Dynamic Absorber
This article deals with the topic of Mechanical Systems with 2 Degrees of Freedom and the Dynamic Absorber. Understanding this subject is essential for technicians who are dedicated to Vibration Analysis.
1. Introduction
Mechanical systems with two degrees of freedom constitute a fundamental extension of systems with one degree of freedom and allow us to understand dynamic phenomena that cannot be adequately described considering only an independent mass and movement.. The introduction of a second mass, connected to the first through elastic and/or dissipative elements, causes coupling phenomena, own modes of vibration, anti-resonances and energy transfer between the different components of the system.
One of the most important applications of this concept is the dynamic vibration absorber, or tuned vibration absorber. This device uses an auxiliary mass, normally connected to the main structure through a spring and, in some configurations, of a shock absorber, to significantly reduce the vibration of a system subjected to periodic excitation.
The principle is particularly important in structures, industrial equipment and systems subject to periodic forces. Instead of trying to indefinitely increase the stiffness or damping of the main structure, the dynamic absorber deliberately introduces a second dynamic into the system, in order to control the vibratory response.
Nomenclature
The following table summarizes the symbols used throughout this article.
| Symbol | Meaning |
| m₁ | Mass of structure/main machine |
| m₂ | Dynamic absorber auxiliary mass |
| k₁, c₁ | Stiffness and damping between the fixed base and m₁ |
| k₂, c₂ | Stiffness and damping between m₁ and m₂ (absorsor) |
| x₁(t), x₂(t) | Instantaneous displacements of m₁ and m₂ |
| F(t), F₀ | Excitation force applied to m₁, and respective amplitude |
| ω | Excitation angular frequency |
| ωₙ | Natural frequency of the system 1 GDL, √(k₁/m₁) |
| ωₐ | Natural frequency of the absorber, √(k₂/m₂) |
| ω₁, ω₂ | natural frequencies (own modes) of the system 2 GDL, com ω₁ < ω₂ |
| m | Mass ratio, m₂/m₁ |
| g₂ | Absorber damping ratio |
| r | Frequency ratio, ω/ωₙ ou ω/ωₐ, depending on the context |
2. One degree of freedom mechanical system
Before analyzing the two degrees of freedom system, consider a mass m₁, connected to a spring of stiffness k₁ and a shock absorber c₁, subjected to a harmonic force:
F(t) = F₀ sin(ωt)
The equation of motion is:
m₁ ẍ₁ + c₁ ẋ₁ + k₁ x₁ = F₀ sin(ωt)
In the absence of damping, the natural frequency of the system is:
ωₙ = √(k₁/m₁)
When the excitation frequency approaches ωₙ, the phenomenon of resonance, characterized by a significant increase in vibration amplitude.
This behavior constitutes the problem that the dynamic absorber seeks to solve.
Figure 1 – Mechanical system of 1 degree of freedom and respective frequency response
3. Introduction of the second degree of freedom
Now consider a second mass m₂, connected to the main mass m₁ through a spring k₂ and a shock absorber c₂.
The system now has two independent displacements:
x₁(t) e x₂(t)
corresponding to the movements of the masses m₁ and m₂.
The basic configuration can be represented by:
Base – (k₁,c₁) – m₁ – (k₂,c₂) – m₂
The mass m₁ represents the main structure or machine, while m₂ constitutes the auxiliary mass of the absorber.
The differential equations of motion are:
m₁ẍ₁ + c₁ẋ₁ + k₁x₁ + c₂(ẋ₁−ẋ₂) + k₂(x₁−x₂) = F₀ sin(ωt)
m₂ẍ₂ + c₂(ẋ₂−ẋ₁) + k₂(x₂−x₁) = 0
These two equations show an essential characteristic: the movements of the two masses are coupled.
The movement of the main mass influences the auxiliary mass and, simultaneously, the auxiliary mass exerts a dynamic force on the main mass.
Figure 2 – Mechanical system of 2 degrees of freedom and respective frequency response
4. Mass matrix, cushioning and stiffness
The matrix formulation allows representing the system in a compact way:
M ẍ + C ẋ + K x = F(t)
com:
x = [x₁ x₂]ᵀ
M = [[m₁, 0], [0, m₂]]
C = [[c₁+c₂, −c₂], [−c₂, c₂]]
K = [[k₁+k₂, −k₂], [−k₂, k₂]]
The matrix representation is particularly important because it forms the basis of the modal analysis of mechanical systems with multiple degrees of freedom..
