Investigation of instability in ultrasonic welding under parametric excitation

2021 ◽  
pp. 002072092098350
Author(s):  
Yongqiang He ◽  
Lewei Zhao
Keyword(s):  
Parametric Excitation ◽  
High Frequency ◽  
Mathieu Equation ◽  
Ultrasonic Welding ◽  
Transition Curve ◽  
Time Varying ◽  
Combination Resonance ◽  
Bending Loads ◽  
Time Dependent Problem ◽  
The Stability

This paper develops a time-varying model for battery tabs based on the parametric excitation of Euler-Bernoulli beams. The instability caused by combination resonance under a high-frequency longitudinal load is considered. A Galerkin procedure is used to discretize the time-dependent problem into the Mathieu equation. The critical axial load is obtained from the transition curve of combination resonance. The effectiveness of the stability analysis was verified by numerical simulations involving longitudinal and bending loads.

Simultaneous Resonance and Anti-Resonance in Dynamical Systems Under Asynchronous Parametric Excitation

2020 ◽  
Vol 15 (9) ◽  
Author(s):  
Artem Karev ◽  
Peter Hagedorn
Keyword(s):  
Dynamical Systems ◽  
Parametric Excitation ◽  
Analytical Method ◽  
Normal Forms ◽  
High Sensitivity ◽  
Combination Resonance ◽  
Concurrent Study ◽  
The Stability ◽  
Analytical Expressions ◽  
Special Case

Abstract Since the discovery of parametric anti-resonance, parametric excitation has also become more prominent for its stabilizing properties. While resonance and anti-resonance are mostly studied individually, there are systems where both effects appear simultaneously at each combination resonance frequency. With a steep transition between them and a high sensitivity of their relative positions, there is a need for a concurrent study of resonance and anti-resonance. The semi-analytical method of normal forms is used to derive approximate analytical expressions describing the magnitude of the stability impact as well as the precise locations of stabilized and destabilized areas. The results reveal that the separate appearance of resonance and anti-resonance is only a special case occurring for synchronous parametric excitation. In particular, in circulatory systems the simultaneous appearance is expected to be much more common.

Stability Analysis of a Rotating System Due to the Effect of Ball Bearing Waviness

2002 ◽  
Vol 125 (1) ◽  
pp. 91-101 ◽  
Author(s):  
G. H. Jang ◽  
S. W. Jeong
Keyword(s):  
Parametric Excitation ◽  
Ball Bearing ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Fourier Series Expansion ◽  
Time Varying ◽  
Algebraic Equations ◽  
Rotating System ◽  
Mathieu’S Equation ◽  
The Stability

This research presents an analytical model to investigate the stability due to the ball bearing waviness in a rotating system supported by two ball bearings. The stiffness of a ball bearing changes periodically due to the waviness in the rolling elements as the rotor rotates, and it can be calculated by differentiating the nonlinear contact forces. The linearized equations of motion can be represented as a parametrically excited system in the form of Mathieu’s equation, because the stiffness coefficients have time-varying components due to the waviness. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as the simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill’s infinite determinant for these algebraic equations. The validity of this research is proven by comparing the stability chart with the time responses of the vibration model suggested by prior research. This research shows that the waviness in the ball bearing generates the time-varying component of the stiffness coefficient, whose frequency is called the frequency of the parametric excitation. It also shows that the instability takes place from the positions in which the ratio of the natural frequency to the frequency of the parametric excitation corresponds to i/2 i=1,2,3,….

Stability of a Time-Delayed System With Parametric Excitation

2006 ◽  
Vol 129 (2) ◽  
pp. 125-135 ◽  
Author(s):  
Nitin K. Garg ◽  
Brian P. Mann ◽  
Nam H. Kim ◽  
Mohammad H. Kurdi
Keyword(s):  
Time Delay ◽  
Parametric Excitation ◽  
Mathieu Equation ◽  
Periodic Coefficient ◽  
Single Element ◽  
New Approach ◽  
Delayed System ◽  
Representative Problem ◽  
The Stability ◽  
Time Periodic

This paper investigates two different temporal finite element techniques, a multiple element (h-version) and single element (p-version) method, to analyze the stability of a system with a time-periodic coefficient and a time delay. The representative problem, known as the delayed damped Mathieu equation, is chosen to illustrate the combined effect of a time delay and parametric excitation on stability. A discrete linear map is obtained by approximating the exact solution with a series expansion of orthogonal polynomials constrained at intermittent nodes. Characteristic multipliers of the map are used to determine the unstable parameter domains. Additionally, the described analysis provides a new approach to extract the Floquet transition matrix of time periodic systems without a delay.

Stabilization of Inverted Coupled Pendula Using Parametric Excitation

1997 ◽  
Author(s):  
Reiko Osada ◽  
Chikara Sato
Keyword(s):  
Parametric Excitation ◽  
Inverted Pendulum ◽  
Mathieu Equation ◽  
Stability Diagram ◽  
Affirmative Answer ◽  
Stable Region ◽  
View Point ◽  
Limited Region ◽  
Resonant Systems ◽  
The Stability

Abstract Parametric stabilization of a single inverted pendulum has been extensively studied using the Mathieu equation and its corresponding stability diagram. The inverted single pendulum may be stabilized using parametric excitation at a specified frequency and amplitude given by a narrow stable region in the Mathieu diagram. Coupled pendula with parametric excitation or corresponding resonant systems have been studied from mathematical view point (Cesari, 1959; Gambill, 1955; Richards, 1983), from electrical view point (Sato, 1962a; Sato, 1962b; Sato, 1971; Sato, 1975) and from mechanical view point (Sato, 1995). Coupled pendula with parametric excitation have been studied within a limited region by some researchers, including the authors. A study of inverted coupled pendula with parametric excitation has not been performed as far as the authors know. Usually it is assumed that inverted coupled pendula are unstable in the absence of any other stabilizing mechanism such as feedback. One question is whether the inverted coupled pendula could be stabilized only by parametric excitation? The present paper gives an affirmative answer to this question in a limited and finite region. The stability is also examined using the differential equations and other methods.

