I attended Professor Nagamatsu's Acoustic Vibration Technology Seminar, which began on March 13, 2002. I believe it contains very useful information that will be beneficial to everyone, so I would like to share some of the key points.
According to Professor Nagamatsu (Professor Emeritus of Tokyo Institute of Technology and Professor of Mechanical Engineering at Hosei University), the most unforgettable question he has received during his 30 years of lectures on vibration is "Why does it vibrate?" While this is generally explained using mathematical formulas, he wanted students to understand what those formulas actually mean, and gave the following explanation.
Objects have a fundamental property of wanting to remain as they are, and mechanically speaking, there are three ways this can occur.
- Things that are moving want to stay moving, and things that are stationary want to stay stationary; they want to remain in their current state. They dislike changes in state, such as moving or stopping. In mechanical terms, a change in state is acceleration.
Everything dislikes acceleration and resists it. This resistance arises from the property of adapting to one's current state, and is therefore called inertia force.
The strength of an object's ability to adapt to a state is expressed by the magnitude of its resistance to a unit acceleration and is called mass. - Solids, being objects with a defined shape, prefer to remain in their original form and therefore dislike changes in shape. A change in shape at a single point within a solid corresponds to displacement. All solids dislike and resist displacement accompanied by deformation. This resistance force arises as an attempt to return to the original shape and is therefore called the restorative force. The strength of this tendency to return to the original shape is expressed by the magnitude of the resistance force to a unit amount of displacement, and this is called stiffness or rigidity.
- Fluids such as air and oil, or solids in contact with or surrounded by a fluid, prefer to remain in their current position and therefore dislike changes in their location. Mechanically, a change in position translates to velocity. Consequently, objects in a fluid dislike and resist velocity. This is because fluids possess a property called viscosity, and the strength of this property is expressed as a resistance force per unit velocity, known as viscosity. The resistance force due to viscosity is called viscous resistance force.
Of the three properties, mass and stiffness are the sources of vibration.
Conversely, viscosity has the effect of suppressing all vibrations, damping and stopping them. Therefore, viscous resistance is also called viscous damping force. As a simple example, let's consider vibrations in a spring system without damping (a one-degree-of-freedom mechanical model).
- An upward external force is applied to the mass at its initial position (equilibrium point), causing it to pull.
When a spring stretches, a restoring force is generated downwards to resist it, and the mass comes to rest when the external force and the restoring force are in equilibrium. If this external force is suddenly released, the point mass (a point where the mass can be thought of as concentrated at one point) begins to move downwards. An inertial force is generated upwards to resist the downward acceleration that occurs with the change from a static to a dynamic state, and this balances the restoring force. In this way, the moment the external force is removed, an inertial force with the same magnitude and direction as the external force is generated and replaces the external force. - As a point mass moves downward and approaches the equilibrium point, the restoring force decreases, and the inertial force balancing it also decreases, causing the acceleration to decrease. However, the acceleration accumulated up to that point is converted into velocity, and the downward velocity increases. Upon reaching the equilibrium point, the principle of wanting to remain in its original shape is satisfied, so the restoring force becomes zero. The inertial force balancing it is also zero, and no acceleration occurs.
- However, a point mass that satisfies the property of wanting to remain in motion will not stop at the equilibrium point, which is the most favorable position for the spring. Instead, it will pass the equilibrium point from top to bottom while moving in a straight line at a constant velocity at its maximum speed. As a result, the spring will begin to compress, an upward restoring force will be generated to resist this, and a downward inertial force will be generated to balance it, resulting in an upward acceleration.
- This upward acceleration consumes the downward velocity, acting as a brake, so the velocity gradually decreases. However, the velocity accumulated up to that point is converted into displacement, resulting in an increase in the downward displacement. This increases the upward restoring force that resists this displacement, and the downward inertial force that balances it increases, causing an upward acceleration that is always in the opposite direction to the inertial force.
- Eventually, when the velocity becomes zero, both the upward restoring force and the downward inertial force reach their maximum. The point mass then begins to move upward with maximum acceleration and returns to the equilibrium point.
- The same phenomenon then unfolds, just with the initial state reversed. The equilibrium point is passed from bottom to top, and eventually the spring reaches its maximum extension.
- This represents one cycle, and when this repeats, it becomes an oscillation.
