Advanced vibrant controller results of an energetic framework structure

Hany Samih Bauomy

doi:10.1515/eng-2024-0055

Artikel Open Access

Advanced vibrant controller results of an energetic framework structure

Hany Samih Bauomy

Veröffentlicht/Copyright: 3. Juli 2024

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Abstract

This research shows the influence of a new active controller technique on a parametrically energized cantilever beam (PECB) with a tip mass model. This article remains primarily concerned with regulating the system’s response using a novel control mechanism. This study describes a novel control mechanism called the nonlinear proportional-derivative cubic velocity feedback controller (NPDCVFC). The motivation of this article is to design a novel control algorithm in order to mitigate the nonlinear vibrations of a parametrically energized cantilever beam with a tip mass model. The proposed controller NPDCVFC incorporates nonlinearly second- and first-order filters into the system. The system is governed by one nonlinear differential equation having both quadratic and cubic nonlinearities within the parametric force. The controller’s efficiency in reducing framework vibrations, managing nonlinear bifurcations, and calming unstable motion is evaluated using numerical simulations of instantaneous vibrations. The perturbation technique is beneficial for solving the current model under the proposed worst resonance case ( Ω ˆ p = 2 ω ˆ 0 ) . In order to choose the optimal controller, we have also added three more controller approaches to the configuration. Integral resonant control, positive position feedback, and nonlinear integral positive position feedback are the three controller approaches that are applied to the structure under consideration. We determine that the NPDCVFC as a new controller is the most effective for lowering the high vibration amplitudes. Over the investigated model, all numerical results were performed using the MATLAB 18.0 programmer software. The stability analysis and the effects of various elements on the controlled structure have been investigated. A comparison with recently published works of a comparable model has also been prepared. Experiment capacities for a PECB with a tip mass are obtainable to validate the results, and they demonstrate good agreement with analytical and numerical results.

Graphical abstract

Keywords: parametrically excited; cantilever beam; perturbation; stability; IRC; PPF; NIPPF; and NPDCVF controllers

1 Introduction

Applying different types of stimulation to dynamic systems is a widespread activity. Direct excitation, sometimes known as forced excitement, is a common type of stimulation. The system itself, damping, and the amplitude of the excitation all have an impact on the vibration amplitudes of a framework under direct excitation. Huge responses can be obtained in directly excited systems when one of the main frequencies of the model is close to the excitation frequency or when there are significant nonlinearities. Damping or the nonlinearity itself can then limit these massive vibration amplitudes. Be that as it may, on the grounds that the excitation looks like time-fluctuating coefficients in the situation of movement, parametrically animated frameworks show aversion to phenomenally huge vibration amplitudes. The interaction of the parametric excitation and system properties, including inherent frequencies, primarily determines the stability of structural systems. In a frequency region near the fundamental parametric resonance, a tiny excitation amplitude can cause a significant response in the system. Large vibrations that follow could damage the system’s components and cause significant dynamic instability before the system collapses. Wind, traffic, and earthquakes are the usual causes of parametric excitation for cable-stayed bridges [1]. Ignoring a substantial correlation between parametric excitation and bridge vibration could have lethal consequences [2,3,4,5]. Occasionally, the roll motion of a ship can cause amplitudes that are dangerously high. At the point when the wave level outperforms a specific limit and the wave excitation recurrence is generally two times that of the boat’s regular recurrence, the biggest amplitudes are conceivable. The ship may capsize as a result of this parametric roll or abrupt change in vibrational amplitude [6].

Parametric excitation is utilized in a variety of applications, such as vibration suppression, signal identification, response amplification, and vibration energy harvesting. Vibration energy harvesters are devices that transform mechanical energy from environmental vibrations into useful electricity. It has been shown that vibration energy collecting, when done properly, can offer a consistent and efficient energy source for everyday electronic device operation [7]. In vibration energy harvesting, the most commonly employed mechanical to electrical transduction mechanisms include electromagnetic [8], electrostatic [9], piezoelectric [10], and magnetostrictive transduction processes. Parametric amplification, the process of introducing parametric excitation to a directly excited system, makes parametric excitation useful for response amplification [11]. Parametric stimulation has also been shown to be an efficient strategy for vibration reduction. The decrease of the vibrational amplitudes of the main (hosting) system is a critical building component of the parametric vibration suppression technique. The pendulum is the most commonly used structure for suppressing parametric vibration [12].

