If you have not changed the engine oil in your car for a long time or are filling up the new engine oil over the old one without checking if it's clean, it can significantly lead to the car shaking when accelerating. Changing engine oil at least once a month is necessary for a healthy ride. So, if this task is overdue and you did not find any of the other 11 signs of shaking, get rid of the old oil & replace it with a new one.