Mechatronics System Design PDF

Department of Mechatronics Engineering Dr. Mahmoud Kaid Mechatronics System Design Chapter 1 Introduction to MSD - Ref: - ch.1, Shetty 1.1. What Is Mechatronics? Mechatronics is a methodology used for the optimal design of electromechani...

Department of Mechatronics Engineering Dr. Mahmoud Kaid Mechatronics System Design Chapter 1 Introduction to MSD - Ref: - ch.1, Shetty 1.1. What Is Mechatronics? Mechatronics is a methodology used for the optimal design of electromechanical products. A methodology is a collection of practices, procedures, and rules used by those who work in a particular branch of knowledge or discipline. Familiar technological disciplines include thermodynamics, electrical engineering, computer science, and mechanical 1 engineering, to name several. Instead of one, the mechatronic system is multidisciplinary, embodying four fundamental disciplines: electrical, mechanical, computer science, and information technology. Many control system engineers are familiar with the quip: Design and build the mechanical system, then bring in the painters to paint it and the control system engineers to install the controls. The mechatronic design methodology is based on a concurrent (instead of sequential) approach to discipline design, resulting in products with more synergy. The difference between a mechatronic system and a multidisciplinary system is not the constituents, but rather the order in which they are designed. Historically, multidisciplinary system design employed a sequential design-by-discipline approach. For example, the design of an 2 electromechanical system is often accomplished in three steps, beginning with the mechanical design. When the mechanical design is complete, the power and microelectronics are designed, followed by the control algorithm design and implementation. The major drawback of the design-by-discipline approach is that, by fixing the design at various points in the sequence, new constraints are created and passed on to the next discipline. Control designs often are not efficient because of these additional constraints. For example, cost reduction is a major factor in most systems. Tradeoffs made during the mechanical and electrical design stages often involve sensors and actuators. Lowering the sensor–actuator count, using less accurate sensors, or using less powerful actuators, are some of the standard methods for achieving cost savings. 1.2. Mechatronics Key Elements 3 Mechatronics is a good design practice. The basic idea is to apply new controls to extract new levels of performance from a mechanical device. Sensors and actuators are used to transduce energy from high power (usually the mechanical side) to low power (the electrical and computer side). The block labeled “Mechanical systems” frequently consists of more than just mechanical components and may include fluid, pneumatic, thermal, acoustic, chemical, and other disciplines as well. Control is a general term and can occur in living beings as well as machines. The term “Automatic control” describes the situation in which a machine is controlled by another machine. The inherent concurrency or simultaneous engineering of mechatronics approach relies heavily on the use of system modeling and simulation throughout the design and prototyping stages. 4  Figure 1-2. Mechatronics Key Elements Mechatronics is the result of applying information systems to physical systems. The physical system (the rightmost dotted block of Figure 1-2) consists of mechanical, electrical, and computer systems as well as actuators, sensors, and real-time interfacing. In some of the literature, this block is called an electromechanical system. 1.2.1. Mechanical Systems 5 Mechanical systems are concerned with the behavior of matter under the action of forces. Such systems are categorized as rigid, deformable, or fluid in nature. Newtonian mechanics provides the basis for most mechanical systems and consists of three independent and absolute concepts: space, time, and mass. A fourth concept, force, is also present but is not independent of the other three. 1.2.2. Electrical Systems Electrical systems are concerned with the behavior of three fundamental quantities: charge, current, and voltage (or potential). When a current exists, electrical energy usually is being transmitted from one point to another. Electrical systems consist of two categories: power systems and communication systems. Communication systems are designed to transmit information as low-energy electrical signals between points. 6 Functions such as information storage, processing, and transmission are common parts of a communication system. Electrical systems are an integral part of a mechatronics application. 1.2.3. Sensors and Actuators Sensors are required to monitor the performance of machines and processes. Using a collection of sensors, one can monitor one or more variables in a process. Sensing systems also can be used to evaluate operations, machine health, inspect the work in progress, and identify part and tools. The monitoring devices are generally located near the manufacturing process measuring the surface quality, temperature, vibrations, and flow rate of cutting fluid. Sensors are needed to provide real time information that can assist controllers in identifying potential 7 bottlenecks, breakdowns, and other problems with individual machines and within a total manufacturing environment. Actuation involves a physical action on the process, such as the ejection of a work piece from a conveyor system initiated by a sensor. Actuators transform electrical inputs into mechanical outputs such as force, angle, and position. Actuators can be classified into three general groups. 1. Electromagnetic actuators, (e.g., AC and DC electrical motors, stepper motors, electromagnets) 2. Fluid power actuators, (e.g., hydraulics, pneumatics) 3. Unconventional actuators (e.g., piezoelectric, magneto strictive, memory metal) There are also special actuators for high-precision applications which require fast responses. They are often applied to controls which compensate for friction, nonlinearities, and limiting parameters. 8 1.2.4. Real-Time Interfacing The real-time interface process falls into the electrical and information system categories but is treated independently as was computer system hardware because of its specialized functions. In mechatronics, the main purpose of the real-time interface system is to provide data acquisition and control functions for the computer. The purpose of the acquisition function is to reconstruct a sensor waveform as a digital sequence and make it available to the computer software for processing. The control function produces an analog approximation as a series of small steps. The inherent step discontinuities produce new undesirable frequencies not present in the original signal and are often attenuated using an analog smoothing filter. Thus, for mechatronic applications, real-time interfacing includes analog to 9 digital (A/D) and digital to analog (D/A) conversion, analog signal conditioning circuits, and sampling theory. 1.2.5. Information Systems Information systems include all aspects of information transmission—from signal processing to control systems to analysis techniques. An information system is a combination of four disciplines: communication systems, signal processing, control systems, and numerical methods. In mechatronics applications, we are most concerned with modeling, simulation, automatic control, and numerical methods for optimization. 1.2.5.1 Modeling Modeling is the process of representing the behavior of a real system by a collection of mathematical equations and logic. The real system/physical system is a system whose behavior is based on matter and energy. Models 10 can be broadly categorized as either static or dynamic. In a static model, there is no energy transfer. Systems, which are static produce no motion, heat transfer, fluid flow, traveling waves, or any other changes. On the other hand, a dynamic model has energy transfer which results in power flow. Power, or rate of change of energy, causes motion, heat transfer, and other phenomena that change in time. Phenomena are observed as signals, and since time is often the independent variable, most signals are indexed with respect to time. In engineering applications, certain conventions in terminology are used. Resources are referred to as design variables, aspects of system behavior as objectives, and system governing relationships (equations and logic) as constraints. 1.2.5.2 Simulation 11 Simulation is the process of solving the model and is performed on a computer. Although simulations can be performed on analog computers, it is far more common to perform them on digital computers. The process of simulation can be divided into three sections: initialization, iteration, and termination. If the starting point is a block diagram-based model description, then in the initialization section, the equations for each of the blocks must be sorted according to the pattern in which the blocks have been connected. The iteration section solves any differential equations present in the model using numerical integration and/or differentiation. An ordinary differential equation is (in general) a nonlinear equation which contains one or more derivative terms as a function of a single independent variable. For most simulations, this independent variable is time. 12 The order of an ordinary differential equation equals the highest derivative term present. The display section of a simulation is used to present and post the output process. Output may be saved to a file, displayed as a digital reading, or graphically displayed as a chart, strip chart, meter readout, or even as an animation. Because the model will be used and altered by engineers from multiple disciplines, it is especially important that it be programmed in a visually intuitive environment. Such environments include block diagrams, flow charts, state transition diagrams, and bond graphs. In contrast to the more conventional programming languages such as Fortran, Visual Basic, C++, and Pascal, the visual modeling environment requires little training due to its inherent intuitiveness. This environment is extremely versatile, low in cost, and often includes a code generator option, which translates the block 13 diagram into a C (or similar) high-level language suitable for target system implementation. Block diagram-based modeling and simulation packages are offered by many vendors, including MATRIXxTM, Easy5TM, SimulinkTM, Agilent VEETM, DASYLabTM, VisSimTM, and LabVIEWTM. 