How to Design a Hoeken’s Straight-Line Linkage in Excel (with VBA Simulator) Figure 1: Geometry of the Hoeken’s straight-line linkage and resulting coupler-point trajectory. The lower portion of the curve approximates straight-line motion over ~180° of crank rotation. The Hoeken’s Linkage is a mechanical engineer's favorite magic trick. It is a four-bar mechanism that converts simple rotational input into a near-perfect straight-line output. Unlike the Watt Linkage (which traces a figure-8), the Hoeken’s Linkage creates a "tear-drop" shape with a long, flat bottom (see Figure 1). This makes it the standard choice for walking robots and intermittent linear actuators. But how do you find the link lengths? If you guess, you get a wobble. This guide provides practical "Golden Ratios" and an Excel VBA tool to simulate the motion path. 1. The Geometry: Practical Design Ratios To achieve a usable straight line, link lengths m...
What is a Column?
In the definition of mechanical engineering, a column does not have to be a vertical pillar like in architecture. A column is defined as any structural member that carries an axial compressive load and tends to fail by elastic instability (buckling) rather than by crushing the material.
This includes connecting rods in engines, hydraulic piston rods, and even truss members in a bridge.
The Phenomenon of Buckling
Buckling (or elastic instability) is a dangerous failure mode. It occurs when the shape of the column is not sufficient to hold itself straight under load.
Unlike "crushing," where the material yields because the stress exceeds its limit, buckling is a geometric failure. At a specific "Critical Load," a sudden, radical deflection occurs. If the load is not immediately removed, the column collapses catastrophically.
Columns that tend to buckle are usually:
- Ideally straight
- Relatively long
- Slender (high slenderness ratio)
Predicting Failure: The Weak Axis
How do we know which way a column will bend? A column will always buckle about the axis that offers the least resistance.
To determine this, we look at three properties:
- Cross-sectional Area (A)
- Moment of Inertia (I): This measures the resistance to bending. We must find the axis where
minMomentOfInertiaoccurs. - Radius of Gyration (r): This is a geometric property that combines Area and Inertia to describe how the mass is distributed.
The Radius of Gyration Formula
The radius of gyration is computed as:
r = √(I / A)
The Golden Rule of Buckling:
A column tends to buckle about the axis where the radius of gyration (r) and the Moment of Inertia (I) are minimum.
Example Analysis (The Ruler Test)
Look at the image at the top of this post. It shows a thin rectangular plate (like a standard ruler) with dimensions h (height) and t (thickness), where t < h.
If you calculate the properties:
- Axis X-X: The inertia is calculated using
hcubed. This results in a largeradiusOfGyrationX. - Axis Y-Y: The inertia is calculated using
tcubed. Sincetis small, this results in a very smallradiusOfGyrationY.
Result: Since r_yy < r_xx, the column will buckle around the Y-Y axis. This is easily proven by pressing on a ruler—it always bows out along its thin face, never the wide face.
Next Step: Calculation Tool
Now that we understand the theory, how do we calculate the critical load using Euler's formula? In the next post, we will build an Excel sheet to solve these problems automatically.
Continue to Part 2:
Column Design: Effective Length and Slenderness Ratio (Part 2)
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