Pendulum Angle Research Articles

A modal momentum method for the identification of structural weak links

In order to meet the demands of weak-link identification of mechanical structures in high-precision equipment, this paper introduces the concepts and equations of modal momentum and modal momentum difference ratio (MMDR), as well as an identification method of weak links based on modal momentum. In this method, the modal data of the mechanical structure is acquired first through experimental modal analysis or finite element analysis. The modal momentum of each measured point is calculated based on the modal data and transformed to the global coordinate system. Then, the MMDRs between the measured point with the largest magnitude of modal momentum and its adjacent measured points are calculated. Weak links are judged if the MMDR exceeds a predefined threshold. Twelve different multi-degrees-of-freedom systems incorporating weak links were designed and then identified using the proposed method. The accuracy of identification is up to 100% for these systems. The method was applied to a double pendulum angle milling head of a five-axis numerically controlled machine tool, and the result reveals that the side plate lacks bending stiffness.

Balance Controller Design for Inverted Pendulum Considering Detail Reward Function and Two-Phase Learning Protocol

As a complex nonlinear system, the inverted pendulum (IP) system has the characteristics of asymmetry and instability. In this paper, the IP system is controlled by a learned deep neural network (DNN) that directly maps the system states to control commands in an end-to-end style. On the basis of deep reinforcement learning (DRL), the detail reward function (DRF) is designed to guide the DNN learning control strategy, which greatly enhances the pertinence and flexibility of the control. Moreover, a two-phase learning protocol (offline learning phase and online learning phase) is proposed to solve the “real gap” problem of the IP system. Firstly, the DNN learns the offline control strategy based on a simplified IP dynamic model and DRF. Then, a security controller is designed and used on the IP platform to optimize the DNN online. The experimental results demonstrate that the DNN has good robustness to model errors after secondary learning on the platform. When the length of the pendulum is reduced by 25% or increased by 25%, the steady-state error of the pendulum angle is less than 0.05 rad. The error is within the allowable range. The DNN is robust to changes in the length of the pendulum. The DRF and the two-phase learning protocol improve the adaptability of the controller to the complex and variable characteristics of the real platform and provide reference for other learning-based robot control problems.

Analysis and Simulation of Permanent Magnet Adsorption Performance of Wall-Climbing Robot

In response to problems such as insufficient adhesion, difficulty in adjustment, and weak obstacle-crossing capabilities in traditional robots, an innovative design has been developed for a five-wheeled climbing robot equipped with a pendulum-style magnetic control adsorption module. This design effectively reduces the weight of the robot, and sensors on the magnetic adsorption module enable real-time monitoring of magnetic force. Intelligent control adjusts the pendulum angle to modify the magnetic force according to different wall conditions. The magnetic adsorption module, using a Halbach array, enhances the concentration effect of the magnetic field, ensuring excellent performance in high-load tasks such as building maintenance, bridge inspection, and ship cleaning. The five-wheel structural design enhances the stability and obstacle-crossing capability, making it suitable for all-terrain environments. Simulation experiments using Maxwell analyzed the effects of the magnetic gap and the angle between the adsorption module and the wall, and mechanical performance analysis confirmed the robot’s ability to adhere safely and operate stably.

Optimization of Pyroshock Test Conditions for Aerospace Components to Enhance Repeatability by Genetic Algorithms

Electronic components assembled in satellites should be able to withstand the vibration, noise, and impact loads generated by space vehicles during launch. To simulate the impact loading in a laboratory environment, a pyroshock test simulates an impact load resulting from explosions during the stage and pairing separation of launch vehicles, which imposes significant stress on the components, potentially leading to failures and damage. To ensure component reliability before the flight model (FM) stage, where components are mounted on the actual launch vehicle and sent into orbit, a pyroshock test is conducted during the qualification model (QM) stage using identical parts and specifications. This process involves measurements of the acceleration induced by pyroshock to calculate the shock response spectrum (SRS) and evaluate the components’ reliability against the required SRS to confirm their ability to endure the shock and operate normally in post-tests. The aerospace developer determines the SRS requirements based on the space launch vehicle and the installation location of the electronic components. Configuring a suitable pyroshock test to meet these requirements typically involves extensive trial and error. This study aims to minimize such trial and error by examination of SRS changes through a numerical approach by table structural vibration analysis. The structure is subjected to in-plane impacts using a steel ball via a pendulum method. Various SRS profiles are calculated by test factors such as the weight of the steel ball, the pendulum angle, and the installation position of the test specimen. Furthermore, a genetic algorithm is utilized to derive the optimal test conditions that satisfy the required SRS. An automated pyroshock test system is developed to enhance repeatability and reduce human errors.

