Rank frequencies for quadratic twists of elliptic curves

We give explicit examples of infinite families of elliptic curves E over Q with (nonconstant) quadratic twists over Q(t) of rank at least 2 and 3. We recover some results announced by Mestre, as well as some additional families. Suppose D is a squarefree integer and let r(E)(D) denote the rank of the quadratic twist of E by D. We apply results of Stewart and Top to our examples to obtain results of the form #{D : \D\ < x, r(E)(D) greater than or equal to 2} much greater than x(1/3), #{D : \D\ < x, r(E)(D) greater than or equal to 3} much greater than x(1/6) for all sufficiently large x.