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Showing posts with the label Newton-Raphson

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Why I Wrote The Sheet Mechanic (And Why Calculations Aren’t Enough)

For engineers who already know the math—but still lose projects. For the last few years, I’ve been sharing technical guides here on Mechanical Design Handbook —how to size a motor, how to calculate fits, and (as you recently read) how to choose between timing belts and ball screws. But after 25 years in industrial automation, I realized something uncomfortable: Projects rarely fail because the math was wrong. They fail because: The client changed the scope three times in one week. A critical vendor lied about a shipping date (and no one verified it). The installation technician couldn’t fit a wrench into the gap we designed. University taught us the physics. It didn’t teach us the reality. That gap is why I wrote my new book, The Sheet Mechanic . This is not a textbook. It is a field manual for the messy, political, and chaotic space between the CAD model and the factory floor. It captures the systems I’ve used to survive industrial projec...
NEW RELEASE: Stop trying to be a Hero. Start being a Mechanic. Get "The Sheet Mechanic" on Amazon »
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Numerical Methods - The Newton-Raphson Method (Part 2)

In the previous post , we talked about several root finding techniques. In this post, we're going to see how we can use Microsoft Excel VBA to find the roots using the Newton-Raphson Method. Advertisement Figure 1: The Newton-Raphson method rapidly converges on a solution using the function's derivative. The Logic As we know, the Newton-Raphson Method is the most widely used of all root-locating formulas. It uses the slope ( first derivative ) of the function to find the root. That means, in the VBA code, we have to calculate the first derivative of the function. We already discussed how to find the first derivative using numerical methods , and we will incorporate that logic here. Figure 2: Geometric interpretation: The tangent line points to the next approximate root. The Iterative Formula: The method uses the Taylor series to approximate the next position (x N ) based on the curre...

Numerical Methods - The Newton-Raphson Method (Part 1)

Machine designers frequently deal with complex equations in their design projects . While some roots can be found directly, many algebraic and transcendental equations require numerical approximation. Advertisement For example, the classical equation f(x) = e -x - x cannot be solved analytically. In these cases, engineers rely on robust Root Finding Algorithms . Figure 1: Numerical methods approximate the point where the function crosses zero. These algorithms generally fall into two categories: Bracketing Methods and Open Methods . 1. Bracketing Methods Bracketing methods require two initial guesses that must "bracket" the root (one positive, one negative relative to the root). They are reliable but often slower. The Bisection Method: An incremental search based on sign changes. It repeatedly cuts the interval in half. Also known as binary chopping or Bolzano's method . The False-Position Me...

Solving Nonlinear Engineering Equations with Excel Goal Seek

Problems in mechanical design often require the solution of nonlinear equations. A classic example is finding the roots of a polynomial stress function: 3x 3 - 20x 2 + 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. Advertisement 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...