Nonlinear Approaches in Engineering Application

Complex dynamic systems are described by differential dynamic equations, mostly nonlinear without closed form analytic solution. To solve them numerically, there are many methods such as Euler’s method, Taylor-series method and Runge-Kutta method, etc., each with advantages and disadvantages. In this chapter, a novel analytical and numerical methodology for the solutions of linear and nonlinear dynamical systems is introduced. The piecewise constant argument method combined with the Laplace transform, makes the new method called Piecewise constant argument-Laplace transform (PL). This method provides better reliability and efficiency for solving coupled dynamic systems. In addition, the numerical solutions of linear and nonlinear dynamic systems can be obtained smoothly and continuously on the entire time range from zero to t. The numerical results of the analytical solution of the method are given and compared with the results of the 4th-order Runge-Kutta (RK4) method, and the accuracy and reliability of the PL method are verified.

In this chapter, a novel predictor-corrector formulation is introduced based on the Bézier curve interpolation. The proposed formulation can be employed to determine numerical integration and solve nonlinear initial value problems with low computational cost while having larger stability region. The stability region of the method is examined in solving benchmark initial value problems, which showed a larger stability region both on the imaginary and real axis compared to some other well-known methods. The results of applying this formulation in solving Duffing and Van der Pol equations as numerical experiments showed that the proposed method is more stable and accurate for a wide range of step size compared to other well-known methods. Considering performance of the method in terms of accuracy, stability, and cost, it is expected that the method to be used is more complicated and applied nonlinear science and engineering problems.

This chapter proposes a simple approach for the control of nonlinear dynamical systems with nonhyperbolic equilibrium points. Such equilibrium points are generally much more difficult to analyze dynamically, and correspondingly the control of nonlinear systems in the vicinity of such points can often become more difficult. The aim is to bring about asymptotically stable behavior of the controlled system in the vicinity of the nonhyperbolic equilibrium point. A new way to control such systems is proposed here through control of their local center manifolds. A simple and effective methodology for doing this is provided, and its advantages are illustrated through several examples.

This book chapter presents a systematic overview of the novel concept of “inertial morphing (IM)”, first introduced by the authors in 2017 and further expanded in their following publications. It involves deliberate changes of the inertial properties of the system for control of the attitude of the spacecraft.

The “inertial morphing” control concept is essentially based on the realisation that the spinning spacecraft can be seen and utilised as gyroscope itself, instead of utilisation of complex, heavy and energy-consuming gyroscopic devices on-board. Utilisation of the concept, therefore, enables reduction of the weight and dimensions of the conventional systems.

It has been discovered and demonstrated via versatile numerical simulations that IM can be used to enable spacecraft with wide range of attitude control capabilities (e.g. 90° and 180° inversions, de-tumbling and controlled agility acrobatic manoeuvrings). Moreover, it has been also discovered that control of very complex manoeuvres can be achieved with a few only controlled inertial morphing actions (two and three morphings correspondingly for 180° and 90° inversions).

The general control methods presented in this chapter are based on the geometric interpretation of the arbitrary 3D rotational motion of the spacecraft, using angular momentum sphere and kinetic energy ellipsoid in the non-dimensional coordinates. The key control strategies involve combination of installing the angular momentum vector into transition polhodes and installing into transition separatrices.

Reduction in weight and dimensions, simplicity of the implementation of the inertial morphing and simplicity of the attitude control, requiring two or three discrete control actions, make this technology attractive for a variety of applications, especially involving autonomous spacecraft.

One of the remarkable features of the IM control is the ability to access a range of solutions between agile (fast) and prolonged (slow) types and select the most appropriate speed of the undertaking attitude manoeuvre. This added variable agility may be useful, for example, to perform for autonomous spacecraft surveillance, landing or manoeuvring. In particular, the IM may foster effective protection of the spacecraft from hostile environments (asteroids, radiation, etc.), as the spacecraft would be able to quickly expose the most protective surfaces to the sources of danger, hence prolonging survivability of the system. In the other cases of capturing the tumbling spacecraft, the prolonged mode can be selected, allowing more time for the capture and handling.

For the practical implementation of the IM concept, this book chapter also presents a range of conceptual mechanical designs. As Euler’s equation for the rotational motion of the rigid bodies paved the way for the development of the theory of gyroscopes and design of various gyroscopic systems, the paradigm of “inertial morphing” may prompt development of new generation of the acrobatic spacecraft with significantly reduced weight and dimensions, reduced cost and enhanced operational capabilities. It may be also possible to design new classes of gyroscopes, possessing an added-on sense of time, which is in contrast to the classical gyroscopes that only possess a sense of orientation.

