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Damping Effects on Shock Response Spectra: 2.5-inch Hard Disk Drives
Half-Sine Acceleration and Modeling the System
The variable m represents the mass of the hard drive and F(t) represents the forcing function. For a half-sine acceleration pulse of duration T:
The equations of motion for the 1DOF system is:
The time domain response of the two equations above can be solved utilizing the Laplace transformation. The solution is a bit lengthy and is outside the purpose of this paper. Once solved, the resulting equations can be used in a computer algorithm, the time domain response can be plotted and maximum displacement—velocity or G level—can be determined. When enough simulations of the HDD response have been conducted and the maximums determined, the graphs found in this report can be generated. A typical acceleration time response for this system is shown below using a system natural frequency of 100 Hz. To illustrate the effect of damping on the time response, two curves are shown. The dashed curve reflects a lightly damped system, and the solid curve reflects a highly damped one.
Figures 3 through 8 on the following pages depict the shock response spectra for three different half-sine shock pulse durations: 0.0005 seconds, 0.001 seconds and 0.002 seconds. For each pulse duration, the peak transmitted G level and deflection are plotted versus system natural frequency. Since the mass is always the same, system natural frequency really represents changing stiffness. Shock Spectrum ResultsShock duration of 0.0005 seconds  Figure 3  Figure 4 Shock duration of 0.001 seconds  Figure 5  Shock duration of 0.002 seconds Figure 6 Figure 7  Figure 8 The high damping response curves shown in the graphs indicate that for a given natural frequency, i.e., stiffness, a system requires less sway space and will obtain a lower G level than with a material with low damping. Note how much the system response changes when the shock duration changes. An optimal solution for one acceleration pulse is not necessarily optimum for another duration pulse (nor for another pulse shape such as a triangular pulse or versed-sine pulse).
Solutions Available E-A-R Specialty Composites offers a variety of grommets that provide isolation and high damping. (See Figure 9.) Custom designs are available as well.  Figure 9: E-A-R Specialty Composites’ grommets for 2.5-inch hard drives There often is little available space for any isolation solution. When this happens, CONFOR CF-EG foam provides a viable solution. CONFOR CF-EG foam is highly damped and can be cut as thin as 1.5 mm. CONFOR CF-EG foam can be compressed to 50 percent of its thickness without a dramatic impact on its stiffness properties. Figure 10 shows a pad of CONFOR CF-EG foam cut to cushion a 2.5-inch hard drive.  Figure 10: CONFOR® CF-EG foam cushion for 2.5-inch hard drive
See also: Click here for information on molding materials |
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, which is the acceleration experienced by the hard drive, usually in units of Gs (1G = 9.8 m/s2). The maximum displacement experienced by the hard drive, called sway space, also is monitored. The variable c traditionally represents viscous damping provided by a dashpot, and k represents the spring stiffness. In an elastomeric spring used for shock protection, c and k combine to form a complex stiffness k*. The damping in the material, called loss factor η, represents the relationship between the real and imaginary components of that complex stiffness. The computer algorithms utilized cannot account for complex stiffness, so viscous damping is used. The variable used to represent viscous damping in the algorithms is ζ, which is called the critical damping ratio. Equating ζ to loss factor η can by achieved with the following expression: 