Simultaneous Resonance and Anti-Resonance in Dynamical Systems Under Asynchronous Parametric Excitation

2019 ◽  
Author(s):  
Artem Karev ◽  
Peter Hagedorn
Keyword(s):  
Dynamical Systems ◽  
Parametric Excitation ◽  
Analytical Method ◽  
Normal Forms ◽  
High Sensitivity ◽  
Combination Resonance ◽  
Concurrent Study ◽  
The Stability ◽  
Analytical Expressions ◽  
Special Case

Abstract Since the discovery of parametric anti-resonance, parametric excitation has also become more prominent for its stabilizing properties. While resonance and anti-resonance are mostly studied individually, there are systems where both effects appear simultaneously at each combination resonance frequency. With a steep transition between them and a high sensitivity of their relative positions, there is a need for a concurrent study of resonance and anti-resonance. The semi-analytical method of normal forms is used to derive approximate analytical expressions describing the magnitude of the stability impact as well as the precise locations of stabilized and destabilized areas. The results reveal that the separate appearance of resonance and anti-resonance is only a special case with synchronous parametric excitation. In particular, in circulatory systems the simultaneous appearance is expected to be much more common.

Stability of Linear Systems With Parametric Excitation

1970 ◽  
Vol 37 (1) ◽  
pp. 228-230 ◽  
Author(s):  
J. R. Dickerson
Keyword(s):  
Linear Systems ◽  
High Frequency ◽  
Mathieu Equation ◽  
Time Varying ◽  
Asymptotically Stable ◽  
System Matrix ◽  
Varying Coefficients ◽  
Stability Matrix ◽  
Time Varying Coefficients ◽  
Asymptotic Stability Conditions

A Lyapunov-type approach is used to develop sufficient asymptotic stability conditions for linear systems with time-varying coefficients. In particular, it is shown that parametric disturbances of high frequency cannot create instability in an already asymptotically stable system. Also it is shown that slowly varying parametric disturbances will not cause instability if the system matrix is a stability matrix for all values of time. The results are applied to the Mathieu equation to illustrate the character of the theorems. This example is chosen because of the availability of its exact stability boundaries.

scholarly journals Delay compensation based controller for rotary electrohydraulic servo system

2021 ◽  
Author(s):  
Zakarya Omar ◽  
Xingsong Wang ◽  
Khalid Hussain ◽  
Mingxing Yang
Keyword(s):  
High Frequency ◽  
Dead Time ◽  
Sliding Mode ◽  
Servo System ◽  
Smith Predictor ◽  
Delay Compensation ◽  
Mechanical Parts ◽  
System Output ◽  
Electrohydraulic Servo System ◽  
The Stability

AbstractThe typical power-assisted hip exoskeleton utilizes rotary electrohydraulic actuator to carry out strength augmentation required by many tasks such as running, lifting loads and climbing up. Nevertheless, it is difficult to precisely control it due to the inherent nonlinearity and the large dead time occurring in the output. The presence of large dead time fires undesired fluctuation in the system output. Furthermore, the risk of damaging the mechanical parts of the actuator increases as these high-frequency underdamped oscillations surpass the natural frequency of the system. In addition, system closed-loop performance is degraded and the stability of the system is unenviably affected. In this work, a Sliding Mode Controller enhanced by a Smith predictor (SMC-SP) scheme that counts for the output delay and the inherent parameter nonlinearities is presented. SMC is utilized for its robustness against the uncertainty and nonlinearity of the servo system parameters whereas the Smith predictor alleviates the dead time of the system’s states. Experimental results show smoother response of the proposed scheme regardless of the amount of the existing dead time. The response trajectories of the proposed SMC-SP versus other control methods were compared for a different predefined dead time.

A new control approach for a class of linear switched systems with time-varying delay

2021 ◽  
pp. 014233122199272
Author(s):  
Abbas Zabihi Zonouz ◽  
Mohammad Ali Badamchizadeh ◽  
Amir Rikhtehgar Ghiasi
Keyword(s):  
Stability Analysis ◽  
Linear Matrix Inequalities ◽  
Linear Matrix ◽  
Switching Systems ◽  
Time Varying ◽  
Matrix Inequalities ◽  
Time Varying Delay ◽  
Linear Switching Systems ◽  
The Stability ◽  
Varying Delay

In this paper, a new method for designing controller for linear switching systems with varying delay is presented concerning the Hurwitz-Convex combination. For stability analysis the Lyapunov-Krasovskii function is used. The stability analysis results are given based on the linear matrix inequalities (LMIs), and it is possible to obtain upper delay bound that guarantees the stability of system by solving the linear matrix inequalities. Compared with the other methods, the proposed controller can be used to get a less conservative criterion and ensures the stability of linear switching systems with time-varying delay in which delay has way larger upper bound in comparison with the delay bounds that are considered in other methods. Numerical examples are given to demonstrate the effectiveness of proposed method.

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