Yes, warming up your car before driving in cold weather can damage the engine over time
Excessive idling can cause engine damage, and experts agree that modern gas-powered engines don’t need to warm up to run effectively.
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How to Clean a Throttle Body
If your car runs rough when idling, the cause might be a dirty throttle body. The throttle body controls the amount of air the engine takes in, and when it gets dirty, the engine can't idle smoothly.