1.2.5.3 Optimization Optimization solves the problem of distributing limited resources throughout a system so that pre-specified aspects of its behavior are satisfied. In mechatronics, optimization is primarily used to establish the optimal system configuration. However, it may be applied to other issues as well, such as: Optimal System Configuration Identification of optimal trajectories Control system design 14  Identification of model parameters Another aspect of optimization is to optimize the overall manufacturing processes from product design to inspection by integrating all of the information into a common database. For example, knowledge of the parts geometry, as contained in the CAD system, can be used to determine the reference values of process variables. Information from various process- related sensors can be integrated to improve the reliability and quality of sensor information. This shared information (such as the data of the geometry of a part and the materials used from CAD/CAM database) can be used in selecting the optimum machining processes, tool selections, and finishing operations. The figure below shows a detailed presentation of mechatronics system components. 15 16 1.3. The Mechatronic Design Process The mechatronic design process is presented in Figure 1-4. Fig.1.4. Mechatronics Design Process 17 The mechatronic design process consists of three phases: modeling and simulation, prototyping, and deployment. All modeling, whether based on first principles (basic equations) or the more detailed physics, should be modular in structure. A first principal model is a simple model which captures some of the fundamental behavior of a subsystem. A detailed model is an extension of the first principal model providing more function and accuracy than the first level model. Connecting the modules (or blocks) together may create complex models. Each block represents a subsystem, which corresponds to some physically or functionally realizable operations, and can be encapsulated into a block with input/output limited to input signals, parameters, and output signals. 18 1.4. Integrated Design Issues in Mechatronics Mechatronics makes the combination of actuators, sensors, control systems, and computers in the design process possible. The integration within a mechatronic system is performed through the combination of hardware (components) and software (information processing). Hardware integration results from designing the mechatronic system as an overall system and bringing together the sensors, actuators, and microcomputers into the mechanical system. Software integration is primarily based on advanced control functions. An important characteristic of mechatronic devices and systems is their built-in intelligence that results through a combination of precision in 19 mechanical and electrical engineering, and real-time programming integrated into the design process. 1.5. Modularity and Re-Configurability Because of their modularity, mechatronic systems are well suited for applications that require reconfiguration. Such products can be reconfigured either during the design stage by substituting various subsystem modules or during the life span of the product. Since many of the steps in the mechatronic design process rely on computer-based tasks (such as information fusion, management, and design testing), an efficient computer-aided prototyping environment is essential. 1.6. Hardware-in-the-Loop Simulation In the prototyping step, many of the non-computer subsystems of the model are replaced with actual hardware. 20 Sensors and actuators provide the interface signals necessary to connect the hardware subsystems back to the model. The resulting model is part mathematical and part real. Because the real part of the model inherently evolves in real time and the mathematical part evolve in simulated time, it is essential that the two parts be synchronized. This process of fusing and synchronizing model, sensor, and actuator information is called real-time interfacing or hardware-in-the-loop simulation, and is an essential ingredient in the modeling and simulation environment. The hardware-in-the-loop model (Figure 1-5) shows the different components of a mechatronic system. There are different ways in which hardware-in-the-loop could be simulated, such as electronics simulation, simulation of actuators and sensors, or simulation of mechanical systems alone. It is possible to simulate the electronics where the actuators, 21 mechanics and sensors are the real hardware. On the other hand, if appropriate models of the mechanical systems, actuators, and sensors are available, the electronics could be the only hardware. Fig 1.5. Hardware in the Loop Model Assuming the following six functions in mechatronics systems: Control: The control algorithm(s) in executable software form. Computer: The embedded computer(s) used in the product. Sensors Actuators 22  Process: Product hardware excluding sensors, actuators, and the embedded computer. Protocol (optional): For bus-based distributed control applications. There are other possibilities summarized in Table 1-1. Table 1-1 Different Configurations for Hardware-In-The-Loop Simulation 1.7. Life Cycle Design in Mechatronics 23 The mechatronic design methodology is not only concerned with producing high-quality products but with maintaining them as well—an area referred to as life cycle design. Several important life cycle factors are indicated. Delivery: Time, cost, and medium. Reliability: Failure rate, materials, and tolerances. Maintainability: Modular design. Serviceability: On board diagnostics, prognostics, and modular design. Upgradeability: Future compatibility with current designs. Disposability: Recycling and disposal of hazardous materials. 1.8. Condition Monitoring in Mechatronics Systems Condition monitoring is defined as the determination of the machine status or the condition of a device and its change with time in order to decide its 24 condition at any given time. The condition of the machines can be determined by physical parameters (like tool wear, machine vibration, noise, temperature, oil contamination, and debris). A change in these parameters provides an indication of the changing machine condition. The condition monitoring systems can be of two types. 1. Monitoring systems that display the machine conditions to enable the operator to make decisions. 2. Automated monitoring of conditions with adaptive control features. If the machine conditions are properly analyzed, they can become a valuable tool in establishing a maintenance schedule and in the prevention of machinery failures and breakdowns. The diagnostic parameters can be measured and monitored continuously at predetermined intervals. In some cases, measurement of secondary parameters such as pressure drop, flow, 25 and power can lead to information on primary parameters such as vibration, noise, and corrosion. The data coming from different levels of the factory provide support for automated manufacturing. Sensors integrated with adaptive processes control capability at the plant level, manufacturing management level, control level, or sensory level and handle the requirements as shown in Figure 1-7. In the case of manufacturing machinery, sensors can monitor machining operations, conditions of cutting tools, availability of raw material, and work in progress. Sensors can assist in the recognition of parts, tools, and pallets. 26 They also can be used on the production floor during pre-process situations or at the time when the manufacturing process is in progress. Figure 1-7 Sensor Distribution at Different Levels of Production 27 The selection of the sensing principle and parameters monitored are shown in Table 1-2. Table 1-2 Examples of Sensing Parameters in Automated Manufacturing 28 1.8.1. Monitoring On-Line The importance of lean production systems has created an opportunity for intelligent autonomous inspection, manufacturing, and decision-making systems that perform tasks without human intervention. Currently, quality is ensured in the product engineering cycle at two distinct levels. At the product design stage: To ensure that quality is designed into the product, using the robust design method. At the final inspection stage: Using statistical process control methods. 1.8.2. Supervisory Control Structure The hierarchical control structure consists of servo, process, and supervisory controls. 29  The lowest level is servo control, where the motion of the cutting tool relative to the workpiece (such as its position and velocity) is controlled. This involves cycle times of approximately 1 millisecond. At the process control level, process variables (such as cutting forces and tool wear) are controlled with typical cycle times of around 10 milliseconds. Control level strategies are aimed at compensating for factors not explicitly considered in the design of the servo and process level controllers. The highest level is the supervisory level, which directly measures product related variables (part dimension and surface roughness). The supervisory level also performs functions such as chatter detection and tool monitoring. The supervisory level operates at cycle times of approximately 1 second. Finally, all of this information can be used to achieve online optimization of the machining process at the shop floor and plant control level. Figure 1-14 30 illustrates a hierarchical control structure where the controller elects’ position and velocity at the machine level, force and wears at the process level, and quality control issues (like dimension and roughness) at the product level. Fig 1.14. Framework for Integration Heterogeneous Systems 31  Chapter 2 Mechatronics System Design Examples 2.1. The Four Questions to be asked when Designing a Mechatronic System When designing mechatronic systems, it is important to consider the following four questions: 1- Is system dynamics important? If the speed of response of a system is critical to its stability and time response specification (e.g. overshoot, rise time, settling time …), then system dynamics is important and has to be considered in the design. In this case, a dynamic model has to be developed for the whole system (including the plant, controller, actuator and sensor). The model is then used to design a suitable controller. 