Phase Space
We will now discuss a topic that is a bit abstract. It's called phase space. It is not like the 3D space in which we live. Instead, it is a mathematical space. By "phase" we don't mean anything like the three common phases of matter - solids, liquids, and gases. Instead think of "phase" more like a way to describe the state of a system uniquely.
It is common to view oscillating systems in phase space. As such, we tend to make plots of the phase (or state) of the system over time. A single point in phase space describes uniquely the state of a system, and all possible states of the system exist within the overall phase space. Each point in the phase space is comprised of the position and the velocity of the system, so that points (which move in time) may be thought of as vectors with coordinates I am using to describe a position in phase space, and not to describe the position of an object. These are entirely different things as you'll soon come to understand if it's not clear already. And 'x' here is the x-component of the actual position of the object. It is assumed that the object here is traveling only along the x direction. If it was an object executing motion in more than one dimension we'd need an additional TWO phase space dimensions for each. That's because we'd need both the y position and the velocity in the y direction. But that would be hard to look at (a 4D diagram) due to our limitations of living in a 3D world.
For a mass on a spring oscillating in one dimension, for example, the x-axis of the phase space will be the position of the oscillator (mass), and the y-axis will be the velocity that it's moving at. Together these two pieces of information are the entire description of the current state of the system. This means that if we know the physics that governs the system, then we can determine all future states of the system given the present state. The point we start with will move as time ticks by, and it will trace a path through phase space that can be instructive to look at. The path through phase space is called the phase trajectory. A series of these trajectories that are formed using different initial conditions forms a phase portrait.
Phase Trajectory of a Simple Harmonic Oscillator
We discussed the simple harmonic oscillator earlier in the chapter. Let's take a look at the phase trajectory of that system. The solution for the motion of the oscillator was The velocity is the time derivative of the position, or Also recall that which we'll use in a moment.
Points in phase space will be given by or just the position of the oscillator for the x-component, and the velocity as the y-component of the phase point. If you have seen parametric functions in math class, you can think of the phase trajectory as a parametric function in t. We can very easily make diagrams of parametric functions in GeoGebra as seen below. The blue point toward the bottom of the diagram is executing the actual motion corresponding to the phase trajectory. It is not part of the phase trajectory.
Phase Space Trajectory of Simple Harmonic Motion
SHM Phase Trajectory is Elliptical
The first thing that becomes apparent is that the phase trajectory is elliptical. For a simple harmonic oscillator it is not approximately elliptical, but exactly elliptical in a mathematical sense.
If you don't recall the equation of an ellipse, it is where is called the semi-major axis and is called the semi-minor axis. The major axis is just the long way across and is twice a, and the minor axis is the short way across the ellipse and is twice b.
If you substitute our phase point's coordinates for x and y, you can see what we should expect the elliptical phase trajectory's major and minor axes to be:
The fact that our phase point's coordinates satisfy the equation means the phase trajectory is an ellipse with the major and minor axes shown above. If you suspect the major and minor axes should have names reversed depending on the value of I understand your concern. It looks like the minor axis is longer when but we really can't compare them since they are not in the same units. That point aside, since we're plotting them parametrically, feel free to relabel a and b if you like, so that the longer one is a.
Interpretation of Phase Trajectory
One fact that is instructive about the phase trajectory is that the motion of an undamped oscillator is repetitive. We will see that while undamped oscillators find themselves in the exact same state once per period, that damped oscillators do not. We will see what the phase trajectory of such oscillators looks like soon.
Another fact that is easy to see from the phase trajectory is the relationship between position and velocity. It is clear from the shape of the ellipse that when velocity is maximum that the position is zero (equilibrium) and that when the velocity is zero, the position is maximum, or at an end point of the oscillation. It is important to see and understand this conceptually. Make sure you can look back at the trajectory and imagine the corresponding position and motion of the oscillator at all points around the ellipse. If not, keep trying until you get it... you can do it. I put the blue dot into the diagram to facilitate this understanding.
A more profound statement about a phase space trajectory is that if you extend a line from the origin to the phase point and watch it as the point is animated, that line sweeps out some area every second. The energy of the oscillator is proportional to the area swept out per unit time. The fact that the undamped oscillator sweeps out the same area cycle after cycle means it's not losing energy. As you might imagine, we'll see that the damped oscillator which must be losing energy over time will have a tendency to sweep out areas smaller and smaller as time goes by.
Lastly, the point about which it looks like the phase point is circling or orbiting is called an attractor. With a damped oscillator the phase space trajectory leads eventually to the attractor. While in both the undamped and damped oscillators the attractor is a single point, there are systems that tend to attract to a certain size and shape of phase space orbit called a limit cycle attractor, rather than just spiraling into some attractor point. This may be the case in a system like a child on a swing being pushed by a parent. The swing might start from rest, and eventually reaches a steady swing amplitude.
