Hey it's a me again @drifter1! Today we continue with Physics and more specifically the branch "Classical Mechanics" to get into the Physics behind Mechanical Gyroscopes that are used in lots of applications. So, without further ado, let's get straight into it!
What they are[Image 2]
Mechanical Gyroscopes are devices that contain a rapidly spinning wheel that is used to detect the deviation of an object from its desired orientation. Optical Gyroscopes are used for the same purpose, containing no moving parts, but only a circulating beam of light. In this article we will only talk about mechanical gyroscopes!
Some HistoryIn the 19th century Jean-Bernard-Léon Foucault, a French physicist, gave the name gyroscope to a wheel (or rotor) that was mounted in gimbal rings. The angular momentum of the spinning wheel/rotor helped the device maintain its attitude even when the gimbal assembly was tilted. Conducting an experiment using such a rotor, during the 1850s he demonstrated that the spinning wheel maintained its original orientation in space regardless of the Earth's rotation. That's exactly when things got interesting and the usage of the gyroscope as a direction indicator was suggested. In 1908 the German inventor H. Anschütz-Kämpfe developed the first gyro-compass for use in submersible. In 1909 American inventor Elmer A. Sperry built the first automatic piloting system to help an aircraft maintain its course. In 1916 a German company installed the first automatic pilot for ships, and during this year a gyroscope was also used in the design of the first artificial horizon for aircraft. After that gyroscopes were also used for automatic steering and correct turn and pitch motion in cruise and ballistic missiles for World War II. During the war, the accuracy of direction definition increased greatly, as they were used in sophisticated control mechanisms, stabilized gun-sights, bomb-sights, platforms for guns and radar antennas etc. Gyroscopes are also used in the inertial guidance systems of orbital spacecrafts and also in the attitude control systems of some satellites.
In modern aircraft, inertial guidance systems (IGS) make good use of gyroscopes to control the orientation of the aircraft in flight. Spinning gyroscopes are kept in special cages that keep their orientation independently of the aircraft in space. This cages have electrical contacts and sensors that relay information to the pilot whenever the plane rolls or pitches. This allows the pilot to "know" the planes current relative orientation in space. The Mars Rover has a set of them, that provide the Rover with stability and also aid with navigation. Gyroscopes also have applications in drone aircraft and helicopters, for the same reason.
How they workTo understand how they work I think I should first inform you that the famous fidget spinner is also based on a gyroscope! If you ever had one or have seen others use them, you might remember that someone can perform lots of "tricks" with them, like balancing on a string or your finger. If you held one in your hand you might have noticed that there is some kind of resistance when you try to change its position.
Well, this phenomena is quite tricky to understand. Gyroscopes seem to defy gravity, because of their angular momentum. Being influenced by torque on a disk, like gravity, they produce something called a precession of the spinning disk or wheel. Thinking about Newton's Second Law, this effect is enhanced the faster the disk or wheel is spinning, as increasing the angular velocity increases the angular momentum. The influence of gravity on the plane of the spinning disk or wheel causes the rotational axis to "deflect". So, the main reason that the gyroscope seems to defy gravity is that the entire rotational axis tries to find some kind of "middle ground" between the influence of gravity and its own angular momentum vector. The fact that they are being stopped from "falling" towards the center-of-gravity is exactly what gives them the fascinating properties that we see in them.
So, let's now get into the Physics more in-depth...
Precession (the gravity-defying part)[Image 3]
In general, this effect works like this:
Suppose that we have a spinning gyroscope and that we try to rotate its spin axis. The gyroscope will instead try to rotate about an axis that's at the right angles to your force axis.
But, why does this happen? Why does a gyroscope display such a behavior? Well, to understand it better we have to think about a gyroscope wheel as two different sections: top and bottom. When a force is applied to the axle, the top section will try to move to the left, whilst the bottom section will try to move to the right. If the gyroscope is not spinning, well then the wheel just flops over. But, when the gyroscope is spinning, well then: Newton's first law of motion. 😅 The two sections of the gyroscope try to continue moving at a constant speed along a straight line, unless an unbalanced force acts upon them. Thinking about this law, when the top point of the gyroscope wheel is acted on by a force applied to the axle then the wheel will begin to move towards the left. So, the different sections of the gyroscope receive forces at one point but rotate to new positions.