32 The following list includes examples of systems for which system dynamics is critical: Quad-rotor Heat seeking missile 33  Inverted pendulum Exothermic reaction (chemical reaction) Nuclear reaction Vehicle active suspension system Space vehicle On other hand, for the following systems, system dynamics is not critical: IRIS scanning system for identification purposes Artificial nose A residential healthy for a storage cylinder A solar tracking photo-voltaic system 2- What type of components to select for each of the four main components for the mechatronic system? 34 The designer of the system will have to make a decision regarding the components of the mechatronic system, as follows: a. Control algorithm: example: PI, PD, PID, lead-lag compensator, Fuzzy controller, neural… genetic, digital PID … b. Physical controller: This is the physical controller within which the control algorithm will be implemented. Examples: PLC, PC/laptop, Raspberry pi and microcontrollers (PIC, AVR), as well as relays and the older type of analogue computers and logic gates. c. Actuator System: The actuator executes the instructions of the controller to realize changes in the plant. The actuator systems comprise the following three sub-components: variable speed drive (optional); actuator (required); mechanical drive (optional). 35 d. Sensor/Transducer/feedback Drivers): The sensor or feedback devices are the eyes and ears of the mechatronic system. They can be thermal, mechanic (translational or rotational) … etc. Where system dynamics is important, it is necessary to find transfer function of the sensor as well, as it forms the H(s) transfer function shown in the figure below. 3- The size of each component: Once the design has selected the components to be used it has to size them. Calculations have to be carried out to find the necessary size. This is especially true of the actuator system (e.g. motor, variable frequency drive, gearbox). 4- Resolution, accuracy and precision: Once the sensor/feedback has been selected, it is necessary to ensure that it can achieve the required accuracy, resolution and precision. 36 The user requirements specify the required accuracy, resolution and precision. In some cases, the controller selection and the actuator selection have to be revised or changed in order to achieve these requirements. The system is shown with multiple inputs 𝑈1, 𝑈2 … 𝑈𝑖; and multiple outputs 𝑦1, 𝑦2 … 𝑦𝑖, to allow for the general case. 2.2. User Requirements Specifications for a Mechatronic System The following categories encompass the types of user requirements for mechatronic systems: 2.2.1. Size/Capacity/Force/Torque/Stroke: This is the most obvious user requirement. It is related to the size of the system, in relationship to what is can do. For example, a bottle filling system will be able to fill 1000 bottles per hour; or on elevator can carry eight persons; or a printer can print 12 pages per minute. 37 2.2.2. Safety/Reliability/Maintainability/Availability. 2.2.3. System dynamics (e.g. rise time; overshoot; settling time). 2.2.4. Resolution/Accuracy/Precision: These characteristics are related to both actuator(s) and transducer(s). Resolution is the smallest change in input that would cause a change in output. Accuracy is freedom from systematic errors. Precision is freedom from random errors. 2.2.5. User Friendliness: The system user interface should be user friendly. It should allow different levels if access. It should also be “fool proof”. 2.2.6. Energy Consumption: In some application, reducing the amount of energy consumed must be restricted/reduced. 38 2.2.7. Cost (Capital cost/Running cost): It is important to consider all costs associated with the system, throughout the lifetime of the system. This includes capital costs (initial cost of the system) as well as running costs (energy costs, maintenance cost). 2.2.8. Space/Size/Weight: In certain cases, there may be constraint on the size of the system and its weight. 2.2.9. Environment: It is important to consider the environment in which the system will be operated. For example, ambient temperature, humidity, dust, and foreign particle and fluid ingress. 2.2.10. Versatility: A versatile system is one that can be used in different ways. 2.3. Examples of Mechatronic Systems 39 The following are examples of mechatronic systems: 1. Home appliances (fridges and freezers, microwave ovens, washing machines, vacuum cleaners, dishwashers, cookers, timers, mixers, blenders, stereos, televisions, telephones, lawn mowers, digital cameras, videos and CD players, camcorders, and many other similar modern devices): Many of the home appliances that are in use today are mechatronics systems. They are manufactured in large numbers end masse and typically require small controllers to be “embedded” within them. 2. ABS (anti-lock braking system) and many areas in automotive engineering: An antilock braking system on a vehicle is a system that prevents the wheels from ceasing up or stopping to rotate when the brakes are suddenly pressed. Another good example of a mechatronics system from automotive engineering is the engine control unit (ECU). 40 3. Elevators and escalators: They have many sensors to detect the position and speed of the elevator car, as well as any calls registered by the passengers. It has many actuators, the most important of which is the main hoist motor. Safety is also paramount in these systems as they carry human beings. 4. Mobile robots and manipulator arms: Robots are widely used today in all domains of life. Robots are generally used for applications that are inaccessible (difficult locations to get too due to height or space), dull (repetitive and tedious tasks), or dangerous (hazardous environments). 5. Sorting and packaging systems in production lines: Mechatronic systems are effectively the basis for modern factory automation. 41 6. Computer Numerically Control (CNC) production machines: CNC machines are critical in modern manufacturing systems. They allow the user to produce a product directly from a computer model of the piece. 7. Aeroplanes and helicopters: These are complex examples of mechatronic systems that incorporate hundreds or even thousands of smaller sub-mechatronic systems. 8. Tank fluid level and temperature control systems: An example is the process used to produce bio-fuels from vegetable oil. 9. Temperature control system in an industrial oven: Many industrial processes require close control of the temperature of the process in order to achieve the exact required outcome. These systems have very long lag times; thus, they take a long time to heat up and cool down. 42 10. Heat-seeking missiles: Heat seeking missiles are complex systems that require extremely fast responses. A poor or slow controller could easily lead to the destruction of the missile. The orientation of the missile will be controlled based on the heat signal received from the target. 11. Packing machines. 12. Using robots for painting windows and doors. 13. Coordinate Measuring Machines (CMM): CMM’s are machines that are used in manufacturing in order to scan the surface of an object to produce a computer aided design (CAD) model of the object. This can be done by direct contact (e.g. by the use of probes) or by the use of contactless methods (e.g. laser range detection). CMM are critical in the areas of reverse engineering and quality control. Activity: 43 Try to identify the four components of the mechatronics systems. Answer Key to Examples of Mechatronics Systems Activity Identifying the four components of the mechatronics systems: 1. The washing machine as an example of a mechatronic system. Let us take one of the systems within the washing machine such as the water heating system. Plant: The water within the washing machine Controller: Embedded controller on an integrated circuit Actuator: Heating element Feedback device: thermal sensing element such as a bimetallic strip or a resistance temperature detector (RTD). 2. In an anti-lock braking system, it is important to prevent the locking of the braking system when the road surface is slippery. 44 Plant: The wheels and the braking system Controller: ABS control module embedded within a microcontroller Actuator: Solenoid valve controlling the hydraulic fluid Feedback device: Wheel speed sensors 3. Elevators: speed control system in an elevator is comprised of the following: Plant: elevator cabin Controller: elevator speed controller system Actuator: induction motor, drive system Feedback device: shaft encoder 8. Tank fluid level systems: Plant: tank and contained fluid Controller: microcontroller 45 Actuator: pump Feedback device: float device or ultrasonic sensor 9. Temperature control system in an industrial over: Plant: oven and product within Controller: microcontroller or programmable logic controller Actuator: heating coil Feedback device: bimetallic strip 10. Heat seeking missiles: Plant: the missile body and any payload Controller: microcontroller embedded within the missile Actuator: jet engine Feedback device: temperature sensors fitted to the body of the missile 11. Packing Machines: 46 Plant: Item to be packed and conveyor carrying it. Controller: PLC (programmable logic controller). Actuator: Hydraulic cylinder. Feedback device: the feedback devices for such a system are usually of three types: a) image processing b) laser range sensors c) ultrasonic sensors 12. Painting robot: Plant: item to be painted (e.g. car frame) Controller: microcontroller Actuator: servo-motors controlling the robot parts Feedback device: camera with image processing software 47 13. Coordinate measuring machine (manipulator arm): Plant: CMM arms and object to be scanned Controller: PC with dedicated software Actuator: Human operator (i.e. not automated!) Feedback device: absolute shaft encoders on the CMM 2.4. Design of Inkjet Printer 2.4.1. Problem Statement You are asked to design an inkjet printer. The inkjet printer has two cartridges internally that are fitted on a horizontal carriage. Each cartridge has a piezoelectric actuator inside in order to dispense the ink. The printer also takes A4 size sheets of paper. Discuss in detail the design of the system, looking at the following actions: 1. Feeding the sheets of paper. 