maximum amplitude expected for the system response and the second the overall mean value of the response. A similar statement has been used in the control techniqueH∞ ̸H2 [10]. Therefore, in this work the performance parameter Pp is defined as the ratioðL∞normÞ ̸ðL2normÞwhere the best control law performance will provide the lowest values for both norms. In this work the performance parameter Pp is defined as:
Pp=maximum receptance amplitude
receptance amplitude norm ð5Þ
Assuming a column vectorA=½ε ε ε⋯A1⋯ε ε ε1 ×N, whereεis a real constant closest to zero andA1a real positive constant, which represents the receptance with one peak only on the frequency spectrum A=A1δð Þf1 , where δð Þf1 is the Dirac function on the frequencyf1, the parameterPp can be written as:
Pp= lim
ε→0Pp= lim
ε→0
A1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N−1 ð Þε2+A21
p =A1
A1
= 1 ð6Þ
The other extreme is a constant amplitude receptance, which are represented by the column vectorA=½ε ε ε⋯ε ε ε1 ×N for which the parameter Pp is written as:
Pp= lim
ε→0Pp= lim
ε→0
ffiffiffiffiffiffiffiffiε Nε2
p = 1
ffiffiffiffiN
p ð7Þ
Equations (6) and (7) give the maximum and minimum values of the perfor- mance parameter as defined in Eq. (5).
3 Numerical Results for a Single-Degree-of-Freedom System
To obtain a reasonable comparison among the obtained results and those of the literature, mass and frequency ratios are the same described by Harris and Piersol [1]. The effectiveness of the proposed TAMD, where the adaptability is obtained by controlling the normal force on the smart friction damper, is evaluated based on mass and frequency ratios variations for each strategy. In this work, the mass ratios studied are ma ̸m1= 0.1, 0.2, 0.3, 0.4, 0.5½ and for the frequency are ωa ̸ω1= 0.1, 0.5, 1½ . These ratios with thefive control strategies are compared to
Tunable Auxiliary Mass Damper with Friction Joint… 53
the correspondent optimum viscous damping AMD, to a well-tuned DVA and to the 1 DOF uncontrolled vibration response.
The numerical results presented in this section have been obtained using the following physical properties values, which represent the parameters from a designed modification of the experimental workbench used on previously works [7,8],m1= 4.14 kg,c1= 3.93 N s ̸m and k1= 70.3 kN ̸m. The physical properties for the secondary systemðma,caandkaÞare deduced from the mass and frequency ratios aforementioned. The contact parameters are the tangential stiffness kT= 1.16 MN ̸m and the friction coefficient μ= 0.33. The auxiliary damping ca
used in the simulations isfixed and equal to 1.0 N s ̸m, a small damping to ensure that most of the damping promoted by the TAMD becomes from the friction damper. And the optimum viscous damping for the AMD is copt= 18.1 N s ̸m obtained following Den Hartog’s procedure as Harris and Piersol [1].
All receptances have been obtained using a harmonic force excitation with an amplitude of 10 N. The excitation force frequency has been swept from 5 Hz up to 100 Hz, in steps of 0.1 Hz. For strategies which use constant normal force value, N1= 20 N has been applied. Again, these values come from previous tests, which also had determined that the ratioN1 ̸Fexc= 2 is the best to be used for constant normal force [7,8].
The responses for all combinations of control strategies, mass and frequency ratios are placed in Fig.2. There are 15 combinations for each strategy, only the best of each is indicated as afilled circle. Additionally, for comparison reasons, the performance parameter for 1 DOF uncontrolled vibration system, the vibratory system coupled to AMD with optimum viscous damping and the system coupled to a well-tuned DVA are also shown in Fig.2as afilled circle.
In Fig.2 the color bar indicates the value of Pp and the arrows indicate the location of the best result for each control strategy, also for the location of the DVA,
Fig. 2 Maximum amplitude and receptance norm chart
54 H. T. Coelho et al.
optimum viscous damping AMD and the 1DOF uncontrolled vibration. The last one refers to the system composed only bym1,k1 andc1.
