LAPPEENRANTA UNIVERSITY OF TECHNOLOGY LUT School of Energy Systems
LUT Mechanical Engineering
PARAMETERIZATION AND REAL TIME SIMULATION OF AN EXCAVATOR
Examiners: Professor Aki Mikkola
D. Sc. (Tech.) Kimmo Kerkkänen
ABSTRACT
Lappeenranta University of Technology LUT School of Energy System
LUT Mechanical Engineering Manouchehr Mohammadi Master’s thesis
2017
79 pages, 59 figures, 5 tables and 8 appendices Examiners: Professor Aki Mikkola
D. Sc. (Tech.) Kimmo Kerkkänen
Keywords: Real-time Simulation, Excavator model, SIM platform, Multibody system, Model parameterization.
This master’s thesis has been done for simulation, Companies working with real-time simulation concept, and training target in a way that a vehicle, an excavator, is developed by parameterization method which obtains a new solution to have a simulated model with a number of customizable parts, values, and bodies. In other meaning, a user can opt her/his favorite part/body based on her/his aim.
From the beginning of this project MeVEA software selected as the real-time simulation software in which all cooperative software should be along MeVEA. The project goal was create a user-friendly way to present a simulation model with ability of being customized. A customized model prepares an opportunity for Companies in this field to analyze new models with a significant spent budget reduction in comparison of previous solutions.
Parameterized simulated model, in this project an excavator, can be used to create a desired model and simulate it, then its results can be analyzable in order to figure out the optimum options of the simulated model for each mission and function. At first, it was decided to create only one way to have a customizable model which was creating an excel file as an interface that the user could select her/his options among all options, then using a python code as a bridge between the excel file and MeVEA, however, in the following one other file created as well.
ACKNOLEDGEMENTS
This thesis has been done at the Laboratory of Machine Design, Department of Mechanical Engineering at Lappeenranta University of Technology (LUT).
I would like to express my sincere gratitude to my Professor Aki Mikkola for his valuable guidance, advice and high level of patience. His comprehensive knowledge about the project in all aspects could help and inspire me to figure out concepts in a best way and overcome difficulties during this master’s thesis. I had a great opportunity to work with him because of his permanent presence with extra-ordinary responsibility in every step of this work.
I want to thank my supervisor Kimmo Kerkkänen, as the second supervisor, about the subject of thesis, appreciable support and constructive feedback which could guide me during my master’s thesis. I also appreciate help which I had from my colleagues in CoSIM project in Machine Design Laboratory. Thanks to MeVEA staff cooperation in format of a number of workshop.
Finally, especial thanks to my dear family who supports me during my life.
Manouchehr Mohammadi Lappeenranta, June 30, 2017
TABLE OF CONTENT
ABSTRACT
ACKNOWLEDGEMENTS TABLE OF CONTENT
ABBREVIATION AND SYMBOL LIST
1 INTRODUCTION ... 10
1.1 SIM Platform – A Glance description ... 10
1.2 Research Questions ... 12
1.3 Aims and objectives ... 12
1.4 Research Methods ... 13
2 METHODS AND METHODOLOGIES ... 15
2.1 Literature Review ... 15
2.1.1 Simulation in researches ... 15
2.2 Principles of a multibody system and its equations ... 21
2.2.1 Global and local coordinates ... 23
2.2.2 Rotational coordinates – Kinematic Constraint Equations ... 23
2.2.3 Kinematic Joints Constraints ... 24
2.2.4 Equations of Motion ... 26
2.2.5 Integration Methods in Dynamic Analysis ... 28
2.3 Simulation in practice ... 34
2.3.1 Simulation ... 34
2.3.2 Simulators ... 34
2.3.3 Marketing ... 35
2.3.4 Customizable Model ... 36
2.3.5 Employed Software ... 37
2.3.6 MeVEA ... 37
2.3.7 CAD Software – SolidWorks ... 39
2.3.8 Blender ... 39
2.3.9 Python and Excel ... 40
2.4 Four-bar Mechanism ... 41
3 CASE STUDY- THE EXCAVATOR ... 46
3.1 Principles of Excavator ... 47
3.2 Simulated industrial vehicle ... 48
3.3 Editable Parameters ... 48
3.3.1 Bucket and lifting system ... 49
3.3.2 Hydraulic circuit system ... 51
3.4 Data Selection ... 54
3.4.1 Assembly files approach ... 54
3.4.2 Coding files approach – User interface ... 57
3.5 Method – A coupler between Excel and MeVEA Modeller ... 58
3.6 Model in MeVEA – Working and Dynamic Simulation Interface ... 59
3.7 Results ... 60
3.7.1 Customization for the Bucket – combination and comparison ... 60
3.7.2 Customization for the Hydraulic Circuits – Combination and Comparison .... 63
4 ANALYSIS ... 67
4.1 Analysis for employment of different Buckets ... 67
4.2 Analysis for employment of different hydraulic circuits ... 71
4.3 Future work ... 75
4.3.1 Using Software and their connections ... 76
4.3.2 User-friendlier interface - Gamification ... 76
4.3.3 Further customizations - Analyze section ... 77
4.3.4 Visualization of models and environments - Environment customization ... 77
5 CONCLUSION ... 78
LIST OF REFERENCES ... 80
APPENDIX
Appendix 1: Concept of the global coordinate system.
Appendix 2: Rotational coordinates in spatial MBS.
Appendix 3: The revolute joints among bodies and their equations.
Appendix 4: The equation of motion for a constrained system.
Appendix 5: The equations related to a four-bar mechanism.
Appendix 6: Detailed data of Volvo excavators.
Appendix 7: The python script code for making the model customized.