Mathematical AnalysisBecause of the influence of gravity there is always a torque about the origin, meaning that the gyroscope falls over when the gyroscope is not spinning. This torque is equal to:
Knowing that the change in angular momentum dL is equal to the total net torque applied, we might now accidentally get into such calculations that would get us stuck. Thinking about the forces acting on a spinning top, the torque produced is of course perpendicular to the angular momentum vector and so the angular momentum of the gyroscope only changes in direction and not magnitude! Making our calculations much simpler, we have:
Let's now think about a gyroscope that we are trying to rotate around an axis perpendicular to the axis of the spinning disk, causing it to precess. Denoting the angle at which the gyroscope precesses as θ the magnitude of the torque applied from the force of gravity is:
Thinking about the change in angular momentum we have:
The angle the top precesses through time dt is:
Thus, the precession angular velocity is:
Knowing that L = Iω we also have:
So, assuming that ωp is much less then ω we have that the precession angular velocity if much less than the angular velocity of the gyroscope disk. This precession adds a small component to the angular momentum along the z-axis. This slight bob up and down is sometimes referred to as nutation.
For even more information you can check out:
Difference to AccelerometersAn accelerometer is a compact device that is designed to measure non-gravitational acceleration. When an objects goes from standstill to any velocity, such a device responds to the vibrations associated with such movement. It uses microscopic crystals, that go under stress when vibrations occur, and is able to create a reading on any acceleration. They are important components that track fitness and other measurements of quantified self-movement.
So, the main difference between them and gyroscopes is simple: one can sense rotation, whereas the other cannot. The accelerometer can gauge the orientation of a stationary item with relation to the Earth's surface, but cannot distinguish the acceleration provided through Earth's gravitational pull. Combining the usability of both of them, someone can detect rotation and acceleration in the same time, having a gyroscope (or a set of them) for rotation readings and a 3-axis accelerometer to detect where the vehicle is heading and how its motion is changing in all three directions.
Mathematical equations used in this article, where made using quicklatex.
Previous articles of the series
- Velocity and acceleration in a rectlinear motion -> velocity, acceleration and averages of those
- Rectlinear motion with constant acceleration and free falling -> const acceleration motion and free fall
- Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
- Rectlinear motion exercises -> examples and tasks in rectlinear motion
- Position, velocity and acceleration vectors in a plane motion -> position, velocity and acceleration in plane motion
- Projectile motion as a plane motion -> missile/bullet motion as a plane motion
- Smooth Circular motion -> smooth circular motion theory
- Plane motion exercises -> examples and tasks in plane motions
Newton's laws and Applications
- Force and Newton's first law -> force, 1st law
- Mass and Newton's second law -> mass, 2nd law
- Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
- Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
- Contact forces and friction -> contact force, friction
- Dynamics of Circular motion -> circular motion dynamics, applications
- Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
- Contact force and friction examples -> exercises in force and friction
- Circular dynamic and vertical circle motion examples -> exercises in circular dynamics
- Advanced Newton law examples -> advanced (more difficult) exercises
Work and Energy
- Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
- Conservative and Non-Conservative Forces -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises
- Potential and Mechanical Energy -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips
- Force and Potential Energy -> Force as Energy Derivative (1-dim) and Gradient (3-dim)
- Potential Energy Diagrams -> Energy Diagram Interpretation, Steps and Example
- Internal Energy and Work -> Internal Energy, Internal Work
Momentum and Impulse
- Conservation of Momentum -> Momentum, Conservation of Momentum
- Elastic and Inelastic Collisions -> Collision, Elastic Collision, Inelastic Collision
- Collision Examples -> Various Elastic and Inelastic Collision Examples
- Impulse -> Impulse with Example
- Motion of the Center of Mass -> Center of Mass, Motion analysis with examples
- Explaining the Physics behind Rocket Propulsion -> Required Background, Rocket Propulsion Analysis
- Angular motion basics -> Angular position, velocity and acceleration
- Rotation with constant angular acceleration -> Constant angular acceleration, Example
- Rotational Kinetic Energy & Moment of Inertia -> Rotational kinetic energy, Moment of Inertia
- Parallel Axis Theorem -> Parallel axis theorem with example
- Torque and Angular Acceleration -> Torque, Relation to Angular Acceleration, Example
- Rotation about a moving axis (Rolling motion) -> Fixed and moving axis rotation
- Work and Power in Angular Motion -> Work, Work-Energy Theorem, Power
- Angular Momentum -> Angular Momentum and its conservation
Final words | Next up
And this is actually it for today's post! Next time we will get into examples around all the stuff that we covered in Angular motion...
Keep on drifting!