An Application of DC Motor Modelled Classical Sliding Mode Control with Moving Sliding Surface to Rotary Inverted Pendulum System

The rotary inverted pendulum system (RIP), which is highly favored in control applications, is examined in this work. By determining the coordinates of the Rip elements' centers of gravity, the system's total kinetic and potential energies were determined. The kinetic and potential energy expressions were used to generate the Lagrangian function. Expressions providing the system's equations of motion were discovered by taking the Lagrangian approach into consideration. The motor's equations, which will initiate the system, have also been considered. Through the use of state variables and a Matlab program, the system's pendulum angle was managed by using a moving sliding surface and the traditional sliding mode control technique. Based on the dynamics, the slip surface's slope was computed. The sliding surface's slope was computed based on the system's dynamics. The genetic algorithm was utilized to determine the ideal values for the coefficients employed in the control structure. The findings showed that the inaccuracy was roughly zero and that the pendulum angle took around 1.5 seconds to reach the intended reference value. Furthermore, it noted that the motor torque and current values are 12 Nm and 2.5 amps, respectively. The findings show that the motor values are reasonably similar to the values seen in real-world applications. Control in real-time applications won't be an issue if the motor is chosen based on these values

Optimized Hierarchical Sliding Mode Control for the Swing-up and Stabilization of a Rotary Inverted Pendulum

This paper presents a study on controlling a rotary inverted pendulum (RIP) system using a hierarchical sliding mode control (HSMC) approach. The objective is to swing up and stabilize the pendulum at a desired position. The proposed HSMC controller addresses the underactuation challenge through a hierarchical structure of sliding surfaces. The particle swarm optimization (PSO) algorithm is used to optimize the controller parameters. Simulations were performed to evaluate the performance of the HSMC controller at different initial pendulum angles, demonstrating its effectiveness in achieving swing-up and stabilization. The integration of the PSO algorithm enhances the controller’s adaptability and robustness, emphasizing the benefits of combining optimization algorithms with controller parameter tuning for underactuated systems like the RIP.

LQR and Fuzzy-PID Control Design on Double Inverted Pendulum

Double inverted pendulum is a non-linear and unstable system. Double inverted pendulum can be stabilized in the upright position by providing control to the system. In this research we compare two types of controllers namely Linear Quadratic Regulator (LQR) and Fuzzy-PID. The objective is to determine the control strategy that provides better performance on the position of the cart and pendulum angle. We modelled the system which is then linearized and given control. From the simulation results, it is proven that LQR and Fuzzy-PID controllers have been successfully designed to stabilize the double inverted pendulum. However, when given a disturbance in the form of noise step, the LQR controller has not been able to achieve the desired reference for up to 20 seconds. In another hand, the Fuzzy-PID controller is able to achieve the desired reference after 8 seconds. Therefore, it can be concluded that the Fuzzy-PID controller when applied to the Double Inverted pendulum system has better performance than the LQR controller.

Revealing the mechanism of crack initiation and propagation in CGHAZ of S690QL welded joints through low-temperature impact tests with gradient setting of pendulum angle

As the most brittle areas of welded S690QL high-strength steel structures, coarse-grained heat-affected zone (CGHAZ) was crucial for the structural integrity of components. This paper designed an innovative low-temperature impact testing method with a gradient setting of pendulum angle to investigate the micro-mechanism of crack initiation and propagation in the CGHAZ of S690QL welded joints. The results reveal that continuous blocky M−A constituents were distributed along prior austenite grain boundaries (PAGBs), leading to stress concentration and crack initiation. During low-temperature impact processes, cracks propagate in a transgranular mode along {100} or {110} crystal planes. Due to the interaction between compressive stress and the crack tip, regions with high kernel average misorientation (KAM) such as PAGBs can divert the crack. The coarsening of prior austenite grains in the CGHAZ leads to a diminished hindering effect of PAGBs on crack propagation. The findings elucidate the complex relationship between microstructure and impact toughness of CGHAZ, providing insights into crack behavior and mechanisms in CGHAZ.

