Some Problems Associated with Digital Control of Dynamical Systems

In the transmission process of gear system, the change of load will make the system in different states of motion, which affects the transmission efficiency of gear system. It is important to investigate the nonlinear dynamic characteristics of gear system under different load states. Using straight cylindrical gears as the object of study, the concentrated mass method is used to establish a dynamic model that takes into account nonlinear factors such as tooth side clearance, time-varying meshing stiffness and transmission errors. The differential equations of the system are solved by the Longe-Kutta method to obtain the bifurcation diagram, the maximum Lyapunov exponent diagram and the phase plane diagram of the gear system to analyze the effect of the meshing damping ratio on the dynamic characteristics of the system under different load states. The results show that the influence of the engagement damping ratio on the dynamic characteristics of the system is greater under light load conditions, showing different states of motion as the engagement damping ratio gradually increases. Under heavy load conditions, the effect of the engagement damping ratio on the dynamic characteristics of the system is small. Appropriately increasing the mesh damping ratio is beneficial to the gear system to avoid the chaotic zone and maintain a stable cyclic motion state. The results of the study provide a reference for the design of gear systems with variable loads.

Experimental study on thermal dynamic characteristics of gear transmission system

Based on energy conservation principle and heat transfer mechanisms, a set of dynamic model was established for an indirect district heating system. Dynamic start-up process under ideal condition was simulated and the influence of heat loss from piping network, make-up water and intermediate heat exchange station, and radiator area surplus on the dynamic characteristics of the system was discussed. Dynamic process without control was simulated under the hourly outdoor temperature measured in the days from February 13 to 19, 2012. To overcome the obvious indoor temperature variation with outdoor temperature, four PI control strategies were proposed. The dynamic characteristic and energy consumption of the system operated under these four control strategies were discussed. It is found that the control strategy 3, controlling the fuel flow rate, and water flow rate in the first loop not only produce stable working performance but also obtain considerable energy saving potentials. These results are very helpful to realize the dynamic characteristics of the indirect district heating system and to select suitable control strategy and optimal performance.

Studies on the planetary gear have attracted considerable attention because of its advantages, such as compactness, large torque-to-weight ratio, vibrations, and high efficiency, which have resulted in its wide applications in industry, wind turbine, national defense, and aerospace fields. We have established a novel dynamic model of the planetary gear transmission by using Newton’s theory, in which some key factors such as time-variant meshing stiffness, phase relationships, and tooth contact characteristics are considered. The influences of gear axial tipping, operating conditions, and the meshing phase on the contact characteristics and the dynamic characteristics were researched systematically. It was found that the contact area of the tooth surface was moved due to the axial gear tipping, which obviously affected the meshing stiffness. With the increase in the inclination angle of the sun gear, the meshing stiffness decreases, which produces an evident influence on the high natural frequency in the planetary transmission system. In terms of the dynamic characteristics of the system, the component of rotating frequency appeared in the dynamic meshing force of the sun gear and the planetary gear. Moreover, the floating track of the center wheel varied significantly and exhibited an oval distribution as the inclination angle of the sun gear changed. When the inclination angle of the sun gear increased, the rotating frequency component increased significantly, but the other meshing frequency components remained unchanged; meanwhile, the deformation of the floating track also increased. If the inclination angle of the sun gear changes, the vibration state of the system and the collision impact could become more serious, and the lifetime of the planetary transmission system will reduce. Furthermore, when the load was increased, we found that the gear-tooth contact zone transformed from line contact to surface contact, the meshing stiffness increased, the effect of high natural frequency on the planetary transmission system became more evident, but its low-order natural frequency remained stable. With regard to the dynamic characteristics of the system, the components of the major frequency at the external gearing remained unchanged, but the rotation frequency of the sun gear and the meshing frequency amplitude increased linearly with the increase in load. In conclusion, the variation in the meshing stiffness of the planetary gear system had minor impact on the low-order natural frequency, but had a significant impact on the high natural frequency of the planetary transmission system due to the phase variation of the gear.

The coupled heave and pitch motions of a ship sailing in head waves affect the stability of the marine rotor-bearing system. Based on the theory of analytical mechanics, this study establishes a dynamic model of the rotor-bearing system subjected to the coupled motions of heave and pitch, considering nonlinear oil film moments produced by the tilting of the rotor in the bearings. The nonlinear dynamic behaviours of the system are analysed using numerical methods to obtain Poincaré sections, bifurcation diagrams, and the largest Lyapunov exponents. The results show that dynamic bifurcation characteristics reveal complex quasi-periodic motion of upper and lower branches after the initial instability of the system, and the speed of second instability increase markedly. At high speeds, the amplitude of the rotor system increases sharply, which can cause the rotor to touch the inner wall of the bearings in the quasi-periodic state and a failure to transition to the chaotic state. Additionally, the effects of heave and pitch amplitude variations on the dynamic characteristics of the system are also discussed.

