CHAPTER ONE: INTRODUCTION

1.1 BACKGROUND OF THE STUDY

Servo system is a feedback control system that is used to follow or reproduce a process accurately. One of the basic requirements in control systems is that they have the ability to regulate the controlled variables to reference commands without a steady-state error against unknown and unmeasurable disturbance inputs. Its main task is to achieve the power amplification, transformation, and control under the command requirements, so that the output torque, speed, and position control is very flexible and convenient. In many cases, the servo system means that output of the system is a feedback control system of mechanical displacement or displacement velocity and acceleration and its role is to enable the mechanical displacement output (or angular displacement) to accurately track the input (or angle). Its structures have no difference with other forms of feedback control systems in principle. The servo system is originally used for national defense, such as gun control, automatic driving ships, aircraft, and missiles. Later it gradually extended to many sectors of the national economy such as automatic machine, wireless tracking control, and semiconductor manufacturing equipment [1–6]. Mechanical resonance is one of the most common problems designers face when trying to maximize either command response or dynamic stiffness [7]. In the case of rotation, the drive and the drive shaft are generally considered rigid. In fact, the mechanical transmission shaft has different degrees of elastic torsion deformation. With the broadening of the frequency band of the system, the effect of the mechanical resonance caused by the elastic deformation on the dynamic characteristics of the system is obviously strengthened. From the transfer function of the system, it often contains a pair of conjugate zeroes and poles, which easily lead to mechanical resonance phenomenon. And this phenomenon is more obvious in the junction of feedback sensor and load connection in servo system [8]. Resonance is harmful in most cases. It causes mechanical deformation and greatly reduces the mechanical structure and even produces environmental pollution, and so on. In order to make the system have good dynamic performance, it is necessary to pay attention to control and eliminate resonance in servo control system. When resonance happened, the work of the excitation input system is balanced with the dissipated work phase, and the shape of the resonance peak is closely related to the damping. From the mechanical point of view, resonance precaution has the following measures: (1) improving the structure or the mechanical excitation to make the natural frequency avoid the excitation frequency; (2) adopting damping devices; (3) passing the resonance fast when there is a mechanical starting or stopping. From the control point, the low-pass or notch filter is often used to reduce the influence of resonance in the system [9–12]. Other theories and approaches such as resonance ratio control [13] and acceleration feedback [14] have been put forward to decrease the influence of resonance mode [15, 16]. The simulation of closed-loop is requisite when analyzing the influence of the resonance modes in servo system. And the mathematical model of the resonance mode must be considered in the simulation; otherwise the closed-loop simulation results will differ from the actual servo system greatly

1.2 STATEMENT OF THE PROBLEM

Resonance is the main mechanism for energy transmission in spatially periodic systems. Most published works on ordered granular media focus on impulsive inputs which often are simplified as prescribed initial conditions. Only a few works have studied the responses of granular particles to harmonic excitations [2, 4, 8, 14]. Recently, Pozharskiy et al. [20] analyzed the dynamics of a harmonically excited granular medium under a frequency-dependent boundary drive with fixed amplitude. Despite the lack of sound propagation in this medium and the lack of a linear resonance spectrum, various types of purely nonlinear waves were observed involving periodic traveling waves, and a mixture of traveling pulses and standing waves, under different excitation frequencies. Focusing on the force transmitted across the harmonically forced medium, local peaks (resonances) and dips (anti-resonances) of force transmission were noticed, and detailed bifurcation diagrams studying the stability of such responses were reported.

1.3 AIM AND OBJECTIVES OF THE STUDY

The main aim of the research work is to carry out a numerical simulation of resonance in a mechanical system. Other specific objectives of the study are:

1. to determine the Mathematical Model of Resonance Mode in a mechanical system
2. to carry out the numerical simulation of Single Resonance Mode in a mechanical system
3. to investigate on the factors affecting the simulation of resonance in a mechanical system
4. to proffer solution to the above problem

1.4 RESEARCH QUESTIONS

The study came with research questions so as to ascertain the above objectives of the study. The research questions for the study are:

1.  What is the Mathematical Model of Resonance Mode in a mechanical system?
2. Is the numerical simulation of Single Resonance Mode in a mechanical system effective?
3. What are the factors affecting the simulation of resonance in a mechanical system?
4. What is the way forward to above problem

1.5 ORGANISATON OF THE STUDY

The chapter one the research work will cover the areas of the background to the study, the problem statement, the aim and objectives of the study, the research questions, the significance of the study, the scope of the study and the definition of terms. The chapter two of the research work will cover the literature review. The chapter three will cover the areas of research methodology. The chapter four of the research work will cover discuss of result and analysis while the chapter five will cover the conclusion and recommendation.

1.6 SIGNIFICANCE OF THE STUDY

The study on numerical simulation of resonance in a mechanical system will be of immense benefit to the physics department in the secondary schools and tertiary institutions in Nigeria in the sense that the study will be able to determine the Mathematical Model of Resonance Mode in a mechanical system. The study will also explain the numerical simulation of Single Resonance Mode in a mechanical system. The study will serve as a repository of information to other researchers that desired to carry out similar research on the above topic. Finally the study will contribute to the body of the existing literature on numerical simulation of resonance in a mechanical system

1.7 SCOPE OF THE STUDY

 The study on numerical simulation of resonance in a mechanical system will cover on the Mathematical Model of Resonance Mode in a mechanical system and also on the numerical simulation of Single Resonance Mode in a mechanical system

1.8 DEFINITION OF TERMS

RESONANCE: resonance is a phenomenon that only occurs when the frequency at which a force is periodically applied is equal or nearly equal to one of the natural frequencies of the system on which it acts.