There seems to be some confusion in some recent postings between the terms “harmonics” and “resonance”, particularly in regard to lathe stability. Let me offer some Engineering Mechanics 101 light on the subject.
Resonance
To define “Resonance” in the technical sense, I will first define something that for lack of a better name, I will call a “resonator”. A resonator in mechanics is a system which when activated by a driving force, periodically transforms its stored energy back and forth between kinetic energy and one or more other forms of energy storage such as a stretched spring or a lifted weight. (It is directly analogous to a resonator in electronics, where energy is periodically exchanged between a magnetic field and an electrical charge on a capacitor.)
A simple example is a pendulum. Given a nudge, the pendulum will swing back and forth, the amplitude of the swing slowly decreasing. The kinetic energy of the pendulum will be at maximum when the pendulum is at the lowest point and the gravitational potential energy will be at maximum at the outer points of the swing. Each of the two quantities will be zero when the other is at maximum.
A somewhat more complicated example is a weight hanging from a spring. Starting with the system at rest, if the weight is pulled down, energy is introduced into to the system in the form of a stretched spring. At this point, the kinetic energy and the gravitational potential energy are both zero. Now, if the weight is released, it will begin to rise, picking up both kinetic and gravitational energy as it rises. At the top of its rise, all of the system energy will be gravitational potential energy. The kinetic energy will have risen and then fallen to zero again, while the energy in the stretched spring will have decreased smoothly to zero.
Having defined resonator, we are in a position to define resonance. In each of the cases described above. When given an initial start, the system will continue to cycle at a characteristic frequency, called the resonant frequency, until somehow energy is removed from the system. In the case of a pendulum, the energy loss mechanism would be mainly air friction and friction in the pivot bearing. In the second system, additional losses would occur from the internal friction incurred in stretching the spring. In physical mechanics, this energy loss is characterized by a quantity called the “damping factor”. It is a measure of by what fraction is the amplitude of the oscillation reduced in each successive cycle. If the damping factor is zero, there is no reduction; if it is large, the oscillation dies out rapidly.
Now consider what would happen if instead of giving the system one nudge and then letting it run, we were to give it similar nudge every time it returned to the same point in the cycle. If the losses in the system were very small (small damping factor), the amplitude of the oscillation would continue to increase until some external factor interceded (e.g., the lathe fell over). This phenomenon, with or without losses, is called “resonance” and is encountered often in woodturning. The work piece, the supporting spindles and other parts of the lathe often form one or more resonant systems, typically of different resonant frequencies. If the rotating work piece is unbalanced, it represents a source of periodic excitation for these resonators.
The critical issues in each case are: What is the resonant frequency, what is the damping factor, and how firmly is the resonator connected to the source of vibration.
The damping factor, In addition to determining how strongly the resonating system will respond to periodic excitation at its resonant frequency, determines how sensitive the system is to small differences between the frequency of the excitation and the resonant frequency. If the damping factor is very small, then the excitation frequency must be very close to the resonant frequency to cause much build-up of vibration. If it is large, the exact frequency of excitation will be much less important, but the build-up at the resonant frequency will be much smaller. This explains the situation often encountered in woodturning where as the system is slowly brought up to speed strong vibration occurs and then disappears as the speed increases.
Harmonics
Before defining “harmonics” I will define “fundamental” or “fundamental frequency”. In woodturning, this usually is the rotational frequency of the lathe since usually the source of the vibration is an unbalanced or uneven work piece. A “harmonic frequency” is any integer multiple of the fundamental frequency. Under certain conditions some of the vibration energy at the fundamental frequency is converted into vibration components at one or more harmonic frequencies. (More about this later.)
Now consider a situation where an unbalanced and perhaps non-round work piece is spinning between centers on the lathe. As the lathe spins, the centrifugal force of the unbalanced component exerts a continuous outward force on the mountings. In the absence of any chatter due to looseness, and assuming that the metal in the lathe is not stressed beyond its yield point, the magnitude of the force acting in any particular direction, say down, will form a perfect sine wave when plotted versus time. That is to say, there are no harmonic components, only the fundamental frequency component. Therefore, in the recurring issue of whether to bolt down the lathe, harmonics are not at factor. Resonances at the fundamental frequency are the primary concern.
When do harmonics become a factor? As far as I can see, the main cause of harmonic vibration will be turning a non-round work piece, either to true it up or as a step in a multi-axis turning project. In these cases, the cutting tool will not exert a constant force on the work piece all the way around. If you plotted the force in the upward direction versus time you would obtain a distorted sine wave. A distorted sine wave can be resolved into a fundamental component and one or more harmonics. Almost always, the fundamental component would remain far larger than any of the harmonics. Furthermore, the harmonics would only be a factor in stability of the lathe if very aggressive cutting were attempted and one of the harmonics excited a resonance with a very low damping factor. Even then, the problem could be corrected by making a small change in lathe speed.