48 2. Moving the paper up and down. 3. Moving the carriage that carries the two cartridges left and right. 4. Printing with a resolution of 600 dpi (dots per inch). 5. Giving warning alarms to the user, such as “no paper” and “cartridge empty”. In your discussion, identify the different control loops in the system. Identify all the components of the mechatronic systems to be designed. Answer: Design of an inkjet printer The inkjet printer has two cartridges: one for black ink, and the other for three colors: cyan, magenta, and yellow. Each cartridge has a piezoelectric actuator that dispenses ink onto the sheet of paper. The inkjet printer has four control subsystems. 1. The vertical motion system that takes the sheet/paper and feeds it through printer. This system also moves the sheet of paper up and down to 49 a certain accuracy and resolution, during the printing process. So, the purpose of this subsystem is to feed the sheet of A4 paper and to move it accurately during the printing process. To do this, it needs an actuator which cloud either be a small d.c or stepper motor. The motor will be geared if a d.c. motor is used. The motor will be connected to a cylinder, over which the paper is fed. Rotational feedback is needed from the cylinder. This is usually in the form of an optical disk (transparent disk with stripes (lines)), where an optical transistor and LED will detect the rotation of the disk, to provide feedback to the controller. The number of lines/stripes on the disk should match the required printing resolution of 600 dots per inch (dpi). A micro-switch is used under the sheets of paper to detect that purpose are present, or if the paper has run out. 50 2. The second system is the horizontal motion system. This system moves the carriage (that holds the two cartridges) left and right in order to position the cartridges over the point where printing needs to take place. To do this movement a cleated/toothed belt shall be used as shown below. In order to provide feedback a disk is used with lines on it with optical feedback. The belt is toothed to prevent slipping. The resolution of optical disk should match the required printing resolution of 600 dots per inch (600 dpi). 51 3. The ink cartridges control system: The cartridges shown above will carry the two inkjet cartridges. A ribbon cable is necessary in order to pass signal to end from the cartridges and controller. No feedback will be used in this system. Each cartridge will have one or more piezoelectric actuator to allow ink to leave the cartridges and go onto the paper. 4. Control and monitoring system: PIC controller will be used to control the three systems above. The controller will also communicate with P.C. with the USB port and also control the on/off light, receive from the input control push-buttons, and sends status signals to LEDs (out of paper, inkjet cartridges empty). It also monitors the paper empty micro-switch. Components of the Mechatronic Systems (copy machine): 1. Physical Controller: PIC micro-controller of microprocessor. 2. Feedback: Optional disk with stripes x2. 52 3. Plant: a. Cylinder and paper. b. Carriage. c. Ink and cartridges. 4. Actuators: a. D.C. motors or stepper motors. b. Piezoelectric actuators. 5. Control algorithm: On-Off controller. 6. Control loops: a. Vertical motion system. (closed loop) b. Horizontal motion system. (closed loop) c. Ink dispensing system (open loop) 2.5. Problems 53 1) What do you understand by the term 'mechatronics'? 2) What are the key elements of mechatronics? 3) Is mechatronics the same as electronic engineering plus mechanical? 4) Is mechatronics as established as electronic or mechanical engineering? 5) List some mechatronic systems that you see every day. 6) Identify the four main components of the following MS systems: Domestic systems, such as air conditioning units, security systems, automatic gate control systems. Office equipment, such as laser printers, hard drive positioning systems, liquid crystal displays, tape drives, scanners, photocopiers, fax machines. Retail equipment, such as automatic labeling systems, bar-coding machines, and tills found in supermarkets. Banking systems, such as cash registers, and automatic teller machines; 54  Manufacturing equipment, such as numerically controlled (NC) tools, pick and-place robots, welding robots, automated guided vehicles (AGVs), and other industrial robots. Aviation systems, such as cockpit controls and instrumentation, flight control actuators, landing gear systems, and other aircraft subsystems. 55  Chapter 3 Modeling of Physical Systems A model of a system is a tool we use to answer questions about the system without having to do an experiment. While there are many types of models, we are concerned with the mathematical models of systems 3.1. Modeling and Simulation of Mixed Systems The majority of technical systems are mixed; i.e. they incorporate components from various fields, such as electronics, mechanics, software and other domains. This raises significant design problems because hitherto design methodologies and the associated design tools have usually been developed for a single field only. This means that the overall function of the system cannot be investigated until the prototype construction phase. However, by the time this stage is reached, changes to the design have 56 already become very expensive and time-consuming. The consideration of virtual prototypes, which allow virtual experiments to be performed on a computer by simulation, offers an elegant solution to the problem described above. 3.1.1. Modeling and Simulation of Mechatronic Systems In mechatronics systems the interaction between the different domains is particularly significant here because the interfaces contribute significantly to system behavior. The problem of the joint simulation of electronics and mechanics must be solved, which throws up a whole range of problems: In the case of mechatronics, the time constants of mechanics and electronics often differ by orders of magnitude. For micromechanics we can expect oscillations of a few (tens of) hertz. In electronics the figure lies four to six orders of magnitude higher. So, we could assume that the dynamic 57 interaction between electronics and mechanics can be disregarded. The opposite is true. For example, a wide range of control algorithms are performed on embedded controllers. Their running time again lies in the millisecond range, so that dynamic feedback between electronics and mechanics very definitely plays a role. This requires the dynamic simulation of the entire system in order to be able to track cyclical dependencies, including those that cross domain boundaries. Another reason for the importance of this is the fact that domain boundaries often also represent the interfaces between design teams working in parallel. For the field of mechanics, precise models that are compatible with an electronics simulator must be prepared. An efficient conversion is of crucial importance for the field of software in particular. Millions of machine instructions are performed in a single second 58 of real time. On the other hand, it is necessary to precisely determine the timing of the functions implemented using software, which requires a precise synchronization between software and electronics. This is indispensable in order to correctly reflect the dynamics between software, electronics and mechanics. In addition, the representation of the results can sometimes be a problem. Of course, we always obtain the values of system variables plotted against time, as is also normal for electronics simulation. In the case of mechanics, however, we would often prefer an animation, in order to be able to evaluate the system behavior at a glance. As far as software is concerned, the typical outputs of an electronics simulator are virtually useless. We would like a debugger, like those used in pure software development, which illustrates 59 the sequence of the software and furthermore permits control of the sequence, perhaps by breakpoints. 3.1.2. Approaches to modeling mechatronic and micro mechatronic systems 1. Transfer the mechanical model into the electronic simulator one possibility is to transfer mechanical models into the form of electronic models (and vice-versa). This permits the consideration of the mechanics in a electronics simulator (and vice-versa. 2. Domain-independent simulators. There are also some approaches that attempt to model the entire electromechanical system as a unit without any preference for electronics or mechanics. These methods include bond graphs, block diagrams, and modelling languages such as Model Ica. Despite the elegance of these description forms it is 60 generally found that neither the electronics nor the mechanics can be modelled with the usual standard procedures. 3. Coupling simulators from different domains: The possibility of coupling together simulators for different domains represents a further approach to solving the problem. This could, for example, occur systematically with the aid of a simulator backplane, as is often created for pure electronics. Typical applications for this are the coupling of circuit and logic simulators or the distribution of simulations on a parallel computer or a cluster of workstations. However, simulator coupling is associated with a whole range of problems: 1- Firstly, the resulting simulator package is unwieldy, it is often difficult to operate 61 2- Licenses are required for all of the individual simulators. 