With a wide spectrum of the presented examples, related to the application of a novel design concept of “inertial morphing”, it is believed that presented concept, modelling and simulation of the spinning systems and attitude control method of the spinning systems will be useful not only for the specialists but for a very wide audience, including engineers, scientists, students and enthusiasts of science and space technology.

Cable network structures, which are a class of nonlinear flexible structures, have been widely used in infrastructures and spacecrafts. In this work, a new form-finding method, namely, the fixed nodal position method (FNPM), is developed for optimal design of geometric configuration and internal force distribution for cable network structures, to meet the operation requirement of high shape/surface accuracy. Different from conventional methods, which usually adopts a stress-first-and-displacement-later procedure in form finding, the FNPM first assigns nodal coordinates for a cable network structure and then determines the internal force distribution of the structure by a nonlinear optimization process. The highlight of the FNPM is that the prescribed nodal coordinates are unchanged during the form-finding process. This unique feature of fixed nodal positions makes it possible to place the nodes of a cable network structure at desired locations, satisfying complicated structural constrains and yielding high shape/surface accuracy as required. As another advantage, the FNPM in form finding undertakes the assignment of geometric configuration (nodal coordinates) and the determination of internal force distribution separately. This translates into significant savings in computational effort, compared with conventional form-finding methods. The new form-finding method is applied to the optimal design of a large deployable mesh reflector of 865 nodes.

Vehicle body is the most integral factor shaping the perception of the design and brand image, and arguably it is the biggest contributor to the vehicle dimensional outcome. The weld process design is one of the key influencers to dimensional quality of the body. An optimum process design will ensure the minimum process-driven dimensional variations and hence improved dimensional quality.

Normally in a body production line, there is one geometry setting station for each segment of the process and then a few re-spot stations to complete the welds as per the product design. In the geometry setting station, adequate number of welds and a well-designed weld distribution (optimum process design) is required to ensure the assembly is dimensionally set and fixed. This is not an easy task in high-rate production lines due to the limited available cycle time. This requires the geometry welds to be characterised and prioritised when designing spot weld process. This challenge is more prominent in large and complicated structures such as car body shell. In this research, we aim to characterise the geometry setting welds of body shell as the most complicated structure in vehicle body. This will enable the optimum process design which delivers the minimum dimensional variation.

We first reviewed the assembly sequence planning problems and the application of different methods in solving these problems. We then assessed the application of genetic algorithm in the field and developed three genetic algorithm models including a simple structure, the bodyside assembly, and finally the body shell. The result was significant, and around 99% of non-viable process designs were eliminated. We concluded that the application of genetic algorithm can replace the existing process design methods which mostly rely on individual’s experience to design an optimum process.

The formal understanding of motion sickness has refined over the years, peculiarly in context of land vehicles. Since land vehicles are on-going a transitional phase in technology the perception of how motion sickness impacted passengers of land vehicles in the twentieth century to the twenty-first century has kept evolving in leaps and bounds. The problems that were previously faced or that had not surfaced up earlier will indeed transgress as we move up in the order or level of automation in land vehicles. Hence, this chapter elaborates how motion sickness was perceived in the early versions of land vehicles and how it modified as and when technology in cars made them more ride comfortable, and what we predict can be faced in context to motion sickness of passengers in Autonomous Cars.

A real suspension system has inherent nonlinearities that are often ignored when modeling, and a more accurate system than the traditional ones must be able to account for these effects. This chapter examines one such nonlinearity which is the possible separation of the tire from the ground. To accomplish this aim, a nonlinear model will be formulated using nondimensionalization which would incorporate the tire-road discontinuities. The number of input dimensionless parameters is minimized by developing the differential equation system. Once the tire loses contact with the ground, the separation time is a crucial criterion for evaluating ride comfort. Thus, the examination of the time response, frequency response, and fraction of the separation time has been sensitively conducted. The separation dynamics are numerically investigated to determine which values of suspension characteristics can postpone and avoid the tire-road separation phenomenon and eliminate the potentially dangerous vibrations.