Appendix 8: Results for the medium and big bucket and medium and big cylinder-piston
SYMBOL AND ABBREVIATION LIST
a - Position vector on a body
A - Rotational transformational matrix 𝐀′- Final rotational matrix
b - Position vector on a body
B - The element of the final rotational transformational matrix C - Constraint Equations
Cq - Jacobian matrix of the four-bar mechanism 𝐂𝑡 - Velocity matrix of the four-bar mechanism D - Jacobian matrix
e - Euler parameter e - Euler vector
E - The element of the final rotational transformational matrix f - Function
f - Force vector g - Generalized force g - Ground acceleration h - The integration step size i - Name of a body
𝑖1 - A name of a particle I - Unit vector
I - Inertia
j - Name of a body J - Global inertia tensor l - Length
m - Mass
M - Mass matrix
M1 – Torque applied on a four-bar mechanism n - Number of coordinates
n1 - Order of a differential equation
𝑛𝑚 - Moments which a body is affected by a force
O - The element of the final rotational transformational matrix P - Name of a point on a body
𝑃′- Order of the error equation pi - Euler parameters in a matrix p - Euler parameter’s matrix q - A generalized coordinate vector r - Vector of position
𝐫̇ - Vector of velocity 𝐫̈ - Vector of acceleration R - Position
S - Function name
s - Position vector from bodies to their connecting joint 𝐬′- Constant Vector
𝐬̇ - Velocity vector from bodies to their connecting joint t - Time
T - Kinetic energy 𝐯 – Velocity vector 𝐯̇ - Acceleration vector V - Potential energy
y - Variable to be integrated ω - Angular velocity
𝛚̇ - Angular acceleration 𝜑 – Angular variable ξ - The body-fixed vectors
Φ - Function for kinematic constraints 𝚽̇ - Constraint of the velocity
𝚽̈ - Constraint function of the acceleration
ɣ - Multiplication of Jacobian matrix and acceleration 𝜆 - Lagrange multipliers
α - Positive constant β - Positive constant
𝜎 - Angular Variable name in a rotational matrix 𝜑 - Angular Variable name
𝛹 - Angular Variable name in a rotational matrix θ - Angular Variable name
𝛼′- Angular position for a four-bar mechanism 𝜃′- Angular position for a four-bar mechanism 𝜑′- Angular position for a four-bar mechanism 𝛼̇′- Angular velocity for a four-bar mechanism 𝜃̇′- Angular velocity for a four-bar mechanism 𝜑̇′- Angular velocity for a four-bar mechanism 𝜏 - Torque
ɛ - Truncation error ɛ𝑔1- Global or total error ζ - The body-fixed vectors η - The body-fixed vectors BEV - Battery electric vehicles CAD - Computer-aided design
DAE - Differential Algebraic Equations DES - Discrete event simulation
FEM - Finite Element Method MBS - Multibody System
MSD - Multibody System Dynamics
LUT - Lappeenranta University of Technology ODE - Ordinary Differential Equation
RLV - Reusable Launch Vehicles SD - System Dynamics
SIM - Sustainable product processes through simulation XML - Extensible Markup Language
3D - Three dimensions s - Spherical
1 INTRODUCTION
Simulation can be helpful to create or improve many models which are so expensive to produce them in reality. In fact simulation is a definition of movements and processes of a body or bodies through the time. In many cases, modifying and changing in a model is not cost-effective, however a precise simulation model can catch a set of data for parameters and gives analysis and results in details. These results can clear that is changing/modifying in any values in the simulation model leads to improvement of its performance or not.
Moreover, simulation has a couple of economic trends. For instance: (Bangsow, 2010, p. 18)
The variation of a product under simulation will increase.
Because of easy-modifying, the demand to upgrade in quality will increase which consequences high quality based on customers’ requirements.
Customization and high flexibility in a model can be impressive to increase the demand of bazaar in order to obtain this approach.
Two striking consequences; life cycle time and also man power will decrease tremendously.
In manufacturing and design methodologies points of view, using simulation in real job have many benefits such as shorter lead time, decreasing man power, decreasing in production’s steps and etcetera. With three major phases, in every simulation model, the questions which arising for the model can be responded; Planning Phase, the first phase, is a phase to make a plan to find out all the possibilities and potentials of a model. Implementation phase, the second one, assists to test the performance and problems of the model during simulation.
Also, the future requirements and limitations of a simulated model can be found in this phase.
Finally, the operational phase which collaborates to test for controlling the alternatives of a model. Moreover, the level of the model’s quality can somehow be traced. (Bangsow, 2010, p. 18)
1.1 SIM Platform – A Glance description
To implement the simulation model, after modelling, a platform is needed to test the capability and movements in a real situation. SIM platform is a capability with a number of
powerful simulators at Lappeenranta University of Technology (LUT). The concept of the SIM Platform is to avoid the outdated and time consuming approaches to obtain the results and trends to the new and practical ways getting aims.
SIM platform has an approach to have a real-time simulation obtaining energy efficient solutions and Also it looks to extend a design simulation ‘’from a single machine to entire production systems’’ which leads to a comprehensive analysis about machines performance in a machines’ complex (LUT, 2016).
Simulators are tools obtaining information from the simulation models. A simulator consists of two parts which are working together; the hardware part which gives a realistic feedback with aim of sound and force feedback simultaneously. The second part is the software parts which for this thesis is the MeVEA software which is a Finnish-based software in Lappeenranta city (About Mevea, 2017) . Also, there are some simulators at LUT (Lappeenranta University of Technology) in the electrical and intelligent machine’s laboratories (Design Laboratory, 2017). Figure 1.1 depicts simulators which are used for this dissertation work.
Figure 1.1. Simulators at the Lappeenranta University of Technology.
One of the striking aspect of the simulation is being used in many fields. Some of them are human-centered simulation process and some of them are free of human (Byrski, 2012). A
simulation is applicable in energy consumption, help to simulate traffic contains BEV (Battery Electric Vehicles) vehicles and also other kinds of vehicles and roads (Pina, 2013, p. 13). In addition, the simulation is using in physics and automotive industry to have a great and new kinds of power transmission for vehicles especially hybrid vehicles (M.Dede, 2014, pp. 4-21). Game industry uses simulation to increase the quality of its game, graphics and reality feeling for its users. With using real graphics, many sensors, and feedbacks such as force feedback, this industry is improving many aspects such as customer’s satisfaction and its market, as the main goal.
1.2 Research Questions
Research questions frequently appear after encountering with a research problem or issue.
The main problem about this project is: An adjustable customization for a simulation model always is time consuming and not cost effective. The MeVEA software uses a readable text file which has all data about a model in it to run a model in its interfaces. There are some options to have a customizable model such as making some alters in its interface, making some changes in mentioned readable text file and etcetera. The following questions are the research questions which come to the mind:
- How can it be possible to get access to all data of the model in a way that it could be changeable and easy to save?
- What is the suitable method to find out desirable data and select among them and make a ready model to run?
- Is it possible to have a customized model that a user could select a wide range of data, instead of only few options, among the sub-assemblies and assemblies and MeVEA software collects them and makes the model without any concerning about their adjustment?
- Is the selected configuration made by the user practicable or reasonable?
- Is there any demand to have a wide range of options for model’s parts?
- Which Parameters or parts in the model can be changeable?
1.3 Aims and objectives
In this dissertation work a simulation model in MeVEA software will be developed to be customizable. At the moment, in simulation point of view, a model has a set of constant parameters to simulate it and extracts the desirable results and analyzes them. Changing
adjustable parameters for a model with old ways is time-consuming, thus it is extremely convenient acquiring a way to create a model which have a capability to change its data or parameters precisely and quickly.