3- The problems associated with synchronization between two very heterogeneous simulator cores are even more severe> 3.2. Mathematical Modeling of Mechatronics systems By mathematical models we mean that the relationships between quantities (distances, currents, flows, unemployment, and so on) that can be observed in the system are described as mathematical relations. 3.2.1. Review of Basic Relations of Electrical and Mechanical Elements The following tables show the basic relations of electrical and mechanical elements. 62 TABLE Voltage-current, voltage-charge, and impedance relationships for capacitors, resistors, and inductors 63 TABLE Force-velocity, force-displacement, and impedance translational relationships for springs, viscous dampers, and mass 64 TABLE 2.5 Torque-angular velocity, torque-angular displacement, and impedance rotational relationships for springs, viscous dampers, and inertia 65 3.3. The Analogy Approach All disciplines of engineering are based on sets of fundamental laws or relationships. Electrical engineering relies on Ohm’s and Kirchoff’s laws, mechanical engineering on Newton’s law, electromagnetics on Faradays and Lenz’s laws, fluids on continuity and Bernoulli’s law, and so on. These laws are used to predict the behavior (both static and dynamic) of systems. Systems may exist completely in one engineering discipline (such as an electric circuit, a gear system, or a water distribution system), or they may be coupled between several disciplines (electromechanical, electromagnetic, etc). Although analytic solutions are appropriate for single discipline static equations it is more often the case that computer-based solution methods are required, especially when dynamics are present in the equations. 66 3.3.1. Potential and Flow Variables, PV and FV Systems consist of components such as springs and dampers in mechanical systems, tanks and restrictions in fluid systems, and insulators and thermal capacitances in thermal systems. When in motion, the energy in a system can be increased by an energy-producing source outside the system, redistributed between components within the system, or decreased by energy loss through components out of the system. In this context, a coupled system becomes synonymous with energy transfer between systems. Since the analogy method was developed for use on analog computers, it is fitting that the approach be described from a basic electrical viewpoint. Electrical systems are based on three fundamental components: Resistor Capacitor Inductor 67 The capacitor and inductor are capable of storing energy. The energy stored in a capacitor is and the energy stored in an inductor is 𝑖. The resistor cannot store energy but can transfer electrical energy into heat energy. In an ideal, lossless LC circuit with nonzero initial energy, all energy remains in the circuit and is transferred back and forth in sustained oscillations between the inductor and capacitor. Addition of a resistor establishes an energy leak to the surrounding air through which heat energy is transferred, causing the oscillations to decay in amplitude and eventually disappear. If the resistor were immersed in a fluid such as water, the temperature of the fluid would rise due to the heat energy transferred to it. In the steady state, all electrical energy in the circuit would be converted to heat energy in the fluid. Further addition of a voltage or current source to the circuit 68 would provide an external source of energy into the circuit. If the source had a nonzero mean value, the heat energy transferred to the fluid would be sustained. Total energy, E, in the LC circuit consists of potential energy, U, and kinetic energy, K. Potential energy is associated with the potential to perform work and kinetic energy with the work to change motion or flow. Based on this association two energy related are defined as. Potential variable = PV Flow variable = FV Example, in an LC circuit, the initial energy may exist in either the capacitor as a potential, in the inductor as a current, or in both. If the potential energy is stored entirely in the capacitor, voltage becomes the natural choice for the potential variable and, current becomes the flow variable. On the other hand, if the potential energy is stored entirely in the inductor, then current 69 may be used as the potential variable and voltage as the flow variable. Since it is natural to picture current as flowing and voltage drops as accumulating through an electrical circuit, the flow variable in an electrical circuit is current, and the potential variable is voltage. 3.3.2. Impedance Diagrams In an electrical circuit the impedance of a component is defined as the ratio of the voltage phasor, v , across the component over the current phasor, 𝑖, through the component. Since voltage and current are complex numbers, the impedance is also a complex number. A complex number consists of a real part and an imaginary part. The placeholder for the imaginary part is j, and no placeholder is required for the real part. The impedance of an electrical circuit element is a complex phasor quantity defined as the ratio of the voltage phasor divided by the current phasor. The 70 impedance phasors for the capacitor, inductor, and resistor are summarized in Figure 2-35 and are shown as bold arrows. Positive phase occurs when the phasor is rotated in the counterclockwise direction beginning from the positive real axis (which is the zero phase direction). Figure 2-35 Impedance Phasors for The Capacitor, Inductor, And Resistor 71 When the phasor is lined up with the positive imaginary axis (vertically upward) 90° of the phase has been accumulated. When the phasor is pointing leftward, 180° of the phases has been accumulated. When the phasor is pointing downward along the negative imaginary axis, 270° or - 90° of the phase has been accumulated. Keeping in mind that impedance is voltage divided by current, a positive imaginary component indicates voltage leading current, and a negative imaginary component indicates voltage lagging current. Because j occurs in the denominator of the capacitor impedance, the capacitor voltage lags its current by 90°. Similarly, because j occurs in the numerator of the inductor impedance, the inductor voltage leads its current by 90°. Consider the sinusoid x(t) = sinwt. If we differentiate x(t) analytically with respect to time, we obtain 72 73 EXAMPLE: Impedance Calculations for a Parallel System This example illustrates how impedances are calculated in a parallel system. The system shown in Figure 2-37 has three impedance’s, three flow variables, and three potential variables. Figure 2-37 Simple Circuit for Impedance Calculations Solution Using Equation 2-1, the impedance’s are calculated as: 74 PV3 is a common potential point in the circuit. It is usually set to either zero or a reference value. Setting PV3 to zero, the impedance equations may be reduced to In many situations, an impedance diagram can be simplified by applying any of six fundamental impedance relationships. These relationships, are based on Ohm’s and Kirchoff’s Laws, are summarized in Table 2-2. 75 Table 2-2 Fundamental Impedance Relationships 76 Parallel and series impedance reductions will be used frequently in our manipulations. The following properties will be used repeatedly. 77  Series Impedance’s Add: The total impedance of a series combination is the sum of the individual impedances. Parallel Impedance’s–Inverses Add: The inverse of the total impedance of a parallel combination is the sum of the inverses of the individual impedances. To illustrate how the impedance relationships are applied, several examples are presented. EXAMPLE: Impedance Diagram Simplification—Simple System This example illustrates how series and parallel reductions can be applied to the previous example to derive a single representative impedance, ZTotal, for the entire system. The system, which is rewritten in Figure 2-38, is reduced in two steps. Step 1. Combine the Z2 and Z3 impedance’s into a single series impedance, Z23. Step 2. Combine the Z1 and Z23 into a single parallel impedance, ZTotal 78  Figure 2-38 Simple Impedance System Solution Step 1. The Z2 and Z3 impedance’s are combined into the single series impedance, Z23, according to the series relationship, Z23 = Z2 + Z3. The impedance diagram is presented in Figure 2-39. Figure 2-39 Series Simplification for The Simple Impedance System Inevitably, some signals are lost as a result of impedance diagram simplifications. In this simplification, we have lost the PV2 signal. 79 Step 2. The Z1 and Z23 impedance’s are combined into the single parallel impedance, ZTotal. This result is important because it is encountered so frequently. It is summarized as the combined impedance of parallel branches is equal to the product of the two impedances’ divided by the sum of the two. It is important to note that the flow through ZTotal is FV1 and not FV2. we have lost the flow variables, FV2 and FV3. Figure 2-40 Parallel Simplification for The Simple Impedance System 80 EXAMPLE; Impedance Diagram Simplification—Complex System This example illustrates how series and parallel reductions can be applied to a more complex system. The system, Figure 2-41, is typical of the type encountered in mechanical systems with several masses. The objective is to reduce the diagram to a single equivalent impedance. Figure 2-41 Complex Impedance System We will solve the problem in the four steps outlined below. Step 1. Combine the Z5 and Z6 into a single series impedance, Z56. Step 2. Combine the Z3 and Z4 into a single series impedance, Z34. 81 Step 3. Combine the Z2, Z34, and Z56 into a single parallel, Z23456. Step 4. Combine the Z1 and Z23456 impedance’s into a single series ZTotal. Solution 82 Step 4. The reduction is completed by combining the series Z1 and Z23456 impedance’s into the single final impedance, ZTotal. The completed impedance diagram is presented in Figure 2-43. Figure 2-43 Final Reduction of Complex Impedance System Not all electrical circuit components have an impedance, for example, an ideal voltage source does not have a fixed impedance. Although the voltage value is constant, the current is determined by the circuit to which the source is connected, making the impedance a variable. The same is true for an ideal current source. 