We deal with the as yet unresolved exponential stability problem for a stretched Euler--Bernoulli beam on a star-shaped metric graph with three identical edges. The edges are hinged with respect to the outer vertices. The inner vertex is capable of both translation and rotation, the latter of which is subject to a combination of elastic and frictional effects. We present detailed results on the asymptotic distribution and structure of the spectrum of the linear operator associated with the abstract spectral problem in Hilbert space. Within this framework it is shown that the eigenvectors have the property of forming an unconditional or Riesz basis, which makes it possible to directly deduce the exponential stability of the corresponding $C_0$-semigroup. As an aside it is shown that the particular choice of connectivity conditions ensures the exponential stability even when the elasticity acting on the slopes of the edges is absent.