The huge vibrational amplitudes obtained at principal parametric resonance, on the other hand, have been extensively employed for sensing purposes [13]. The important parametric reverberation is actuated at two times the framework’s normal recurrence on the off chance that how much the parametric excitation is adequately enormous to beat energy dissemination in the framework. At long last, the nonlinearities of the framework hose vibrations. The powerful behavior of parametrically energized frameworks has been extensively investigated in a range of applications using parametrically invigorated cantilever radiates [14,15,16,17,18,19,20]. Methods of perturbation, such as the method of multiple scales (MMS) [21,22,23,24,25,26] and averaging approaches [27,28], have frequently been used to investigate the energetic performance of parametrically simulated models. These strategies have been demonstrated to be effective at predicting how such systems will react, particularly in the frequency region near the major parametric resonance [29]. Conventional MMS has been demonstrated to accurately forecast response only under basic parameters of modest system characteristics, moderate excitations, and confined frequency ranges around the central parametric resonance [15,29,30,31,32,33,34,35,36]. Furthermore, a variety of control mechanism solutions have been explored and demonstrated to lessen the harmful vibrations caused by various nonlinear systems [37,38,39,40,41,42,43,44,45]. A linear proportional-derivative (PD) controller is implemented to eliminate the nonlinear behaviors and to reduce the lateral oscillations of an asymmetric horizontally supported nonlinear rotor system. The proposed controller is joined to the rotor system through an electromagnetic actuator with four poles. One pair of poles manages system vibrations horizontally, while the other pair handles vertical oscillations, as mentioned in Saeed et al. [46]. Recently, research studies [47,48] presented an eight-pole actuator as an alternative to the four-pole actuator for reducing nonlinear vibrations in a Jeffcott system. In their initial research [47], the authors utilized a PD control algorithm with the eight-pole actuator to dampen unwanted vibrations in a nonlinear Jeffcott model suspended vertically. In their later study [48], they employed a proportional integral resonant controller (PIRC) to reduce the nonlinear oscillations in the model analyzed by Saeed et al. [47]. The analysis showed that combining the PIRC control with the eight-pole actuator stabilizes the unstable motion seen with the PD controller. The performance of the positive position feedback (PPF) controller is improved by suppressing its two peaks to acceptable levels, where that could be done by coupling additional nonlinear saturation controllers to the rotating compressor blade system to impose a V-curve at each one of the peaks. Using multiple time-scale method, the approximate solutions were derived, and a stability analysis was achieved in Kandil and Eissa [49]. Also, a macro fiber composite (MFC) system is applied an active control algorithm to mitigate the unwanted vibrations of a rotating blade via MFC sensors and actuators by applying the PPF algorithm [50]. Formerly, a horizontally supported car’s motion has been modeled and controlled under the effect of a nonlinear spring, a damper, and a harmonic excitation external force. The car’s oscillations were controlled via an integral resonant controller, which was built on a linear variable differential transformer and a servo-controlled linear actuator [51].

This article aims to examine a novel active controller approach for the structure model offered within parametric excitation in the worst-case scenario, which serves as the primary parametric resonance situation ( Ω ˆ p = 2 ω ˆ 0 ) using a perturbation technique and numerical methods. The motivation of this article is to design a novel control algorithm in order to mitigate the nonlinear vibrations of a parametrically energized cantilever beam using a tip mass model. The proposed controller, nonlinear proportional-derivative cubic velocity feedback controller (NPDCVFC), incorporates nonlinear second- and first-order filters into the system. The system is governed by one nonlinear differential equation having both quadratic and cubic nonlinearities within the parametric force. The controller’s efficiency in reducing framework vibrations, managing nonlinear bifurcations, and calming unstable motion is evaluated using numerical simulations of instantaneous vibrations. The requirement for another dynamic regulator approach for this model is to lessen as many hazardous and dangerous vibrations caused by the structure as possible. To avoid these vibrations, the author tried other regulators such as NPD, PPF, and IRC, but found that the NPD was inadequate. Thus, he increased the nonlinear terms in NPD by connecting them through the NCVF regulator, which reduces vibrations more than other regulators. With the help of a corresponding subsidiary regulator and negative cubic speed criticism, the new dynamic control has shown an expansion in nonlinear boundaries and a more noteworthy number of ends of the related system’s perilous vibrations. As a new nonlinear control method, NPDCVFC, the explored controller uses NPD plus NCVFC. A numerical comparison is made with various nonlinear controllers that have an impact on the system. The key finding from the numerical result shows that the new controller, NPDCVFC, has the most influence in lowering and getting rid of the high vibrations on the model. MATLAB 18.0 software was used to compute the stability study and the effects of various framework parameters both mathematically and numerically. There has also been a comparison with other recent works on the same concept.

2 Description of the model

Figure 1 describes a cantilever beam with a tip mass m under parametric excitation, where L marks the length of the cantilever, w(t) is the displacement of the clamped end giving parametric excitation, and u ( s , t ) ν ( s , t ) are the cantilever’s axial and transverse deflections, respectively. As illustrated in Aghamohammadi et al. [16], s is the coordinate along the middle plane of the cantilever representing the arc length, ρ c is the density of the cantilever, b c signifies the width, t c is the thickness, A c is the cross-sectional area, E c designates the elastic modulus, I c is the area moment of inertia, ρ is the radius of curvature, and θ is the angle of rotation. The tip mass m is defined as a point mass added at the cantilever tip ( s = L ) . It is presumed that its moment of inertia is negligible. It is believed that the beam is uniform. Furthermore, the cantilever beam is treated as an Euler–Bernoulli beam, with its thickness assumed to be minimal in relation to its length. Shear deformation and rotating inertia’s effects are therefore disregarded. Moreover, straight gooey damping with a coefficient c is accepted to be the damping in the framework. Furthermore, it is expected that the cantilever’s neutral axis is inextensible.

Figure 1

A parametrically energized cantilever framework with a tip mass.

