Channel balance difficulties

I have a heavy tilt sash window (44" wide) that I took out for cleaning. I wanted to see what kind of channel balance I would need if I ever wanted to replace it. I was surprised to have discovered that there are no channel balance. All I could see were tilt lock shoes which, when unlocked, would just freely slide up and down the track. There were also plastic guides inside the track that were permanently attached. I expected to see a spiral spring or something like a pulley to counteract the weight of the window but there was no such a thing. How is this possible? I thought that channel balances are used in all types of sash windows, including tilt windows.

Is my window "unusual", not normal? I am hoping that perhaps you could answer my question.

Thanks, Andre

PS. The windows were all installed in 2006 and the brochure that came with them says that they were made by Home Craftsman, a Texas company. I was unsuccessful trying to reach their customer support.

I have many other identical windows in my house and they all look the same (no visible balancer!) yet they seem to have a balancer system because they lift effortlessly. Only this one does not. It stopped working only after we tried to open it and take the sash out. Then when we put it back the balancers don't balance, on both sides.

I am attaching the picture of the window showing the track and the shoe at the bottom.

This single hung window, like the rest of them in my house, was made by Home Craftsman (Dallas, TX). They were installed in 2006.

The sticker info:

Manufacturer Stipulates Conformance to:
NFRC 100/200
AAMA/NWWDA 101/I.S.2-97
ASTM F 588
Code H-713-1
Rating H-R35/50
Series 8110SH
MST 4060/3060
NFRC
ALE-1-044
05-M

Thanks, Tom. I've made a picture of the pivot shoe that is on another exactly the same window but this one is not broken. I have also removed the coil from the window that is broken. It is 1/2" wide. The coil itself seems fine, the stamp says *5.5 and 35 (would it mean 5.5lbs per coil and 35" coil length?).

I have attached the photos of the pivot shoe and the coil. Please, let me know if you have a pivot shoe that would attach to the coil, or would I need to replace all coils and pivot shoes.
Thank you!