The solution is a better antenna. As an easy first project, I selected a ground plane antenna from the my public library's copy of the The ARRL Handbook for Radio Amateurs.
Here are the sources for the parts and additional tools I needed.
From Amazon
BCP 5pcs UHF Female Jack Solder SO-239 4-Hole Chassis Mount Coax RF Connector by BCP
$6.89
MPD Digital smaF/pl259f SMA Female to UHF SO239 PL259 Female RG58 Adapter Cable, 3-feet
$19.59
DHT Electronics Handheld Antenna Cable, Adapter, SMA Female to UHF SO-239 Female Connectors, 3-Feet
$7.00
From McMaster-Carr
1 89015K176 6061 Aluminum Sheet, 0.05" Thick, 8" x 8",
8.19 each
1 8859K164 Ultra-Formable 260 Brass, 3 Feet Long Rod, 1/8" Diameter, packs of 5
17.69 per pack
2 6750K11 Hard Anodized 6061 Aluminum Rod, 1/8" Diameter, 3 Feet Long
4.63 each
From the local big-box hardware store.
10-24 screws, nuts, and washers.
4-40 screws and nuts - 2-56 would have been better but they didn't have any.
3/4" PVC Pipe.
Three 3/4" PVC tees.
Four 3/4" PVC caps.
From Harbor Freight
Step Drill Bit
The first order of business was to mount the SO-239 connector on to the aluminum sheet. The logical way to do this would be to use a 5/8" Greenlee chassis punch, but when I saw that it costs almost $60, I looked for an alternative. What I found was called a step drill bit that lets you drill various large holes and costs less that $10. After drilling the mounting holes, I bent the corners of the aluminum sheet to 45 degrees.
Next step was to figure out how to mount the round aluminum rods onto the flat aluminum plate. Pounding the ends flat with a hammer was crude but effective. As I hammered the ends would curl to the left or the right depending on which side was stretching more. If it curled to the left, I would have to pound more on the left side causing that side to stretch more and correct the curl. When the flattened end was wide enough for a 4-40 clearance hole I filed off any remaining black anodizing and drilled the holes. If I do this again, I'll procure 2-56 screws, because drilling such big holes in the end of the rod leaves it weaker than I would have liked. Finally, I used a cut-off wheel on my rotary tool to cut the rod length to spec. This process was repeated four times.
Now for the brass vertical element. It's got to go into the solder cup on the SO-239 jack, however, the end of the brass rod needs to be ground down to fit into the cup. For this I used the rotary tool again, this time with a cylindrical grinding burr. It was a tedious process, but when the end of the rod was ground down sufficiently, I was able to solder it into the cup with a liberal amount of flux. After the rod is connected to the jack, the jack can be connected to the plate with 4-40 screws.
To support the antenna, I made a fixture from PVC pipe. In the tee in the middle, I used the step drill bit to make a hole so I could run the cable through the pipe. I ended up not using any solvent glue because the fit was pretty tight, and the 10-32 screws I used to bolt the fixture to the aluminum sheet hold it together pretty well.
Here's a photo of the finished project mounted on a bar stool (not its permanent home).
Now I'm able to reach the repeaters, but to really tell how much better it works I'll need a field strength meter. Maybe that'll be my next project.