Design and Test of Disturbed Fertilizer Strip-Ejection Device with Vertical Pendulum Bar Based on Discrete Element Method

Fertilizer can improve the yield of crops per unit area, and uniform fertilizer discharge can improve the fertilizer utilization rate. Therefore, it is meaningful to improve the performance of fertilizer-discharge devices in order to improve the modernization level of crop field fertilizer management. To address the problems of operational smoothness, stability and poor uniformity of fertilizer discharge, and other difficult problems encountered with strip fertilizer-discharge devices, this study designs a disturbed fertilizer strip-discharge device with a vertical pendulum. The main factors affecting the performance of fertilizer discharge were the wedge angle of the push-disturbing main pendulum bar (PMPB), the inclination angle of the aided-stirring pendulum pick (APP), the flow gap of the pendulum bar (FGPB), and the operation frequency of the swing-rod combination (SRC). The discrete element method (DEM) was used to establish a simulation model of the fertilizer device to explore the influence of the main factors on the performance of fertilizer discharge, with the coefficient of variation (CV) of fertilizer discharge uniformity and fertilizer discharge accuracy (FDA) used as the evaluation indices. The results show that the factors affecting the CV of fertilizer discharge uniformity and FDA were, in order of priority, the operation frequency of the SRC, the FGPB, the wedge angle of the PMPB, and the inclination angle of the APP. The optimal parameters after rounding were as follows: the wedge angle of the PMPB was 45°, the inclination angle of the APP was 46°, the operation frequency of the SRC was 188 times/min, and the FGPB was 4.5 mm. At this point, the model predicted that the CV of fertilizer discharge uniformity would be 10.53%, and that the FDA would be 3.19%. Using the optimal parameters for bench test verification, it was found that the wedge angle of the PMPB was 45°, the inclination angle of the APP was 46°, the operation frequency of the SRC was 188 times/min, the FGPB was 4.5 mm, the CV of the uniformity of the fertilizer discharge was 11.06%, and the FDA was 3.51%. In the test, the fertilizer-discharge device was stable and had good adaptability to different fertilizers. The results of this study can provide a theoretical reference for the development of precision strip-fertilizer application devices.

An Extension of the Feedback Linearization Method in the Control Problem of an Inverted Pendulum on a Wheel

This paper continues previous studies on designing stabilizing control laws for a mechanical system consisting of a wheel and a pendulum suspended on its axis. The control objective is to simultaneously stabilize the vertical position of the pendulum and a given position of the wheel. The difficulty of this problem is that the same control is used to achieve two targets, i.e., stabilize the pendulum angle and the wheel rotation angle. Previously, the output feedback linearization method was applied to this problem. The sum of the pendulum angle and the wheel rotation angle was taken as the output. For the closed loop system to be not only asymptotically stable in the output but also to have asymptotically stable zero dynamics, a dissipative term was added to the output-stabilizing control law. Below, a two-parameter modification of this law is described. Along with the dissipative term, we introduce a positive factor. The more general parameterization allows stabilizing this system in the cases where the control law proposed previously appeared ineffective. The properties of the new control law are investigated, and the attraction domain is estimated. The estimation procedure is reduced to checking the feasibility of linear matrix inequalities.

Analysis of the dynamic characteristics of an eccentric cylinder in axial flow

The simple pendulum model is a fundamental and widely studied concept in physics, its movement follows a tendency to stabilize towards the side with lower gravitational potential energy. Consequently, the simple pendulum structures have been extensive applications in petroleum, aerospace machinery, and others, particularly in gravity induction and actuator systems. The utilization of this model offers versatile solutions for complex engineering challenges. The research object of this study is an eccentric cylindrical stabilized platform. A nonlinear dynamic model of the eccentric cylindrical stable platform in axial flow with variable damping is established. Due to the complexity of the analytical solution of the simple pendulum motion, we adopt the method of numerical analysis to building the simulation and analysis model. Combined with the fluid calculation and dynamics simulation methods, the motion characteristics of the eccentric cylindrical model under steady and unsteady flow fields are analyzed by adjusting the rotational speed, inclination angle, and initial pendulum angle. The experimental platform is built to analyze the motion law of the eccentric cylinder under the action of axial flow, and the experimental results of the motion law of the eccentric cylindrical platform are consistent with the numerical calculation results.

On the dynamics of spring-pendulum system: an overview of configuration space and phase space

The dynamics of the spring-pendulum system with two degrees of freedom were studied. The motion of this system is restricted to be in a vertical plane so that the chosen generalized coordinates are the increased length of the spring and the swing angle of pendulum . Hamiltonian of the system is obtained from the Legendre transformation of Lagrangian. Hamilton’s equation yields four differential equations that represent the dynamic of the system. The obtained results were visualized in configuration space and phase space trajectories. It is shown that generally the greater the initial swing angle, the more complex pattern will occur followed by the appearance of chaotic phenomena.

