Every linear motion design starts with the same choice: How do you convert rotary motor motion into linear travel? The two most common answers are the Lead Screw (simple, cheap, friction-based) and the Ball Screw (complex, expensive, rolling-based). Making the wrong choice here is costly. Use a lead screw where you need precision, and you get backlash. Use a ball screw in a vertical lift without a brake, and your load crashes to the floor. In this guide, we compare them side-by-side. Table of Contents 1. The Physics: Sliding vs. Rolling 2. Efficiency & The "Back-Driving" Danger 3. Accuracy and Backlash 4. Selection Table Advertisement 1. The Physics: Sliding vs. Rolling The fundamental difference is friction. Lead Screws rely on Sliding Friction . The nut (often bronze or plastic) slides directly against the steel screw threads. This generates heat and wear. Ball Screws re...
Problems in mechanical design often require the solution of nonlinear equations. A classic example is finding the roots of a polynomial stress function:
3x3 - 20x2 + 1000x + 12000 = 0
These are known as root solving problems: finding the value of x where f(x) = 0. While you could solve this by trial and error (guessing values until the result is zero), or use the mathematical Newton-Raphson method, there is a faster way for the practicing engineer.
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In this post, I will share a simple technique to solve these problems instantly on your desktop using Microsoft Excel Goal Seek. No VBA programming or expensive MATLAB license is required.
Step-by-Step: Using Goal Seek for Engineering
1. Setup the Spreadsheet
Open Excel and set up two cells: one for your input variable (x) and one for your equation (f(x)). The formula cell must mathematically reference the input cell.
2. Locate the Tool
In modern Excel versions, navigate to: Data Tab > Forecast Group > What-If Analysis > Goal Seek...
3. Configure the Parameters
A dialog box will appear. To find the root of the equation, we want to force the result to be zero.
- Set cell: The cell containing your equation (f(x)).
- To value: Enter 0.
- By changing cell: The cell containing your variable (x).
4. The Solution
Click OK. Excel iterates through values instantly. It will find the exact value of x that satisfies the equation. Note that due to computer floating-point precision, the result might appear as "0.0001", which is effectively zero for engineering tolerances.
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Application: Shaft Design for Fatigue Loading
Real-world mechanical design is rarely linear. Consider designing a transmission shaft under fatigue loading. The relationship between diameter (d) and safety factor (n) often follows the rigorous ASME Elliptic or DE-Goodman criteria.
Solving for the diameter (d) algebraically is difficult because it appears inside cubic roots and squares. Here is the standard equation form:
1/n = (16 / Ï€d3) × [ (4(Kf Ma)2 + 3(Kfs Ta)2)1/2 / Se + (4(Kf Mm)2 + 3(Kfs Tm)2)1/2 / Sut ]
Instead of rearranging this massive equation to isolate d, you can simply:
- Set up cells for all inputs (Ma, Ta, Se, etc.).
- Create a cell for the Left-Hand Side (1/n).
- Create a cell for the Right-Hand Side (The stress terms).
- Create a "Check Cell":
= LHS - RHS. - Use Goal Seek to force the "Check Cell" to 0 by changing the Diameter cell.
This reverse-engineering approach gives you incredible flexibility. You can just as easily fix the diameter and solve for the maximum allowable Moment (Mm) without rewriting a single formula.
Comments
I need to generalize this equation solver thing for some calculations, so that, i can get the solution every time on changing value of coefficient.
for example;
in a cubic equation, if i change coefficient a,b,c,d then also, i can get the solution.
Please help.
Thanks !!