The overseeing condition of movement for the cantilever beam can be acquired by applying the extended Hamilton variation principle, following a series of simplifications and taking into account the homogenous boundary conditions [14,52,53,54]:

(1) ρ c A c ν u + c ν t + E c I c ( ν s s s s + ν s 2 ν s s s s + 4 ν s ν s s ν s s s + ν s s 3 ) − ρ c A c ν s w t t + ν s s ∫ L s w t t d ζ + 1 2 ρ c A c ν s ∫ 0 s ( ν γ 2 ) t t d γ + ν s s ∫ L s ∫ 0 ζ ( ν γ 2 ) t t d γ d ζ − m δ D ( s − L ) ν s w t t + ν s s ∫ L s w t t d ζ + 1 2 m δ D ( s − L ) ν s ∫ 0 s ( ν γ 2 ) t t d γ + ν s s ∫ L s ∫ 0 ζ ( ν γ 2 ) t t d γ d ζ = 0 ,

where δ D represents the Dirac delta function and the subscripts t and s symbolize the derivatives with respect to these variables, respectively. Presenting

(2) Y = E c I c ρ c A c L 4 ,

and the non-dimensional parameters

(3) s ˆ = s L , ν ˆ = ν L , w ˆ = w L , γ ˆ = γ L , ζ ˆ = ζ L , t ˆ = t Y , c ˆ = c ρ c A c Y , ω ˆ 0 = ω 0 Y , m ˆ = m ρ c A c L ,

where ω 0 is the framework’s lowest un-damped natural frequency. One way to express Equation (1) is in the dimensional form

(4) ν ˆ t ˆ t ˆ + c ˆ ν ˆ t ˆ + ν ˆ s ˆ s ˆ s ˆ s ˆ + ν ˆ s ˆ 2 ν ˆ s ˆ s ˆ s ˆ s ˆ + 4 ν ˆ s ˆ ν ˆ s ˆ s ˆ ν ˆ s ˆ s ˆ s ˆ + ν ˆ s ˆ s ˆ 3 − ν ˆ s ˆ ( 1 + m ˆ δ D ( s ˆ − 1 ) ) w ˆ t ˆ t ˆ − ν ˆ s ˆ s ˆ ( 1 + m ˆ δ D ( s ˆ − 1 ) ) ∫ 1 s w ˆ t ˆ t ˆ d ζ ˆ + ν ˆ s ˆ ( 1 + m ˆ δ D ( s ˆ − 1 ) ) ∫ 0 s ˆ ( ν ˆ t ˆ s ˆ 2 + ν ˆ s ˆ ν ˆ t ˆ t ˆ s ˆ ) d γ ˆ + ν ˆ s ˆ s ˆ ( 1 + m ˆ δ D ( s ˆ − 1 ) ) ∫ 1 s ˆ ∫ 0 ζ ˆ ( ν ˆ t ˆ s ˆ 2 + ν ˆ s ˆ ν ˆ t ˆ t ˆ s ˆ ) d γ ˆ d ζ ˆ = 0 .

There is no closed-form solution to the nonlinear governing differential Equation (4).

The transverse deflection ν ˆ ( s ˆ , t ˆ ) is anticipated by way of a direct mix of the commitments from N vibration modes in order to make a diminished request model for the parametrically energized cantilever. As a result, ν ˆ ( s ˆ , t ˆ ) can be expressed as [55], which is

(5) ν ˆ ( s ˆ , t ˆ ) = ∑ r = 1 N ψ r ( s ˆ ) Z r ( t ˆ ) ,

where ψ r ( s ˆ ) and Z r ( t ˆ ) are the direct mass-standardized mode shape capabilities for the Euler–Bernoulli cantilever pillar with a tip mass and the non-layered uprooting reaction of the r th vibration mode, individually. Depending simply on the principal vibration mode (N = 1) in Equation (5) and disposing of addendum 1 for clarity, which turns out as expected when the excitation recurrence is essentially lower than the framework’s second normal recurrence ψ ( s ˆ ) , is addressed as

(6) ψ ( s ˆ ) = R ψ ¯ ( s ˆ ) ,

where

(7) R = 1 ∫ 0 1 ( ψ ¯ ( s ˆ ) ) 2 d s ˆ + m ˆ ( ψ ¯ ( s ˆ = 1 ) ) 2 ,

(8) ψ ¯ ( s ˆ ) = ( cos ( λ s ˆ ) − cosh ( λ s ˆ ) ) + Q ( sin ( λ s ˆ ) − sinh ( λ s ˆ ) ) ,

(9) Q = sin ( λ ) − sinh ( λ ) + λ m ˆ ( cos ( λ ) − cosh ( λ ) ) cos ( λ ) + cosh ( λ ) − λ m ˆ ( sin ( λ ) − sinh ( λ ) ) .

In Equations (8) and (9), the eigenvalue of the first vibration mode for which the relation holds is represented by

(10) λ = ω ˆ 0 ,

Additionally, λ satisfies the characteristic equation

(11) 1 + cosh ( λ ) cos ( λ ) + m ˆ λ ( cos ( λ ) sinh ( λ ) − sin ( λ ) cosh ( λ ) ) = 0 ,

It is believed that the cantilever beam’s clamped end moves axially in a harmonic manner with an acceleration amplitude a p , i.e., w t t is symbolized by way of

(12) w t t = a p cos ( Ω p t ) ,

where Ω p is the parametric excitation frequency that has a non-dimensional expression

(13) Ω ˆ p = Ω p Y .

