I love this time of year in New England–the weather, apple picking, the colors as the leaves turn. But what I don’t love is the half a trillion leaves that fall into my garden that I then have to blow/rake and pick up. But they don’t just fall on the ground; they fall into the gutters causing clogs and mini floods the next time it rains. So last weekend I braved up and got out the ladder to clear them out.
I’m in a new house, so it’s the first time I have done these gutters, and I was struck by how my extension ladder could be both as rigid as a rock or as flexible as rubber (okay maybe a slight exaggeration), depending upon the angle of the ladder and how much it was extended. Idle speculation lead me down to my basement office, and in a few short minuntes, I had a test model ready to go in SolidWorks Simulation. In order to get a more
representative result, I also created a model of myself on the ladder, to add some extra weight.
As you can see I used weldments for both the ladder and myself, with my weight set to 190 lbs [you wish - Ed.]. This is because I am not interested in the detailed stresses in the ladder; rather, I want to know why it become so flexible as it extends. The stiffness of the ladder is related to its natural mode of vibration, so my first test was to see what effect extending the ladder had while keeping the maximum elevation at 4m.
Setting up the problem in SolidWorks Simulation was very simple; all you have to do is restrain the ends of the ladder and hit run. And this is where it gets interesting. If you look at the first natural mode for the ladder (see below) we can see as the ladder extends the frequency drops, indicating a reduction in stiffness.
|No Extension||5.0 Hz||Twisting|
|Half Extension||4.0 Hz||Twisting|
|Full Extension||2.7 Hz||Hogging-Sagging|
But the real analysis ‘gold’ comes from looking at the mode shapes. For the ladder with no or only half extension, the first natural mode is a twisting mode, and for the fully extended ladder it is a hogging/sagging mode.
The reason this is important is because this mode is excited by the movement of a person up the ladder. So the greater the force in the excitation direction, the greater the movement of the ladder. The amount of twisting force applied as you go up a ladder is relatively small as at the most you weight moment arm is half the width of the ladder, but the force in the last case is well aligned to the natural mode so even if you do not move up the ladder one step every 1/3 of a second to induce resonance, the ladder will tend to move a lot mode due to this force alignment.
So what would happen if I moved up this ladder by two rungs at one rung per second?? To understand this I need to carry out a dynamic analysis.
And the results are telling; even though I am away from the first natural mode, the ladder will flex at a peak of 35mm, but when I stop moving, the movement quickly dies away.
So what have I learned from this simple test? Firstly, don’t just consider strength; if the operational load moves or varies, look at the flexibility of you design. A frequency study can qualitatively tell you how flexible your design is, but a dynamic analysis will tell you exactly how your design will perform under variable loads.
I also learned that I hate ladders, and that the local handyman can clear out my gutter next time..
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