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Fig. 3 a Uncontrolled system frequency-response curve at different levels of the excitation force, b uncontrolled system force- response curve at different values of the excitation frequency Time-delayed control to suppress a nonlinear system vibration utilizing

Figure 3 a Uncontrolled system frequency-response curve at different levels of the excitation force, b uncontrolled system force- response curve at different values of the excitation frequency Time-delayed control to suppress a nonlinear system vibration utilizing

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Fig. 1 a Cantilever beam system under transversal excitation, and b deformed and undeformed differential element denotes the moment of inertia. Substituting Eqs. (4) and (5) into the Hamilton principle assuming that F is the external excitation force yields the following partial differential equation:
Fig. 1 a Cantilever beam system under transversal excitation, and b deformed and undeformed differential element denotes the moment of inertia. Substituting Eqs. (4) and (5) into the Hamilton principle assuming that F is the external excitation force yields the following partial differential equation:
Time-delayed control to suppress a nonlinear system vibration utilizing Substituting Eqs. (7) and (8) into Eq. (6), considering the first mode only (1.e., n = 1), we can get the nonlinear dynamical equation that governs the first-mode vibrations of a cantilever beam system as given in Refs. [8, 10, 13] as follows:
Time-delayed control to suppress a nonlinear system vibration utilizing Substituting Eqs. (7) and (8) into Eq. (6), considering the first mode only (1.e., n = 1), we can get the nonlinear dynamical equation that governs the first-mode vibrations of a cantilever beam system as given in Refs. [8, 10, 13] as follows:
where a,(t) and 4(t) are two real functions on the time that represent vibration amplitude and the modified phase angle of the system, respectively, at superharmonic resonance setting dj = 6 = 0 into Eqs. (42) and (43) yields two nonlinear algebraic equations. By solving these equations numerically utilizing one of the system parameters as a bifurcation control parameter, we can obtain the bifurcation diagram given in Sect. 3.2. Following the same procedure given in the previous section, one can obtain the amplitude-phase modulating equations as:
where a,(t) and 4(t) are two real functions on the time that represent vibration amplitude and the modified phase angle of the system, respectively, at superharmonic resonance setting dj = 6 = 0 into Eqs. (42) and (43) yields two nonlinear algebraic equations. By solving these equations numerically utilizing one of the system parameters as a bifurcation control parameter, we can obtain the bifurcation diagram given in Sect. 3.2. Following the same procedure given in the previous section, one can obtain the amplitude-phase modulating equations as:
where a2(t) and y(t) are two real functions on the time that represent vibration amplitude and the modified phase angle of the system, respectively, at subharmonic resonance setting d2 = y = 0 into Eqs. (46) and (47), yields two nonlinear algebraic equations. By solving these equations numerically utilizing one of the system parameters as a bifurcation control parameter, we can obtain the bifurcation diagram given in Sect. 3.3.
where a2(t) and y(t) are two real functions on the time that represent vibration amplitude and the modified phase angle of the system, respectively, at subharmonic resonance setting d2 = y = 0 into Eqs. (46) and (47), yields two nonlinear algebraic equations. By solving these equations numerically utilizing one of the system parameters as a bifurcation control parameter, we can obtain the bifurcation diagram given in Sect. 3.3.
Fig. 4 a Controlled system frequency-response curve at different values of the position gain 6; when 62 = t = O, and b controlled system frequency-response curve at different values of the velocity gain Bz when Bj = t = 0
Fig. 4 a Controlled system frequency-response curve at different values of the position gain 6; when 62 = t = O, and b controlled system frequency-response curve at different values of the velocity gain Bz when Bj = t = 0
Fig. 5 a Stable and unstable solution region in 6; — t plane when f = 0.06, c= 6B» = 0, b the system frequency-response curve at B; = 1, Bo =0, f = 0.06 at different time-delay values, c the system frequency-response curve at 8; = —1, Bo = 0, f = 0.06 at different time-delay values, and d the system frequency-response curve at 62 = 0, f = 0.06 at different position gain and time-delay
Fig. 5 a Stable and unstable solution region in 6; — t plane when f = 0.06, c= 6B» = 0, b the system frequency-response curve at B; = 1, Bo =0, f = 0.06 at different time-delay values, c the system frequency-response curve at 8; = —1, Bo = 0, f = 0.06 at different time-delay values, and d the system frequency-response curve at 62 = 0, f = 0.06 at different position gain and time-delay