The slender type tool usually can be used in milling the thin-walled components in a narrow and deep processing space, and the tool and the workpiece are usually deemed a system because the stiffness of both is closed. The milling stability is influenced by both the coupling deflection and dynamic characteristics of the tool-workpiece system. In order to investigate the time variation of the coupling deflection and dynamic characteristics of the tool-workpiece system, a new computational model was established in this paper. The workpiece was divided into several milling units, and the whole milling process was divided into a series of corresponding sub-milling processes. The coupling deflection of the tool-workpiece system in the sub-milling process was induced by milling force, and the milling force in the next sub-milling process was reversely induced by the coupling deflection of the tool-workpiece system, and by this analogy. Therefore, the thicknesses of units were different under the interaction between the milling forces and the coupling deflections. The changes of dynamic characteristics of the tool-workpiece system were induced by the changes of thicknesses of units. Based on the influences of the coupling deflection and the dynamic characteristics of the tool-workpiece system, a new stability lobe diagram which can show different stability zones and chatter zones in different process positions was obtained. Experimental testing has been conducted to validate the newly established model.

STRUCTURAL IDENTIFICATION FROM AMBIENT VIBRATION MEASUREMENT USING THE MULTIVARIATE AR MODEL

In this study, the dynamic model for the herringbone planetary gear transmission system is established by the lumped parameter method based on the system dynamics and the Lagrange equation, and the impact of the support stiffness and the torsional stiffness on dynamic characteristics is studied. The research results have a guiding significance for the design of the herringbone gear transmission system. In this model, the herringbone gear is treated as a special gear coupled by 2 opposite helical gears, where the stagger angle, comprehensive meshing error, support stiffness, support damping, and load inertia are considered in the analysis of dynamics. Moreover, the dynamic characteristic of the carrier is considered as well. By calculating the meshing force curve of the transmission system, the impact of the stagger angle, supporting stiffness, and the torsional stiffness on meshing force and load sharing coefficient is analyzed. The results show that the stagger angle has an obvious impact on load sharing coefficient while it has little impact on maximum meshing force. And the support stiffness has a more obvious impact on the dynamic characteristics of the system. The recommendary support stiffness of the system is that all of the support stiffness of the sun gear, planetary gear, ring gear, and carrier is 107 N/m. The torsional stiffness has little impact on the dynamic characteristics of transmission system, except the torsional stiffness of planetary gear, and carrier has an obvious impact on load sharing coefficient. The commercial software ADAMS carried out dynamics analysis of the transmission system to verify the necessity validity of the theoretical analysis.

Modern power systems are so large that it is impossible to design a single central control system that would handle the overall control job. A development of techniques for the design of multivariable control systems is of considerable practical importance. The systems have a great variety of control methods, i.e., fuzzy control, neural control, multichannel control, etc. This paper reports the hierarchy power system's dynamic characteristics, accompanied by the synthesis of the interconnected power system and load frequency control method of weakly decoupling state feedback integrator.

The article provides a detailed analysis of brushless motors (BM) within the context of automated control, focusing on dynamic modes and mathematical modeling of the electric drive. It addresses the dynamic characteristics, transfer function development, and specific operational aspects of BM in systems with regulated rotational speed. The article presents experimental data and computer simulation results that validate the effectiveness of the selected models. Experimental studies conducted on a laboratory model confirm the accuracy of the developed mathematical model, particularly demonstrating the feasibility of approximating the motor's transfer function in a linear form to simplify control system analysis and tuning. Computer simulations are conducted in MATLAB using SIMULINK, which enables precise and illustrative visualization of transient processes in the BM. The article also introduces a subordinate control principle within a closed-loop control system, which ensures control stability and accuracy by dividing into speed and current loops. The simulation results demonstrate the potential of utilizing the developed models for further research and the design of efficient and reliable control systems for brushless motors in industrial settings. The proposed modeling approach paves the way for the creation of energy-efficient electric drives with high-precision control.

As for the vibration problem of rotor system for large rotating machinery such as aero engines, a finite element model for dynamic similarity experiment system of rotor is established based on finite element theory. The effect of support stiffness and support position on the dynamic characteristics of the rotor system have been emphatically analyzed. Moreover, the effect of support stiffness on the dynamic characteristics of rotor system have been verified on the designed test rig. The results show that, under different support stiffness intervals, each order natural frequency of the rotor system has a different variation tendency with the support stiffness or position. In addition, the variation of the mode shapes with the support stiffness or position are closely related to the variation of the natural frequencies with the support stiffness and position. Combining with the practice, the natural frequencies of the rotor system can be reasonably designed and the harmful mode shapes can be avoided by adjusting the supporting characteristics of the rotor system.

