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Figure 16. Schematics of linkage spring ATVA.

Figure 16 Schematics of linkage spring ATVA.

Related Figures (22)

Figure 1. Tuned vibration absorber used as a vibration neutraliser.
Figure 1. Tuned vibration absorber used as a vibration neutraliser.
Figure 2. Dynamics of pinned shallow arch with end longitudinal restraints of finite stiffness. and Pp, = 7° EI/L? is the | Fiuler critical load. ] Equations (2.1) and (2.2) and the / subsequent analysis assume the arch is shallow, i.e. (w/)’, (wp, )? <1, where ()'=d()/da. For small vibrations v= w—wp, about the preloaded configuration wp, (x) (figure 2c), the neutral surface of the beam is strained by an amount depending on the stiffness Kg of the longitudinal end restraints. In reference Bonello et al. (2004) it is shown that, for vibrations symmetrical about the mid- section, the equation of motion is given by
Figure 2. Dynamics of pinned shallow arch with end longitudinal restraints of finite stiffness. and Pp, = 7° EI/L? is the | Fiuler critical load. ] Equations (2.1) and (2.2) and the / subsequent analysis assume the arch is shallow, i.e. (w/)’, (wp, )? <1, where ()'=d()/da. For small vibrations v= w—wp, about the preloaded configuration wp, (x) (figure 2c), the neutral surface of the beam is strained by an amount depending on the stiffness Kg of the longitudinal end restraints. In reference Bonello et al. (2004) it is shown that, for vibrations symmetrical about the mid- section, the equation of motion is given by
Figure 3. Variation of natural frequency with curvature of a shallow pinned arch of rectangular section 2h with no preload and Kg=.
Figure 3. Variation of natural frequency with curvature of a shallow pinned arch of rectangular section 2h with no preload and Kg=.
Figure 4. Two alternative designs for a shallow arch ATVA.
Figure 4. Two alternative designs for a shallow arch ATVA.
Figure 5. Performance curves for shallow arch ATVA with fixed H, variable Hp, (figure 4a).
Figure 5. Performance curves for shallow arch ATVA with fixed H, variable Hp, (figure 4a).
Figure 6. Performance curves for shallow arch ATVA with fixed Hp,, variable H (figure 40)
Figure 6. Performance curves for shallow arch ATVA with fixed Hp,, variable H (figure 40)
Figure 7. Cantilever-type ATVA with anticlastic curvature control.
Figure 7. Cantilever-type ATVA with anticlastic curvature control.
Figure 9. (a) Tuning characteristic for cantilever type TVA in figure 7 estimated from equations (2.10)-(2.15) for a 2-layer cross-ply cross-section with E,, Fy>=130, 10 GPa, v;2=0.3, 2h=2X107? m, 2L=0.24 mm; (b) Edge force required to achieve cross-section curvature in figure 9(a). in tuned frequency are possible, provided that the actuator can provide the necessary changes to the subtended angle a. However, the slope of the graph in figure 9b is seen to be enormous, indicating that stretching the composite cross- section is not a practical way of varying the cross-section.
Figure 9. (a) Tuning characteristic for cantilever type TVA in figure 7 estimated from equations (2.10)-(2.15) for a 2-layer cross-ply cross-section with E,, Fy>=130, 10 GPa, v;2=0.3, 2h=2X107? m, 2L=0.24 mm; (b) Edge force required to achieve cross-section curvature in figure 9(a). in tuned frequency are possible, provided that the actuator can provide the necessary changes to the subtended angle a. However, the slope of the graph in figure 9b is seen to be enormous, indicating that stretching the composite cross- section is not a practical way of varying the cross-section.
Figure 10. Vibration absorber with stiffness element formed from parallel curved beams.
Figure 10. Vibration absorber with stiffness element formed from parallel curved beams.
Figure 11. Finite element model of clamped-free beam with a pressurised fluid-filled cavity separating a pair of laminae of orthotropic material with material axes at right angles to each other (thickness of each lamina is 0.25 mm). separated by a fluid filled cavity. Fibre reinforcements joining the upper and lower surfaces prevent them from bulging. Figure 11 shows the FE model of the deformation resulting from the application of a 1 bar pressure. For the dimensions shown, and a Poisson ratio (vj2=0.3, the tip deflections are 0.13 mm, 0.145 mm, 0.274 mm, respectively, for major and minor Young moduli (£\, E2) of (180 GPa, 10 GPa), (10 GPa, 5 GPa), (100 GPa, 5 GPa). If such a beam were incorporated into the devices of figure 4 and figure 10 then these deflection values would be the changes in the crown height H for a 1 bar change in pressure. Hence, it is expected that a pressure increase of, say, 20 bar would result in significant variation to the tuned frequency. MTL. 06h RR el , i ee: Se a, ee nD Ee Sy es fe ns fe
Figure 11. Finite element model of clamped-free beam with a pressurised fluid-filled cavity separating a pair of laminae of orthotropic material with material axes at right angles to each other (thickness of each lamina is 0.25 mm). separated by a fluid filled cavity. Fibre reinforcements joining the upper and lower surfaces prevent them from bulging. Figure 11 shows the FE model of the deformation resulting from the application of a 1 bar pressure. For the dimensions shown, and a Poisson ratio (vj2=0.3, the tip deflections are 0.13 mm, 0.145 mm, 0.274 mm, respectively, for major and minor Young moduli (£\, E2) of (180 GPa, 10 GPa), (10 GPa, 5 GPa), (100 GPa, 5 GPa). If such a beam were incorporated into the devices of figure 4 and figure 10 then these deflection values would be the changes in the crown height H for a 1 bar change in pressure. Hence, it is expected that a pressure increase of, say, 20 bar would result in significant variation to the tuned frequency. MTL. 06h RR el , i ee: Se a, ee nD Ee Sy es fe ns fe