5. Own vibration modes
A system with two degrees of freedom has, in general, two natural frequencies and two vibration modes.
In the absence of damping and external forces:
M ẍ + Kx = 0
Assuming a harmonic solution:
x = φe^(jωt)
you get:
(K − ω²M)φ = 0
For a non-trivial solution to exist:
the(K − ω²M) = 0
This equation gives two natural frequencies:
ω₁ and ω₂
com:
ω₁ < ω₂
Each frequency is associated with its own mode and a specific form of relative movement between the two masses..
Each eigenmode corresponds to an eigenvector (fixed ratio between amplitudes x₁/x₂) and at its own frequency (eigenvalue) of the system. Any free system vibration, no matter how complex it may seem, can be written as a linear combination of these two modes.
| Way 1 — symmetrical (in phase), ω₁ (lowest frequency) The arrows above m₁ and m₂ point in the same direction and have the same length: the two masses oscillate in unison, with the same amplitude and the same phase. As the central spring k₂ practically does not deform (the distance between m₁ and m₂ remains almost constant), this mode has the lowest natural frequency. | Way 2 — antisymmetric (phase opposition), ω₂ (higher frequency) Arrows point in opposite directions: when m₁ moves to one side, m₂ moves to the opposite side. Here the coupling spring k₂ is the one that deforms the most, what “hardens” the e mode results in a higher natural frequency. |
Figure 3 — Modes specific to a system with 2 GDL
6. Dynamic coupling
The fundamental aspect of a two-degree-of-freedom system is dynamic coupling..
When the main mass starts to vibrate, the auxiliary mass does not necessarily remain stationary. On the contrary, develops its own movement determined by the excitation frequency and the characteristics of m₂, k₂ e c₂.
Vibratory energy can, therefore, be transferred from the main mass to the auxiliary mass.
This phenomenon constitutes the physical basis of the dynamic absorber.
Under certain conditions, the auxiliary mass can develop a dynamic force on the main mass that opposes the excitation force. The result can be a very significant reduction in the vibration amplitude of the main structure.
Figure 4 – Dynamic coupling in a system with 2GL
The coupled system
The main mass m₁, supported on k₁ e c₁ on a fixed support, subject to external excitation force F(t). The auxiliary mass is mounted on m₁ m₂, linked by k₂ e c₂ — the three parameters that, determine “the movement itself” that m₂ develops when m₁ starts to vibrate. Two annotations directly translate the key ideas of the text: One purple curved arrow, dashed, leaves the interior of m₁ and enters m₂, labeled “vibratory energy transfer” — represents the energy that passes from the main mass to the auxiliary mass through the k₂/c₂ coupling. To set F₂ (red), not a lot of m₁, points in the opposite direction to F(t) (orange) — represents the dynamic force that m₂ develops on m₁, “which opposes the excitation force”, as the text says.
The practical consequence
A graph of amplitude X₁ as a function of frequency ω compares two situations: The curve dashed gray (“without absorber”) shows a single high resonance peak — the behavior of the isolated main mass, nor m₂. The curve solid green (“with absorber”) shows the same frequency range, but with two lower peaks and a deep valley exactly at ω = ω₂ (the absorber tuning frequency) — the classic signature of dynamic coupling at work: in that area, The amplitude of m₁ fell drastically.
7. The dynamic absorber
A conventional dynamic absorber consists of:
- an auxiliary mass m₂;
- a k₂ spring;
- possibly a shock absorber c₂.
Its own natural frequency is approximately:
ωₐ = √(k₂/m₂)
The operating principle consists of tuning this frequency with the excitation frequency to be attenuated..
Like this:
ωₐ ≈ ωexc
When this condition is satisfied, the auxiliary mass enters a particularly effective movement condition to generate a dynamic opposing force on the main mass.
| Dynamic absorber principle | Photograph of a dynamic absorber applied to a pipe to reduce vibrations |
Figure 5 – The dynamic absorber
8. How vibration reduction occurs?
Consider a structure subjected to a periodic force:
F(t) = F₀ sin(ωt)
Without absorber, the response of the structure can present a large amplitude when:
ω ≈ ωₙ
When installing a tuned absorber, a second natural frequency is introduced into the system.