Nonlinear Model Predictive Controllers are among the most popular techniques for developing intelligent control systems in automotive applications, including Adaptive Cruise Control Systems. Nonlinear Model Predictive Controllers, however, can be computationally expensive and many studies in the last decade have been dedicated to decrease their computational time. In this regard, by applying the so-called indirect optimization approach of Generalized Minimal Residual method to the Adaptive Cruise Control problem for Toyota Plug-in Hybrid Prius, this chapter contributes to the literature in three ways; first, it provides a comparative study of indirect real-time optimization methods in the context of Nonlinear Model Predictive Adaptive Cruise Control. These methods are implemented by the MPsee tool, which facilitates automatic code generation, implementation, and calibration of Nonlinear Model Predictive Controllers for MATLAB and Simulink users. Second, real-time implementability of proposed real-time optimization approaches is investigated by conducting Hardware-in-the-Loop experiments. Finally, this study argues that utilizing a Perturbed Chord modified Newton method instead of a Standard Newton method improves the computational speed up to 50%.

The dynamic vehicle roll study requires incorporating ideas from both the physics and mathematics. Applying artificial intelligence (AI) driver algorithms in autonomously controlled vehicles enables them to determine and negotiate corners effectively. The relational concepts vary when comparing the value of vehicle turn angles and their combined active contribution in turning the vehicle automatically.

The planar mathematical theory model for autonomous vehicles was initially developed. This theory was developed for use with 4-wheel steering vehicles but also works for 2-wheel steering vehicles. This theory extends the roll model by using the parameters of the angular velocity of a vehicle; that is, roll φ, pitch θ, yaw ψ, roll rate p, pitch rate q and yaw rate r. However, a roll model that uses forward, lateral, yaw and roll velocities is more exact and effective compared to this planar model.

Autodriver algorithm was introduced as a path-following algorithm for autonomous vehicles which is using road geometry data and planar vehicle dynamics. An autonomous vehicle can follow a given road if it turns about its centre of curvature at a correct moving position equal to the radius of the curvature of the path. The autodriver algorithm is improved according to practical implications, while a more realistic vehicle model (roll mode) is used, which considers roll degree of freedom in addition to a planar motion. A ghost-car path-following approach is introduced to define the desired location of the car at every instance. Finally, simulations are performed to analyse the path-following performance of the proposed scheme. The results show promising performance of the controller both in terms of error minimisation and passenger comfort.

Vehicle rollover is an important road safety problem world-wide. Although rollovers are relatively rare events, they are usually deadly accidents when they occur. The roll stability loss is the main cause of rollover accidents in which heavy vehicles are involved. In order to improve the roll stability, most of modern heavy vehicles are equipped with passive anti-roll bars to reduce roll motion during cornering or riding on uneven roads. However, these may be not sufficient to overcome critical situations. The active anti-roll bar system is considered as the most common method in order to improve the roll stability of heavy vehicles. In this chapter, the authors are interested in the effects of the internal leakage inside the electronic servo-valve on the performance of the active anti-roll bar system of heavy vehicles. Hence, the main contents are summarized in the following points:

Creating and maintaining thermal comfort for people who are working and living inside the buildings is the primary target of engineers who are designing building environment control system. Since thermal comfort by its definition is the state of mind and is different for different people, calculating, measuring, and controlling thermal comfort for everybody with similar and unique values are not sufficient. Following the current applicable thermal comfort standard as it is done in the professional industry will lead to setting and using similar temperature and relative humidity values for all people. Therefore, this is done without implementing direct comfort preferences of actual people who are working and living in the space. In this chapter, we introduce a novel method for calculating and maintaining the maximum possible thermal comfort for all the occupants in a building. We use the rules of game theory to propose a new way to collect, analyze, calculate, and maintain the maximum possible thermal comfort for all the actual occupants in one building with a holistic approach. In this approach, we categorize all the building occupants into a few subgroups of the people with similar thermal comfort preferences and use these actual preferences along with the game theory rules to find best responses of all occupants. The environment control system then will be programmed to operate based on these best response values and not based on same temperature and humidity ratio at all time as it is being done currently.

In this chapter, we will review the fundamental science related to wind origin and variability and the modeling of such variability based on empirical and fundamental laws. We will describe the currently accepted methodologies to assess wind energy potentials based on the literature review of key past research findings on this topic with implications on the current practices. We will focus on wind variability on the long and medium time scale of years and wind statistics based on the medium-short-term average (1 h). We will describe the use of specific software as well as current statistical and physical models together with common assumptions, which underpin the wind energy sector to determine wind farm siting adequacy. We will show the applications of this wealth of knowledge to specific case studies and provide insight into the currently accepted methods to determine wind energy generation based on medium-term time series data average. We will identify current gaps in this knowledge about short-term time variability (minutes) and implications to wind energy production for grid uptake in the conclusions.