The main objective of this thesis is to make an adjustable simulation model with user’s selections. In the other words, there is a simulation model in MeVEA software which collects data from a user and makes a model, in this case an excavator, according to given data. The point is, data are coming from some assemblies and sub-assemblies and will use to make the model and all these assemblies must be well-matched to each other that the user could feel the effects of her/his selections.
For this dissertation work, the alterable attributes for a simulated excavator model are:
Dimensions of the bucket in visual and collision modes and its mass accordingly.
Dimension of the cylinder and piston for the dipper arm.
The amount of the nominal flow rate going into the dipper arm cylinder.
The parameters which a user will select can cause a simulation model with a capability to do a task fast in comparison of a normal real model, however with high fuel consumption. On the contrary, it can be a model with less fuel consumption but slower than before. All graphics for the environment, customizable parts of the model are adjusted with other fixed parts.
Future aims is to have more customizations parameters and parts of a model, like the excavator, with a way to analyze them based on some practical parameters. In the other words, it will be a model and a way to analyze which model is cost-effective, based on fuel consumption and working hour time, or which one is more reliable based on wearing, depreciation, maintenance and easy to work. Future aims will be discussed at the end of this dissertation work in detail.
1.4 Research Methods
Tools which are used in this dissertation work are the MeVEA software as a real-time simulation to create a simulation model with other software as its assistants. It should be
noticed that during creation, other software such as SolidWorks, Blender, Python and Excel are involved. Afterwards, the simulation model can be run.
In the design section, MeVEA creates an XML file for data of the model and there are some ways to edit that file in order to have changing capability of the model, however, creating an XML code with some assemblies in it, (which will be explained in next chapters), and also writing a python code and run it with a excel code are the most feasible ways to reach to the bottom line of this dissertation work.
As the Procedure of this thesis, there is a simulation model which has fixed parts and parameters, and modifiable ones. With the ways noted in design section, the model will be editable. Simultaneously, the visualization and collision graphics will adapt with assist of SolidWorks and Blender software and finally the whole model can be run in the MeVEA platform. Figure 1.2 demonstrates the story line and procedure of this research.
Figure 1.2. The procedure of building and running a simulation model.
As in figure 1.2 has shown, there is an editable model which collects data from the user side and has interaction with the graphics part and uses appropriate graphics and creates the ready model. In this dissertation work there are three options for the bucket and also three options for the hydraulic circuit system.
Gathering Data / Editing Tools
Editable Model
Graphics
Simulate Concluded
Model
2 METHODS AND METHODOLOGIES
The first part of this chapter is literature review which is about previous researches in real- time simulation field in order to have a new concept or new idea about customization. Then principles of a MBS will be discussed. Moreover, simulations and simulators will be under consideration in being practical point of view. Finally, a four-bar mechanism and its equation will be explained.
2.1 Literature Review
When a new project and idea comes in mind, it is always logical to have a look and review to previous researches beforehand to see what researches have already done and what are their approach to solve a problem. With a literature review previous researches’ results, analysis can be found and it is possible to have a comparison among them and figure out their overlapping at work, their idea about the project and title which is under consideration.
Moreover, by a careful literature review it can be possible to find barriers and limitations in front of previous researches.
In this chapter, previous works in the field of simulation, real-time simulation and earlier efforts can be discussed, however, because the simulation are very practical in a tremendous amount of fields, it is rational to have a review in real-time simulation about industrial and widely-used vehicles, especially in past efforts about parameterization.
2.1.1 Simulation in researches
For the review in previous researches, a practical database have been used which is LUT FINNA – Wilma. It has covered enormous amount of articles, conferences and books and has reviewed most scientific databases and journals such as publications in Lappeenranta University of Technology, Springer, and so on. One of the effort to have done by Steffen Bangsow in which to aim to the simulation solution, he suggested that steps formulation of problems and targets, data collection, modelling and running the model, and analyze the results is a rational chart to solve the simulation problem. (Bangsow, 2010, p. 2). Figure 2.1 shows the steps of Steffen Bangsow, however in his research it did not mentioned a way to figure out the customization of a simulated model.
Figure 2.1. Phases of a simulation problem based on Steffen Bangsow research.
Edward Robert Comer and his colleagues have patented an approach in real-time simulation in training part which all of their system’s parts have interconnection together. As figure 2.2 has demonstrated, there is a data-driven simulation kernel including some sections. In fact this patent is invented ’’ for training technical skills on equipment, machinery, and software- based systems’’. In this creation they tried to have a training environment with help of simulation which is realistic and reliable. (Comer, 2005)
Figure 2.2. Patented application for training with help of simulation (Comer, 2005).
Comer has tried to make connections between the core of simulation training model, such as XML interface and simulation data, and a simulation client. This aspect which can get access to some important parts of a model is valuable however the drawback of its system is having
limitation in order to work with it. In the other words, it cannot emerge from training part and extend to other parts which can be useable for not-trained user.
Based on previous researches, there are two simulation approaches, discrete-event simulation and system dynamics, which are called DES and SD simultaneously. Discrete- event simulation and also system dynamics approaches are based on development performance of a simulated vehicle through time. Furthermore, they can identify some improvement for models which can be done in the future of a model. System dynamics approach is created based on differential equations. (Tako , 2010, p. 784)
J A Ninan has tried to have a customizable model with help of internet. Figure 2.3 illustrates a chart to obtain a feasible and practical model. In this method data gathers from a user and creates the CAD model. In this implementation method, it considers two steps to check feasibility of the model, one of them is after building FE model and analyze it which if it was feasible it can generate CAD model and extract results. The other consideration is after creating FEA model and analyze its practicality that if it was not functional, it should be terminated.
Figure 2.3. Implementation to finalize a CAD model ready to analyze (Ninan, 2006, p. 533).
A practical attempt to consider customer as an effective option during design is working on the internet-based framework by J A Ninan. With help of computer-aided design (CAD) and finite element method (FEM), he tried to allow to customers doing some customization via internet. (Ninan, 2006, p. 529) He also mentioned that before his research there were some researchers such as Gilmore and Pine who they have researched about mass customization and they provided four approaches to make a customizable model; collaborative, adaptive, cosmetic, and transparent approaches. The research of J A Ninan has some drawbacks such as being time-consuming in order to analyze in finite element model, though is a kind of inspiring research about making a model customizable. As figure 2.4 has showed, he have tried to use FE Analysis and other optimization tools with interaction via internet with customers to create more appropriate and reliable results which can be modified based on feedbacks from customers and design sections. (Ninan, 2006, p. 531)
Figure 2.4. Phases in Mass Customization model of Ninan (Ninan, 2006, p. 531).