Instead of having a single dominant resonance region, the system now presents two natural frequencies close to the absorber tuning frequency.
Between these two frequencies an anti-resonance arises, in which the movement of the main mass can be theoretically reduced to zero in the ideal undamped and perfectly tuned case.
This is one of the most important results of the dynamic absorber theory.
9. Anti-resonance
Anti-resonance is particularly interesting because it corresponds to a frequency at which the main mass response presents a very pronounced minimum..
Physically, the force transmitted by the absorber has a component that opposes the excitation force.
The phenomenon can be interpreted through dynamic balance:
Fexcitement + Fabsorsor ≈ 0
e, consequently,
x₁ ≈ 0
Although the main mass has a very small amplitude, the auxiliary mass may have a relatively high amplitude.
This means that the absorber does not eliminate vibrational energy. It modifies the path through which this energy is stored and transferred in the system.
Figure 6 – The placement of the dynamic absorber generates anti-resonance
10. Relationship between resonance and anti-resonance
The frequency response of a system with an absorber presents, typically, the following sequence:
resonance → anti-resonance → resonance
The system that originally had a dominant resonance now presents two resonances separated by a region of very reduced response..
This characteristic is often called resonance splitting. (resonance splitting).
The objective of absorber design is to properly position these frequencies and control the maximum amplitude of the response..
| Only the main mass m₁, supported at k₁ and c₁ on the fixed base. As the system is in resonance and there is nothing to counteract the excitation, m₁ oscillates like very large amplitude — visually the block goes up and down almost until it touches arrow F(t) and almost to the floor. | The same mass m₁, but now with the auxiliary mass m₂ mounted on top, connected by k₂ and c₂. Under the same F(t), m₁ barely moves (almost imperceptible oscillation), while m₂ oscillates with amplitude grande, in phase opposition — when F(t) push down, m₂sobe, and vice versa. It's this oscillation of m₂, transmitted back to m₁ via spring k₂, which generates the force that cancels F(t) and keeps the main structure almost still. |
Figure 7 – The dynamic absorber in operation
11. Undamped dynamic absorber
In the ideal model, considered:
c₂ = 0
The response can exhibit extremely deep anti-resonance when the absorber is perfectly tuned..
Yet, This system has an important limitation: the effectiveness is very dependent on the excitation frequency.
If the machine frequency moves away from the tuning frequency:
ωexc ≠ ωₐ
absorber efficiency decreases.
Consequently, an undamped absorber is particularly suitable for applications in which the excitation frequency is practically constant.
12. Damped dynamic absorber
In practice, a shock absorber can be introduced:
c₂ > 0
Damping alters the system's response and reduces the absorber's sensitivity to variations in the excitation frequency..
There is, Yet, a project commitment.
- Damping that is too low produces a very narrow high efficiency region.
- Excessive damping can reduce the depth of antiresonance.
The project must, therefore, look for a damping value that allows you to obtain an adequate reduction of the response in a given frequency band.
Figure 8 – The influence of dynamic absorber damping
12.1 Optimal tuning and damping (fixed point method)
For a damped absorber (c₂ > 0) mounted on an undamped main frame (c₁ = 0), the classical theory of dynamic absorbers — developed by J. P. Den Hartog — shows that there is an optimal combination between the tuning ratio and the absorber damping ratio that minimizes the maximum amplitude of the main mass, for a given mass ratio μ = m₂/m₁.
The optimal tuning ratio is:
f_opt = ωₐ/ωₙ = 1 / (1 + m)
that is, the absorber should be slightly detuned to a frequency just below ωₙ. The corresponding optimal damping ratio is:
g₂,opt = √[ 3m / (8(1+m)³) ]
and the maximum dimensionless amplitude that the main mass still reaches, on the two remaining peaks, is:
(X₁k₁/F₀)_max = √(1 + 2/m)
These expressions define the so-called “fixed points” frequency response: two points on the amplitude curve that, by property of the transfer function, always have the same amplitude, whatever the value of ζ₂. The optimal damping is precisely that which makes these two fixed points coincide with the two local maxima of the response curve — the condition illustrated in panel C (“great cushioning”) from Figure 8.