Consequently, taking into account Equation (3), the clamped’s non-dimensional acceleration w ˆ t ˆ t ˆ is calculated as

(14) w ˆ t ˆ t ˆ = a ˆ p cos ( Ω ˆ p t ˆ ) ,

where a ˆ p is the non-dimensional acceleration amplitude defined as

(15) a ˆ p = a p L Y 2 .

Consequently, Equation (5) for the first vibration mode is inserted into Equation (4) to produce the final, reduced-order equation of motion for the system, which is then multiplied by ψ ( s ˆ ) and integrated over the non-dimensional length as [56]

(16) Z ̈ + β Z ̇ + ω ˆ 0 2 ( 1 + P cos ( Ω ˆ p t ˆ ) ) Z + η Z 3 + α ( Z ̇ 2 + Z Z ̈ ) Z = 0 ,

where the derivatives are with respect to t ˆ , and β , P , η , and α are the coefficients of damping, parametric excitation amplitude, Duffing-type nonlinearity, and nonlinear inactivity term, respectively, which are characterized as

(17) β = c ˆ , P = h 2 ω ˆ 0 2 h 1 a ˆ p , η = h 3 h 1 , α = h 4 + h 5 h 1 ,

where

h 1 = ∫ 0 1 ψ 2 d s ˆ , h 2 = ∫ 0 1 ( ( 1 − s ˆ ) ψ ψ s ˆ s ˆ − ψ ψ s ˆ ) d s ˆ − m ˆ ψ ( 1 ) ψ s ˆ ( 1 ) ,

h 3 = ∫ 0 1 ψ ( ψ s ˆ 2 ψ s ˆ s ˆ s ˆ s ˆ + 4 ψ s ˆ ψ s ˆ s ˆ ψ s ˆ s ˆ s ˆ + ψ s ˆ s ˆ 2 ) d s ˆ , h 4 = ∫ 0 1 ψ ψ s ˆ ∫ 0 s ˆ ( ψ γ ˆ ( γ ˆ ) ) 2 d γ ˆ d s ˆ ,

h 5 = ∫ 0 1 ψ ψ s ˆ s ˆ ∫ 1 s ˆ ∫ 0 ζ ˆ ( ψ γ ˆ ( γ ˆ ) ) 2 d γ ˆ d ζ ˆ d s ˆ .

This part produced the mathematical model for a PECB with a tip mass. In this section, we use the first-order approximation of the multiple time-scale technique to get approximate solutions for Equation (16). Equation (16) is scaled as follows, assuming that the system parameters counting β , P , η , and α to be minor and of a similar order ε

(18) Z ̈ + ε β ε Z ̇ + ω ˆ 0 2 ( 1 + ε P ε cos ( Ω ˆ p t ˆ ) ) Z + ε η ε Z 3 + ε α ε ( Z ̇ 2 + Z Z ̈ ) Z = 0 ,

where β = ε β ε , P = ε P ε , η = ε η ε , α = ε α ε .

3 Cantilever beams with various control kinds mathematically

After including several controllers, the improved model system Equation (18) was explored as follows [16]:

(19) Z ̈ + ε β ε Z ̇ + ω ˆ 0 2 ( 1 + ε P ε cos ( Ω ˆ p t ˆ ) ) Z + ε η ε Z 3 + ε α ε ( Z ̇ 2 + Z Z ̈ ) Z = F c ( t ) ,

where F c ( t ) is the control input, and it can be stated as follows using different types of controls to lessen the vibrations that occur in the principal parametric resonance case ( Ω ˆ p = 2 ω ˆ 0 ) as follows:

First type: IRC

(20) v ̇ 1 + σ v 1 v 1 = λ v 1 Z F c ( t ) = ε k v 1 v 1 .

Second type: PPF

(21) u ̈ 1 + ε μ 1 u ̇ 1 + ω 1 2 u 1 + ε δ 1 u 1 3 = ε λ u 1 Z F c ( t ) = ε k u 1 u 1 .

Third type: Nonlinear integral positive position feedback (NIPPF)

(22) u ̈ 1 + ε μ 1 u ̇ 1 + ω 1 2 u 1 + ε δ 1 u 1 3 = ε λ u 1 Z v ̇ 1 + σ v 1 v 1 = λ v 1 Z F c ( t ) = ε k u 1 u 1 + ε k v 1 v 1 .

Fourth type: Nonlinear proportional-derivative (NPD) within NPDCVFC

(23) Z ̈ + ε β ε Z ̇ + ω ˆ 0 2 ( 1 + ε P ε cos ( Ω ˆ p t ˆ ) ) Z + ε η ε Z 3 + ε α ε ( Z ̇ 2 + Z Z ̈ ) Z = F c ( t ) , F c ( t ) = − ε ( p 1 Z + d 1 Z ̇ + α 1 Z 3 + α 2 Z 2 Z ̇ + α 3 Z Z ̇ 2 + G 1 Z ̇ 3 ) ,

where − ( p 1 Z + d 1 Z ̇ ) , is the linear control force, − ( α 1 Z 3 + α 2 Z 2 Z ̇ + α 3 Z Z ̇ 2 ) is the non-linear control force, and G 1 , G 2 are the gains.