Fig. 6 a Stable and unstable solution region in 62 — t plane when f = 0.06, c= 6, = 0, b the system frequency-response curve at B2 = 0.1, 61 = 0, f = 0.06 at different time-delay values, c the system frequency-response curve at B2 = —0.1, 6; = 0, f = 0.06 at different time-delay values, and d the system frequency-response curve at Bj = 0, f = 0.06 at different velocity gain and time-delay Time-delayed control to suppress a nonlinear system vibration utilizing
Fig. 6 a Stable and unstable solution region in 62 — t plane when f = 0.06, c= 6, = 0, b the system frequency-response curve at B2 = 0.1, 61 = 0, f = 0.06 at different time-delay values, c the system frequency-response curve at B2 = —0.1, 6; = 0, f = 0.06 at different time-delay values, and d the system frequency-response curve at Bj = 0, f = 0.06 at different velocity gain and time-delay Time-delayed control to suppress a nonlinear system vibration utilizing
Fig. 7 a Stable and unstable solution region in 6; = £2 = 6 — Tt plane when f = 0.06,0 = 0, b the system frequency- response curve at Bj = Bo = 0.2, f = 0.06 at different time-delay values, and ¢ the system frequency-response curve at Bi = Bo = —0.2, f = 0.06 at different time-delay values
Fig. 7 a Stable and unstable solution region in 6; = £2 = 6 — Tt plane when f = 0.06,0 = 0, b the system frequency- response curve at Bj = Bo = 0.2, f = 0.06 at different time-delay values, and ¢ the system frequency-response curve at Bi = Bo = —0.2, f = 0.06 at different time-delay values
Fig. 8 a the system time history at 61 = 62 = 0.2, f = 0.06,0 = Oat rt = 1.5, 2 (ie., corresponding to Fig. 7b at o = 0), and b the system time history at 6; = Bo = —0.2, f = 0.06, 0 = Oat t = 0.5, 1 (i.e., corresponding to Fig. 7c ato = 0)
Fig. 8 a the system time history at 61 = 62 = 0.2, f = 0.06,0 = Oat rt = 1.5, 2 (ie., corresponding to Fig. 7b at o = 0), and b the system time history at 6; = Bo = —0.2, f = 0.06, 0 = Oat t = 0.5, 1 (i.e., corresponding to Fig. 7c ato = 0)
Fig. 9 a the system time history, b phase trajectory, and ¢ frequency spectrum at 6; = 62 = —0.2, f = 0.06,0 =O att = 0.3 (i.e., corresponding to Fig. 7a)
Fig. 9 a the system time history, b phase trajectory, and ¢ frequency spectrum at 6; = 62 = —0.2, f = 0.06,0 =O att = 0.3 (i.e., corresponding to Fig. 7a)
Fig. 10 a the system time history, and b phase trajectory at 6} = Bo = 0.5, f = 0.06,0 = Oat t = 0.5 (i.e., corresponding tc Fig. 7a) Time-delayed control to suppress a nonlinear system vibration utilizing
Fig. 10 a the system time history, and b phase trajectory at 6} = Bo = 0.5, f = 0.06,0 = Oat t = 0.5 (i.e., corresponding tc Fig. 7a) Time-delayed control to suppress a nonlinear system vibration utilizing
Fig. 11 The system superharmonic-response curves before control a the system frequency-response curve at different level of the excitation force, b the system force-response curve at different values of the excitation frequency
Fig. 11 The system superharmonic-response curves before control a the system frequency-response curve at different level of the excitation force, b the system force-response curve at different values of the excitation frequency
Fig. 12 The system superharmonic-response curves after control a controlled system frequency-response curve at different values of the position gain 6; when £2 = t = 0, and b controlled system frequency-response curve at different values of the velocity gain By when B} = t = 0
Fig. 12 The system superharmonic-response curves after control a controlled system frequency-response curve at different values of the position gain 6; when £2 = t = 0, and b controlled system frequency-response curve at different values of the velocity gain By when B} = t = 0
Fig. 13 The system superharmonic-response curves after control a stable and unstable solution region in 6; — t plane when f = 1.8, 01= B2 = 0, and b the system frequency-response curve at 6, = 1, B2 = 0, f = 1.8 at different time-delay values
Fig. 13 The system superharmonic-response curves after control a stable and unstable solution region in 6; — t plane when f = 1.8, 01= B2 = 0, and b the system frequency-response curve at 6, = 1, B2 = 0, f = 1.8 at different time-delay values
Fig. 14 a The system time-history, b phase-trajectory, and ¢ frequency-spectrum at B; = 1, bp = 0, f = 1.8,0, =0.5 att =0 (i.e., corresponding to Fig. 13b at oj = 0.5, t = 0) Time-delayed control to suppress a nonlinear system vibration utilizing
Fig. 14 a The system time-history, b phase-trajectory, and ¢ frequency-spectrum at B; = 1, bp = 0, f = 1.8,0, =0.5 att =0 (i.e., corresponding to Fig. 13b at oj = 0.5, t = 0) Time-delayed control to suppress a nonlinear system vibration utilizing
Fig. 15 a The system time-history, b phase-trajectory, and ¢ frequency-spectrum at 6} = 1, Bo = 0, f = 1.8,0; = 0.5 at Tt = 1.5 (ie., corresponding to Fig. 13b at oy = 0.5, t = 1.5)