Part 1 Computers in control systems: historical development digital computer interfacing structure of a digital control loop interface units data representation and aliasing. Part 2 The z-transformation: definition of the z-transform z-transform properties inverse z-transform pulse of z-transfer functions of systems discrete approximations numerical integration pole-zero mapping modified z-transform multi-rate sampling systems analysis of multi-rate systems fast-slow multi-rate systems closed-loop multi-rate systems. Part 3 Analysis of sampled-data control systems z-transformation of systems stability in discrete systems time domain analysis root locus analysis frequency domain analysis. Part 4 Digital compensator design: continuous domain design digital design digital root locas design state feedback design digital PID control disign deadbeat response design optimal control design controller design in the presence of noise. Part 5 Real-time computer control systems: hardware requirements the digital computer and peripheral equipment the real-time clock interface units software aspects controller implementation programming languages system design co-ordination sampling rate selection and quantization errors computer/device communications interrupt handling multi-tasking and background/foreground. Part 6 Advances in computer control: artificial intelligence expert systems examples of expert systems knowledge based systems in control engineering conventional controller design methods manual control methods fuzzy sets and rule-based control basic fuzzy operations fuzzy relations fuzzy control algorithms parallel processing techniques parallel computers algorithms for parallel (MIMD) computers programming parallel computers performance of parallel computers parallel processing in control engineering matrix algebra numerical integration fault tolerance. Appendices: Table of laplace and z-transforms continuous second-order systems response specifications frequency response characteristics the transputer and occam occam overview transputer systems and occam configuration vibration control of a flexible cantilever system simulation modelling and state estimation controller design user interface solutions.

The study on the position-dependent dynamic characteristics of a vertical ball screw feed system without counterweight is an important step in the enhancement of the structural performance of mini-type vertical milling machines. The ball screw is generally driven by a servomotor, which converts a rotary motion into a linear motion through a screw–nut pair. To assess the position-dependent dynamic characteristics of a vertical ball screw feed system subjected to the influence of the screw–nut joint stiffness, a variable-coefficient lumped parameter model of the system is developed. This model is established taking into account the screw–nut joint stiffness under three different strategies: (1) considering the preload and the weight of the spindle system, (2) considering the elastic deformation but ignoring the effect of the weight, and (3) a perfectly rigid model. The differences between the three models in predicting the position-dependent dynamic characteristics of the system are compared, revealing that the stiffness of screw–nut joint greatly affects the vibratory behavior of the spindle system in the transmission direction. A set of conducted experimental results demonstrate that the stiffness model under the preload and the weight of the spindle system is the most accurate model for the prediction of the position-dependent natural frequency and displacement response of the system with the spindle system position. Therefore, it is more suitable for structure design, performance simulation, and evaluation of a vertical ball screw feed system without counterweight.

China Spallation Neutron Source (CSNS) is a high intensity proton accelerator-based facility. Its accelerator complex includes two main parts: an H- linac and a rapid cycling synchrotron (RCS). The RCS accumulates an 80 MeV proton beam and accelerates it to 1.6 GeV, with a repetition rate of 25 Hz. The AC dipole of the CSNS/RCS is operated at a 25 Hz sinusoidal alternating current which causes severe vibration. The vibration will influence the long-term safety and reliable operation of the magnet. The CSNS/RCS AC dipole-girder system takes vibration isolator to decrease the vibratory force and the vibration amplitude of the dipole. For the long-term safety and reliable operation of the dipole, it is very important to study the dynamic characteristics of the dipole-girder system. This paper takes the dipole-girder as a specific model system. A method for studying the dynamic characteristics of the system is put forward by combining theoretical calculation with experimental testing. The modal parameters with and without vibration isolator of the dipole-girder system are obtained through ANSYS simulation and testing. Then, the dynamic response of the system is calculated with modal analysis and vibration testing data. With the simulation and testing method, the dynamic characteristics of the AC dipole-girder are studied.

Algorithms of exact restoration in an analytical form of dynamic characteristics of non-stationary dynamic systems are constructed. Non-stationary continuous dynamical systems modeled by Volterra integral equations of the first kind and non-stationary discrete dynamical systems modeled by discrete analogues of Volterra integral equations of the first kind are considered.The article consists of an introduction and three sections: 1) The exact restoration of the dynamic characteristics of continuous systems, 2) The restoration of the transition characteristics of discrete systems, 3) Conclusions. The introduction provides a statement of the problem and provides an overview of dynamical systems for which algorithms for exact reconstruction in ananalytical form of the impulse response (in the case of continuous systems) and the transition characteristic (in the case of discrete systems) are constructed. In the first section, the algorithm is constructed for the exact reconstruction of the impulse response of an non-stationary continuous dynamic system from three interconnected input signals. The first signal may be arbitrary, the second and third signals are associated with the first signal by integral operator. The exact formula for the Laplace transform of the impulse response, represented by an algebraic expression from the Laplace transform of the system output signals, is given. A model example illustrating the effectiveness of the algorithm is given. The practical application of the presented algorithm isdiscussed. In the second section, an algorithm is constructed for the exact reconstruction of the transition response of a non-stationary discrete dynamical system from three input signals that are interconnected. The first signal may be arbitrary, the second and third signals are associated with the first summing operator. The exact formula of the Z-transform of the transition characteristic is presented, which is represented by an algebraic expression from the Z-transform of the system output signals. A model example is given. The “Conclusions” section provides a summary of the results presented in the article and describes the dynamic systems to which the proposed algorithms can be extended.