- Natural frequency is the frequency at which an object can vibrate freely without external force, maintaining a balance between inertial and restoring forces, and satisfying the law of conservation of energy, as illustrated in the examples above.
It can be said that objects and vibrations are inextricably linked.
It goes without saying that vibration is closely related to human life. Also, as we hear of terms like metal fatigue, objects repeatedly vibrate, and over long periods of time, they become fatigued, developing subtle cracks. If this progresses, it can lead to fatigue failure, making vibration a very important topic.
That concludes our discussion of inertial force, restoring force, and viscous damping force. Next, I will give you an overview of "why resonance occurs with natural frequencies."
The balance of the three forces described above is mathematically expressed as an equation of motion. The solution to this equation of motion can only be a periodic function. This can be intuitively understood from the fact that vibrations are periodic movements. Since vibration waveforms can be expressed as a superposition of harmonic waveforms (waveforms represented as sine functions), if we investigate the system's response to a harmonic excitation force, we can find out the response to an arbitrary external force by superimposing these waveforms.
In a one-degree-of-freedom mechanical model where a weight (mass m) is attached to a spring (stiffness k), under undamped free conditions, it will always undergo periodic motion (oscillation) at a single angular frequency, and no other periodic oscillations will ever occur. m acts to slow down the oscillation, and k acts to speed it up, meaning there is one speed at which these forces are balanced. This frequency is called the undamped natural frequency.
In mathematical terms, natural frequency is a condition for the equations of motion to have a dynamic solution. In physical terms, from the perspective of force equilibrium, it is the frequency at which free vibration is possible while maintaining force equilibrium. Furthermore, in terms of energy, it is the frequency at which free vibration is possible while satisfying the law of conservation of energy.
Viscous damping c is lazy and unwilling to move. It has two effects: it reduces vibrations over time and slows down the vibration frequency.
Let's return to the 1-degree-of-freedom undamped model and consider resonance.
The phenomenon where the amplitude reaches its maximum is called resonance. When a periodic excitation force f is applied to m, the intuitively understandable and well-known aspects are as follows:
- At low frequencies, the weight moves along with the force applied.
- When resonance occurs, the amplitude increases.
- After the resonance, the amplitude gradually decreases, and eventually it stops moving.
It becomes so tiny that it's imperceptible. - From the balance of forces
Excitation force f = Fsinωt (harmonic excitation force with amplitude F and period ω)
Displacement x = Xsin (ωt - Φ)
(X is proportional to t, and the displacement is delayed by Φ phase relative to the force.)
Phase difference Φ = 0° before resonance, -90° at resonance, -180° after resonance
Inertial force fm = mω² * x (proportional to the square of ω and the displacement)
Restoring force fk = -kx (acts at a displacement of -180°)
fm + fk + f = 0 (equilibrium of forces)
At the start of excitation, forced vibration and free vibration occur simultaneously. The former is an exchange of energy between the excitation source and the system, while the latter is an exchange of energy between the spring and mass within the system. The former is forced from the outside; when the excitation source injects energy into the system, the system repels it and pushes back the injected energy. The repeated injection and push-back of energy constitutes forced vibration. The latter, on the other hand, is a spontaneous phenomenon that occurs naturally within the system, and the system does not resist it. When the frequencies of the two vibrations are different, they proceed as completely separate and unrelated phenomena.
However, when the frequencies become the same (ω = natural frequency), the system can no longer distinguish between the two vibrations, and it no longer resists forced vibrations, accepting all the energy injected into the system by the excitation source. The system then tunes to the free vibrations and continues to absorb the periodically injected energy, causing the energy of the free vibrations to increase. This increase in energy manifests as an increase in amplitude, which increases linearly in proportion to time. This is resonance in an undamped system.
When considering the resonance of a 1-degree-of-freedom undamped model,
- At low frequencies, the weight moves along with the force applied.
- When resonance occurs, the displacement amplitude increases.
- After resonance, the displacement amplitude gradually decreases, and eventually it stops moving.
It becomes so tiny that it's imperceptible. - From the balance of forces
Excitation force f = Fsinωt (harmonic excitation force with amplitude F and period ω)
Displacement x = Xsin (ωt - Φ)
(X is proportional to t, and the displacement is delayed by Φ phase relative to the force.)