Fig. 15 a The system time-history, b phase-trajectory, and ¢ frequency-spectrum at 6} = 1, Bo = 0, f = 1.8,0; = 0.5 at Tt = 1.5 (ie., corresponding to Fig. 13b at oy = 0.5, t = 1.5)
Fig. 16 a The system time-history, and b phase-trajectory at Bj = 1, Bo = 0, f = 1.8,0, =0.5 at tr = 0.3 (Le., corresponding to Fig. 13b at oj = 0.5, t = 0.3)
Fig. 16 a The system time-history, and b phase-trajectory at Bj = 1, Bo = 0, f = 1.8,0, =0.5 at tr = 0.3 (Le., corresponding to Fig. 13b at oj = 0.5, t = 0.3)
Fig. 17 The system superharmonic-response curves after control a stable and unstable solution region in 62 — t plane when f =1.8,01= B, =0, and b the system frequency-response curve at Bj = 0, B2 = 0.1, f = 1.8 at different time-delay values
Fig. 17 The system superharmonic-response curves after control a stable and unstable solution region in 62 — t plane when f =1.8,01= B, =0, and b the system frequency-response curve at Bj = 0, B2 = 0.1, f = 1.8 at different time-delay values
Fig. 18 a Stable and unstable solution region in 6} = 62 = 6 — Tt plane when f = 1.8, 0, = 0, and b the system frequency- response curve at 6; = 62 = —0.25, f = 1.8 at different time-delay values
Fig. 18 a Stable and unstable solution region in 6} = 62 = 6 — Tt plane when f = 1.8, 0, = 0, and b the system frequency- response curve at 6; = 62 = —0.25, f = 1.8 at different time-delay values
Fig. 19 The system subharmonic-response curves before control a the system frequency-response curve at different level of the excitation force, b the system force-response curve at different values of the excitation frequency
Fig. 19 The system subharmonic-response curves before control a the system frequency-response curve at different level of the excitation force, b the system force-response curve at different values of the excitation frequency
Fig. 20 The system subharmonic-response curves after control a controlled system frequency-response curve at different values of the position gain 6; when £2 = t = 0, and b controlled system frequency-response curve at different values of the velocity gain By when B; = t = 0
Fig. 20 The system subharmonic-response curves after control a controlled system frequency-response curve at different values of the position gain 6; when £2 = t = 0, and b controlled system frequency-response curve at different values of the velocity gain By when B; = t = 0
Fig. 21 The system subharmonic-response curves after control a stable and unstable solution region in 6, — t plane when f =5,02= 1, By = 0, and b the system frequency-response curve at 6; = 0.5, B2 = 0, f = 5 at different time-delay values
Fig. 21 The system subharmonic-response curves after control a stable and unstable solution region in 6, — t plane when f =5,02= 1, By = 0, and b the system frequency-response curve at 6; = 0.5, B2 = 0, f = 5 at different time-delay values
Fig. 22 The system subharmonic-response curves after control a stable and unstable solution region in 62 — t plane when f =5,02= 1, B, = 0, and b the system frequency-response curve at Bz = 0.1, 6; = 0, f =5S at different time-delay values
Fig. 22 The system subharmonic-response curves after control a stable and unstable solution region in 62 — t plane when f =5,02= 1, B, = 0, and b the system frequency-response curve at Bz = 0.1, 6; = 0, f =5S at different time-delay values
Fig. 23 The system subharmonic-response curves after control a stable and unstable solution region in 6} = 62 = 6 — Tt plane when f = 5, 02, and b the system frequency-response curve at Bj = B2 = —0.2, f =5 at different time-delay values
Fig. 23 The system subharmonic-response curves after control a stable and unstable solution region in 6} = 62 = 6 — Tt plane when f = 5, 02, and b the system frequency-response curve at Bj = B2 = —0.2, f =5 at different time-delay values
Fig. 24 a The system time-history, b phase-trajectory, and ¢ frequency-spectrum at 6, = 0.5, B2 = 0, f =5,02 =lattr =0 (.e., corresponding to Fig. 21b at 07 = 1, t = 0)
Fig. 24 a The system time-history, b phase-trajectory, and ¢ frequency-spectrum at 6, = 0.5, B2 = 0, f =5,02 =lattr =0 (.e., corresponding to Fig. 21b at 07 = 1, t = 0)
Fig. 25 a The system time-history, b phase-trajectory, and ¢ frequency-spectrum at B; = 0.5, bp = 0, f =5,02 =latt =1.5 (i.e., corresponding to Fig. 21b at 02 = 1, t = 1.5) Time-delayed control to suppress a nonlinear system vibration utilizing
Fig. 25 a The system time-history, b phase-trajectory, and ¢ frequency-spectrum at B; = 0.5, bp = 0, f =5,02 =latt =1.5 (i.e., corresponding to Fig. 21b at 02 = 1, t = 1.5) Time-delayed control to suppress a nonlinear system vibration utilizing
Fig. 26 a The system time-history, b phase-trajectory, and ¢ frequency-spectrum at 6; = 0.5, bp = 0, f =5,02 = latt =0.5 (.e., corresponding to Fig. 21b at 07 = 1, t = 0.5)
Fig. 26 a The system time-history, b phase-trajectory, and ¢ frequency-spectrum at 6; = 0.5, bp = 0, f =5,02 = latt =0.5 (.e., corresponding to Fig. 21b at 07 = 1, t = 0.5)
Table 1 Optimum time delay values
Table 1 Optimum time delay values
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