This paper proves that any initial condition in the energy space for the plate equation with square root damping z''- r Delta z' + Delta^2 z' = u on a smooth bounded domain, with hinged boundary conditions z=Delta z=0, can be steered to zero by a square integrable input function u supported in arbitrarily small time interval [0,T] and subdomain. As T tends to zero, for initial states with unit energy norm, the norm of this u grows at most like exp(C_p /T^p) for any real p>1 and some C_p>0. Indeed, this fast controllability cost estimate is proved for more general linear elastic systems with structural damping and non-structural controls satisfying a spectral observability condition. Moreover, under some geometric optics condition on the subdomain allowing to apply the control transmutation method, this estimate is improved into p=1 and the dependence of C_p on the subdomain is made explicit. These results are analogous to the optimal ones known for the heat flow.