Coriolis acceleration and critical slowing-down: A quantitative laboratory experiment

We experimentally investigate the motion of a pendulum on a turntable. The dynamics of this conical pendulum experiment are very rich and can be studied both at the undergraduate and graduate levels. At low rotational frequency of the turntable, we measure the Coriolis acceleration. Increasing the rotational frequency, we experimentally demonstrate a supercritical pitchfork bifurcation: above a critical rotational frequency, the pendulum arm spontaneously rises up. Beyond the characterization of the equilibrium pendulum angle, we evidence the so-called critical slowing down corresponding to the increase in the pendulum period when approaching the critical rotational frequency. Bifurcation and critical slowing down are key concepts in the study of critical phenomena that are seldom illustrated experimentally. All our experimental measurements are in excellent quantitative agreement with the theory we provide.

Gain-Scheduled Model Predictive Control for Cart–Inverted-Pendulum with Friction and Disturbances

The cart–inverted pendulum system (CIPS) is a typical example of underactuated mechanical systems. For the CIPS with friction and disturbances, a gain-scheduled model predictive control method is proposed to achieve the upright stabilization objective of the single inverted pendulum (SIP) while controlling the cart to reach a desired new position. To this end, first, a dynamic equation of the CIPS with friction and disturbances is formulated based on the Newton–Euler equation. On the basis of the dynamic equation of the CIPS, its motion characteristics and control process are analyzed. Next, the given dynamic equation of the CIPS is linearized to obtain a series of linearized models at seven different pendulum angles. Then, seven model predictive controllers (MPCs) are designed based on the above-linearized models, respectively. Introducing the idea of the gain-schedule, a gain-scheduled MPC (GSMPC) is designed to switch one of these seven MPCs to realize the control objective of the CIPS, according to the actual pendulum angle of the SIP during the control process. Finally, multi-group simulations that consider the friction and disturbances of the CIPS are implemented to demonstrate the effectiveness of the proposed gain-scheduled model predictive control method.

Research on anti-swing control of tower crane with lifting weight swing function

In order to ensure the accurate positioning of the assembly building, a swing trolley with swing function is designed. In order to solve the swing angle difference generated in the process of adjusting and positioning of the swing trolley, a simplified model is established to analyze the force and establish the relationship between the angular acceleration of the lower lifting beam and the angle of the swing. Secondly, the kinematic analysis is carried out for the rotating pendulum trolley, and its motion model is analyzed under uniform acceleration braking, uniform speed running and uniform deceleration braking. Finally, the angle difference of the rotating pendulum is gradually increasing in the case of different angular acceleration by example. And the variation of the rotating pendulum angle difference in different processes, laying the foundation for subsequent control studies.

FIV and energy harvesting using bluff body piezoelectric water energy harvester with different cross-sections

Research utilizing piezoelectric fiber composites to power network nodes or micro-sensors by capturing energy from the surrounding environment is becoming a hot spot, and the structural design to efficiently improve the performance of energy harvesting is very important. The performance of a vertical cantilever piezoelectric water energy harvester (PWEH) with three different cross-sections including circular, square and equilateral triangle, each with a mass ratio of 4.8, was investigated at Reynolds number within 770–8800. Emphasis is placed on the effects of the bluff body cross-section, load resistance, and flow velocity on the flow-induced vibration (FIV) and energy response of the vertical cantilever beam of the PWEH. Except for the decreasing amplitude in the lower branch of the circular cylinder water energy harvester (CWEH), there is no distinction in the vibration response branches compared to the classical elastically supported circular cylinder. The resistance has little effect on the vibration response, while it has a significant effect on the voltage and power response. The maximum amplitude and pendulum angle of the square cylinder water energy harvester (SWEH) are slightly larger than those of the CWEH. The maximum voltage and power are smaller for the SWEH than for the CWEH. The equilateral triangular cylinder water energy harvester (TWEH) demonstrates excellent performance. It has a maximum amplitude of 3.8 D, 3.4 times that of the CWEH. The peak pendulum angle of TWCH is 13.9°, 3.2 times that of CWEH. It has a voltage of 0.057 V at a resistance of ∞, which is 3.4 times that of CEWH. In addition, the TWCH has an output power of 0.38 nW at a resistance of 106Ω, which is 9.5 times that of the CWEH. This study suggests that the best energy efficiency of TWEH is achieved by coupling vortex-induced vibration (VIV) and galloping excitation.

