Dissertation
Essay Mechanical Engineering Design for Control systems
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Control design and analysis of closed-Loop Dynamic Responses for Rectilinear Vibration Systems.
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ENGD3113Control design and analysis of closed-Loop Dynamic Responses for Rectilinear Vibration
Systems
Johnson Akanbi
P2591881
BEng Mechanical Engineering
Control
2022-2023
,Table of Contents
Abstract ........................................................................................................................................................... 3
Introduction...................................................................................................................................................... 3
Theory ............................................................................................................................................................. 4
Task 1: Identify the system parameters through the experimental data ............................................................... 6
Task 2: Modelling the system using Transfer Fcn and Simscape Fcn on MATLAB Simulink ............................. 9
Modelling by transfer function............................................................................................................... 10
Modelling by Simscape function............................................................................................................ 11
Task 3: Qualitative P Control Design and Tests............................................................................................... 14
When Proportional KP is 1.......................................................................................................................... 16
When Proportional KP is 15 ........................................................................................................................ 17
When Proportional KP is 100 ...................................................................................................................... 18
Task 4: Qualitative PID Control Design and Tests........................................................................................... 21
Task 4.1: PI controller- using proportional gain and changing the integral gain for system outputs. ............. 21
Task 4.1(b): Theoretical analysis on the effect of the integral item on the steady-state tracking error ....... 23
Task 4.2: PID controller- using proportional gain, integral gain and changing the Derivative gain ............... 24
Conclusions.................................................................................................................................................... 25
References ..................................................................................................................................................... 26
Appendix ....................................................................................................................................................... 26
Task 3........................................................................................................................................................ 26
Figure (3a) ............................................................................................................................................ 26
Figure (3b) ............................................................................................................................................ 27
MATLAB code used to get Pole-Zero graph .......................................................................................... 27
Task 4.1..................................................................................................................................................... 28
Figure (4a): graph showing when KI is 1 system output position ............................................................. 28
Figure (4b): Control input force of position when KI is 1 ........................................................................ 28
Figure (4c) graph showing when KI is 15 system output position. ........................................................... 29
Figure (4d) Control input force of position when KI is 15. ...................................................................... 29
Figure (4e) graph showing when KI is 35 system output position. ........................................................... 30
Figure (4f) Control input force of position when KI is 35........................................................................ 30
Task 4.2..................................................................................................................................................... 31
Figure (4g) graph showing when KD is 1 system output position ............................................................ 31
Figure (4h) Control input force of position when KD is 1 ........................................................................ 31
Figure (4i) graph showing when KD is 5 system output position ............................................................. 32
Figure (4j) Control input force of position when KD is 5......................................................................... 32
Figure (4k) graph showing when KD is 10 system output position. .......................................................... 33
Figure (4l) Control input force of position when KD is 10 ....................................................................... 33
Figures ...................................................................................................................................................... 34
Tables........................................................................................................................................................ 34
2
,Abstract
The control design and analysis of closed-loop dynamic responses for rectilinear vibration
systems are covered in this report. Rectilinear vibration systems are widely used in a variety
of applications and controlling them is critical for minimising unwanted vibration and
improving system performance and robustness. The report describes how to design a
proportional-integral-derivative (PID) controller for a rectilinear vibration system, including
how to determine the system transfer function and tune the controller parameters. The
frequency response and time-domain analysis methods are also used to analyse the system's
closed-loop dynamic response. The report concludes that control design and analysis of
closed-loop dynamic responses are critical for rectilinear vibration systems to achieve stable,
fast, and accurate performance.
Introduction
Many engineering applications, from automotive and aerospace to industrial and robotic
systems, rely on control systems. They are intended to regulate a system's behaviour by
continuously adjusting the inputs based on the system's output. Control systems come in a
variety of configurations, including proportional-integral-derivative (PID) controllers,
adaptive controllers, and model-based controllers.[1]
Mechanical vibration systems are one type of system that requires precise control.
Mechanical vibration systems are commonly used in a wide range of industrial and
engineering applications, including automotive suspensions, aerospace structures, and robotic
systems. They involve the oscillatory movement of a mass attached to a spring and damper
system. Mechanical vibration control is critical for minimising unwanted vibration,
improving system performance, and increasing system robustness against uncertainties.
Controlling mechanical vibration systems, for example, is required in automotive suspensions
to provide a comfortable ride for passengers while maintaining good handling and stability
[2]. Mechanical vibration systems must be controlled in aerospace structures to ensure
structural integrity and aircraft safety. Control of mechanical vibration systems is required in
robotic systems to improve the accuracy and reliability of the robot's movements.
The significance of mechanical vibration system control stems from their potential negative
impact on system performance, safety, and lifespan. Unwanted vibration can damage
structures, reduce efficiency, increase energy consumption, and cause component failure.
Controlling mechanical vibration systems effectively can reduce these negative effects,
improve system performance, and extend system lifespan.
3
, Theory
Figure 1 depicts a schematic representation of a mechanical system with one degree of
freedom. Aside from the external force F(t), the trolley of mass m experiences elastic force
generated by an attached coil spring with stiffness coefficient k and linear viscous friction
generated by an attached dashpot or damper with coefficient c. Newton's second law can be
used to write the equation of motion for the forced damped system:
1
m x¨ + c x˙+ kx = F(t) ⇒ x¨ + 2ζωn x˙+ ω2 x = 𝐹 (𝑡), (1)
𝑚
𝑐 𝑘
where ζ =2√𝑘𝑚 is called the damping ratio and ωn =√𝑚 is called the natural frequency.
Derive the following transfer function model of the above system by applying the Laplace
transform to equation (1) and assuming zero initial conditions equation. This model can be
used for theoretical analysis of the open-loop and closed-loop system dynamics. See figure 1
Figure 1: Forced 1-DOF rectilinear vibrating system
𝑋(𝑠) (1/𝑚) 𝐾𝜔𝑛 2
𝐺 (𝑠) = = = 2 (2)
𝐹(𝑠) 𝑠 2 +(𝑐/𝑚)𝑠+(𝑘/𝑚) 𝑠 2 +2𝜁𝜔𝑛 𝑠+𝜔𝑛
The damping ratio ζ and the natural undamped frequency ωn determine the shape of
its step response.
The parameters listed below characterise various aspects of the system's response and are
frequently used to define the design criteria for a controller to be designed.
4
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