Since the fourth control kind is the most effective at suppressing vibrations in the worst resonance condition, we have only included it in this area of the mathematical study. Figure 2 displays the model with the controller under consideration.

Figure 2

A schematic diagram model of the cantilever beam with excitation via new controller NPDCVFC.

3.1 Perturbation and stability analysis

The response is approximated as [57] in the first-order approximation of the multiple time scale

(24) Z ( t ˆ , ε ) = Z 0 ( T 0 , T 1 ) + ε Z 1 ( T 0 , T 1 ) + O ( ε 2 ) ,

where T 0 = t and T 1 = ε t represent time scales. Substituting Equation (24) into Equation (19) within Equation (23) and merely accepting the terms of the order O ( ε 0 ) and O ( ε 1 ) taking into consideration the equation’s outcomes

(25) D 0 2 Z 0 + ε ( D 0 2 Z 1 + 2 D 0 D 1 Z 0 ) + ε ( D 0 2 Z 1 + 2 D 0 D 1 Z 0 ) + ε β ε ( D 0 Z 0 ) + ε η ε Z 0 3 + ω ˆ 0 2 ( 1 + ε P ε cos ( Ω ˆ p t ˆ ) ) ( Z 0 + ε Z 1 ) + ε α ε Z 0 ( ( D 0 Z 0 ) 2 + Z 0 ( D 0 2 Z 0 ) ) + ε p 1 ( Z 0 + ε Z 1 ) + d 1 ( D 0 Z 0 ) + α 1 ( Z 0 3 ) + α 2 ( ( Z 0 + ε Z 1 ) 2 ( D 0 Z 0 ) ) + α 3 Z 0 ( D 0 Z 0 ) 2 + G 1 ( D 0 Z 0 ) 3 = 0 ,

Wherever, the derivatives are with respect to t ˆ . Associating terms of the same order of ε in Equation (25) yields

(26) O ( ε 0 ) : ( D 0 2 + ω ˆ 0 2 ) Z 0 = 0 ,

(27) O ( ε 1 ) : ( D 0 2 + ω ˆ 0 2 ) Z 1 = − 2 D 0 D 1 Z 0 − β ε D 0 Z 0 − η ε Z 0 3 − α ε Z 0 ( D 0 Z 0 ) 2 − α ε Z 0 2 D 0 2 Z 0 − 1 2 ω ˆ 0 2 P ε ( e i Ω ˆ p T 0 + e − i Ω ˆ p T 0 ) Z 0 − p 1 Z 0 − d 1 ( D 0 Z 0 ) − α 1 Z 0 3 − α 2 Z 0 2 ( D 0 Z 0 ) − α 3 Z 0 ( D 0 Z 0 ) 2 − G 1 ( D 0 Z 0 ) 3 .

The solution for Equation (26) is stated in the form

(28) Z 0 = A ( T 1 ) e i ω ˆ 0 T 0 + c c .

Substituting Equation (28) into Equation (27) yields

(29) ( D 0 2 + ω ˆ 0 2 ) Z 1 = − 2 i ω ˆ 0 D 1 A − i ω ˆ 0 β ε A − 1 2 ω ˆ 0 2 P ε A ¯ e i ( Ω ˆ p − 2 ω ˆ 0 ) T 0 − 3 η ε A 2 A ¯ + 2 α ε ω ˆ 0 2 A 2 A ¯ − p 1 A − i ω ˆ 0 d 1 A − 3 α 1 A 2 A ¯ − 3 i α 2 A 2 A ¯ − α 3 ω ˆ 0 2 A 2 A ¯ − 3 i G 1 ω ˆ 0 3 A 2 A ¯ e i ω ˆ 0 T 0 + NST + cc ,

where cc stands for the complex conjugate of the preceding terms and NST denotes non-secular terms. Taking into account the principal parametric resonance scenario Ω ˆ p = 2 ω ˆ 0 , the frequency detuning parameter σ ˆ p for the parametric excitation frequency is characterized by way of

(30) Ω ˆ p − 2 ω ˆ 0 = ε σ ˆ p .

The solvability conditions in Equation (29) produce when terms that produce secular terms are rejected using Equation (30). Considering Equation (30), eliminating the secular terms in Equation (29) yields

(31) 2 i ω ˆ 0 D 1 A + i ω ˆ 0 β ε A + 1 2 ω ˆ 0 2 P ε A ¯ e i σ ˆ p T 1 − 3 η ε A 2 A ¯ + 2 α ε ω ˆ 0 2 A 2 A ¯ − p 1 A − i ω ˆ 0 d 1 A − 3 α 1 A 2 A ¯ − 3 i α 2 A 2 A ¯ − α 3 ω ˆ 0 2 A 2 A ¯ − 3 i G 1 ω ˆ 0 3 A 2 A ¯ = 0 .