Having an opportunity to work with a model in a way that more users can have accessibility to demonstrate their favorite options and specifications is one of the imperative aims for many kinds of customization and simulation. For instance, Scott Fortmann-Roe have tried to explain a kind of access to users to present its idea and model, to a client which can display and simulate the model as well.
Figure 2.5 depicts this idea schematically. (Fortmann-Roe, 2014, p. 32) Despite of not exactly customization in simulation model, the aim of this effort could lead a kind of remote control via internet and share the results of each simulated model.
Figure 2.5. Presence of a client which can simulate and display results based on users model (Fortmann-Roe, 2014, p. 32)
Moreover other researchers have worked on parameterization issue; Schwarz Bachinger prepaid and demonstrated a unique way to parameterize ‘’all types of gear transmission topologies’’ (Schwarz, 2015, p. 1). They have tried to provide a customizable model for a drivetrain model in its friction elements, clutches, figure 2.6.
Figure 2.6. A block diagram of inputs and outputs of a drivetrain model to parameterize the friction elements (Schwarz, 2015, p. 1).
For model shown in figure 2.6, the parameterization is just considered slipping and open modes for clutches and they have written a Java script as the environment, which means all specifications needed to model a drivetrain. (Schwarz, 2015, pp. 1-2)
A Kaylani and his colleagues have introduced a NASA approach in producing generic model. In order to cut cost, NASA has worked on a project named RLV, Reusable Launch Vehicles, which provides an opportunity to launch more flights per year for shuttles. With help of a kind of simulation method named DES, discrete event simulation, the functional performance of a launch vehicle can be analyzed and moreover, it can assess that is a parameterized parts in a launch vehicle effective to decrease the amount of budget of a flight or not? After discussion and consideration of model fidelity, generalization in model, and function ability of the model (means that the model should be easy to customize and configure), they have shown a story line to attain a generic model in figure 2.7. (Kaylani, 2007)
Figure 2.7. Steps to attain to a generic model in RLV approach (Kaylani, 2007, p. 4)
Ren, Q and D.A.; Morris, worked on design for an EV, electric vehicle, to figure out the effect of variety kinds of transmission on an electric vehicle performance. At first they tested an EV with a generic motor with the power of 40 kW in some kinds of power transmission;
single transmission ration, continuously variable gearing mode, and a multispeed gearbox.
They have changed crucial parameters such as total vehicle mass, wheel diameter, rolling
resistance coefficient et cetera to analyze the effect of these changing in an EV. Employment of any kinds of power transmission for an electric vehicles has some advantages and disadvantages in fuel consumption. Figure 2.8 demonstrates a comparison among three kinds of power transmission in separate driving cycle. It should be noticed that, at this level, discussing about kinds of driving cycles and their differences is not requirements of this dissertation work.
Figure 2.8. A comparison among three kinds of power transmission used in an electric vehicle (Ren, 2016, p. 1264).
Figure 2.8 illustrates comparison among mentioned power transmissions and it shows using any of power transmission has its benefits. At the moment using a customizable simulation model with a capability of easy to switch among different kinds of power transmission with accurate analyze is needed to depict the advantages and drawbacks of usage of each power transmission.
After a couple of reviewing in other researchers about real-simulation and editable models, there is a lack of a comprehensive approach to satisfy all vital issues was felt. An approach which can take care of some major issue such as not being time-consume, easy to use, high feasibility, and easy to save. Using MeVEA software as a real-time simulation software interacting with other software creates a customizable model which is reliable and cost- effective and can be helpful for any companies in design, manufacturing and test sections.
2.2 Principles of a multibody system and its equations
This chapter presents a general point of view of the concepts of multibody System Dynamics (MSD), global and local coordinates, kinematic constraint equations and equations of
motion. Many signs from the history of the mechanical engineering root illustrate that knowledge multibody system dynamics is founded on classic mechanical mechanism, satellites and robots. ‘’Multibody system dynamics is characterized by algorithms or formalism, respectively, ready for computer implementation.’’ Precise and practical interaction with CAD software, parameterization, real-time simulation, joints and connection among the components, control mechatronic systems and the analysis of the whole multibody are the main perspectives and concepts of Multibody System Dynamics (MSD). Moreover, in analysis of MSD’s treatment, there is an interest in using reduction methods to have a too far precise results of integration codes for ODE, Ordinary Differential Equations, and DAE, Differential-algebraically equations. (Schiehlen, 1997, p. 149)
The equations which explain a motion for multibody systems are known as Newton-Euler equations. The principle of equation of motion for MSD will be explained later in this report.
In 1788, Lagrange presented an analysis of the system of mechanical constraints. DAE and ODE are Lagrange’s equations which explain the total kinetic and potential energy of the system. In this system, the constraints and generalized coordinates should be taken into account. (Schiehlen, 1997, pp. 1-2) In a clear way, a multibody system (MBS) has connections with two groups of vital characters; first, mechanical components which can illustrate displacements and the second is relations among bodies, constraints, with the kinematic joints. ‘’ In the other words, a multibody system encompasses a collection of rigid and/or flexible bodies interconnected by kinematic joints and possibly some force elements’’. Based on the demands of a MBS, the body for a multibody system can be described as a rigid or flexible body. With using six generalized coordinates with six degrees of freedom, DOF, the motion of any rigid body can be demonstrated while 3D space has defined. (Flores, 2015, p. 1)
After definition and consideration of joints among parts in a MBS system, in a spatial case, the number of the degree of freedom will reduce. (Flores, 2015, p. 5) Figure 2.9 illustrates types of coordinates which are common explaining the MBS. (Flores, 2015, p. 7)
Figure 2.9. Frequently used coordinates in MBS (Flores, 2015, p. 7).
2.2.1 Global and local coordinates
Three variables which are independent from each other can describe the displacement of a free moving particle ‘’i1’’ in 3D space and the vector ‘’r’’, vector of position, is (Flores, 2015, p. 11):
𝐫𝑖1 = {𝑥𝑖1 𝑦𝑖1 𝑧𝑖1}𝑇 (2.1)
With the same way, above definition can be used for a rigid body and its location. Also its orientation can be explained with respect of a reference system. (Flores, 2015, p. 12) In appendix 1, the concept of global coordinate system can be found.
2.2.2 Rotational coordinates – Kinematic Constraint Equations
There are some approaches to explain rotational coordinates in 3D MBS which are Euler Angles, Bryant Angles and Euler Parameters. With aid of six coordinates, three translational and three rotational ones, the location of any rigid body can be discovered. In appendix 2, all the steps for rotational coordinates in spatial MBS can be found.
