1 Introduction
2 Analytical modeling of a VFP-TMD
2.1 The variable friction concept
2.2 The general VFP-TMD model
2.2.1 Kinematic relations
2.2.2 Dynamic equations
2.2.3 Constitutive equations
2.2.4 Integration of the equations of motion
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Equations (12)–(14) and (16) then provide \(V_{i\, }\text {cos}\varphi _{iV}\), \(V_{i}\,\hbox {sin}\varphi _{iV}\), \(V_{i\, }\rho _{iV}\), and \(F_{i}\) as known functions of \(\theta \), and \({\dot{\theta }}\), making the TMD system completely described by Eqs. (7.1), (7.2), (7.3), (7.4), (7.5), (7.6), (7.7), (7.8), and (7.9).
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Denoting the state variables as \(q_{\mathrm {1}} = \theta \) and \(q_{\mathrm {2}} = {\dot{\theta }}\) and adding the further equation \({\dot{q}}_{1} =q_{2} \), a system of 16 equations in 16 variables is obtained. The 16 equations are nonlinear w.r.t. the states, \(q_{\mathrm {1}}\) and \(q_{\mathrm {2}}\), but linear w.r.t. the state derivatives, \({\dot{q}}_{1} \) and \({\dot{q}}_{2} \), as well as w.r.t. the remaining following 14 variables: \(N_{i\, }\text {cos}\varphi _{iN}\) and \(N_{i\, }\)sin\(\varphi _{iN}\) (repeated 4 times, once for each interface), and \(N_{\mathrm {3}}\), \(V_{\mathrm {3}}\), \(M_{\mathrm {3}}\) (repeated twice, once for each bearing).
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Algebraically solving the linear system provides \({\dot{q}}_{1} \), \({\dot{q}}_{2} \) and the 14 variables as nonlinear functions of \(q_{\mathrm {1}}\) and \(q_{\mathrm {2}}\) and as linear functions of the acceleration input. The knowledge of \({\dot{q}}_{1} \) and \({\dot{q}}_{2} \) feeds the dynamic integration algorithm. The knowledge of the 14 variables provides \(N_{i}\) \(\hbox {and}_{\mathrm {\, }}\varphi _{iN}\) and ultimately \(\sigma _{\mathrm{im}}\) and \(\varphi _{\mathrm{im}}\) for the next step.
2.3 Reduced VFP-TMD models
2.3.1 Reduced model \(\texttt {M}_{1}\): infinitely distant bearings
2.3.2 Reduced model \(\texttt {M}_{2}\): infinitely distant symmetrical bearings with massless slider
2.3.3 Reduced model \(M_3\): rocking P-TMD
2.3.4 Integration of the equations of motion for the reduced models \(\texttt {M}_2\) and \(\texttt {M}_3\)
2.4 The mechanical model of the VFP-TMD on an MDOF structure
2.5 The simplified TMD model on a generalized SDOF structure
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The slider angular amplitude \({\bar{\varphi }}_{1} \) and the rotation \(\theta \) are assumed very small, so only first-order terms are retained in their nonlinear expansions. Accordingly, any spherical region degenerates in its horizontal projection, so that \(\rho _{\mathrm {1}V} = R_{\mathrm {1}}\).
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The normal contact force \(N_{\mathrm {1}}\) is supposed to be centered on \(A_{\mathrm {1}}\) (i.e., \(\varphi _{\mathrm {1}N} = \varphi _{\mathrm {1}m} = 0\)), as if the slider height were null (\(h_{\mathrm {1}} = 0\)).
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The vertical acceleration input is neglected (\(a_{z} = 0\)), as well as the slider inertia (\(J_{\mathrm {1}} = 0\)).
3 Design and simulation of the device
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\(c_{F1} =2c_{Feq} =4\zeta _{Feq} \omega _{Feq} m^{*}\), with \(\zeta _{Feq} =-\frac{\ln e_{Feq} }{\sqrt{\pi ^{2}+\ln ^{2}e_{Feq} } }\), where \(e_{Feq} \) is the elastic restitution coefficient [35], here taken as 0.5.
