If the vibration problem Mÿ+ Bý + Ky = 0,y(t0 ) = y0 , ý(t0) = ý , is cast into state-space form x = Ax,x(t0) = x0 , so far only two-sided bounds on x(t) could be derived, but not on the quantities y(t ) and ý(t) . By means of new methods, this gap is now filled by deriving two-sided bounds on y(t ) and ý(t) ; they have the same shape as those for x(t ) . The best constants in the upper bounds are computed by the differential calculus of norms developed by the author in earlier work. As opposed to this, the lower bounds cannot be determined in the same way since || y(t )||2 and || ý(t)||2 have kinks at their local minima (like | t |1/2 at t = 0 ). The best lower bounds are therefore determined through their local minima. The obtained results cannot be obtained by the methods used so far.