A car or car's engine may vibrate while idling for various reasons, including that the motor or transmission mounts have a problem. When a car is stopped at a red light or turned on in park, a noticeable vibration may occur that signals damaged or broken transmission mounts.
Switch the gear to neutral. If the vibrating ceases somewhat, the engine's motor mounts are having issues. In this case, it is best to take the car to see a certified mechanic who can assess and fix the problem.
Another reason why a car may be vibrating is a problem with its timing belt. Belts can become loose and damaged. When this happens, the corresponding components may not rotate or operate properly, leading to harsh vibrations and strange noises. To avoid this, belts should regularly be checked for any cracks or ribbing damage and should always be wound tight.
Also, similar problems can arise with any hoses on the engine. If they become loose or disconnected, this can cause the engine to vibrate and shake considerably. It's best to inspect the engine to see if any hoses have come loose and re-tighten them appropriately.
Engine vibration can also be caused by a poorly adjusted fuel intake system. Sometimes, this can be easily fixed. For instance, the idle on the carburetor may have to be adjusted. This allows fuel to flow more smoothly through the engine and facilitate better combustion.