It is possible to observe that the performance parameter is efficient in locate the responses with the lowest norms and smaller maximum amplitudes, i.e., that are located in the lower left corner in thefigure. At the enlargement is noted that the best responses from strategies S3 and S4 are better than the other approaches and much better than the DVA and the 1 DOF uncontrolled vibration response. It is also possible to see that the optimum viscous damping AMD is nowadays the most efficient passive approach.
In Fig.3 the responses for the best of each control strategy are presented and compared with the best optimum viscous damping AMD result, the well-tuned DVA and the 1 DOF uncontrolled vibration response.
The best receptances are for S3½ and S4½ , they present, at the same time, the lowest peaks and the lowest norm. Based on these receptance results, it can be concluded that the proposed semi-active suspension almost entirely suppressed the resonance peaks. Should be also observed that the receptances for S3½ and S4½ had almost their entirely values at the same levels or lower than the static response, which is a great advantage for the proposed TAMD.
These results demonstrate that the TAMD can be effective in a wide frequency range, since all strategies promote an improvement in the attenuation of the reso- nant peaks as well as in the L2norm value. They are also promoting a better response than that obtained by optimal viscous damped AMD. Strategies [S1], [S2]
and [S5] also present good receptances, almost as good as the optimum damping AMD, their highest values for Pp are due to the maximum amplitude of the receptance, which are a little higher than [S3] and [S4] maximum amplitude. It should be noticed that all strategies give better results than the well-tuned DVA.
Table2 summarizes the mass and frequency ratio combinations on which the lowest values forPp for each strategy were obtained. This table also presents the
Fig. 3 Receptance results for the best of each strategy
Tunable Auxiliary Mass Damper with Friction Joint… 55
ratios used to tune the DVA, those that presented the best optimum viscous damping AMD response and their respective performance parameter value. The 1 DOF uncontrolled vibration also had its receptance quantified by using the per- formance parameterPp.
As observed, strategies [S3] and [S4] presented excellent results, however with mass ratioma ̸m1= 0.5, which is too much mass to be added. Strategies [S2] and [S5] have their best performance for mass ratio ma ̸m1= 0.1 and frequency ratio ωa ̸ω1= 1. So, for this ratio combination thePp values for the optimum viscous damping AMD, [S1], [S3] and [S4] were 0.15, 0.17, 0.11 and 0.12 respectively, which are excellent values when compared to the mass reduction that was possible to achieved. Note that strategies [S3] and [S4] performance parameter values remain smaller than for [S2] and [S5] presented on Table2.
Here it becomes clear that there is a compromise solution between the mass ratio and the performance parameter. Changing the mass ratio for 10% and recalculating again the receptances it is possible to verify that strategies [S3] and [S4] remain the better ones, as shown in Fig.4.
The symbol *ð Þ in the legend indicates the optimum viscous damping AMD receptance previously obtained with a mass ratioma ̸m1= 0.5. Besides the good aspect of the receptances, presented in the Fig.4, they are little worse than those presented in Fig.3. Concerning the compromise solution between the mass ratio and the performance of the proposed TAMD, the worsening of the receptances is justified by the great reduction in the mass to be added in the system. Besides, strategies [S3] and [S4] receptance remains better than optimum viscous damping AMD in almost one order of magnitude attenuation for the receptance maximum value on the chosen ratios.
The natural frequency of the TAMD is 20.73 Hz, which is close to the resonant frequency of the original vibratory system. The TAMD acts similarly to a well-tuned DVA; however, introducing two new resonant peaks less significative than as is expected in the application of DVAs. The discontinuity in the receptance shows that the selected physical properties for the auxiliary system makes more difficult for the TAMD to work against the resonance frequency and maintain lower amplitude levels.