Damped Oscillators
In the macroscopic world oscillators are always damped. That just means that they contain friction that slows the oscillation over time. The idea of friction is that energy is shared by the system with other parts of the surroundings.
The only instance in nature where we see undamped harmonic motion is in very simple molecular oscillators like diatomic molecules. On the other hand, when we play with actual springs we always measure some damping even when nothing was deliberately done to the system to add friction. If you hang a mass from a spring in lab and pull on it to start it oscillating, it will eventually stop. This is due to two forms of damping - air drag and hysteresis. Air drag you know about. Hysteresis is a consequence of microscopic structural rearrangement of the molecules in a spring as the spring is stretched and compressed. The rearranging of molecules takes away from macroscopic energy of motion and turns that energy into heat. This is not deformation of the spring. To the eye the spring still looks the same since you can't see molecules.
In many cases in engineering practice, such as automotive suspension design, damping is deliberately added to a system. A car rides on a set of springs that make road imperfections less jarring on passengers. If, however, a car did not also have shock absorbers - which serve as damping - you would hit a bump and continue riding down the street bobbing up and down for a long, long time. You will occasionally see cars that desperately need new shocks that actually do this. It is not only annoying but also dangerous because if the wheels start bobbing up and down too rapidly they can lose contact with the ground, which is obviously a bad thing!
Shock absorbers in cars are placed in parallel with the springs, and contain movable pistons that force fluid through small holes as they move due to road imperfections. That design leads to damping that is linearly dependent on the velocity of the piston. The damping, of course, opposes the piston's (and wheel's) up and down motion. We can write this damping force as where is the damping coefficient and the velocity is describing the rate at which the suspension is being compressed or extended. In either case, because of the minus sign, the damping opposes it.
To solve for the motion of a damped oscillator we just write out Newton's second law with both Hooke's law and the damping force along what I'll call the x-axis and get: Next we rewrite the acceleration and velocity as derivatives to get Lastly we solve for acceleration and put all terms on one side of the equation, which is just the standard way of writing the differential equation. This becomes
This differential equation is one that you will study extensively both in math classes on differential equations and in engineering-specific courses dealing with harmonic analysis of structures and circuits. Solving this differential equation for position versus time gives us:
The symbol is the Greek lower case zeta, sounds like our letter 'z', and stands for the damping ratio in the system, or just This seemingly arbitrary combination of constants tells us something important about the level of damping in the system. The expression results from a condition that leads to a damped frequency The condition of zero frequency is no oscillation at all. Let's discuss what tells us.
In the event that the system is over-damped and will only very slowly approach equilibrium and will never oscillate. In a mass/spring system with a shock (damper), the spring has a very hard time moving the shock, and does so very slowly. If the system is under-damped and will oscillate perhaps a little or a lot on its way to equilibrium, depending on how small zeta is. The dividing line is a critically damped system in which Such systems don't oscillate, but instead reach equilibrium as quickly as possible without ever oscillating.
This critical damping condition is the goal of automotive suspension systems - with minor exceptions to account for cargo, passengers and sometimes for front to rear bobbing. Without worrying about those exceptions, a car's suspension system is designed to allow the suspension to compress while absorbing the impact of some road imperfection, and then, without bouncing, returns the car as quickly as possible to the correct ride height. That way the suspension will be ready for the next bump.
Numerical Solution of Damped Oscillator
I don't expect you to know how to arrive at the analytic solution for the damped oscillator for this course since you will learn the required techniques in a future math class. We discussed the solution because we can learn something from looking at the analytic solution.
On the other hand, you know already how to deal with solving sets of differential equations numerically. Let's discuss briefly how to solve the system of the damped oscillator using GeoGebra. As we have discussed before, we need to write our second order differential equation as a set of two first order ones that only contain first derivatives. Recall that the second order equation for the damped oscillator is
In GeoGebra, the variables 'x' and 'y' take on special meaning of the x and y coordinates, so we need to use something else to describe the position. In the past we used 'r' and that's what I'll do here. The two equations we need are
r'(t,r,v)=v, and v'(t,r,v)=-c/m*v-k/m*r. After that we need to decide on initial position and velocity and solve the equations. If we assume initial position is 0.3m (arbitrary choice) and initial velocity is zero, and if we want to solve for the first 20 seconds of motion, we write NSolveODE[{r',v'},0,{0.3,0},20]. As usual you will be asked to create sliders for the undefined constants which you should allow. Keep in mind that you should really constrain the values of 'k' and 'c' to be positive constants when you choose the range of the sliders. The solution along with a corresponding phase diagram for the damped oscillator is below.