3.1 Design methodology
3.2 Simulations of the simplified VFP-TMD model on an SDOF structure
3.2.1 Perfectly homogeneous TMD
3.2.2 Actual VFP-TMD
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Case I: \(\mu _{1}^{a} = 0\) & \(\theta _{F\mathrm {1}} \rightarrow \infty \)
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Case II: \(\mu _{1}^{a} = 0\) & \(\theta _{F\mathrm {1}} = 2{\bar{\varphi }}_{1} \)
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Case III: \(\mu _{1}^{a} = \mu _{1}^{b} \)& \(\theta _{F\mathrm {1}} \rightarrow \infty \)
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Case IV: \(\mu _{1}^{a} = \mu _{1}^{b} \)/10 & \(\theta _{F\mathrm {1}} \rightarrow \infty \)
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Cases I and II substantially ensure the same performance of a homogeneous TMD as long as \(f_{s0} /(2\mathrm{mg}{\bar{\varphi }}_{1} ) \le \) 1.15, i.e., as long as \(| \theta |\le 2{\bar{\varphi }}_{1} \). In this range, the nonlinear dependence of \(\mu _{\mathrm{eff}1} \) on \(| \theta |\) does not practically affect effectiveness.
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As \(| \theta |\) exceeds \(2{\bar{\varphi }}_{1} \), case I undergoes a gradual performance degradation, as \(\mu _{\mathrm{eff}1} \) can no longer increase beyond \(\mu _{1}^{b} \) and the TMD equivalent damping ratio decreases. Case II, instead, undergoes a sudden degradation as the slider collides against the restrainer, worsening with increasing excitation. Softening the restrainer and/or increasing its activation angle will reduce and/or delay this degradation, but will also increase the bearing dimensions and the slider aspect ratio. This trade-off should be carefully considered in design.
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Case III shows the limitations of a uniform-friction TMD. Unrestrained as case I, case III achieves a similar performance only at a specific input amplitude (i.e., at \(f_{s0} /(2\;\mathrm{mg}{\bar{\varphi }}_{1} )\approx 1\)), proving otherwise less effective, either because insufficiently damped (at larger amplitudes) or excessively damped (at smaller amplitudes). Under a certain input threshold, the friction force exceeds the inertial force on the TMD, which remains stuck to the structure.
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Case IV shows pros and cons of a plausible realization of VFP-TMD. It is nearly as effective as case I over a wide range of force amplitudes and still quite effective under amplitudes at which case III is totally useless. The complete inefficiency of the device occurs only at a very small input amplitude, 100 times smaller than that which activates the restrainer.
3.3 Simulations of the rigorous VFP-TMD model on an SDOF structure
Configuration | \({\bar{\varphi }}_{1} \) | \(a_{R}\) | \(\eta \) | \({\tilde{\mu }}_{\mathrm{eff}1} \) | \(\mu _{1}^{b} \) |
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(\(^\circ \)) | (–) | (–) | (–) | (%) | |
Case 1 | 1 | 0.079 | 1.0014 | 0.194 | 0.532 |
Case 2 | 5 | 0.391 | 1.0353 | 0.188 | 2.58 |
Case 3 | 10 | 0.772 | 1.1547 | 0.168 | 4.62 |
Case 4 | 15 | 1.132 | 1.4142 | 0.138 | 5.66 |
Case 5 | 20 | 1.462 | 2.0000 | 0.0972 | 5.33 |
3.4 Simulation of the device on an MDOF structure
3.4.1 The structure
3.4.2 The wind load
3.4.3 The VFP-TMD
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Case 1: \(\mu _{1}^{b} =\frac{\pi }{2}{\tilde{\mu }}_{\mathrm{eff}1} {\bar{\varphi }}_{1} = 7.04{\%}\) & \(\mu _{1}^{a} =\) 0,
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Case 2: \(\mu _{1}^{b} =2{\tilde{\mu }}_{\mathrm{eff}1} {\bar{\varphi }}_{1} =\) 8.97% & \(\mu _{1}^{a} =\) 0,
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Case 3: \(\mu _{1}^{b} =4{\tilde{\mu }}_{\mathrm{eff}1} {\bar{\varphi }}_{1} =\) 17.94% & \(\mu _{1}^{a} =\) 0,
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Case 4: \(\mu _{1}^{b} =\frac{10}{11}2{\tilde{\mu }}_{\mathrm{eff}1} {\bar{\varphi }}_{1} =\) 8.15% & \(\mu _{1}^{a} =\frac{1}{10}\mu _{1}^{b} =\) 0.815%.
