_h and m_l. Lumped stiffness and damping coefficient of HDD rubber mounts are denoted by k_h and c_h. Also, lumped stiffness and dampingcoefficient of laptop rubber feet are shown by k_l and c_l, respectively.

Fig. 1 HDD supported by rubber mounts [10]

Fig. 2 Laptop rubber feet

Fig. 3 Two degrees of freedom model of the laptop body and HDD

In order to start mathematical modeling, Newton’s law is employed to derive equations of motion for 2DOFs model subjected to base shock excitation. In current model, the equations of motion are derived as

 (1)

For convenience, the following change in variables is applied by defining

               (2)

After substituting new variables in equations of motion, the new rearranged equations are written as

                      (3)

where

                                (4)

In addition, ω_h and ζ_h are undamped natural frequency and damping ratio of the HDD, ω_l and ζ_l are undamped natural frequency and damping ratio of the laptop body, respectively and are defined as

           (5)

After obtaining the maximum value of HDD absolute acceleration and other useful parameters from Eq. (3), the optimization problem should be determined.

3 Optimization of rubbers lumped modal parameters

Optimization is an efficient procedure to improve operational conditions and performance of most engineering systems [15-17]. In particular, it is always an important necessity for HDDs to have low absolute acceleration and consequently less mechanical damages (better digital performance), so the target of optimization problem used here is to minimize the absolute acceleration of HDD. In addition, a suitable objective function is defined and some practical constraints are mentioned. After that, suitable optimization methods applicable for this problem are selected.

3.1 Optimization problem statement

In order to reach the optimum value of lumped modal parameters for rubber feet and rubber mounts, the nonlinear optimization problem should be considered which is defined as follows.

Search for design variable vector (X) which minimizes f(X) subject to

,i=1, 2, …, n                      (6)

where f and g_i denote the objective function and the ith constraint, respectively and n refers to number of constraints.

3.1.1 Objective function

As was argued above, the operational conditions and read/write performance of a HDD are very sensitive to shock and vibrational excitations. Hence, the HDD should be protected against external excitation sources specially, external impacts which are simulated by shock impulse. Therefore, the target of optimization in this research is to minimize the maximum value of HDD absolute acceleration caused by base shock excitation and also to obtain optimum value of lumped modal parameters, so the objective function is stated as

                             (7)

where X is the vector of design variables which should be determined in current optimization problem and could be stated as

                         (8)

X₁=ω_l, X₂=ω_h, X₃=ζ_l, and X₄=ζ_h. The optimum value of X is used to minimize mentioned objective function while all of constraints are satisfied.

3.1.2 Constraints

In the present work, the optimization problem includes inequality constraints which satisfy targets of design. First constraint indicates that deflection of the HDD relative to the laptop body (z_rel=x_h-x_l) should be less than 2 mm. This constraint is because of the factory geometrical limitation in initial gap between HDD and the laptop body frame which is about 2 mm or less (depending on laptop characteristics). Thus, the first constraint is satisfied when the peak of relative deflection becomes less than 2 mm:

g₁=max-0.002≤0                        (9)

It should be noted that z_rel can be obtained after solving Eq. (3). All other constraints imply that lumped modal parameters (design variables) and damped natural frequency of rubber feet and rubber mounts should be between allowable upper and lower bounds which are available in related investigations [10, 18]. These constraints of optimization problem are defined as

                       (10)

                       (11)

                     (12)

                     (13)

                               (14)

                               (15)

                          (16)

                          (17)

                           (18)

                          (19)

where a and b are constant values which change based on mass of HDD. These constant values are listed in Table 1 for three different HDD types.

Table 1 Upper and lower bounds for damped natural frequency of HDD

3.2 Sequential quadratic programming

Before starting the optimization process, a suitable optimization method should be selected and applied. SQP [19, 20] is a high-efficiency and widely-used method which is considered in this optimization problem by PLBA algorithm, which has been extended to include Hessian updating and potential set strategy. In the SQP method, the objective function and constraints are approximated by quadratic models. Suitable starting point should be chosen to find the solution of optimum problem. On the other hand, the solution may depend on the selected starting points, so it is not any assurance that the solution results are global minimum or not. Consequently, multiple starting points should be selected by using trial-and-error approach to overcome this weakness. In this optimization algorithm, the basic optimization problem is replaced with a quadratic model in each iteration as follows.

Search for X which minimizes subject to

, i=1, 2, …, n        (20)

where H_k denotes an approximation for the Hessian matrix of the Lagrangian function. Furthermore, using quasi-Newton method facilitates updating process of H_k in each iteration.

The Lagrangian function is defined as

                  (21)

where λ_i denotes the Lagrangian multiplier. Finally, the present optimization problem should be converted to the standard design optimization model in order to stand in required frameworks of the PLBA algorithm.