A constraint always embeds a kind of limitation or restriction in the degree of freedom for one or more bodies. With assistance of the concept of generalized coordinates, the location and orientation of bodies can be defined and now it will be known with a vector,𝐪 = {𝐪1, 𝐪2, 𝐪3, … , 𝐪𝑛}𝑇 that n is the number of coordinates. In this report, Φ depicts the constraint with a denoted parameter and a number. The parameter illustrates the type of
constraint and the number is the number of equation. For instance, 𝚽(s,2) shows that there is a spherical constraint with two equations. (Flores, 2015, p. 31) The kinematic equation based on the vector of body-coordinates can be shown as (Flores, 2015, p. 33)
𝚽 ≡ 𝚽(𝐪) = 0 (2.2)
Where q shows the vector 𝐪𝑖 = {𝐫𝑖 𝐩𝑖}𝑇 which has ri, which includes three translational coordinates,𝐫𝑖 = {𝑥𝑖 𝑦𝑖 𝑧𝑖}𝑇, pi is Euler parameters, and i is a body name. With a derivation of Φ, the velocity constraints will appear (Flores, 2015, p. 33)
𝚽 = 𝐃𝐯 = 0̇ (2.3)
D is the Jacobian matrix. v is the below equation (Flores, 2015, p. 28)
𝐯𝑖 = {𝐫𝑖̇
𝛚𝑖} (2.4)
While the omega is the angular velocities vector (Flores, 2015, p. 28)
𝛚𝑖 = {𝛚𝑥 𝛚𝑦 𝛚𝑧}𝑖𝑇 (2.5)
The second derivative of Φ is (Flores, 2015, p. 33)
𝚽̈ ≡ 𝐃𝐯̇+𝐃̇𝐯=0 (2.6)
Derivative of the velocity is the derivative of the equation a2.1 in appendix 2. Term 𝐃𝐯̇ is denoted as ɣ. (Flores, 2015, p. 33)
2.2.3 Kinematic Joints Constraints
With three kinematic joints, the spherical, revolute and spherical-spherical joints, a tremendous amount of 3D MBS can be studied in real simulation point of view. For a spherical joint, figure 2.10, it allows three relative rotations. ‘’Therefore, the center of the spherical joints has constant coordinates with respect to any of the local coordinates systems
of the connected bodies, i.e., a spherical joint is defined by the condition that the point Pi on body i coincides with the point Pj on body j. This condition is simply the spherical constraint, which can be written in a scalar form as”: (Flores, 2015, p. 43)
𝚽(s,3) ≡ 𝐫𝑗𝑃− 𝐫𝑖𝑃 = 𝐫𝑗+ 𝐬𝑗𝑃− 𝐫𝑖− 𝐬𝑖𝑃 = 0 (2.7)
Figure 2.10. Spherical joint between two bodies, i and j (Flores, 2015, p. 44).
The first derivative of the Eq. 2.7 describes the equation of the velocity constraint (Flores, 2015, p. 44):
𝚽̇(s,3) = 𝐫̇𝑗+ 𝐬̇𝑗𝑃− 𝐫̇𝑖− 𝐬̇𝑖𝑃 = 0 (2.8)
The second derivative of the Eq. 2.7 illustrates the equation of the acceleration constraint (Flores, 2015, p. 44):
𝚽⃗⃗⃗ ̈(s,3) = 𝐫̈𝑗− 𝐬̇̃𝑗𝑃𝛚𝑗− 𝐬̇̃𝑗𝑃𝛚̇𝑗− 𝐫̈𝑖+ 𝐬̇̃𝑖𝑃𝛚𝑖 + 𝐬̃𝑖𝑃𝛚̇𝑖 = 0 (2.9)
The revolute joints between two bodies and their equations, can be found in appendix 3.
2.2.4 Equations of Motion
Before the inception of this section it should be mentioned that in appendix 4, the equation of motion for constrained system can be found. In this part, a main process for dynamic analysis of MBS will explain. This process is based on the standard Lagrange multipliers method. At first, the equation of motion for a constrained MBS based on Newton-Euler concepts are written as below which g is the generalized force vector. (Flores, 2015, p. 61) 𝐌𝐯̇ − 𝐃𝑇𝜆 = 𝐠 (2.10)
With consideration of constraint equations at the acceleration level with the differential equations at the same time, the dynamic analysis can be accomplished. Hence, 𝐃𝐯̇ can be written as (Flores, 2015, p. 61)
𝐃𝐯̇ = ɣ (2.11)
‘’ Equation 2.10 can be appended to Eq. 2.11, yielding a system of differential algebraic equation (DAE). This system of equations is solved for accelerations vector,𝐯̇ and Lagrange multipliers, λ. Then, in each integration time step, the accelerations vector,𝐯̇, together with velocity is integrated in order to obtain the system velocities and positions for the next time step.’’ (Flores, 2015, p. 61) For launching any dynamic simulation a set of initial conditions, such as velocity or position is needed. Equations 2.10 and 2.11 can be written such below in a matrix form (Flores, 2015, p. 62);
[𝐌 𝐃𝑇
𝐃 0 ] {𝐯̇𝜆} = {𝐠
𝜆} (2.12)
For this level, the equation of motions can be analytically considered and solved. In order to do that, the equations 2.10 can be written as below in which the acceleration vector is put (Flores, 2015, p. 62).
𝐯̇ = 𝐌−1(𝐠 + 𝐃𝑇𝜆) (2.13)
In order to have the inverse matrix for M, it is supposed that there is no null inertia (or mass) in the MBS matrix (Flores, 2015, p. 62).
𝜆 = [𝐃𝐌−1𝐃𝑇]−1(ɣ − 𝐃𝐌−1𝐠) (2.14)
If equation 2.14 is used in the equation 2.13, then the below equation can be obtained (Flores, 2015, p. 62):
𝐯̇ = 𝐌−1𝐠 + 𝐌−1𝐃𝑇{[𝐃𝐌−1𝐃𝑇]−1(ɣ − 𝐃𝐌−1𝐠)} (2.15)
Figure 2.11 illustrates a flowchart which obtains the algorithm of a standard solution of the equation of motion. The following steps explain the algorithm:
- 𝑡0, 𝐪0 𝑎𝑛𝑑 𝐯0 are initial values.
- The mass matrix, M, should be assembled. The Jacobian matrix should be evaluated.
The constraint equations should be constructed. Ɣ, the right-hand side of the accelerations, should be determined and the force vector g, should be calculated.
- To obtain values for 𝐯 ̇ and λ, the linear set of equations of motion for a constrained MBS should be solved.
Figure 2.11. The flowchart for dynamic analysis of MBS base on the standard Lagrange multipliers method (Flores, 2015, p. 63).
In simulations, the main equations of the constraint begin to be broken because of integration process. In order to solve this problem and keep the constraint break under control, a method called the Baumgarte stabilization method can be helpful. The main goal of this method is substitute differential equations which have been used up to now with following equation.
