That is an important question, and I wish I knew the full answer. Since the whole class of organisms represented by the dinosaurs is extinct, we might surmise that they must not have been on the ark. Perhaps God in his mercy chose to have this class of organisms, among the most dangerous to man, and the largest terrestrial organisms, not be a part of man's world after the flood. Man was in a very different condition, shorter in stature, shorter in lifespan, and weakened in every way from those humans who lived before the flood. Such creatures could have posed a real threat to human culture. On the other hand, perhaps there were dinosaurs on the ark, but they were not able to survive the climatic changes this side of the flood. It does appear they were homeothermic (constant body temperature), and were probably, like all other reptiles, ectotherms (dependent on environment for warmth), so one could easily imagine them dying out during the Pleistocene cool down (ice age), because the environment was so cool they could no longer function. Hope this helps.
2010 Arthur V. Chadwick, Ph.D.