designing your own iPhone case
The primary objective of this project is to minimize shock to mobile device screens and prevent failure after drop from a given height H. We predict the relevant design variables to be the thickness of the designed case, surface area of contact with a rigid surface, and the case material properties (density, Young’s Modulus, and Poisson’s ratio).
For initial analysis for material choice, we assume contact surface area for impact to be the bottom surface of the case (see Fig 1), the rigid impact surface to be concrete, and that we can neglect energy loss from heat and friction.
Figure 1: Visualization of assumed contact surface area
In the instance of a solid case design composed of one material, the he phone can be modeled using a mass-spring-damping model in which the case acts as a spring and damper with spring constant and damping constant, respectively. Dropping the phone can be modeled after a modified ball bounce physics setup.
For one degree of freedom linear dynamic systems, the modeling equation is as follows:
where m = mass, c = viscous damping constant, k = spring constant, g = acceleration of gravity.
Converting the first degree of freedom modeling equation to a form with damping ratio and natural frequency we can get:
where ζ = damping ratio and ω = natural frequency.
We can find damping constant, spring constant, damping ratio and natural frequency by measuring time of contact of initial bounce, mass of the phone, height of drop, and a few other parameters.
First, we must measure the dT, or the time of contact the phone has with the ground when it bounces. Using a slow motion camera, we can see how long the phone remains contacted to the ground. The DT can be then used to calculate e, coefficient of restitution.
By definition, .
To empirically find e, we will use the following equation with n -> 0 due to only concentrating on the first bounce.
Where h0 is initial height. Plugging in our contact time, we can find the coefficient of restitution, which allows us to find the natural frequency, spring constant, and damping constant.
Using the damping constant and the spring constant, we can find the damping ratio
Thus, we have all the components of our modeling equations in both forms. We can solve for the differential equation. The solution becomes:
Impact conditions are a function of both the case material and the contact surface (concrete). For calculations, we assume that in comparison to the soft polymeric materials we will be working with, the spring constant and damping factor for concrete is negligible (See Fig 2.)
Figure 2: Mass-spring damping model
Impact velocity for the phone is calculated based on kinematic equations to be
V = sqrt( 2gH)
The stress/ strain relationship is given by: (where Y is Young’s Modulus and S is cross-sectional area, and L is displacement- compression of the solid material). An elastic solid as the ones we will use can be modeled as a bundle of springs, and displacement per unit length will indiciate compliance of the solid.
Once we have prototypes, we can compare the theoretical impact velocity to actual by standardized drop test procedures.
Alternate Structural Designs
We expect impact behavior of the solid case designs to differ than those of hollow, porous, or other mechanically structured forms. In order to evaluate the efficacy of such proposed models, we seek to find models of re-calculating the physical properties such as Young’s Modulus and Poisson’s ratio (See paper by Lauge Nielson, University of British Columbia, “Elasticity and Damping of Porous Materials and Impregnated Materials”, in which the author proposes a predictive method of calculating elastic modulus of “isotropic 2 phase materials” and porous materials of given void fraction).
Additionally, another structure that can be explored is the truss structure, which may provide better energy dissipation than a solid block of case material. Holnicki-Szulc et.al suggest based on numerical modelling that a cellular truss-like microstructure can be effective in impact resistant design. They state that their approach can be applied to both macro and micro structures. (See paper: High Performance Impact Absorbing Materials - the concept, design tools, and applications)
Additionally, another design we can pursue is the integration of polymeric foams into the case bumper designs. Due to their low density, high energy absorbance, cheapness and flexibility, these materials are often used for safety applications in automobiles and seat cushions. In those applications, the protected object we seek to avoid fracture for is usually the human body, but these properties could also be applied to protecting a phone screen. [http://www.sciencedirect.com/science/article/pii/S0734743X00000609]
Drawbacks of Manufacturing by 3D Printing
There is evidence of reduction of fatigue life of elastomers due to introduction of voids and imperfections when processed by additive manufacturing.
[https://sffsymposium.engr.utexas.edu/Manuscripts/2012/2012-49-Moore.pdf]
Alternative processing methods considered: injection molding