83 3.4. The Modified Analogy Approach The modified analogy approach is a process which allows you to convert an illustration of a physical system to a block diagram model. The approach is based on the electrical notion of impedance and a four-step conversion process explained in this section. The difference between the modified analogy approach and the basic analogy approach is the manner in which nonlinearities are handled. The basic analogy approach presented in many texts is restricted to linear applications. If a nonlinearity exists, it must be linearized prior to incorporating it into the model. Linearization provides only an approximation to the behavior of the nonlinearity; the difference between the linearized and actual behavior becomes an undesirable modeling error. 84 This results in a more accurate model with better predictive capability and less modeling error. Given a system illustration, analogies are first established for the PV and FV. The following four-step procedure is applied to obtain the block diagram model. Step 1. Create and (if possible) simplify the impedance diagram using the manipulations presented in Table 2-2. Simplifications of this nature include minor parallel and series branches which can be easily reduced to single equivalent branches. Step 2. Circle all nodes (FV and PV) in the impedance diagram and label all signals entering and leaving these nodes. A FV node is a point in the impedance diagram where three or more branches intersect. A PV node occurs when two or more impedance elements exist in series. The PV node relates the individual PV drops of the elements to a single overall PV drop. 85 Step 3. Construction of the block diagram is initiated by representing select nodes (PV and FV) from the previous step as summing junctions with inputs and outputs labeled according to signals from the impedance diagram. In general, it usually is not necessary to implement all PV and FV nodes, because often they are dependent upon one another. Select the output of each summing junction such that, when it is applied to the corresponding impedance block, a causal operation (either an integration or multiplication by a gain) results. It should be noted that in some situations it will not be possible to create a block diagram with only gain or integral causality. In these situations, we 86 either attempt to differentiate the non-causal elements directly or modify the model to achieve causality. Step 4. The block diagram is completed by placing each component impedance from the impedance diagram onto the block diagram and connecting them with signals from either summing junctions or other impedances. EX: Block Diagram Construction—Parallel Resonant Electrical Circuit The parallel resonant circuit exhibits a controllable resonant peak suitable for notch filtering applications. Notch filters are used to remove unwanted frequencies from a signal leaving the other frequencies unaltered. The parallel resonant circuit diagram with the resistance lumped in the inductor branch is presented in Figure 2-44. Figure 2-44 Parallel Resonant Circuit 87 FV= current and PV= voltage. Capacitor: 88 Step 2. Identify all independent nodes (FV and PV) in the impedance diagram and label all signals. A FV node is a point in the impedance diagram where three or more branches intersect. A PV node relates the individual PV drops over a series of impedance’s to an overall PV drop. Our diagram has one FV node and one PV node as shown in Figure 2-47. Step 3. Represent select nodes as a summing junction, and select the output of the summing junction such that either gain or integral causality results. The two nodes in our impedance diagram produce the two summing 89 junctions shown in Figure 2-48. It has arbitrarily selected the summing junction output in step 3. Step 4. Add the impedance blocks; connect and create all necessary intermediate and output signals to complete the block diagram. Figure 2-47 Nodes in The Parallel Resonant Circuit Impedance Diagram Figure 2-48 Partial Block Diagram Representation of The Parallel Resonant Circuit 90 Noting that IR _ IL and that vout=vc , the block diagram is constructed by first adding the three impedance blocks. Next, the appropriate signal connections are made using wires. Luckily, it has selected the summing junction outputs which provide integral causality, so no modifications are needed in step 3. The completed block diagram is presented in Fig 2-49. Fig 2-49 Completed Block Diagram Representation of The Parallel Resonant 91 The system equations can be derived by simplifying the block diagram. For example, the transfer function relating the input current to the output voltage is presented in Equation 2-2. 3.4.1. Electrical Systems Electrical circuits rely on two variables, voltage and current, to transport energy. Since current flows through an electrical circuit, it is natural to associate current with the flow variable and voltage with the potential variable. Using this convention, the impedances of six basic ideal circuit components are discussed: the resistor, capacitor, inductor, voltage source, current source, and transformer. The impedances of these components will provide the fundamental analogies for components in 92 other disciplines. Of the six basic electrical components, only the resistor, capacitor, and inductor have impedance’s which are not functions of the circuit to which they are attached. The resistor, capacitor, and inductor impedance characteristics are summarized in Table 2-3. Table 2-3 Resistor, Capacitor, And Inductor Impedances The remaining three components have impedances which are functions of the circuit to which they are attached. The ideal voltage source is used to 93 create a specified potential at any point in a circuit. The potential exists between the two terminals of the voltage source. The current which passes through the voltage source is determined by the circuit to which the source is connected. Due to the current being an unknown, it is not possible to write the impedance relationship for the voltage source without knowledge of the rest of the circuit. Sometimes the voltage value for the source will be a function of another variable of the circuit (such as a current or voltage). In this situation, the voltage source is called dependent, since it’s value is dependent on another signal in the circuit. The ideal current source is used to create a specified current at any point in a circuit. The voltage which exists between the two terminals of the current source is determined by the circuit to which the source is connected. Due to the voltage being an unknown, it is not possible 94 to write the impedance relationship for the current source without knowledge of the rest of the circuit. Similar to the voltage source, sometimes the value for the current source will be a function of another variable of the circuit (such as a current or voltage). In this situation, the current source is called dependent, since it’s value is dependent on another signal in the circuit. A transformer is a magnetically coupled electrical device consisting of two coils wound along each side of a closed conducting core. One winding is called the primary (winding 1) and the other winding called the secondary (winding 2). The number of windings in the primary and secondary coils are N1 and N2, respectively. The impedance characteristics of the ideal transformer are dependent on the circuit to which it is connected. The impedance characteristics of the voltage source, current source, and transformer are presented in Table 2-4. 95 To illustrate how the analogy approach is applied to electrical circuits to create block diagrams, two examples are presented: a bridge circuit and a transformer circuit. Bridges can be constructed entirely of resistors or capacitors depending on the quantity being measured. The transformer is an important electric circuit component, because (as will be seen later) it is analogous to gear trains in mechanical rotational systems and lever arms in mechanical translation systems. Transformers have many applications, including impedance matching, voltage step up, and voltage step down. Electric power-transmission systems rely heavily on step-up and step-down transformers to efficiently send electricity over large distances. 96 Table 2-4 Voltage Source, Current Source, And Transformer Impedances 97 EXAMPLE: Bridge Circuit System A thermistor is a semiconductor device whose resistance changes with temperature. Temperature readings in terms of voltage can be obtained by installing the thermistor as one of the resistances in a bridge circuit. A typical configuration is shown in Figure 2-50. Figure 2-50 Bridge Circuit for Temperature Measurement When a constant voltage is applied to the circuit, V, heat source variations cause the thermistor resistance to change, thus creating a potential difference between points A and B, which is proportional to temperature. 98 The objective of this example is to apply the analogy method to develop a block diagram model of the bridge circuit. Solution Step 1. Create/simplify the impedance diagram. The first step of the procedure is the construction of the impedance diagram. This is relatively straight forward. All flow paths, potentials, and branches remain intact; the only difference is the replacement of each component with its associated impedance. The impedance diagram is presented in Fig 2-51. Figure 2-51 Impedance Diagram for Temperature Measurement Circuit 99 Step 2. Identify all independent nodes (FV and PV) in the impedance diagram and label all signals. The impedance diagram has one FV node and two PV nodes. The node equations are given as Step 3. Represent select nodes as a summing junction, and select the output of the summing junction such that (when it is connected to its associated impedance blocks) either gain or integral causality results. The initial 100 Fig 2-52 Summing Junctions for Temperature Measurement Circuit Block block diagram is constructed with two summing junctions to model the two PV nodes, Figure 2-52. Step 4. Add the impedance blocks; connect and create all necessary intermediate and output signals to complete the block diagram. 