To separate the averaging conditions that administer the elements of Equation (31), let express A and A ¯ are exposed in the next polar formulas

(32) A = 1 2 a ( T 1 ) exp [ i θ ( T 1 ) ] , A ¯ = 1 2 a ( T 1 ) exp [ − i θ ( T 1 ) ] ,

where a and θ are the steady-state amplitudes and phases, respectively. Changing Equation (32) hooked on Equation (31), we obtain

(33) − i ω ˆ 0 a ′ − 1 2 ω ˆ 0 a ( σ ˆ p − γ ′ ) + 1 2 i ω ˆ 0 β ε a + 1 4 ω ˆ 0 2 a P ε ( cos γ + i sin γ ) − 3 8 η ε a 3 + 2 8 α ε ω ˆ 0 2 a 3 − 1 2 p 1 a − 1 2 i ω ˆ 0 d 1 a − 3 8 α 1 a 3 − 3 8 i α 2 a 3 − 1 8 α 3 ω ˆ 0 2 a 3 − 3 8 i G 1 ω ˆ 0 3 a 3 = 0 ,

where γ = σ ˆ p T 1 − 2 θ , separating real and imaginary elements, we obtain

(34) a ′ = 1 2 d 1 − β ε − 1 2 ω ˆ 0 P ε sin γ a + 3 8 α 2 ω ˆ 0 + G 1 ω ˆ 0 2 a 3 ,

(35) a γ ′ = σ ˆ p − 1 2 ω ˆ 0 P ε cos γ + p 1 ω ˆ 0 a + 1 4 3 ω ˆ 0 η ε − 2 ω ˆ 0 α ε + 3 ω ˆ 0 α 1 + α 3 ω ˆ 0 a 3 .

For steady-state responses ( a ′ = γ ′ = 0 ) , the periodic solution corresponding to Equations (34) and (35) is given by

(36) d 1 − β ε − 1 2 ω ˆ 0 P ε sin γ a + 3 4 α 2 ω ˆ 0 + G 1 ω ˆ 0 2 a 3 = 0 ,

(37) σ ˆ p − 1 2 ω ˆ 0 P ε cos γ + p 1 ω ˆ 0 a + 1 4 3 ω ˆ 0 η ε − 2 ω ˆ 0 α ε + 3 ω ˆ 0 α 1 + α 3 ω ˆ 0 a 3 = 0 .

Subsequently, the steady-state responses can be derived from the algebraic equations by the use of the Newton–Raphson approach and MATLAB software. The stability of the steady-state shell system is assessed by finding the right-hand side eigenvalues of the Jacobian matrix at Equations (34) and (35) using the Lyapunov first approach

(38) a ′ γ ′ = R 11 R 12 R 21 R 22 a γ ,

where

R 11 = ∂ a ′ ∂ a = 1 2 d 1 − β ε − 1 2 ω ˆ 0 P ε sin γ + 9 8 α 2 ω ˆ 0 + G 1 ω ˆ 0 2 a 2 , R 12 = ∂ a ′ ∂ γ = − 1 4 ω ˆ 0 a P ε cos γ

R 21 = ∂ γ ′ ∂ a = 1 a σ ˆ p − 1 2 ω ˆ 0 P ε cos γ + p 1 ω ˆ 0 + 3 4 3 ω ˆ 0 η ε − 2 ω ˆ 0 α ε + 3 ω ˆ 0 α 1 + α 3 ω ˆ 0 a ,

R 22 = ∂ γ ′ ∂ γ = 1 2 ω ˆ 0 P ε sin γ

The following determinant in the preceding matrix must be solved in order to determine the stable regions of the controlled model

(39) R 11 − λ R 12 R 21 R 22 − λ = 0 .

Then,

(40) λ 2 + ρ 1 λ + ρ 2 = 0 ,

where λ indicates the Jacobian matrix’s eigenvalue, ρ 1 = − R 11 − R 22 and ρ 2 = R 11 R 22 − R 12 R 21 . The Routh–Hurwitz criteria state that the following are ρ 1 > 0 , and ρ 2 > 0 , necessary and sufficient requirements for the structure to be stable. If the real parts of the eigenvalues are negative, the system is stable; if not, it is unstable. In frequency response curves, stable and unstable periodic responses are represented by solid and dotted lines, respectively.

3.2 Experimental model study

A vibration controller, power amplifier, shaker, cantilever beam, signal analyzer, data collection computer, and two accelerometers make up the experimental apparatus are used to conduct the studies. This configuration, which uses the novel controller NPDCVFC, is roughly block diagrammed in Figure 3 and illustrated in Figure 4 as described in Aghamohammadi et al. [16]. The numerical results and their corresponding numerical outcomes are presented in the following section.

Figure 3

An experimental set-up schematic block design.

Figure 4

Design of a parametrically energized cantilever framework with a tip mass.

4 Outcomes and simulation discussions

4.1 Numerical simulation with time history

Equations (19)–(23) just illustrated the nonlinear dynamical structure. Four different types of control techniques (IRC–PPF–NIPPF–NPDCVFC) were then attached, and the 0 18.0 computer programmer was used to numerically simulate them in order to determine which control would minimize the oscillations in the model. The time history of Figures 5–9 is used to illustrate the parameter values as follows:

$Figure 5 Uncontrolled framework vibration amplitude on the measured principal parametric resonance case ( Ω ˆ p = 2 ω ˆ 0 ) ({\hat{\Omega }}_{\text{p}}=2{\hat{\omega }}_{0}) .$

Figure 5

Uncontrolled framework vibration amplitude on the measured principal parametric resonance case ( Ω ˆ p = 2 ω ˆ 0 ) .