3.3 Genetic algorithm

It is noteworthy to mention that there are various choices of traditional numerical methods to optimize engineering design problems, but they are often gradient based and difficult to solve. The genetic algorithm (GA) [21] is a strong and intelligent random search method specially to find out the optimal solution in wide space optimization cases. This method was first invented by HOLLAND [22] and is inspired by the procedures in natural evolution. GA has many advantages in comparison with traditional methods, for instance, GA just works with values of functions and is independent from evaluation of the function gradient, it can easily deal with special objective functions (nonlinear, discrete, non-differentiable or stochastic functions), it has ability to escape from local optimal solutions and it works with set of distributed points at one time unlike the traditional methods. Moreover, GA transforms the design space to a genetic space, so design variables should be encoded by using binary coding. GA is an iterative process that modifies a population of individual solutions. In each iterate, this algorithm selects individuals randomly from the current population to be parents and uses them to produce children for the next generation. There are three main rules that are applied in GA at each step to reach the next generation from current population: Selection, crossover and mutation operator.

Selection (survival) operator selects good individuals which contribute to the population at the next generation. Crossover operator performs key-role in success of the GA and has the highest percentage of reproduction probability. This operator combines two parents to produce children for the next generation. Mutation operator applies random changes to individual parents to form children. The target of mutation is to improve diversity of chromosomes in the population. For above operations, participants in the operation should be selected randomly, somehow the better solutions have more chance to be selected. The general procedure of genetic algorithm is shown as a flowchart in Fig. 4.

4 Results and discussion

In the present investigation, a practical optimization problem is considered as a numerical example to reach the values of optimum modal parameters. In order to solve the PLBA algorithm as a nonlinear optimization problem, the MATLAB software was used. In this optimization problem, the PLBA algorithm uses the numerical solution to obtain 26 unknown variables from 26 equations.

As mentioned above, the values of objective function and constraints should be determined in each iteration of optimization process. Despite Eqs. (10)-(19), the first constraint and the objective function are difficult to be obtained analytically. Thus, Eq. (3) and its derivative should be solved numerically in each iteration, then the maximum value of HDD relative displacement and its absolute acceleration are detected in order to find out values of objective function and first constraint. In addition, Eq. (3) and its derivative are ordinary differential equations (ODEs) which are solved numerically by the fourth-order Runge-Kutta method [23]. In order to prepare equations for the standard form of Runge-Kutta method, they should be transformed to the first-order ODEs. So, the state space conversion of Eq. (3) is employed as follows:

 (22)

where

The same equations are also employed for derivative of Eq. (3) in order to reduce its order.

Fig. 4 Flowchart diagram of genetic algorithm

On the other hand, there are some practical shock motions defined in the literature which fall in good fit within the actual shock motions in the nature, including the acceleration impulse and the acceleration step. The half-sine, versed-sine, rectangular, dual quadratic and triangular are examples of practical shock motions which are mentioned by HARRIS and PIERSOL [24]. In this work, the base shock excitation is considered as a half-sine pulse of acceleration with peak value of 120g and duration of 2 ms. Thus, the following base shock excitation is

               (23)

This shock excitation is widely-used in simulating actual shocks which excite electronic systems such as laptop HDDs [10].

The mass of laptop body is assumed 2 kg (without HDD) which is a typical mass in most of laptop bodies. The optimization problem is analyzed by three widely-used cases of HDD which are more common in laptop production. Technical details and mass of studied HDD cases are listed in Table 2.

Table 2 Technical details and properties of studied HDD cases

In order to reach the optimum design variables by use of the PLBA algorithm, suitable starting point should be specified for design variables to compute the objective function and the inequality constraints with their gradients in the first iteration. In the present work, the initial guess (X₀) is specified for each HDD case separately, as given in Table 3.

Table 3 Initial guess of PLBA algorithm for each HDD case

Table 3 contains starting points for undamped natural frequency of the laptop body and HDD (ω_l, ω_h) and also damping ratio of the laptop body and HDD (ζ_l, ζ_h). The values of maximum constraint violation, convergence parameter, penalty parameter and γ constant are presented in Table 4 and should be determined before starting the PLBA algorithm. Then, the PLBA algorithm is solved separately for three considered cases of HDD and optimum values of lumped modal parameters (design variables) are determined.

Table 4 Parameter values of PLBA algorithm

Furthermore, GA was solved by using the MATLAB software in order to verify results of PLBA algorithm. The required parameters of genetic algorithm are considered as population size N=100, initial penalty of 10 and penalty factor of 100. In addition, related stopping criteria are presented in Table 5. In order to select parents, roulette wheel method (probability method) was considered as a selection function, so that the higher the probability p(x_i) is, the more the chromosome x_i is selected. Crossover fraction specifies the fraction of the next generation that crossover produces which is considered 0.85 in this optimization problem. Also, mutation probability is considered 0.13 for the next generation. The crossover and mutation functions are scattered and adaptive feasible, respectively. Scattered creates a random binary vector so that 1 and 0 are representatives of gene selection from the first and second parents, respectively.

