Explore the contents of this article with a free Wolfram SystemModeler trial. In 1869, Rankine extended Euler and Bernoulli's century-old theory of lateral vibrations of bars to an understanding of rotating machinery that is out of balance. Classical dynamics had a new branch: rotor dynamics. Machine vibration caused by imbalance is one of the main characteristics of machinery in rotation. All structures have natural frequencies. The critical speed of a rotating machine occurs when the rotational speed matches one of these natural frequencies, often the lowest. Until the end of the nineteenth century the primary way of improving performance, increasing the maximum speed at which a machine rotates without an unacceptable level of vibration, was to increase the lowest critical speed: rotors became stiffer and stiffer. In 1889, the famous Swedish engineer Gustaf de Laval pursued the opposite strategy: he ran a machine faster than the critical speed, finding that at speeds above the critical threshold, vibration decreased. The trick was to accelerate fast through the critical speed. Thirty years later in 1929, the American Henry Jeffcott wrote the equation for a similar system, a simple shaft supported at its ends. Such a rotor is now called the de Laval rotor or Jeffcott rotor and is the standard rotor model used in most basic equations describing various phenomena.

Background

Explore the contents of this article with a free Wolfram SystemModeler trial. Today, many helicopters launch from and land on ships at sea. Some are conventional helicopters, both commercial and military, and some are drones. In Wolfram SystemModeler, we now have a system for simulating helicopter landings and launches that includes waves and ships. The models have been used for the design of mechanical parts, autopilots, landing criteria, and operational limits.

Major components of the system

The aim has been to develop a model with an accurate depiction of the waves, ship motion, and helicopters in such a way that the results can be used not only qualitatively but also quantitatively in real industrial applications. The first task is to calculate the motion of the landing platform mounted on the ship's deck. There is commercially available historical wave data for different seas and oceans. Since access to this data is expensive, we will instead describe the waves mathematically. A model of the forces on the ship's hull was developed with classical analytical theory. With the waves and ship hull forces, the motion of the ship's landing platform can be calculated. If we assume that the helicopter landing does not influence the landing platform motion, the system is simplified. We speed up the simulation by storing the motion in a database for the different wave heights, lengths, and directions, and the ship's speed. Typically the database will include wave heights of 1, 2, 3, and 4 m; wave directions 0, 30, 60, 90, 120, 150, and 180 degrees; wave lengths 100, 150, and 200 m; and ship speeds of 5 and 10 knots. The helicopter was modeled with the MultiBody library. It includes mechanical parts such as rotors with gyroscopic effects and landing gear with hydraulic dampers. Friction models for wheel-deck interface and flexible beams for the rotor blades have been developed. We have also developed a simple autopilot where the landing algorithm is implemented and tested. For one application, the model has been run with the actual autopilot as hardware in the loop.