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Doc

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- TL;DR Summary
- Stress minimization in design of spring clip

Hi all,

(Please move to general or mechanical engineering sub-forum if more appropriate over there. I put this here as it is essentially a mathematics problem.)

Broken into sections:

- problem categorization (what

- the question,

- specifics (description of the design task and all variables),

- solution attempt.

I have a design task that I believe to be a multivariable constrained optimization problem.

I have a stress induced in a piece of glass by a spring clip σG(D,w), how do I minimize this stress in the context of multivariable optimization? Or is there a more appropriate way to solve this problem?

I need to design a spring clip (three identical clips will be used) that will retain a piece of glass in a mount, Figure 1.

Figure 1, spring clip shown retaining cut-away section of glass.

The clip needs to retain the glass with a set preload,

Figure 2.

The stress induced in the glass is given by:

[tex] \sigma_G = 0.798 \sqrt { \frac {K_1 \frac {F} {w} } {K_2} } [/tex]

Where

[tex] K_1 = \frac {1} {D} [/tex]

And:

[tex] K_2 = \left[ \frac {(1- \nu^2_G)} {E_G} \right] + \left[ \frac {(1- \nu^2_M)} {E_M} \right] [/tex]

ν is the material Poissons ratio and E is the material Young's modulus. The subscripts

Therefore the stress in the glass,

[tex] \sigma_G (D,w) [/tex]

The procedure, as I understand it, is to:

1. reduce the equation or quantity that I want to optimize (minimize in this case) and reduce it down to a single variable,

2. differentiate,

3. set equal to 0,

4. solve for the variable.

In order to achieve step 1 I will simply say that, for practicality in using the clip after manufacture, D > w. To keep things simple I'll say that D = 2w.

Recall the equation for stress:

[tex] \sigma_G = 0.798 \sqrt { \frac { \frac {1} {D} \frac {F} {w} } {K_2} } [/tex]

I can then substitute D and rewrite the stress equation:

[tex] \sigma_G = 0.798 \sqrt { \frac { \frac {1} {2w} \frac {F} {w} } {K_2} } [/tex]

Combining terms and simplifying gives:

[tex] \sigma_G = 0.798 \sqrt { { \frac {F} {2w^2 K_2} } } [/tex]

Taking the constants out of the root gives:

[tex] \sigma_G = 0.798 \sqrt{ \frac {F} {2w^2} } (w^-1) [/tex]

Taking the derivative gives:

[tex] \sigma_G = -0.798 \sqrt{ \frac {F} {2w^2} } (w^-2) [/tex]

I got to here and then started getting from what I could tell to be nonsensical answers.

Would be much appreciated if somebody could let me know where I'm going wrong. Apologies, it's been sometime since I last used this stuff.

Regards,

Doc

(Please move to general or mechanical engineering sub-forum if more appropriate over there. I put this here as it is essentially a mathematics problem.)

Broken into sections:

- problem categorization (what

*type*of problem I*think*I have),- the question,

- specifics (description of the design task and all variables),

- solution attempt.

**Problem categorization**I have a design task that I believe to be a multivariable constrained optimization problem.

**Question**I have a stress induced in a piece of glass by a spring clip σG(D,w), how do I minimize this stress in the context of multivariable optimization? Or is there a more appropriate way to solve this problem?

**Specifics**I need to design a spring clip (three identical clips will be used) that will retain a piece of glass in a mount, Figure 1.

Figure 1, spring clip shown retaining cut-away section of glass.

The clip needs to retain the glass with a set preload,

*F.*The clip preload will induce a compressive stress in the glass,*σg*, and this should be minimised so as not not damage the glass. Figure 2 shows a diagram of the spring clip with its associated dimensions, ignore*t*and*L*for now.Figure 2.

The stress induced in the glass is given by:

[tex] \sigma_G = 0.798 \sqrt { \frac {K_1 \frac {F} {w} } {K_2} } [/tex]

Where

*w*is the width of the spring clip and:[tex] K_1 = \frac {1} {D} [/tex]

And:

[tex] K_2 = \left[ \frac {(1- \nu^2_G)} {E_G} \right] + \left[ \frac {(1- \nu^2_M)} {E_M} \right] [/tex]

ν is the material Poissons ratio and E is the material Young's modulus. The subscripts

*G*and*M*refer to glass and metal, respectively.Therefore the stress in the glass,

*σg*, is a function of the diameter, D, of the rounded end of the clip and the width of the clip: the rest of the components may be considered constants. So I have:[tex] \sigma_G (D,w) [/tex]

**Solution attempt**The procedure, as I understand it, is to:

1. reduce the equation or quantity that I want to optimize (minimize in this case) and reduce it down to a single variable,

2. differentiate,

3. set equal to 0,

4. solve for the variable.

In order to achieve step 1 I will simply say that, for practicality in using the clip after manufacture, D > w. To keep things simple I'll say that D = 2w.

Recall the equation for stress:

[tex] \sigma_G = 0.798 \sqrt { \frac { \frac {1} {D} \frac {F} {w} } {K_2} } [/tex]

I can then substitute D and rewrite the stress equation:

[tex] \sigma_G = 0.798 \sqrt { \frac { \frac {1} {2w} \frac {F} {w} } {K_2} } [/tex]

Combining terms and simplifying gives:

[tex] \sigma_G = 0.798 \sqrt { { \frac {F} {2w^2 K_2} } } [/tex]

Taking the constants out of the root gives:

[tex] \sigma_G = 0.798 \sqrt{ \frac {F} {2w^2} } (w^-1) [/tex]

Taking the derivative gives:

[tex] \sigma_G = -0.798 \sqrt{ \frac {F} {2w^2} } (w^-2) [/tex]

I got to here and then started getting from what I could tell to be nonsensical answers.

Would be much appreciated if somebody could let me know where I'm going wrong. Apologies, it's been sometime since I last used this stuff.

Regards,

Doc