Figure 2.9 illustrates open and close loop of control systems (Flores, 2015, p. 64).
Figure 2.9. Open and closed loop for control systems (Flores, 2015, p. 64).
𝛗̈ + 2𝛼𝛗̇ + 𝛽2𝛗 = 0 (1.61)
Alpha and Beta are constant, positive constant. ‘’ The principle of the method is based on the damping of acceleration of constraint violation by feeding back the position and velocity of constraint violations, as illustrated in figure 12’’. (Flores, 2015, p. 64) In close loop system, φ and its differential do not move toward zero, it means that the system is not stable.
By using the Baumgarte method (Flores, 2015, p. 64):
[𝐌 𝐃𝑇 𝐃 0 ] {𝐯̇
𝜆} = { 𝐠
ɣ − 2𝛼𝛗̇ − 𝛽2𝛗} (1.62)
2.2.5 Integration Methods in Dynamic Analysis
This section explains the utilization of some practical integration algorithms in the resolving in the equation of motion. ‘’ Particular emphasis is paid to the Euler method, Runge-Kutta approach and Adam Predictor-corrector method that allows for the use of variable time steps
during the integration process.’’ (Flores, 2015, p. 67). In regular, the equations of motion for MBS is based on two main methods which are Newton-Euler method and the augmentation one. Translational and rotational motions are interpreted by Newton-Euler method, while to link the constraint equation of MBS, the augmentation method have been used. Using numerical integration algorithms is considerably beneficial to solve ODE, hence, in this dissertation work the DAE are changed to ODE. The number of n1 second-order differential equations can be converted to 2n1 first-order equations can be seen as below (Flores, 2015, p. 68):
𝑦̈1 = 𝑓(𝑦1, 𝑦̇1, 𝑡) (2.16)
𝑦̇1 = 𝑦2 (2.17) 𝑦̇2 = 𝑓(𝑦1, 𝑦2, 𝑡) (2.18)
Methods Euler, Rung-Kutta and Adams predictor-corrector are mostly used numerical integration methods. Although these methods have been used for many years, more than 100 years about Rung-Kutta, availability of computers helped enormously to understand a tremendous amount of ways to utilize them. ‘’ The discrete points may have either constant or variable spacing as ℎ𝑖1 = 𝑡𝑖1+1− 𝑡𝑖1, where ℎ𝑖1 is ‘’the integration step size’’ for any discrete 𝑡𝑖1. At each 𝑡𝑖1, the solution y (𝑡𝑖1) is approximated by a number 𝑦𝑖1. Since no numerical method is capable of finding y (𝑡𝑖1) exactly, the below quantity, Eq. (2.19), represents the global or total error at t=𝑡𝑖1’’. (Flores, 2015, p. 68)
ɛ𝑔𝑖11 = |𝑦(𝑡𝑖1) − 𝑦𝑖1| (2.19)
The occurred errors has two different components, first one is a kind of truncation error and other one is the round-off error. The truncation error which is a kind of inherent error, happens because this error is related to nature of numerical algorithms while analyzing 𝑦𝑖1. Finite word length in a computer can cause the round-off error. There is a method called single step methods which is a type of method for progressing to solve an equation of motion for a MBS which needs data from problem to solve it. For this dissertation work, in this
thesis, the solution method for the equation of motion is Runge-Kutta method which is a single step method. The algorithm for the single step methods, called multistep methods which is Adams predictor-corrector method. A crucial point about the numerical integration method is it require some function evaluation. For instance, for 4-order Runge-Kutta method, 4 function evaluation are needed. The numerical task has a relation with an initial value’s integration which can be seen as below equation (Flores, 2015, p. 68):
𝑦̇1 = 𝑓(𝑦, 𝑡) (2.20)
Initial condition for the Eq. 2.20 is: 𝑦(𝑡0) = 𝑦0 where ‘’y is the variable to be integrated and function f (t, y) is defined by the computational sequence of the selected algorithm’’. (Flores, 2015, p. 69)
Euler approach is one of the best and also simple approach integrators. Euler method can solve the differential equations in one single step (Flores, 2015, p. 69):
𝑦𝑖1+1= 𝑦𝑖1+ ℎ𝑓(𝑦𝑖1, 𝑡) (2.21)
‘’ Where variable h is the integration step size h=𝑡𝑖1+1− 𝑡𝑖1, for i which is a non-negative integer.’’ (Flores, 2015, p. 69). Figure 2.12 illustrates the Euler method in a basic type. Curve y=y (t) is the solution of the Eq. (2.20) which can be seen that it passes through the point P.
The height RQ, value of y1=y0+Δy should be found. There is no data or information about the curve’s points, however the slope of the curve is equal to f (t,y) and it means the differential equation based on the geometric interpretation. So the equation 𝑦̇0 = 𝑓(𝑡0, 𝑦0) is the slope of the tangent at point P. Length PS does not have a big deviation from the curve PQ if h is not big. So, 𝑅𝑆 ≅ 𝑅𝑄. RS is equal to ℎ𝑦̇0.
Figure 2.12. ‘’Geometric interpretation of the Euler integration method’’ (Flores, 2015, p.
70).
With help of the Taylor series about t=𝑡𝑖1, y (t) can be expanded at t=𝑡𝑖1+1 (Flores, 2015, p.
70).
𝑦(𝑡𝑖1+1) = 𝑦(𝑡𝑖1) + ℎ𝑓(𝑡𝑖1, 𝑦𝑖1) + 𝑂(ℎ2) (2.22)
The truncation error is in this equation is given by (Flores, 2015, p. 70)
ɛ𝑙 = 𝑂(ℎ2) (2.23)
The accuracy of the method is related to the order of that method and can explain the truncation error. So in a scalar equation (Flores, 2015, p. 70):
ɛ𝑙 = 𝑂(ℎ𝑃′+1) (2.24)
The order for this equation is 𝑃′th order. The Euler method is a first order method. If h is too big, the accuracy while computing will decrease and in order to very high amount of oscillation in motion, there will be very fast changes in the derivatives of the function.
(Flores, 2015, p. 70)
The difference between y (𝑡𝑖1) and 𝑦𝑖1 can assist to find the whole (global) truncation error.
It should be noticed that this calculation is in the absence of other error, round-off error (Flores, 2015, p. 70).