$Figure 6 Controlled structure vibration amplitude inside the measured principal parametric resonance item ( Ω ˆ p = 2 ω ˆ 0 ) ({\hat{\Omega }}_{p}=2{\hat{\omega }}_{0}) via IRC.$

Figure 6

Controlled structure vibration amplitude inside the measured principal parametric resonance item ( Ω ˆ p = 2 ω ˆ 0 ) via IRC.

$Figure 7 Controlled framework oscillation amplitude at the restrained principal parametric resonance case ( Ω ˆ p = 2 ω ˆ 0 ) ({\hat{\Omega }}_{\text{p}}=2{\hat{\omega }}_{0}) applying PPF.$

Figure 7

Controlled framework oscillation amplitude at the restrained principal parametric resonance case ( Ω ˆ p = 2 ω ˆ 0 ) applying PPF.

$Figure 8 Controlled structure vibration amplitude organized on the dignified principal parametric resonance item ( Ω ˆ p = 2 ω ˆ 0 ) ({\hat{\Omega }}_{\text{p}}=2{\hat{\omega }}_{0}) via NIPPF.$

Figure 8

Controlled structure vibration amplitude organized on the dignified principal parametric resonance item ( Ω ˆ p = 2 ω ˆ 0 ) via NIPPF.

$Figure 9 Controlled framework vibration amplitude concluded the measured principal parametric resonance item ( Ω ˆ p = 2 ω ˆ 0 ) ({\hat{\Omega }}_{\text{p}}=2{\hat{\omega }}_{0}) via NPDCVFC.$

Figure 9

Controlled framework vibration amplitude concluded the measured principal parametric resonance item ( Ω ˆ p = 2 ω ˆ 0 ) via NPDCVFC.

β ε = 0.2 , ω ˆ 0 = 1.5 , P ε = 2.15 , Ω ˆ p = 2 ω ˆ 0 , η ε = 1.15 , α ε = 1.02 , σ v 1 = 0.66 , λ v 1 = 0.7 , k v 1 = 0.5 , μ 1 = 2.0 , ω 1 = 1.6 , δ 1 = 0.5 , λ u 1 = 1.75 k u 1 = 0.4 , p 1 = 0.2 , d 1 = 0.3 , α 1 = 0.4 , α 2 = 0.35 , α 3 = 0.74 , G 1 = 3.6 , ε = 0.25

and zero initial conditions. Figure 5 has attained the basic steady-state amplitude Z ( t ) before starting any controller at the principal parametric resonance item Ω ˆ p = 2 ω ˆ 0 as the worst resonance case of the system. The outcomes for adding the controllers (IRC, PPF, NIPPF, and NPDCVFC) are displayed in Figures 6–9, allowing you to choose the most effective way to lower the high vibration amplitudes.

Based on Figures 6–9, we may conclude that the new controller NPDCVFC is the best. This section discusses the construction of the controller for the active vibration of the measured cantilever beam is covered in this section (Figure 9). As mentioned earlier, NPD and the NCVFC algorithm are used to provide active vibration control. For this structural model, the depreciation rate features are enhanced by this controller rule. It is evident that this straightforward strategy may work well for reducing the infinite norm of vibration amplitudes. The findings demonstrate the effectiveness of the optimization strategy in reducing vibrations and the speed at which vibrations in the measured cantilever beam were suppressed by angularly orienting actuators and sensors in the proper places. Thus, in order to explore the controlled model and examine the acts of different controlled parameters on the framework, we shall select and statistically evaluate it.

4.2 Simulation of stability and the impact of different regulated model coefficients

This section investigated the influence of various parameters on the controlled structure in Equations (36)–(40) besides numerically demonstrated stable and un-stable regions. The helpful case a 1 ≠ 0 , a 2 ≠ 0 is investigated in order to obtain a large number of parameter impacts. As can be seen in Figures 10–18, all curves exhibit only stable sections with no instability zonoutcome for any vibrating systemes when NPDCVFC is indicated to the system. This is another justification for adding this additional controller to the structure, which is a desirable outcome for any vibrating system. Solid curves reflect responses that are stable. The solid line represents the stability regions shown graphically by (ـــــــــــ). Figure 10 depicts the performance of the amplitude–frequency response steady-state response curves. As illustrated in Figure 10, the backbone curve is acquired as a basic case of a against which shows stable region with no instability zones. As damping coefficient β ε decreased the amplitude a is increased and the stability region is increased as appeared in Figure 11. By way of the excitation frequency ω ˆ 0 is increased the steady-state amplitude a is decreased and all regions are stable as shown in Figure 12. Moreover, the diagrams of the nonlinear coefficient α ε is shifted to right when the values are increased with no instability regions as depicted in Figure 13. Also, as the parametric force coefficient P ε is increased the steady-state amplitude a increased with increasing in stability regions as displayed in Figure 14. On the other hand, when the linear control force coefficient d 1 increased the steady-state amplitude a has decreased and all regions are stable as plotted in Figure 15. Besides, the curve of the linear control force coefficient p 1 is shifted to right (S.R) when the values of parameter p 1 are increased with no instability regions as described in Figure 16. Also, as shown in Figure 17, the values of the nonlinear control parameter α 2 is decreased when the amplitude is increased with increasing the stable regions. In the last, as the gain coefficient G 1 increased the steady-state amplitude a 1 be small with stability regions as presented in Figure 18.

