A mast is usually stayed with cables or rods holding it up :

http://www.lusas.com/case/civil/cablemasts.html

Masts can be designed in such a way as to not require stays.

Not sure I have got the hang (?!) of your context, but if the mast is suspended, then "free(ly suspended" may be better than unstayed, or even simply suspended mast.

- piece of balance, acrobatic moving circus equipement.

I found the only example of this construction in a description of theatre play by Christophe Huysman "Les Hommes penches" =

http://www.les-subs.com/0809/en/matitube-christophe-huysman-...

"Le Mâtitube ? C’est un mât "libre" soutenu par quatre pieds. Mettez-y du monde, des gens, enfin des personnes assez habiles et souples pour y trouver un point d’équilibre, pendant qu’évidemment le mât toujours titube.Trois saltimbanques, trois hommes très doux sont en colère. Leur maison s’écroule, ils volent, marchent sur l'air, l'air du temps, ses mots, sa grammaire, sa vulgarité, ses paroxysmes et ses questions. Un trio émotif, emporté dans un enfer métronomique. Le Mâtitube est une pièce de place publique. Un manège parlant, une expérience plastique et sonore. "

If "mât libre" is in fact similar to "mât oscillant", then "sway pole" may be the term you are looking for.

See this FMC guide.
Chinese Pole Act
An act created and devised in China where acrobats perform a number of synchronized leaps from one vertical pole to another.
Chinese Pole Act Mât Chinois
Sway Pole Act
A display in which a daredevil performs a series of breath-taking stunts atop a tall (up to 20 meters) flexible pole that “sways” back and forth throughout the routine.
Sway Pole Act Numéro de mât oscillant
http://www.circusfederation.org/uploads//news/pdf/FMCInterna...

Chinese pole can be adapted into a sway pole act, where the performer must scale a tall swaying steel pole and balance precariously in a crows nest on the tip.
http://www.fusecircus.co.nz/acts-for-hire/bouncy-wobbly.html

If it is in fact not fixed to the ground, it could refer to something like this, a sort of mobile pole with a base:
Le Mâtitube ? C’est un mât « libre » soutenu par 4 pieds.
http://maisonfoliemons.be/Le-Matitube

I'm not sure precisely how this would be described in English - free-standing pole? Unsupported pole? The only similar thing in the FMC guide is:
Ladder Act
An acrobatic balancing act where one or more acrobats climb and balance atop a freestanding, unsupported ladder.
Ladder Act Echelle libre, Echelle acrobatique

--------------------------------------------------
Note added at 2 days23 hrs (2012-05-14 08:06:06 GMT)
--------------------------------------------------

Looking at this again, without being swayed by the description you posted (the writer himself seemed unsure exactly what was involved in "mât libre" and I think the suggestion that it might somehow be suspended was a bit of a red herring), I'm now fairly sure that this must be something along the lines of the Mâtitube as I suggested as an alternative above. If you do a search for "mât libre", you find that it isn't a very common expression in French either. Apart from the various references to the Mâtitube, the two names that keep cropping up are Gert de Cooman and Kenzo Tokuoka. If you then search for information on their circus company Le Carré Curieux, you will find this:
Le mât chinois libre :
D'origine asiatique, le mât chinois est composé d'un poteau en métal généralement fixé verticalement. Le mât chinois est utilisé autant en art de rue qu'en salle ou en chapiteau et fait partie des numéros aériens.
Sa taille peut varier de 3 à 9 mètres de haut. Un ou plusieurs acrobates y grimpent et y évoluent, accomplissant différentes figures, rattrapages et sauts.
L'utilisation du mât dans le spectacle « Le Carré Curieux « est différente du mât chinois traditionnel car il n'est pas fixé au sol. Il permet donc aux artistes de dynamiser et diversifier les figures grâce au mouvement créé par le mât tout en demandant une technique et une force physique considérable.
http://www.ac-bordeaux.fr/ia40/fileadmin/pedagogie/circonscr...

On their own website in French, Dutch and some English, there are also the following references:
Gert De Cooman - aerial tissue and chinese pole
Gert De Cooman - tissu aérien et mât Chinois
DE COOMAN Gert Tissu aérien et mât
a pole that refuses to remain vertical
mât fuyant la verticale
http://www.carrecurieux.be/spectacle-fr.html

This of course still leaves you with the problem of exactly what to say in English. I think my suggestion would now be free-standing Chinese pole.


--------------------------------------------------
Note added at 2 days23 hrs (2012-05-14 08:10:43 GMT)
--------------------------------------------------

I meant also to include this additional reference to the Mâtitube:
De vertige, il sera aussi question avec le " Mâtitube ", nouvel agrès du spectacle éponyme de la compagnie Les Hommes Penchés, qui permet de repousser les limites du spectaculaire. Il s'agit d'un mât chinois mobile et libre, soutenu par quatre pieds
http://webcache.googleusercontent.com/search?q=cache:W-XjEzx...