3.4.4 Results
Configuration | \(u_{sN,\mathrm{max}}\) | \(u_{sN,rms}\) | \(a_{x,\mathrm{max}}\) | \(a_{x,rms}\) | \(u_{max}\) | \(u_{rms}\) | \(\frac{V_{1,\max } }{m^{*}g}\) | \(\frac{F_{1,\max } }{m^{*}g}\) | \(\frac{\varphi _{1N} }{{\bar{\varphi }}_{1} }\) |
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(cm) | (cm) | (cm/\(\hbox {s}^{\mathrm {2}})\) | (cm/\(\hbox {s}^{\mathrm {2}})\) | (cm) | (cm) | (–) | (–) | (–) | |
Uncontrolled | 15.8 | 5.0 | 47.4 | 13.3 | – | – | – | – | – |
Linear TMD | 11.4 | 3.1 | 32.1 | 8.6 | 18.7 | 5.9 | – | – | – |
VFP-TMD – Case 1 | 11.0 | 3.1 | 32.3 | 9.8 | 17.5 | 5.9 | 0.018 | 0 | 0.01 |
VFP-TMD – Case 2 | 10.3 | 3.1 | 33.1 | 9.8 | 14.7 | 5.2 | 0.020 | 0 | 0.01 |
VFP-TMD – Case 3 | 11.0 | 3.4 | 41.4 | 10.5 | 9.9 | 3.5 | 0.027 | 0 | 0.01 |
VFP-TMD – Case 4 | 12.6 | 3.8 | 39.9 | 10.5 | 10.6 | 2.4 | 0.020 | 0 | 0.01 |
Configuration | \(u_{sN,\mathrm{max}}\) | \(u_{sN,rms}\) | \(a_{x,\mathrm{max}}\) | \(a_{x,rms}\) | \(u_{\mathrm{max}}\) | \(u_{rms}\) | \(\frac{V_{1,\max } }{m^{*}g}\) | \(\frac{F_{1,\max } }{m^{*}g}\) | \(\frac{\varphi _{1N} }{{\bar{\varphi }}_{1} }\) |
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(cm) | (cm) | (cm/\(\hbox {s}^{\mathrm {2}})\) | (cm/\(\hbox {s}^{\mathrm {2}})\) | (cm) | (cm) | (–) | (–) | (–) | |
Uncontrolled | 73.8 | 26.7 | 176.0 | 65.3 | – | – | – | – | – |
Linear TMD | 35.5 | 11.8 | 85.3 | 28.2 | 99.2 | 33.0 | – | – | – |
VFP-TMD – Case 1 | 37.3 | 11.5 | 113.2 | 29.0 | 87.7 | 33.0 | 0.073 | 0.64 | 0.40 |
VFP-TMD – Case 2 | 34.6 | 12.0 | 96.2 | 30.7 | 78.1 | 29.1 | 0.086 | 0 | 0.05 |
VFP-TMD – Case 3 | 43.9 | 15.1 | 133.5 | 38.6 | 55.5 | 19.5 | 0.144 | 0 | 0.08 |
VFP-TMD – Case 4 | 34.9 | 12.4 | 95.6 | 30.9 | 81.2 | 27.6 | 0.079 | 0 | 0.04 |