Note that these formulas assume c₁ = 0; in systems with significant structural damping in the main mass, the optimal values must be obtained by numerical optimization.
13. Mass ratio
An important parameter in the design of an absorber is the mass ratio:
μ = m₂/m₁
The greater the auxiliary mass relative to the main mass, The greater the capacity of the absorber to modify the dynamic response of the system tends to be.
Yet, increasing m₂ implies mass penalties, space, support structure and costs.
Therefore, In practical engineering, the aim is to obtain the greatest possible reduction using an auxiliary mass compatible with the physical restrictions of the application..
Figure 9 – The influence on the vibration amplitude of the mass ratio (µ=m2/m1) of the dynamic absorber
14. Absorber tuning
The basic tuning condition is:
ωₐ = √(k₂/m₂)
If the desired excitation frequency is known:
ωexc = 2π fexc
Then the stiffness can be selected using:
k₂ = m₂ωexc²
This equation shows that, auxiliary mass chosen, the required stiffness can be calculated directly.
15. Energy interpretation
A particularly useful way of understanding the dynamic absorber is to analyze the energy.
- The excitation force introduces energy into the system.
- Without absorber, a significant part of this energy can be stored and dissipated through the dynamics of the main structure, leading to large vibration amplitudes.
- With the absorber installed, part of the energy is transferred to the auxiliary mass.
- The system switches to distributing energy between m1 and m₂.
- The auxiliary mass may present large movements, while the main mass remains with reduced amplitude.
Like this, the absorber functions as a dynamic reservoir of vibratory energy.
16. Dynamic Absorber Limitations
Despite its high effectiveness, a conventional dynamic absorber has some limitations.
- Frequency dependence: an absorber tuned to a certain frequency may lose effectiveness when the machine speed varies significantly.
- Large auxiliary mass amplitudes: in the tuning region, the auxiliary mass may present amplitudes greater than those of the main structure. It is necessary to ensure that there is sufficient physical space for your movement.
- Changing natural frequencies: installing the absorber modifies the dynamic characteristics of the original system.
- Temperature sensitivity and aging: in systems that use elastomeric elements or other components whose stiffness varies with temperature, the tuning frequency may shift.
- Need for maintenance: mechanical elements, springs, shock absorbers and connection components may wear out.
17. Long band absorbers
When the machine operates over a significant range of speeds, a fixed frequency absorber may not be enough.
In these situations, you can use:
- damped absorbers;
- multiple absorbers;
- tuned mass absorbers;
- pendulum absorbers;
- adaptive absorbers;
- semiative absorbers;
- variable frequency absorbers.
The use of several absorbers allows the control action to be distributed over different frequencies.
18. Adaptive absorbers
Our modern systems, the absorber can have a variable tuning frequency.
If the machine's excitation frequency varies from f₁ to f₂, the control system may modify the effective stiffness or other properties of the absorber to maintain:
fₐ ≈ fexc
This approach is particularly interesting on variable speed machines..
Can be used in systems in which the dominant excitation frequency follows the rotation speed.
19. Conceptual example
Consider a pipe with: m₁ = 1000 kg
and an excitation frequency: fexc = 20 Hz
It is intended to use an absorber with: m₂ = 50 kg
The excitation angular frequency is:
ωexc = 2p(20) ≈ 125,66 rad/s
For optimal tuning:
k₂ = m₂ωexc²
resulting in approximately:
k₂ = 50(125,66)² ≈ 790 kN/m
The absorber has, like this, a natural frequency close to the 20 Hz.
This example highlights an important feature: a relatively small auxiliary mass can significantly modify the dynamic response of the main structure when it is correctly tuned.