Resonance
To define “Resonance” in the technical sense, I will first define something that for lack of a better name, I will call a “resonator”. A resonator in mechanics is a system which when activated by a driving force, periodically transforms its stored energy back and forth between kinetic energy and one or more other forms of energy storage such as a stretched spring or a lifted weight. (It is directly analogous to a resonator in electronics, where energy is periodically exchanged between a magnetic field and an electrical charge on a capacitor.)
A simple example is a pendulum. Given a nudge, the pendulum will swing back and forth, the amplitude of the swing slowly decreasing. The kinetic energy of the pendulum will be at maximum when the pendulum is at the lowest point and the gravitational potential energy will be at maximum at the outer points of the swing. Each of the two quantities will be zero when the other is at maximum.
A somewhat more complicated example is a weight hanging from a spring. Starting with the system at rest, if the weight is pulled down, energy is introduced into to the system in the form of a stretched spring. At this point, the kinetic energy and the gravitational potential energy are both zero. Now, if the weight is released, it will begin to rise, picking up both kinetic and gravitational energy as it rises. At the top of its rise, all of the system energy will be gravitational potential energy. The kinetic energy will have risen and then fallen to zero again, while the energy in the stretched spring will have decreased smoothly to zero.
Having defined resonator, we are in a position to define resonance. In each of the cases described above. When given an initial start, the system will continue to cycle at a characteristic frequency, called the resonant frequency, until somehow energy is removed from the system. In the case of a pendulum, the energy loss mechanism would be mainly air friction and friction in the pivot bearing. In the second system, additional losses would occur from the internal friction incurred in stretching the spring. In physical mechanics, this energy loss is characterized by a quantity called the “damping factor”. It is a measure of by what fraction is the amplitude of the oscillation reduced in each successive cycle. If the damping factor is zero, there is no reduction; if it is large, the oscillation dies out rapidly.
Now consider what would happen if instead of giving the system one nudge and then letting it run, we were to give it similar nudge every time it returned to the same point in the cycle. If the losses in the system were very small (small damping factor), the amplitude of the oscillation would continue to increase until some external factor interceded (e.g., the lathe fell over). This phenomenon, with or without losses, is called “resonance” and is encountered often in woodturning. The work piece, the supporting spindles and other parts of the lathe often form one or more resonant systems, typically of different resonant frequencies. If the rotating work piece is unbalanced, it represents a source of periodic excitation for these resonators.
The critical issues in each case are: What is the resonant frequency, what is the damping factor, and how firmly is the resonator connected to the source of vibration.
The damping factor, In addition to determining how strongly the resonating system will respond to periodic excitation at its resonant frequency, determines how sensitive the system is to small differences between the frequency of the excitation and the resonant frequency. If the damping factor is very small, then the excitation frequency must be very close to the resonant frequency to cause much build-up of vibration. If it is large, the exact frequency of excitation will be much less important, but the build-up at the resonant frequency will be much smaller. This explains the situation often encountered in woodturning where as the system is slowly brought up to speed strong vibration occurs and then disappears as the speed increases.
Harmonics
Before defining “harmonics” I will define “fundamental” or “fundamental frequency”. In woodturning, this usually is the rotational frequency of the lathe since usually the source of the vibration is an unbalanced or uneven work piece. A “harmonic frequency” is any integer multiple of the fundamental frequency. Under certain conditions some of the vibration energy at the fundamental frequency is converted into vibration components at one or more harmonic frequencies. (More about this later.)
Now consider a situation where an unbalanced and perhaps non-round work piece is spinning between centers on the lathe. As the lathe spins, the centrifugal force of the unbalanced component exerts a continuous outward force on the mountings. In the absence of any chatter due to looseness, and assuming that the metal in the lathe is not stressed beyond its yield point, the magnitude of the force acting in any particular direction, say down, will form a perfect sine wave when plotted versus time. That is to say, there are no harmonic components, only the fundamental frequency component. Therefore, in the recurring issue of whether to bolt down the lathe, harmonics are not at factor. Resonances at the fundamental frequency are the primary concern.
When do harmonics become a factor? As far as I can see, the main cause of harmonic vibration will be turning a non-round work piece, either to true it up or as a step in a multi-axis turning project. In these cases, the cutting tool will not exert a constant force on the work piece all the way around. If you plotted the force in the upward direction versus time you would obtain a distorted sine wave. A distorted sine wave can be resolved into a fundamental component and one or more harmonics. Almost always, the fundamental component would remain far larger than any of the harmonics. Furthermore, the harmonics would only be a factor in stability of the lathe if very aggressive cutting were attempted and one of the harmonics excited a resonance with a very low damping factor. Even then, the problem could be corrected by making a small change in lathe speed.