101 The FV node equation was not directly implemented using a summing junction; however, since ZR1 and ZR2, both have the same flow, FVA, and since ZR3 and ZRth have FVB flowing through them, the following two constraint relationships are written. Figure 2-53 Summing Junctions for Temperature Measurement Circuit Block Diagram With Slight Modification 102 The final block diagram, Figure 2-54, is constructed by adding these two relationships to the block diagram to define the PVR1 and PVR2 signals. From the revised block diagram, the system equations may be derived after substituting the appropriate resistance values and noting that V= PV1, VA= PVA, VB= PVB, we have As written, the system equation represents the output voltage as a function of the thermistor resistance and the input voltage; V0=V0(RTH, V) With a 103 constant input voltage, the output voltage becomes only a function of the thermistor resistance. Figure 2-54 Block Diagram for Temperature Measurement Circuit EXAMPLE: Transformer System The basic transformer circuit with input, V, and output, i2, is shown in Figure 2-55. 104  Figure 2-55 Basic Transformer Circuit Voltage, V1, is applied to the transformer primary side coil which consists of a series resistance and inductance, R1 and L1. The secondary side coil of the transformer consists of a load impedance, Zload. Again, the objective of this example is to develop the block diagram model for the transformer circuit. Solution Step 1. 105 Create/ simplify the impedance diagram. The impedance diagram for the transformer is created by replacing each element of the circuit with its associated impedance. The impedance diagram is presented in Fig 2-56. Figure 2-56 Basic Transformer Impedance Diagram Step 2. Identify all independent nodes (FV and PV) in the impedance diagram and label all signals. The impedance diagram has two PV nodes which represent the 106 potential drops around the primary winding and secondary winding loops. These equations are summarized here. In addition to the loop equations, the auxiliary transformer equations which relate PV and FV across the transformer ratio are: Step 3. Represent select nodes as a summing junction and select the output of the summing junction such that (when it is connected to its associated impedance blocks) either gain or integral causality results. The block 107 diagram construction is initiated with the primary winding loop PV equation and presented in Figure 2-57. Step 4. Add the impedance blocks; connect and create all necessary intermediate and output signals to complete the block diagram. The block 108 diagram is completed by incorporating these definitions and presented in Fig 2- 58. We have assumed that the load impedance has current causality in the formulation. If this were not the case, for example, if it had voltage causality, the diagram would need to be modified. many system relationships can be computed from the block diagram. For example, Equation (2-3) relates input voltage to secondary current 109 3.4.2. Mechanical Translational Systems Mechanical translation system analysis is based on Newton’s law, which states: The vector sum of all forces applied to a body equals the product of the vector acceleration of the body times it’s mass. The equation for Newton’s law is presented in Equation 2-4.. F=m.a where the units in the British system are F = total force, newtons, N a = total acceleration, m/S2 Two elements typically encountered in mechanical systems are the linear damper and the linear spring. The linear damper produces a force proportional to the applied velocity, and the linear spring produces a force proportional to the applied displacement. 110 Depending on the system, either velocity or displacement may be used as the PV. Regardless of the choice of PV, force is used for the FV. Table 2-5 summarizes the impedance’s of the three mechanical translation system components for both analogies. Table 2-5 Mechanical System Impedance Analogies 111 Mass–Damper System The basic mass–damper system is modeled in this example. Selection of logical PV and FV variables will create a causality problem which is also discussed. An illustration of the mass–damper system is shown in Figure 2- 59. Since the input, , and output, , of the system are both velocities and no springs are involved, velocity is the logical choice for the potential variable. The flow variable is force. Figure 2-59 Mass–Damper System Illustration Solution Step 1. Create/simplify the impedance diagram. 112 The impedance diagram for the mass–damper system is created by replacing each element of the circuit with its associated impedance. The impedances are defined as Figure 2-60. Mass-Damper System Impedance Diagram 113 Step 2. Identify all independent nodes (FV and PV) in the impedance diagram and label all signals. The impedance diagram consists of one PV node represented by the following equation. Step 3. Represent select nodes as a summing junction and select the output of the summing junction such that (when it is connected to its associated impedance blocks) either gain or integral causality results. Integral causality for the ZM element requires that FV be its input. Our strategy is to model the PV 114 node equation such that PVZB is the output. The damper, which has no causality problems because the potential variable is velocity, is used to create the FV required as input to the ZM block. Step 4. Add the impedance blocks, the resulting block diagram is presented in Figure 2-61. The output velocity, 𝑦, is computed by reducing the block diagram and substituting for the two impedances as: Figure 2-61 Mass–Damper Block Diagram 115 The force flowing through the system, FV, may also be computed from the block diagram as: One also could solve this problem using displacement instead of velocity as the potential variable. The input and output variables become x and y. Since displacement is the integral of velocity and integration is represented in operator notation as 1/D, the impedances in the displacement–voltage analogy system are equivalent to the impedances of the velocity–voltage system multiplied by 1/D. These impedances become: Because the system is linear, the transfer function relating 𝑦 is: 116 We can compute the transfer function from x to y by integrating both sides. This is analogous to division by the D operator. The resulting transfer function becomes: This is no surprise, however. Suppose we were confronted with the task of modeling the system with displacement used as the potential variable. Causality now becomes an issue. For integral causality, both elements ZB and ZM must have an FV input. Investigation of the PV node equation for this system reveals that this is not possible; however, all is not lost. 117 We recognize that the real problem is that the only causality independent element capable of converting a PV to an FV signal in this situation is the spring, which is not present in our diagram. This problem is solved by using an approximate system which includes an additional spring with its stiffness set to a very large value. The approximate system will be of integral causality and will approximate the actual response closer and closer as the spring stiffness is increased. Setting this limit, the original transfer function will result. The approximate system block diagram is presented in Figure 2-62. The added spring is placed just to the right of the PV node summing junction to produce the required FV output. 118  Figure 2-62 Approximate System Block Diagram Since we are interested in computing the system transfer function from x to y, it is beneficial to redraw the block diagram before any reductions are performed, as in Figure 2- 63. Figure 2-63 Redrawn Approximate System Block Diagram Since displacement is the PV, the impedance’s are: 119 Reducing the block diagram and substituting the impedance relationships yields the following transfer function. As the spring stiffness is made very large, the transfer function approaches the expected transfer function as: Problems of this nature are often found in real systems and with proper attention, integral causality can be maintained. 120 3.4.3. Mechanical Rotational Systems Mechanical rotational system analysis also is based on Newton’s Law; however, the law is slightly modified to account for rotation instead of translation. The law states: The vector sum of all moments applied to a body equals the product of the vector angular acceleration of the body times it’s inertia. A rotational system obeys Equation 2-5. 𝝉 = 𝒋𝜽̈ (2.5) where 𝜏 = Total Torque, N-m J= body inertia about it’s center of mass, 𝜃̈= angular acceleration, rad/sec2. 121 Two elements typically encountered in mechanical rotational systems are the linear torsional damper and the linear torsional spring. The damper produces a torque proportional to the applied angular velocity, and the spring produces a torque proportional to the applied angle. An analogy similar to that used for translation systems exists for rotational systems—except angle replaces displacement, angular speed replaces velocity, and torque replaces force. Also, mass becomes inertia, the translational spring constant becomes a torsional spring constant, and translational damping becomes rotational damping. The impedance analogies are identical in form to those used in translational systems. The flow variable is defined as torque, and the potential variable is defined as either angular velocity or angle. The analogies and impedances for rotational systems are summarized in Table 2-7. 