Figure 13. Variation of curvature to free strain ratio with actuator to beam thickness ratio for b, = by and £,=59.5 GPa, ds, = —212X107'*mV—!.
Figure 13. Variation of curvature to free strain ratio with actuator to beam thickness ratio for b, = by and £,=59.5 GPa, ds, = —212X107'*mV—!.
Figure 14. Minimum voltage versus subtended angle a for b= by, and 2h=0.5X107* m, S=0.05 m, E,=59.5 GPa, dy;=—212X10° mV.
Figure 14. Minimum voltage versus subtended angle a for b= by, and 2h=0.5X107* m, S=0.05 m, E,=59.5 GPa, dy;=—212X10° mV.
eee en Me eee Se MN eg, EE! FR MS ME Le Se ae Fe The force required from the actuator to control the stiffness is minimal but the actuator needs to have a large stroke. A further requirement for the actuator is that its longitudinal stiffness should be sufficiently high as to prevent variation in the distance between the cantilevers as a result of vibration-induced deformation. Hence, a suitable actuating mechanism for the device would be a screw driven by a servomotor, like that used in Kidner & Brennan (2002). Alternatively, it might be possible to use the gel actuator developed in associated research (Santulli et al. 2002) forming part of the same multi-disciplinary project as the research presented in this paper. This would be in keeping with the musculo-skeletal analogy behind the device in figure 16, since the gel actuator can be considered as the engineering analogue of a muscle. Se ee ee ee ee eo The effectiveness of the device is limited to frequencies for which inertia effects of the components forming the linkage are negligible. For low values of 4) the device is prone to ‘snap through’ to its lower stable equilibrium position at high amplitudes of vibration (Brennan et al. 2003). Moreover, for a given level of vibration, nonlinear effects are expected to be more prominent at low values of M9, as shown in figure 17, which compares the nonlinear expression in equation (3.22) with the linear (tangential) relationship of equation (3.23) for three different values of #9. It is also seen from figure 17 that, for deflections from the stable equilibrium position, the spring hardens in tension (F'>0) and softens in compression (F'<0).
eee en Me eee Se MN eg, EE! FR MS ME Le Se ae Fe The force required from the actuator to control the stiffness is minimal but the actuator needs to have a large stroke. A further requirement for the actuator is that its longitudinal stiffness should be sufficiently high as to prevent variation in the distance between the cantilevers as a result of vibration-induced deformation. Hence, a suitable actuating mechanism for the device would be a screw driven by a servomotor, like that used in Kidner & Brennan (2002). Alternatively, it might be possible to use the gel actuator developed in associated research (Santulli et al. 2002) forming part of the same multi-disciplinary project as the research presented in this paper. This would be in keeping with the musculo-skeletal analogy behind the device in figure 16, since the gel actuator can be considered as the engineering analogue of a muscle. Se ee ee ee ee eo The effectiveness of the device is limited to frequencies for which inertia effects of the components forming the linkage are negligible. For low values of 4) the device is prone to ‘snap through’ to its lower stable equilibrium position at high amplitudes of vibration (Brennan et al. 2003). Moreover, for a given level of vibration, nonlinear effects are expected to be more prominent at low values of M9, as shown in figure 17, which compares the nonlinear expression in equation (3.22) with the linear (tangential) relationship of equation (3.23) for three different values of #9. It is also seen from figure 17 that, for deflections from the stable equilibrium position, the spring hardens in tension (F'>0) and softens in compression (F'<0).
Figure 17. Nonlinear relationship between F and y for various b/L values.
Figure 17. Nonlinear relationship between F and y for various b/L values.
Figure 18. Prototype ATVA with variable curvature beams as stiffness element.
Figure 18. Prototype ATVA with variable curvature beams as stiffness element.
Figure 19. Tuning characteristic for prototype ATVA.
Figure 19. Tuning characteristic for prototype ATVA.
Figure 20. Demonstrator for manually tuned linkage spring device.
Figure 20. Demonstrator for manually tuned linkage spring device.
Figure 21. Transmissibility plots at low amplitude (Ab/L= b/L— bjom/L, Onom/L= 0.8).
Figure 21. Transmissibility plots at low amplitude (Ab/L= b/L— bjom/L, Onom/L= 0.8).
Figure 22 compares the predicted and measured tuning characteristics for the nominal configuration byom/L=0.8, where the correlation between simulation and measurement is seen to remain reasonably good at high amplitude. The maximum variation in f, is 70% (—40 to +30%) for a total actuator movement of 1.6 cm.
Figure 22 compares the predicted and measured tuning characteristics for the nominal configuration byom/L=0.8, where the correlation between simulation and measurement is seen to remain reasonably good at high amplitude. The maximum variation in f, is 70% (—40 to +30%) for a total actuator movement of 1.6 cm.
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EngineeringVibration ControlSmart StructuresDynamic Vibration AbsorberVibration Absorber
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