ɛ𝑔𝑖11 = |𝑦(𝑡𝑖1) − 𝑦𝑖1| (2.25)
If the precise express is needed, then the Runge-Kutta method, which is a second-order algorithm, can help (Flores, 2015, p. 70),
𝑦𝑖1+1 = 𝑦𝑖1 +ℎ2(𝑓1+ 𝑓2) (2.26)
The function, 𝑓1 , (Flores, 2015, p. 70)
𝑓1 = 𝑓(𝑡𝑖1+ 𝑦𝑖1) (2.27)
𝑓2 = 𝑓(𝑡𝑖1+ ℎ, 𝑦𝑖1+ ℎ𝑓1) (2.28)
For this approach, in any time step, two function evaluations are needed. The Rung-Kutta method can be interpreted as figure 2.13 geometrically.
Figure 2.13. Runge-Kutta method in geometric interpretation (Flores, 2015, p. 71).
𝑓1 does not depend on 𝑓2 and on 𝑦𝑖1+1. It should be noticed that for more accuracy in bigger time steps, the fourth-order of Runge-Kutta method can be used. (Flores, 2015, p. 71) 𝑦𝑖1+1 = 𝑦𝑖1 + ℎ𝑓5 (2.29) Where
𝑓5 = 16(𝑓1+ 2𝑓2+ 2𝑓3+ 𝑓4) (2.30)
𝑓1 = 𝑓(𝑡𝑖1, 𝑦𝑖1) (2.31)
𝑓2 = 𝑓(𝑡𝑖1+ℎ2, 𝑦𝑖1+ℎ2𝑓1) (2.32)
𝑓3 = 𝑓(𝑡𝑖1+ℎ2, 𝑦𝑖1+ℎ2𝑓2) (2.33)
𝑓4 = 𝑓(𝑡𝑖1+ ℎ, 𝑦𝑖1 + ℎ𝑓3) (2.34)
Figure 2.14 illustrates the interpretation of fourth-order Runge-Kutta method geometrically.
‘’ The local error of this method is of order ℎ5, which is relatively small even for larger time steps. The major disadvantage of this method is that the function f (t, y) needs to be evaluated four time at each time step.’’ (Flores, 2015, p. 72)
Figure 2.14. ‘’Geometric interpretation of the fourth-order Runge-Kutta method’’ (Flores, 2015, p. 72).
2.3 Simulation in practice
In addition of scientific targets, simulation has to be functional in other areas such as marketing. To reach to this objective, a comprehensive simulated model is required which can satisfy requested expectations. This chapter introduces simulation and simulators, then simulation will be discussed from marketing aspect. Then, customizable models and employed software will be discussed. Finally, design and results for a four-bar mechanism will be presented.
2.3.1 Simulation
Real time simulation is a functional tool which prepares a vast angle about behaviors of variety vehicles and responses of their parts in diverse situations. Based on real inputs and data for all components, with a detailed and correct design in simulation, real output can be achievable and useable. Another striking benefit of real-time simulation is being cost effective. To design and build a real machine for obtaining results, a significant amount of budgets have to be costed, however, with real simulation, this aim can be achievable.
Moreover, there is a considerable chance to modify and make parameterization for a simulated vehicle.
A drawback of simulation is that for some parameters and situations there are always some estimation data which can distance from obtaining real results. The number of assumptions and simplifications should be minimal in order to decrease the errors in result data.
2.3.2 Simulators
Simulators play a crucial role to provide real feeling to user (customer) while using them.
With help of accurate software and hardware which are working simultaneously, simulator’s output is enough accurate. For this master’s thesis a kind of simulator, illustrated in figure 2.15, have been used. The software installed on it is MeVEA. Moreover, with a motion platform which has employed four hydraulic cylinder-pistons, the feeling of movement can be transferred to the user.
Figure 2.15. SIM STUDIO with motion platform in order to obtain real feeling to users.
This motion platform has two joysticks and also a steering wheel, which obtains real feeling about moving, lifting sands, et cetera. Also, this simulator has sound effect beside real movements.
2.3.3 Marketing
Simulations and simulators have a tremendous amount of possibilities in markets. There are considerable number of companies which can utilize simulators and achieve its benefits. In order to decrease testing budget, a company can use a simulator which is appropriate to its research and extract results without spending significant amount of money. One of the most remarkable point of simulation for marketing is parameterization. It means the amount of spending money can be raised if the company wants to change some parts of a machine and figures out new results. With a parameterized simulation model, any type of changing are unchallenging. Not only simulating of a machine in simulator is cheaper than manufacture it, but also substituting some parts in simulation has no more charge. That is why more and more companies are figuring on the advantages of simulations and simulators.
2.3.4 Customizable Model
Preventing of wasting time and budget are the main purposes of producing a generic model in simulation. There are noticeable number of possibilities for a user, based on the concept of parameterization, to build a simulation model and extract it results. Suppose a simulation model with many assemblies such as engine assembly, hydraulic assembly, power transmission assembly, et cetera, which can be selected by a user. Each of these assemblies has their own sub-assemblies. Based on user’s options, there are a large number of combinations for assemblies and sub-assemblies which will present different results and analysis. Figure 2.16 depicts a schematic of a simulation model with its assemblies and sub- assemblies. As the figure 2.16 shows, there are many options and combinations for building one model which every single changing in sub-assemblies will affect to other parts and data, so when a user finishes his/her selections, a new model is created which are totally different from other models.
Figure 2.16. Simulation model made by assemblies and sub-assemblies
Each sub-assembly has its own code, based on a software which has been used to write the code. Then, all assemblies will gather and the real-simulation model will be ready to run.
2.3.5 Employed Software
In this dissertation work as it mentioned before, all real-time simulation issues is handled in MeVEA software. In addition, for other aspects, such as graphics, other software are used. Blender, Python, SolidWorks, and Excel. The duty of each software will be explained in next chapters.
2.3.6 MeVEA
MeVEA does the real-time simulation task with some rational simplifications. There is no graphics while modelling in MeVEA. Objects in MeVEA do not have any shape so their collision graphics and graphics, connections and environments must be defined in other software. In order to reach faster and easier analysis during simulation, these simplifications are very important. MeVEA has two interfaces, one is to create a simulation model and other one is to run the model and get all results. Figures 2.17 and 2.18 show the work interface and dynamic simulation one, respectively.
Figure 2.17. Working interface in MeVEA software to create a model, like excavator.
Figure 2.18. Dynamic simulation interface in MeVEA software.
As it is shown in figure 2.17 in MeVEA, the bodies, constraints among them, all graphics of bodies and environment, data related to movements, hydraulic model, inputs and outputs, virtual sensors, and other data can be modeled and implemented.
In dynamic simulation interface, a user can run the simulation model and see does it work properly or not and extract desired results and plots. Because of simplification, graphics in working interface do not have all details, on the other hand in dynamic simulation interface, graphics are different and with details.