122 Table 2-7 Impedance Analogies for Rotational Systems EXAMPLE 2-17 Elevator System A cable-driven elevator hoistway system consists of a drive pulley (drive sheave) attached to a gearbox powered by an electric motor. The drive sheave is wrapped (usually six or more times to prevent slippage) with a 123 cable—one end of which is attached to a counterweight and the other end to the elevator cab. the elevator hoistway system is shown in Fig 2-72. Figure 2-72 Geared Elevator Hoistway System Illustration The cable is assumed to act as a spring with no damping. For modeling, the cable weight on either side of the pulley is halved. One half is lumped as part 124 of the pulley weight, and the other half lumped into the car weight and counterweight, respectively. The radius of the drive sheave is designated as r, and the gear ratio as 1:N (N motor revolutions to 1 drive sheave rev). the hoistway system contains springs, the logical choice for the potential variable is displacement. Solution Step 1. Create/simplify the impedance diagram. The following impedance diagram (Figure 2-73) is constructed. The force due to gravity has been included on both the counterweight and the car and on the direction results from the definition of the car and counterweight directions. The variable x denotes the linear displacement of the drive sheave and is related to Ѳ by: 125 Figure 2-73 Geared Elevator Hoistway System Impedance Diagram The impedances in Figure 2-73 are listed as: 126 Step 2. Identify all independent nodes (FV and PV) in the impedance diagram and label all signals. The impedance diagram has six nodes. Four of these nodes are FV nodes, and two are PV nodes. The node equations are given as: Several auxiliary equations pertaining to the gear ratios are also necessary and listed as: Gear ratio: T1 = N Tm ,Drive sheave ratio: F = T2/r 127 Step 3. Represent select nodes as a summing junction and select the output of the summing junction such that either gain or integral causality results. Construction of the block diagram begins by implementing the FV and PV equations as summing junctions. We also include the auxiliary equations. The initial block diagram is presented in Figure 2-74. Figure 2-74 Geared Elevator System Summing-Junction Block Diagram 128 Step 4. Add the impedance blocks; connect and create all necessary intermediate and output signals to complete the block diagram. Substitution of the three mass impedances give Replacing ZK = 1/K the spring impedances, allows us to complete the block diagram. The completed block diagram is presented in Figure 2-75. Figure 2-75 Geared Elevator System Block Diagram 129 In Example 2-17, the reaction torque of the car and counterweight to the motor have been excluded. The effect is important, as it models the effect of load or reaction torque on the motor. The effect can be added easily once two fundamental electromechanical relationships—Lorentz’s law and Faraday’s law—are presented. 3.5. Automobile Suspension System The suspension system of a car can be modeled on a per-wheel basis as a two-mass system: the car mass and the wheel mass. The tire behaves as a spring, and the connection between the tire and the car is a spring shock absorber (damper) assembly. The road roughness provides the input to the system as a displacement. The outputs are the axle displacement and the vehicle displacement. An illustration of the suspension system is shown in Fig2-64. 130 Suspension Mechanical Diagram Suspension System Illustration Solution Step 1. Create/simplify the impedance diagram. The impedances that will be used in the impedance diagram are listed here. 131  Figure: Suspension System Impedance Diagram Step 2. Identify all independent nodes (FV and PV) in the impedance diagram and label all signals. The impedance diagram may be reduced by first combining the parallel spring–damper into equivalent impedance defined as ZKB. With this reduction, the impedance diagram has one FV node at y2 and two PV nodes over ZKtire and ZKB. The node equations are summarized here. 132 Step 3. Represent select nodes as a summing junction and select the output of the summing junction such that (when it is connected to its associated impedance blocks) either gain or integral causality results. These summing junction representations of the node equations are in Fig 2-67. 133 Step 4. Add the impedance blocks; connect and create all necessary intermediate and output signals to complete the block diagram. For integral causality, the inputs to the and blocks must be FV signals. Since only one FV node equation is present, we must use the ZKB block to produce the additional FV signal required for ZM2. For brevity, the general PV and FV notation is dropped, and the completed block diagram shown in Figure 2-68 uses the problem variables. Figure: Suspension System Block Diagram 134 135 3.6. Electromechanical System Case Study It is required to find the transfer function for the electromechanical system consisting of a shaker table that runs on a pair of slider rails shown in Figure 5.1. The table shall be design to move back and forth in a sinusoidal manner using a direct current motor. The transfer function relates the voltage applied to the DC motor to the velocity of the table. Figure 5.1. bSketch of shaker Table System This problem can be broken down as follows: 136 (1) Derive the math model of the table, using any of the four methods (2) Derive the math model of the motor and transmission. (3) Derive the math model of the motor controller. 3.6.1. Deriving the Math Model of the Table The table can be modeled as a combination pure translational mass and pure translational damper. Figure 5.2a shows our symbolic model and Figure 5.2b the associated circuit diagram. We want to find a relationship between the velocity of the table v and the applied force F using any of the following methods: (1) Path-Vertex-Elemental Equation Method (2) Impedance Method (3) Operational Block Diagram Method (4) Free-body Diagram Method. 137  Figure 5.2. System Diagrams Path-Vertex-Elemental Equation Method From the circuit diagram, we can write the elemental equations 138 Substituting (5.2) and (5.3) into (5.1) gives Impedance Method Again, use the circuit diagram in Figure 5.2b and replace each elemental component with its impedance as in Figure 5.2c. Since the impedances are in parallel, we write Or 139 Free-body Diagram Method Using the free-body diagram shown in Figure 5.2d, we can write: Equations (5.8) and (5.9) are static equations because the bearings hold the table to the rails. These equations simply tell us that Fb3 and Fb4 are equal and that their sum equals the weight of the table. Equation (5.7) reveals the assumption we are making regarding the resistive bearing forces. That are assuming: Fb1 + Fb2 = bv (5.10) Substituting equation (5.10) into (5.7) and rearranging terms gives: 140 Block Diagram Method We start constructing the block diagram with the desired response variable as an output of an integrator placed on the right side of the diagram. The velocity of the table is the response variable and the applied force is the forcing variable. We rewrite equation (5.2) as, Figure 5.3a And rewrite equation (5.1) as, Figure 5.3b: Fa = F - Fb (5.13) Finally, draw the block diagram for equation (5.3) as shown in Figure 5.3c, and connect the individual block diagrams together as shown in Figure 5.3d. The block diagram can be reduced to a single transfer function by first write the equation for the output of the summer as: 141  e = F - Fb (5.14) Then write the equation for v as: Substitute (5.14) into (5.15) to eliminate e and rearrange the result Which can be written using the time constant as: 142 Where: The math model of this mechanical system and that of the electrical system discussed so extensively e are identical. So the mechanical circuit will act like a low-pass filter (mechanical style filter). 143 The table in essence will move with a velocity equal to F/b when a low- frequency sinusoidal force is applied. As the frequency of the force increases past the break frequency (f = 1/2πt), the velocity will become smaller than F/b. 144 3.6.2. Derivation of Math Model of Motor and Transmission Figure 5.11 shows a sketch of one way to make the DC motor drive the shaker table. The table is attached at two ends with a taut, inelastic cable wound around an idler pulley at one end and a drive pulley at the other. The following equations (5.25) and (5.26) are fundamental transmission equations. They provide a mathematical relationship to convert angular 145 speed into linear speed and vice versa, and to convert linear force into rotational torque and vice versa. Q=rF (5.26) and v=rw (5.25) Current flowing through the motor armature creates a motor torque Qm ft- lbs Qm = Km ia 146 The bearings in the motor and pulley act as a rotational damper having a damping coefficient, B ft-lb / rad / sec. From the free-body diagram we can write: Where: Jm: ft-lb-sec2: the motor moment of inertia QB: ft-lbs: bearing damper torque given by QB = B w (5.28) QL: ft-lbs: load placed on the motor output shaft given by where r is the radius of the pully and F is the linear force acting on the Table. Combining equations (5.27) and (5.28) and then rearranging gives: Substituting the F value calculated from the table model gives: 147 Since v= r w or in terms of the time constant of the motor-table system Which is the block diagram of the motor-table system given in Figure 5.14. We will use this block diagram to connect to the electrical portion of the motor model next. Figure 5.14. Block Diagrams of Motor-Table System 148 3.6.3. Derivation of Math Model of Motor Controller A motor is an electromechanical component, the mechanical output generated by an electrical input. We will derive the transfer function for one particular kind of electromechanical system, the armature-controlled dc servomotor. Lorentz and Faraday’s laws are the physical laws applied. Figure. Physical laws applicable to the DC motor. 149 The motor’s schematic is shown in Figure 2.35(a), and the transfer function we will derive appears in Figure 2.35(b). FIGURE 2.35 DC motor: a. schematic;12 b. block diagram Lorentz’s Law In Figure 2.35(a) a magnetic field is developed by stationary permanent magnets or a stationary electromagnet called the fixed field. A rotating circuit called the armature, through which current 𝑖a(t) flows, passes 150