Table 2 Ratios combination andPpvalue for the bests of each strategy
Control strategy Ratios combination Pp ωa ̸ω1 ma ̸m1
1 DOF ð Þ− ð Þ− 0.57
DVA 1.0 0.1 0.36
AMD 0.5 0.5 0.09
Strategy S1 0.5 0.5 0.11
Strategy S2 1.0 0.1 0.15
Strategy S3 0.1 0.5 0.08
Strategy S4 0.1 0.5 0.08
Strategy S5 1.0 0.1 0.18
56 H. T. Coelho et al.
To evaluate the robustness of the TAMD with the chosen mass and frequency ratios, a numerical study with other types of excitation force for strategies [S3] and [S4] were also performed. These are clearly better than strategies [S1], [S2] and [S5]. Figure5 presents the system’s time response of m1 displacement to a 10 N impact excitation applied at 0.5 s to the 2 DOF presented in Fig.1a.
It can be noted that strategies [S3] and [S4] presents the lowest settling time showing an excellent damping capability. They show better responses than Fig. 4 Receptance results forma ̸m1=0.1 and ωa ̸ω1=1. Optimum viscous damping AMD response with a mass ratio ma ̸m1=0.5 (AMD*), optimum viscous damping AMD response (AMD), TAMD response using strategy S3 (S3) and TAMD response using strategy S4 (S4)
Fig. 5 Time response to an impact excitation for ma ̸m1=0.1 and ωa ̸ω1=1. Uncontrolled vibration response of without auxiliary mass (1 DOF), well-tuned DVA response (DVA), optimum viscous damping AMD response (AMD), TAMD response using strategy S3 (S3) and TAMD response using strategy S4 (S4)
Tunable Auxiliary Mass Damper with Friction Joint… 57
optimum viscous damping AMD, which in turn is better than the DVA’s response.
Figure6 presents the time response to a random excitation varying up to 10 N.
Normally it is too difficult to deal with random excitations due its nature, especially for semi-active systems, which try to tune the vibratory system to the excitation force, maximizing the performance of the TAMD. It can be noted that the DVA presents a better response than 1 DOF system’s uncontrolled vibration, with RMS amplitude of 45.1μm and 162.2 μm, respectively. A good improvement is obtained with optimal viscous damped AMD response with RMS amplitude of 27.2 μm and even better responses can be observed for the results obtained with [S3] and [S4]. The RMS values for these responses are 19.8μm and 21.2μm, respectively for S3½ and S4½ , which are less than half of the DVA’s RMS and eight times smaller than the 1 DOF system’s uncontrolled vibration RMS.
The time response of each strategy and DVA for chirp excitation with its fre- quency sweeping from 5 Hz up to 100 Hz, changing in a ratio of 19 Hz ̸s, are presented in Fig.7. The 1 DOF response presents a maximum amplitude of 2.28 mm near to 1.0 s, which is bigger than DVA’s response with 1.48 mm around 1.5 s. The optimal viscous damped AMD maximum amplitude is 0.82 mm. The strategies [S3] and [S4] presents better results than AMD’s response with the lowest amplitudes, in which the passage through resonance is almost imperceptible, with maximum amplitudes of 0.37 mm and 0.39 mm, respectively. This last fact could be useful for applications in rotating machines, permitting a smoother passage through critical speeds.
Therefore, in this numerical study of the TAMD it was also verified that the proposed control strategies’ efficiencies are independent of the excitation force nature. It was also verified that strategies [S3] and [S4] present better results than Fig. 6 Time response to a random excitation for ma ̸m1=0.1 and ωa ̸ω1=1. Uncontrolled vibration response of without auxiliary mass (1 DOF), well-tuned DVA response (DVA), optimum viscous damping AMD response (AMD), TAMD response using strategy S3 (S3) and TAMD response using strategy S4 (S4)
58 H. T. Coelho et al.
the optimal viscous damped AMD, which is nowadays some of the best design alternatives for this vibration attenuation approach.
Since they make use of friction dampers, strategies [S3] and [S4] can be employed in heavy and big structures applications, even under low velocities of vibration, were viscous dampers are inefficient or unfeasible due to their size.