$Figure 10 Influence response curves a a versus σ \sigma of the controlled system.$

Figure 10

Influence response curves a versus σ of the controlled system.

$Figure 11 Impact of damping coefficient β ε {\beta }_{\varepsilon } .$

Figure 11

Impact of damping coefficient β ε .

$Figure 12 Effect of excitation frequency ω ˆ 0 {\hat{\omega }}_{0} .$

Figure 12

Effect of excitation frequency ω ˆ 0 .

$Figure 13 Influence of nonlinear coefficient α ε {\alpha }_{\varepsilon } .$

Figure 13

Influence of nonlinear coefficient α ε .

$Figure 14 Influence of parametric force coefficient P ε = 3.5 , 2.15 , 1.5 {P}_{\varepsilon }=3.5,\hspace{.25em}2.15,\hspace{.25em}1.5 .$

Figure 14

Influence of parametric force coefficient P ε = 3.5 , 2.15 , 1.5 .

$Figure 15 Influence of the linear control force coefficient d 1 {d}_{1} .$

Figure 15

Influence of the linear control force coefficient d 1 .

$Figure 16 Influence of the linear control force coefficient p 1 {p}_{1} .$

Figure 16

Influence of the linear control force coefficient p 1 .

$Figure 17 Influence of nonlinear control parameter α 2 {\alpha }_{2} .$

Figure 17

Influence of nonlinear control parameter α 2 .

$Figure 18 Influence of nonlinear control gain G 1 {G}_{1} .$

Figure 18

Influence of nonlinear control gain G 1 .

4.3 Comparison of the same model with more recent studies

A parametrically energized cantilever beam (PECB) with a tip mass model system akin to Equation (18) was examined in previous studies [14,16,17]. However, they applied the multiple time-scale process within the principal parametric resonance item to study the behavior of the structure under mixed parametric and harmonic excitations, eliminating the need for a controller. The development of the model given by Aghamohammadi et al. [16] is examined in the current work. I add a variety of control strategies to the vibrating structure system’s modified system in order to determine which one reduces the framework structure’s risk of vibration. Additionally, the upgraded system’s new controller (NPDCVFC) is examined in this article. The results of this research show that the novel controller less than the other controllers decreases the high vibrational amplitude of the model exposed to parametric stimulation inside the principal parametric resonance, as shown in Section 4.1. The perturbation approach is used to aid in the acquisition of analytical solutions. Plotting of the frequency response graphs occurs at different framework parameter levels. We end with a numerical validation of the obtained results. The comparisons show that the current approach produces findings that are remarkably similar to those found in Aghamohammadi et al. [16] and that the discrepancies are less than 1%. The influence response on recent works [14,16,17] with and without a controller is also compared to the current study’s NPDCVFC controller, as displayed in Figure 19.

Figure 19

A comparison of the current work’s results with those from earlier studies [14,16,17] both with and without a controller.

5 Conclusion

This work aims to present a new controller, NPDCVFC, that operates on (PECB) with a model of the tip mass. To raise the nonlinear coefficients, it blends NPD with the inclusion of NCVFC. The perturbation strategy is suitable for approximating the solution of the considered controlled model. The item that is regarded as the worst is the primary parametric resonance item ( Ω ˆ p = 2 ω ˆ 0 ) . To determine which controller design method reduces high amplitude vibrations at the major parametric resonance case the best, a comparison of several controller design methods (PPF control, NIPPF control, IRC control, and NPDCVFC as novel control techniques) has been made. It has been shown that the novel controller, NPDCVF, is very effective in reducing vibrations in the structure close to the resonance case under consideration. For every frequency response curve, the unstable and stable zones were obtained through the execution of a numerical stability study. The numerical clarification of the model’s amplitude variation inside frequency response curves under the new controller has been considered. The results demonstrate how remarkably well the unique measured controller lowers the vibrations of the system under investigation. The lack of zones of instability in the response curves is another unique result of the novel controller that has been demonstrated. Numerical research has been done on the impacts and implications of each coefficient on the changed controlled system inside the frequency response curves.

Acknowledgment

The author extends his appreciation to Prince Sattam bin Abdulaziz University for funding this research work through Project number PSAU/2023/01/25363.

Funding information: This work was supported by Prince Sattam bin Abdulaziz University in Alkharj (Grant No. PSAU/2023/01/25363).
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.
Conflict of interest: The author states no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2024-02-21

Revised: 2024-05-17

Accepted: 2024-06-06

Published Online: 2024-07-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/eng-2024-0055

Schlagwörter für diesen Artikel

parametrically excited; cantilever beam; perturbation; stability; IRC; PPF; NIPPF; and NPDCVF controllers

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