Enhancing stability control of inverted pendulum using Takagi–Sugeno fuzzy model with disturbance rejection and input–output constraints

The Takagi–Sugeno (T–S) fuzzy model is a versatile approach widely used in system control, often in combination with other strategies. This paper addresses key control challenges linked to the T–S system and presents important considerations to ensure its successful application using the Lyapunov theorem. One crucial aspect is determining the optimal number of premise variables and selecting accurate fuzzy rules for the T–S model. Additionally, the theorem based on Linear Matrix Inequality (LMI) is developed to enable effective disturbance rejection. To enhance stability control, constraints are imposed on the output angle and control input of a rotary inverted pendulum (RIP). By integrating T–S fuzzy control, disturbance rejection, and input/output constraints, robust stability in controlling the RIP is achieved. Extensive simulations are performed to showcase the efficiency of the suggested method, and the simulation results are thoroughly discussed and analyzed to verify the efficacy of the control method.

Implementation of Sliding Surface Moving Anfis Based Sliding Mode Control to Rotary Inverted Pendulum

This study covers the control of the pendulum angle by taking into account the dynamic equations and motor dynamics of the rotary inverted pendulum system, with the help of state variables in the Matlab program, by using the sliding mode control method with sliding surface moving and the adaptive neural fuzzy inference system together. The sliding mode control method with a changing sliding surface is a part of the control structure. The slope of the sliding surface was calculated using the adaptive neural fuzzy inference technique. The optimum values of the coefficients in the adaptive neural-fuzzy inference system structure have been calculated by genetic algorithm. The finding of the coefficients, the sum of the squares of the errors chosen as the objective function. The input of the adaptive neural fuzzy inference system structure consists of the error of the pendulum and the derivative of the error of the pendulum. The gradient of the sliding surface of the sliding mode control structure is the output of the adaptive neural fuzzy inference system structure. According to the findings, the pendulum angle achieved the appropriate reference value after 1.5 seconds, with an error of around zero. It obtained that the engine torque value reaches up to 50 Nm. From here, it is seen that the motor torque values used in practical applications and the motor torque values as a result of this study overlap.

Lagrangian Equation of Coupled Spring-Pendulum System

A coupled spring-pendulum system in a conservative field was studied where the equation of motion of the system using Lagrangian and Hamiltonian equation were obtained. The equation of motion represented by a second-order differential equation from the three generalized coordinate were used. The potential energy equal to zero when the system is in its equilibrium position. The generalized coordinate that being used were the angle of the first pendulum θ_1, the angle of the second pendulum θ_2, and the increase in the length of the spring x. The resulting equation of motion can be used to determine the dynamics behavior of the system at any time. Students' understanding is expected to be more complete by providing a procedure to derive the Lagrangian equation of motion.

Optimal Design and Experiment of Corn-Overlapped Strip Fertilizer Spreader

As the complex terrain in hilly areas is not conducive to corn mid-tillage precision fertilization, a corn-overlapped strip fertilizer spreader was designed without an external power source. By configuring a passive overlapping spreading method with a three-branch split chamber structure, the uniform spreading of fertilizer in strips was achieved. A horizontal and vertical movement model of fertilizer spreading was developed to determine the angle of the fertilizer extending tube, the width of fattening small plates, and the height of the fertilizer spread as the main factors affecting the fertilizer distribution pattern. The single-factor ternary orthogonal rotational combination response surface simulation test was carried out with pendulum angle, width, and height as test factors and the transversal fertilizer uniformity coefficient and longitudinal fertilizer uniformity coefficient as test indicators. The test results showed that the pendulum angle, height, and width had significant effects (p < 0.05) on the transversal fertilizer uniformity coefficient, and the pendulum angle and width had a considerable impact (p < 0.05) on the longitudinal fertilizer uniformity coefficient. In the optimal combination of parameters, swing angle 52°, height 400 mm, and width 50 mm operation, the coefficients of uniformity of both the transversal fertilizer uniformity coefficient and longitudinal fertilizer uniformity coefficient were less than 0.15%. A verification test was carried out under the optimal combination of parameters for the simulation tests with the simulation conditions as the standard. The test results were consistent with the simulation results within the error range. The deviation values of the transversal fertilizer uniformity coefficient and longitudinal fertilizer uniformity coefficient were 8.11% and 9.01%, respectively. The corn-overlapped strip fertilizer spreader was able to complete the fertilizer spreading operation smoothly. This study provides evidence for further optimizing the performance of the corn mid-tillage fertilizer applicator.