User can run the model and control it with joysticks and a steering wheel, if needed, or with keyboard, figure 2.19. As figure below depicts, user can have plotting diagrams while simulating. These plots demonstrate behavior of each element during running a simulation model. In the other words, plots are the main section of final results in MeVEA. In plot tab in the dynamic simulation interface, user can opt specifications which she/he wants to see its behavior at the end of simulation.
Figure 2.19. Running situation with the keyboard control and a plot.
2.3.7 CAD Software – SolidWorks
To model some parts of the graphics in my case study, an excavator, SolidWorks played a crucial role because of its precision in 3D assemblies. With a user-friendly environment, SolidWorks is the first option for designing. In graphics point of view, MeVEA can read .stl files format, however 3ds file format is more compatible with MeVEA. So beside SolidWorks, presence of another software for other graphics is needed.
2.3.8 Blender
With blender all environment graphics, collision graphics and some vehicle’s parts can be modeled and it is a strong software for making visualization extremely professional.
Moreover, Blender is an open-source software which can be utilized easily and has some striking features such as fast modeling, photorealistic rendering, and preparing real feeling about materials (Blender, 2017). The output file of the Blender can be 3ds format which is readable in MeVEA software. In MeVEA two types of graphics are needed, one of them is visualization graphic and the other one is collision graphic. File format for Collision graphics in MeVEA is only .3ds (Mevea, 2017). Clearly, visualization graphic is to see the model and the environment and the collision graphic is for parts of a model which must have collision with other parts of body and also environment. For instance, in the excavator model, the
bucket should dig the sand, so it must have collision graphics beside its visualization one. In blender the collision and visualization graphics can be seen and modified simultaneously in order to have a realistic actions during running a simulation model.
Figure 2.20-a, illustrates a visualization graphic of a bucket and in figure 2.20-b the visualization and collision graphics can be seen simultaneously. Moreover, scales, colors and textures can be set and manipulated in blender.
Figure 2.20. a) Visualization graphic, b) Visualization and collision graphics in blender.
2.3.9 Python and Excel
With Python and excel software, parameterization target can be reachable. Python is a coding software which can be used as an open-source software with a practical database (Python, 2017).
With an Excel file as an interface file, a user can choose an option among available options for a model for instance, and can create the vehicle based on his/her customization.
To implement this customization, a code-based software is needed to play as a bridge. With writing a script in Python software, all selected options can be readable in MeVEA and after running the simulated vehicle, the desired options in the vehicle can be seen in the MeVEA environment. Creating Python code and excel file will be explained in the case study chapter.
2.4 Four-bar Mechanism
To evaluate and figure out the real-simulation mechanism, there is a four-linked model is considered, figure 2.21. In order to find out behavior of a mechanism, position analysis, the velocity analysis, the Lagrangian formulation and the equation of motion will be explained.
Figure 2.21. Four-Bar Mechanism
For the four-bar mechanism, position analysis, equations of the close loop for the mechanism’s dimensions, figure 2.22, can be written as below.
Figure 2.22. 4-bar mechanism for analysis
Table 2.1 demonstrates the dimensions for 4-bar mechanism, shown in figure 2.22. It should be noticed that values in the table are considered in center to center way.
Table 2.1. Values for parameters shown in figure 2.22.
Parameter 𝑙0 𝑙1 𝑙2 𝑙3 𝜃0′ 𝛼0′ 𝜑0′ 𝑚1 𝑚2 𝑚3
Value (unit)
117.9 cm
50 cm
100 cm
80 cm
90° 15.68° 106.1° 5 kg
10 kg
8 kg −𝑙1cos 𝜃′− 𝑙2cos 𝛼′+ 𝑙0+ 𝑙3cos 𝜑′ = 0 (3.1) −𝑙1sin 𝜃′− 𝑙2sin 𝛼′+ 𝑙3sin 𝜑′ = 0 (3.2)
The final equation for position:
𝛼′(𝜃′, 𝜑′) = 𝑡𝑎𝑛−12(−𝑙1sin 𝜃′+ 𝑙3sin 𝜃′, 𝑙0− 𝑙1cos 𝜃′+ 𝑙3cos 𝜑′) (3.6)
The equation of the velocity,
[𝛼̇′
𝜑̇′] = [𝑆1(𝜃′, 𝛼′, 𝜑′)
𝑆2(𝜃′, 𝛼′, 𝜑′)] 𝜃̇′ (3.10)
Kinetic energy:
𝑇 =12(𝑚1𝑙𝑐12 (𝜃̇′)2+ 𝐼1(𝜃̇′)2) +12(𝑚2𝑙12(𝜃̇′)2+ 𝑙𝑐22 (𝛼̇′)2+ 2𝑙1𝑙𝑐2cos(𝜃′− 𝛼′) 𝜃̇′𝛼̇′+ 𝐼2(𝛼̇′)2) +12(𝑚3𝑙𝑐32 (𝜑̇′)2+ 𝐼3(𝜑̇′)2) (3.11)
Where T is the kinematic energy, m is the mass, I is inertia and 𝑙𝑐 is l/2.
Potential energy:
V𝑚1𝑔𝑙𝑐1sin 𝜃′+ 𝑚2𝑔𝑙1sin 𝜃′+ 𝑙𝑐2sin 𝛼′+ 𝑚3𝑔𝑙𝑐3sin 𝜑′ (3.12)
V is the potential energy and g is the ground acceleration.
The final equation of motion:
𝐌(𝜃′)(𝜃̈′)2+ 𝑉(𝜃′, 𝜃̇′) = 𝜏𝜃 (3.21)
The constraint equations, Jacobian matrix, Newton difference for position analysis and velocity analysis for the four-bar mechanism can be found in appendix 5.
Results of 4-bar-mechanis:
After modelling in MeVEA, figure 2.23, the result of simulation can be found below;
Figure 2.23. Dynamic simulation for 4-bar mechanism
Figure 2.24 depicts the total torque versus time in this mechanism while there is a torque on the revolute joint between stand1 and left. Link.
Figure 2.24. Total torque for 4-bar-mechanism.
Clearly, figure 2.24 shows the torque is increased through the time with a fluctuation.
Figure 2.25 illustrates the angular velocity in y and z direction for the middle link in x while simulating.
Figure 2.25. The angular velocity in y and z direction for the Middle-link.
As figure 2.25 illustrates, the angular velocity in z direction is 1.03e3 deg/s in its maximum position while in y direction is 1.03e-11 simultaneously and they are repeating these data through the time because of the constant torque applying to the system. In figure 2.26, the local joint force for the left-link body on the constraint between left-link and the middle-link is shown. In this figure, the Fxl force is increasing through the time, however, the Fyl force is almost is repeated.
Figure 2.26. The local joint force in joint between the left-link and the middle-link.