Table 5 Stopping criteria of GA

After solving optimization problems, numerical results of GA and PLBA algorithm are presented in Table 6 for three considered cases of HDD. Also, Table 7 gives optimum values of lumped spatial parameters obtained for rubbers by using SQP and GA results.

In addition, the convergence history of objective function during performance of PLBA algorithm is presented in Fig. 5 for three optimization problems solved in this article. Also, active constraints (constraints which have zero value in last iteration) are identified by substituting optimum design variables in Eqs. (9)-(19). In this optimization problem, the fourth constraint is activated for all three cases of HDD.

Finally, the optimum peak values of HDD absolute acceleration (objective function) should be analyzed in order to investigate efficiency of designed rubbers in compensating base shock excitation. Figure 6 depicts the optimum response of system (HDD absolute acceleration) for three various types of HDD cases. As shown in Fig. 6, the peak value of acceleration is reduced by increasing HDD mass. Consequently, light-weight HDDs are more sensitive to shock excitation. So, their rubbers should be designed by more accuracy.

Table 6 Optimum values of design variables using SQP and GA

Table 7 Optimum values of lumped spatial parameters obtained for rubbers by using SQP and GA results

Fig. 5 Convergence history of objective function during performance of PLBA algorithm for three optimization problems

In order to have a better sense from efficiency of proposed rubbers, output response of optimum system is compared with responses of system while various widely-used rubbers are separately mounted on system. Table 8 presents six widely-used rubber mounts and a sample of rubber feet which are mentioned by PARK et al [10]. After determination of output response for these six sets of rubbers and comparing with a pair of optimum-based designed rubbers, a great drop in peak values of HDD absolute acceleration is obvious which is shown in Fig. 7 for all types of HDD cases, respectively. Therefore, the read/write performance of HDDs could be improved through a good selection of rubbers by laptop manufacturers so that the rubbers should be designed by the target of reaching optimum spatial properties.

Fig. 6 Acceleration response of optimum systems for three HDD cases

Table 8 Spatial parameters for each of widely-used rubber samples

It is perceived from Fig. 7 that the pair of R4 rubber mount sample and RF rubber feet is the most efficient rubber set in reducing mechanical effects of shock excitation among other widely-used samples, regardless of optimum-based designed rubbers which are proposed in this article. The set of R4 rubber mount sample and RF rubber feet show 26.4%, 41.8% and 79.7% reduction in the peak value of HDD absolute acceleration in comparison with input shock excitation for Case 1, Case 2 and Case 3, respectively. However the pair of R4 rubber mount sample and RF rubber feet show good compensation, but they are not as efficient as the pair of optimum-based designed rubbers because the proposed pair of rubbers show 91.2%, 91.3% and 92.1% reduction in the peak value of HDD absolute acceleration in comparison with input shock excitation for each of HDD cases, respectively.

Fig. 7 Absolute acceleration of HDD cases by considering various rubber samples in comparison with optimum-besed designed rubbers:

5 Conclusions

In the foregoing investigation, the laptop hard disk drive (HDD) was precisely protected against base shock excitations. The 2DOFs system was extended in order to construct a model from the laptop and its HDD. The SDOF to 2DOFs expansion of system model and subsequently involving effects of rubber feet, facilitated the performance of vibration compensation. Effects of rubber mounts and rubber feet were considered by lumped stiffness and damping coefficient. In addition, the base shock excitation was considered as half-sine pulse of acceleration. Then, the nonlinear optimization problem was solved for three widely-used commercial cases of HDD by using the SQP method to minimize peak of HDD absolute acceleration. Also, the genetic algorithm was used to verify results of PLBA algorithm and a good agreement was observed between results of GA and PLBA algorithm.

From numerical point of view, results of optimization indicate that by using the obtained values for k_h, c_h, k_l and c_l, designers can reduce the shock response up to 64.8%, 49.5% and 12.4% for each of HDD cases, respectively in comparison with the best set among widely-used samples (pair of R4 and RF rubbers). Thus, the optimum lumped spatial parameters of rubber mounts and rubber feet were obtained for three widely-used cases of HDD. The obtained numerical results are applicable in production of new rubbers to design high-protected laptops against base shock excitation. For future work, it could be a good suggestion to consider a random impulse as base shock excitation instead of deterministic impulses. Another useful suggestion is to find an applicable control method and subsequently absorb the base shock excitation. Indeed, an optimization problem would be extended to find optimum control performance.

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(Edited by YANG Bing)

Cite this article as: Seyed Rashid Alavi, Mehdi Rahmati, Saeed Ziaei-Rad. Optimization of passive control performance for different hard disk drives subjected to shock excitation [J]. Journal of Central South University, 2017, 24(4): 891-899. DOI: 10.1007/s11771-017-3491-z.

Received date: 2015-12-24; Accepted date: 2016-04-11

Corresponding author: Seyed Rashid Alavi; Tel: +98-3133915244; Fax: +98-3133912628; E-mail: r.alavi@me.iut.ac.ir