19.1 Example extension: optimal absorber damping
Continuing the previous example, the mass ratio is μ = m₂/m₁ = 50/1000 = 0,05 (5%). Applying the formulas in the section 12.1:
f_opt = 1/(1+0,05) ≈ 0,952
g₂,opt = √[3(0,05) / (8(1,05)³)] ≈ 0,127 (12,7%)
Taking ωₐ ≈ ωexc = 125,66 rad/s (the nominal tuning calculated in section 19), the corresponding optimal damping of the absorber is approximately:
c₂,eight = 2ζ₂,opt m₂ωₐ ≈ 2(0,127)(50)(125,66) ≈ 1,60 kN·s/m
and the maximum dimensionless amplitude that the main mass would still present at the two peaks of the optimal response would be:
(X₁k₁/F₀)_max = √(1 + 2/0,05) = √41 ≈ 6,40
This result is revealing: the undamped absorber (section 11) can cancel x₁ exactly at the tuning frequency, but this anti-resonance is “sharp” and fragile to any deviation from ωexc. The optimally damped absorber exchanges this perfect zero for much more robust behavior — the maximum amplitude is limited to about 6,4 times the static deflection over the entire relevant frequency range, and not just at a single point.
20. Dynamic absorber versus increased stiffness
An important question in the project is whether it is preferable to increase the rigidity of the structure or install an absorber.
Increasing stiffness changes the natural frequency:
ωₙ = √(k/m)
and can shift the resonance outside the operating zone.
An absorbent, on the other hand, seeks to control the response without necessarily drastically altering the main structure.
Like this:
- Increased stiffness: shift the resonance.
- Dynamic Absorber: control the response in the region of interest.
In so many cases, the second solution may be simpler and more economically favorable.
21. Dynamic absorber versus increased damping
Another strategy is to increase the damping of the structure.
Damping reduces the resonance amplitude, but may require materials, significant devices or structural changes.
The dynamic absorber uses another mechanism:
energy transfer + dynamic interference
He can, therefore, produce a very high reduction in vibration at a specific frequency.
The choice between structural damping and dynamic absorption depends on the required bandwidth, operating frequency, available space, permissible mass and stability of operating conditions.
A third alternative, not developed in detail in this article, is vibration isolation: instead of modifying the dynamic response of the structure itself, a resilient element is interposed (spring and/or shock absorber) between the machine and the foundation, reducing the force transmitted to the outside. Unlike the dynamic absorber — which acts on the vibration amplitude of the main mass itself —, insulation acts on the transmissibility of force to the support structure. The two strategies are complementary, and not competitors: in many practical applications they are used together.
22. Mechanical Systems with 2 Degrees of Freedom and the Dynamic Absorber – Conclusion
Mechanical systems with two degrees of freedom constitute one of the fundamental models of structural dynamics and vibration analysis. The introduction of a second mass allows us to understand phenomena such as modal coupling, energy transfer, division of resonances and anti-resonances.
The dynamic absorber constitutes a direct application of these principles. By introducing an auxiliary mass, a spring and, when necessary, a shock absorber, it is possible to deliberately modify the dynamic response of a machine or structure.
The central idea is not simply to “eliminate” the excitation force, but create additional dynamics capable of generating a response that opposes the vibration of the main structure.
Under conditions of adequate tuning, the amplitude of the main mass can be drastically reduced, even when the excitation force remains practically unchanged.
Like this, the study of systems with two degrees of freedom is not just an academic step in the theory of vibrations: constitutes a practical tool for designing, diagnose and control real mechanical systems, establishing the link between modal analysis and modern passive and adaptive vibration control techniques.
Formula summary
| Greatness | Formula |
| Natural frequency (1 GDL) | ωₙ = √(k₁/m₁) |
| Natural frequency of the absorber | ωₐ = √(k₂/m₂) |
| Tuning condition | ωₐ ≈ ωexc ⇒ k₂ = m₂ωexc² |
| Mass ratio | μ = m₂/m₁ |
| Optimal tuning ratio | f_opt = ωₐ/ωₙ = 1/(1+m) |
| Optimal absorber damping | g₂,opt = √[3m / (8(1+m)³)] |
| maximum amplitude (excellent) | (X₁k₁/F₀)_max = √(1+2/m) |
| Own frequencies (2 GDL) | the(K − ω²M) = 0 → ω₁, ω₂ |
References
The Hartog, J. P. (1956). Mechanical Vibrations (4.ª ed.). McGraw-Hill.
Rao, S. S. (2011). Mechanical Vibrations (5.ª ed.). Prentice Hall.
Inman, D. J. (2014). Engineering Vibration (4.ª ed.). Pearson.
Kelly, S. G. (2012). Mechanical Vibrations: Theory and Applications. Cengage Learning.
