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.
4 Numerical Results for a Multiple-Degree-of-Freedom
m1 0 0 0
0 ma 0 0
0 0 m2 0
0 0 0 m3
2 66 64
3 77 75
ẍ1
ẍa
ẍ2
ẍ3
8>
>>
<
>>
>: 9>
>>
=
>>
>;
c1+ca+c2 −ca −c2 0
−ca ca 0 0
−c2 0 c2+c3 −c3
0 0 −c3 c3
2 66 64
3 77 75
ẋ1
ẋa
ẋ2
ẋ3
8>
>>
<
>>
>: 9>
>>
=
>>
>;
+
k1+ka+k2 −ka −k2 0
−ka ka 0 0
−k2 0 k2+k3 −k3
0 0 −k3 k3
2 66 64
3 77 75
x1 xa
x2 x3
8>
>>
<
>>
>: 9>
>>
=
>>
>;
= 1
−1 0 0 2 66 64
3 77 75Ff+
1 0 0 0 2 66 64
3 77 75Fexc
ð8Þ
The complementary physical properties values are presented in Table3. The physical properties for the auxiliary system ðma and kaÞ are deduced from the aforementioned mass and frequency ratios, i.e.,ma= 0.41 kg,ka= 7.03 kN ̸m and ca= 1 N s ̸m. The contact parameters remain the same, the tangential stiffness is kT= 1.16 MN ̸m and the friction coefficient isμ= 0.33.
Figure9 show the receptances obtained as in previous section, however now with excitation force frequency sweeping from 5 Hz up to 50 Hz, also in steps of 0.1 Hz. For the multi degree of freedom case, only results obtained with the strategies S3 and S4 are presented. Once again, these were the best strategies to modulate the normal force in a TAMD.
As can be observed in Fig.9, the use of the proposed TAMD and strategies promote excellent results suppressing the second resonant peak of the MDOF vibratory system. This happens due to the natural frequency of the TAMD, which is 20.73 Hz, is close to one of the resonant frequencies of the original MDOF
k1 kT
c1
m1 μ N
ma
ka ca
xa
x1
k2
c2
m2 x2
k3
c3
m3 x3
Fexc
Fig. 8 Schema of the studied three-DOF vibratory system with the TAMD
Table 3 Complementary physical properties values of the three-DOF system
Physical properties 1st DOF 2nd DOF 3rd DOF Mass (kg) m1= 4.14 m2= 1.97 m3= 0.93 Stiffness (kN/m) k1= 70.3 k2= 9.67 k3= 24.44 Damping (N s/m) c1= 3.93 c2= 2.35 c3= 0.99
60 H. T. Coelho et al.
vibratory system. Acting similarly to a well-tuned DVA, however, without intro- ducing two new resonant peaks as is expected in the application of DVAs. Over again the discontinuity in the receptance shows the TAMD working against the resonance frequency to maintain lower amplitude levels.
The semi-active friction damper is an energy sink in the proposed TAMD, dissipating the energy in the frequency range for which the TAMD has been previously adjusted.
The proposed TAMD do not present a significant attenuation of the resonant peaks around 8 and 32 Hz. This occurs due to the TAMD’s natural frequency, as mentioned before. Someone can design the parameters of the TAMD using an optimization procedure aiming to widen the frequency band in which the TAMD is efficient.
To evaluate the robustness of the TAMD in a MDOF vibratory system, when it is positioned at m1, the system was subjected to other types of excitation force.
Figure10 presents the envelope of the absolute time response to a 10 N impact force applied at 0.5 s onm1.
It can be noted in Fig.10 that the TAMD placed on m1 can be extremely effective. The amplitude of the mass m1 has been reduced from 11 to 9μm, the vibration damping was impressive, but it was reduced to insignificant levels in fractions of second. The RMS amplitude levels were reduced from 3.4 to 0.6μm.
The proposed TAMD can also promote a good attenuation in the displacement of m3 with the maximum amplitude coming from 10 to 9μm and RMS amplitude levels from 2.2 to 1.7μm. Also, a little vibration suppression was observed form2 with the maximum amplitude value coming from 6.1 to 5.9μm and RMS levels from 1.58 to 1.53μm.
Fig. 9 Receptance results for the three-DOF system with ma ̸m1=0.1 and ωa ̸ω1=1.
Uncontrolled vibration response (3 DOF), TAMD response using strategy S3 (S3) and TAMD response using strategy S4 (S4)
Tunable Auxiliary Mass Damper with Friction Joint… 61
Figure11 presents the envelope of the absolute time response to a random excitation varying up to 10 N. It can be observed that the proposed TAMD achieves again an excellent performance form1, with the RMS value coming from 0.12 to 0.02 mm. A reasonably attenuation of the displacement ofm3is also observed, with RMS level reducing from 0.06 to 0.05 mm and maintaining the RMS levels ofm2. The envelope of the absolute time responses of each strategy to a chirp excitation with its frequency sweeping from 5 Hz up to 50 Hz, changing in a ratio of 9 Hz ̸s, Fig. 10 Envelope of the absolute time response to an impact excitation for the three-DOF system with ma ̸m1=0.1 and ωa ̸ω1=1. Uncontrolled vibration response (3 DOF), TAMD response using strategy S3 (S3) and TAMD response using strategy S4 (S4)
Fig. 11 Envelope of the absolute time response to a random excitation for the three-DOF system with ma ̸m1=0.1 and ωa ̸ω1=1. Uncontrolled vibration response (3 DOF), TAMD response using strategy S3 (S3) and TAMD response using strategy S4 (S4)
62 H. T. Coelho et al.
are presented in Fig.12. Over again the efficiency overx1is excellent,m1is almost insensitive to the passage through the resonances, the maximum displacement value is reduced from 3.0 to 0.5 mm, and the RMS level is reduced from 0.89 to 0.13 mm, using the proposed TAMD. Despite the maximum displacement value was not significantly reduced for m3, the TAMD is effective, especially after 2.0 s, as it reduces the RMS value from 0.60 to 0.48 mm. And form2, although the maximum displacement value was maintained the same, a small attenuation of the RMS levels, from 0.44 to 0.42 mm, is achieved.
Although the proposed TAMD could not prevent the energy to achieve the others DOF, it was extremely effective in attenuating the vibration amplitude from the DOF in which it was attached. The TAMD removes a significative amount of energy fromm1, but not enough to present the same efficient in the others DOF.