Rain began battering the densely populated coast ahead of the storm`s landfall, while fierce winds uprooted trees and electricity pylons. Flights at Chennai airport were delayed or cancelled.
Vardah is forecast to weaken gradually as it moves west but the National Disaster Management Authority (NDMA) said wind speeds had picked up to around 140 kmph (87 mph), with heavy rain expected when the storm reaches land.
A "storm surge" will be about one metre (three feet) high.
More than 23,000 people in Tamil Nadu have been moved to relief centres, with plans for tens of thousands more to be evacuated, a senior state official, K. Satyagopal, told Reuters.
More than 10,000 people from two districts in Andhra Pradesh have also been moved, its disaster management commissioner, M.V. Seshagiri Babu, said.
The NDMA warned fishermen not to venture to sea for the next 36 hours, and urged residents to stay in safe places.
Navy ships and aircraft, as well as 30 diving teams, were on standby to help move people and deliver aid if needed, a navy spokesman said.
India`s cyclone season usually runs from April to December, with storms often causing dozens of deaths, evacuations of tens of thousands of people and widespread damage to crops and property.
Wind speeds topped 300 km per hour (186 mph) in an Indian "super-cyclone" that killed 10,000 people in 1999, while a cyclone packing speeds of more than 200 kph (124 mph) lashed the east coast in 2013.
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