It sounds to me like I would suspect a defective or even broken motor mount. When you accelerate the torque of the engine trying to move the car actually causes the engine to rotate or lift in its mounts. If a mount is defective or broken it allows the engine to move beyond design dimensions. This allows some part of the engine to come in contact with some other fixed part in the engine bay. That is usually what causes a vibration or similar noise as you noted.
Sometimes you can confirm this by first raising the hood. Then start the car. With the brake firmly applied and nothing and no one in front of the car, put it in a forward gear and apply some pressure to the gas pedal. Someone can watch by standing beside one of the front fenders to see if the engine lifts or moves noticeably. Some movement is normal but it should not look as though the engine is trying to climb out of the car.