Phase difference Φ = 0° before resonance, -90° at resonance, -180° after resonance
Inertial force fm = mω² * x (proportional to the square of ω and the displacement)
Restoring force fk = -kx (acts at a displacement of -180°)
fm + fk + f = 0 (equilibrium of forces)
When we examine the equations for the equilibrium of forces along ω, they take the form shown in the following diagrams.
(From "Introduction to Modal Analysis" by Professor Nagamatsu, p. 47, Figure 2-13)
supplement:
Restoring force fk = -kx
Inertial force fm＝ω2mx
Excitation force (constant) = f = Fejωt
Displacement x＝Xej(ωt＋Φ)
Static displacement Xst = F/K
fm always acts in the same direction as the displacement x.
In the range where ω is small, the "spring (fk)" plays the main role, while in the range where ω is large, the "mass (fm)" takes center stage.
① When ω is small
fm can be approximated as 0, and only fk occurs in the system, inversely to f.
Here, as with static deformation, the spring plays the main role, and mass does not participate in the vibration.
fk is the resistance to displacement and always acts in the opposite direction to x, so as a result, f and x move in the same direction and Φ = 0. (They move together with the force.)
(2) As ω approaches the resonant frequency,
Since fk is slightly larger than fm, Φ = 0. However, fm is proportional to ω^2, so it becomes a fairly large value. fk is almost the same magnitude as fm, but since the difference between fk and fm must be f, fk also needs to be large, and as a result the response amplitude x becomes extremely large. (Understanding why x becomes large)
(3) When ω is in a resonant state (ω = natural frequency)
fm and fk have the same magnitude and act in opposite directions. Within the system, fm and fk cancel each other out, and there are no internal forces that oppose f anywhere.
The system loses its means of resisting the excitation force f and moves at the mercy of the excitation force, with the amplitude increasing indefinitely. Because it moves at the will of the excitation force, the direction of movement, i.e., the direction of velocity (phase), coincides with the direction of the excitation force. Since the displacement occurs 90° behind the velocity, it also lags 90° behind the excitation force. (fm = fk)
④At the point where ω has slightly passed the resonance point
Since fm is slightly larger than fk, fk + f cancels out fm, resulting in equilibrium. The response x, which is always opposite to fk, is also opposite to f, resulting in Φ = -180°. Beyond resonance, the system can no longer keep up with the speed of excitation, and the response lags behind. At this point, even though fm and fk are still almost equal in magnitude, the difference between their magnitudes must be a constant value f. Therefore, both fm and fk must be very large (because the difference between fm and fk is small, fm,
(Unless fk is large, fm - fk = f) The response x, which is proportional to both of these, is extremely large. (Before resonance, fm < fk; after resonance, fm > fk)
⑤ When ω exceeds the resonant frequency
fm becomes significantly larger than fk. Therefore, for the same reason as above, Φ = -180°. However, since the difference between fm and fk must be a constant value f, the magnitudes of both fm and fk must gradually decrease, and x decreases. (As ω increases, fm should increase, but the difference between fm and fk increases, so in order to keep that difference f constant, the magnitudes of fm and fk gradually decrease.)
This is an extremely rapid excitation. Despite ω being extremely large, fm = mω^2 * x must not become larger than f due to the balance of forces, so it approximates as x = 0. At this time, the spring hardly deforms, and the mass plays the main role in the vibration. Furthermore, there is no fk in the system, only fm exists, canceling out f. Since x is in the same direction as fm, it is in the opposite direction to f, so Φ = -180°.
A system consisting only of mass, such as a sphere, succinctly illustrates this. To make a sphere move in a harmonic (sinus function) manner, a force must always be applied in the opposite direction to the displacement; otherwise, it will simply fly off in an arbitrary direction. In a system where mass is the main component, the excitation force does work on the mass and generates acceleration, so the excitation force and acceleration are in the same direction. The displacement always lags behind the acceleration by 180°, so Φ = -180°.
Exciting an object with an impulse hammer or vibrator and determining its frequency response with an FFT analyzer is equivalent to finding the solution to the equation of motion and determining its dynamic characteristics. The FFT analyzer is excellent because it displays the solution not as a mathematical formula, but by quantifying and graphing the formula.
I think it would be difficult to understand from a text-only explanation. Professor Nagamatsu's lectures are
We discussed many other things, and I found it very informative.
I recommend that everyone attend a seminar at least once.
(Excerpts from email newsletters issued on March 22, April 19, and May 24, 2002)
