  In present paper, we study the normalized solutions $(\lambda_c, u_c)\in
\R\times H^1(\R^N)$ to the following Kirchhoff problem $$
-\left(a+b\int_{\R^N}|\nabla u|^2dx\right)\Delta u+\lambda
u=g(u)~\hbox{in}~\R^N,\;1\leq N\leq 3 $$ satisfying the normalization
constraint $ \displaystyle\int_{\R^N}u^2=c, $ which appears in free vibrations
of elastic strings. The parameters $a,b>0$ are prescribed as is the mass $c>0$.
The nonlinearities $g(s)$ considered here are very general and of mass
super-critical. Under some suitable assumptions, we can prove the existence of
ground state normalized solutions for any given $c>0$. After a detailed
analysis via the blow up method, we also make clear the asymptotic behavior of
these solutions as $c\rightarrow 0^+$ as well as $c\rightarrow+\infty$.
