Wind Turbine Project and Competition

timer Asked: Nov 4th, 2016

Question description

Wind Turbine Project and Competition

Design Project #1 Wind Turbine Project and Competition ENGR-111 Fall 2016 Project Overview The broad context of this project entails the concepts of energy, energy conversion and energy conversion efficiency (and closely related power, power conversion and power conversion efficiency). These concepts, introduced in more detail in the Reference section, are so fundamental to engineering that they will certainly be relevant to your future profession, regardless of what type of engineering you are going to pursue. More specifically, the project focuses on wind power generation which utilizes the renewable resource of winds on our planet in order to generate clean electricity. This technology is one of the fastest growing sources of renewable energy in recent years. This rapidly growing trend is clearly shown in Figure 1. Note that the capacity increased almost tenfold in the last ten years. Still, the current share of the total generated electricity is well below 5%. The wind power technology is described in more detail in the Reference section. Figure 1 Global wind power cumulative capacity. Source: Global Wind Energy Commission. The basic schedule of this 4-week endeavor is summarized in Table 1. Table 1 Schedule Week Topics Intro Introduction Overview, Energy, Energy Conversion, Truss towers Learn the project goals, equipment, and underlying physics. Identify energy conversion devices and define their efficiency. Introduce/assign HW1 & 2 (efficiency & tower design). Module 1: Fan Learn how to measure electrical power generated. Measure power conversion efficiency of a fan/turbine system. Module 2: Wind Turbine (for Competition) Build a wind turbine. Measure and maximize the generated power and the efficiency of power conversion. Module 3: Turbine Tower (for Competition) Build the supporting tower. Ensure proper load support while minimizing the tower weight. Competition Teams from all sections which meet at the same time will compete for fame and prizes. Kirkbride Lobby. 1 1, 2 and 3 4 Specific Goals Since there is no wind in our engineering labs, you are going to generate some. The basic schematic of the system used in both Module 1 and Module 2 is presented in Figure 2 below. The energy-converting engineering components of the system are: electric motor, fan, wind turbine, electric generator, and electric load (resistor). Electricity Source Electric Generator Electric Motor Electric Load (Resistance) Fan Wind Blades Turbine Blades Figure 2 Schematic of energy conversion sequence The sequence of energy conversion processes of Figure 2 (with the energy flowing from left to right) is summarized in Table 2. In short, the left side of the system generates wind using electricity and the right side of the system generates electricity using wind energy. Naturally, in the actual wind power applications only the right side is present. Table 2 Sequence of conversion processes Component Input Energy (Power) Output Energy (Power) Electric Motor Electrical Mechanical (shaft rotation) Fan Blades Mechanical (shaft rotation) Mechanical (air motion) Wind Turbine Mechanical (air motion) Mechanical (shaft rotation) Electric Generator Mechanical (shaft rotation) Resistance Electrical Electrical Heat A sequence of 5 energy conversion processes can be seen in Figure 2 and Table 2. For each conversion process, the conversion efficiency  can be defined as useful power output Pout divided by the required power input Pin   Pout / Pin For example, incandescent light bulbs convert less than 10% of the input electric power into visible radiation, and the remaining 90% into heat. Thus, their efficiency is less than 10%. In this project the overall system efficiency (from the electric energy driving the fan to the electric energy dissipated in the resistor) will be determined experimentally. The project also addresses another topic related to wind power: the construction of a supporting tower. You are going to perform some basic structural analysis, design the tower and build it to load specifications, while keeping its weight low. The subsequent sections describe in detail several activities, including:       Measuring electrical power consumption Determining the system efficiency through measurement Using the fan as a turbine Building a larger wind turbine Building a wind turbine tower The competition Important: Before working on detailed tasks, please read carefully the Competition Rules and the entire Reference section. Important: This project deals with rotating (and possibly flying) parts. Safety glasses are required in the labs at all times. Introduction The goals of the Project Introduction are to:         Form teams of 3 (4, if needed) Understand the project goals thoroughly Understand the fundamentals of energy and its conversion (e.g., from electrical to mechanical). Understand the conversion losses and conversion efficiency. Learn about various devices that convert energy from one form to another (combustion; electric motor and generator; dry-cell battery and rechargeable battery; fan and wind turbine; pump and hydraulic motor; solar cell and light bulb; fuel cell and hydrogen production; human body). Emphasize the huge size range of the devices. Learn basic engineering formulae related to wind turbines Learn basic structural calculations for a truss tower Introduce Project 1 assignments Become familiar with all the components, instrumentation and materials used in the project. Module 1: Power Conversion Efficiency of Fan/Turbine Systems An electric fan is a device that consists of an electric motor with a set of fan blades attached to it. The motor converts electrical energy to shaft (mechanical) energy and the fan blades convert the shaft energy to kinetic (mechanical) energy of the moving air. The goal of Module 1 is to determine power conversion efficiency of the fan/turbine system through measurement. The block schematic of the Module 1 activity is presented in Figure 3. An electric DC motor is powered by a set of dry cell batteries (in series) and spins the fan blades (of your design), generating the flow of air. The air then impacts a PASCO ET-8783 AC Generator, which is an instrumented AC generator driven by a small wind turbine. In order for the generator to produce power, an electric load must be added. In Module 1 the load is a 100 ohm resistor. Dry Cell Batteries (DC Voltage) DC Motor Wind Turbine with Electric Generator PASCO ET-8783 Resistor (100 ohm) Fan blades disc Figure 3 Block diagram of the fan system used in Module 1 The sequence of energy conversion processes present in the system shown in Figure 3 is very similar to the sequence presented earlier in Table 2. One can add one more energy conversion process that takes place in the dry cell batteries. Can you identify the two types of energy involved? Activity 1. Measurement of consumed and generated electric power The equipment available for this activity is listed in Table 3. Secure the fan blade disc on the shaft of the DC motor using the available coupling. Mount the fan/motor unit on the available rod stand. This unit will generate a stream of moving air to drive the wind turbine on the PASCO ET-8783 AC Generator (see Figure 3). The DC motor is powered by 3 D-size batteries. Measurement of power consumed by the electric motor: 1. Mount fan disc on the DC motor and on a stand 2. Connect DC motor to the 3 D-size batteries via an ammeter (in series) to measure current I (Figure 4) 3. Connect voltage meter across the motor terminals (in parallel) to measure voltage V (Figure 4) 4. Calculate the consumed power P as P  V I . Record the calculated value (you will need it in Activity 2) Measurement of power generated by PASCO turbine generator: 1. Measure the actual resistance R of the 100 ohm (nominal) resistor. 2. Connect the resistor to the PASCO Generator terminals 3. Connect voltage meter across the generator terminals (in parallel) to measure voltage V (Figure 5) 4. Spin the turbine blades by blowing air (you can use a fan) 5. Measure voltage V 6. Calculate the generated power P as P  V 2 / R . Record the calculated value. Figure 4 Figure 5 Table 3 Equipment used for Module 1 fan activities Equipment Quantity Pasco Turbine / Generator 1 Digital Multimeter 1 DC Motor / Generator 1 Fan blade disc 1 Base / Rod Stand 1 D Batteries with battery holders 4 100 ohm resistor 1 Scissors 1 Wire – black, red, green 2’ each 3/16” shaft coupling – to attach the fan 1 Activity 2. Determine the power conversion efficiency of a fan/turbine system Position the fan/motor unit from Activity 1 in front of the PASCO generator making sure that the two devices are separated by 6 inches. Determine the electrical power produced by the generator and the system efficiency   Pgenerated / Pconsumed  100% . Use the value of consumed power from Activity 1. Repeat the measurements for the separation distance of 4 inches, 2 inches and 1 inch. Record the results in Table 4 below. Table 4 Generated Power and Conversion Efficiency as a Function of Distance Separation Distance 6 in. 4 in. 2 in. 1 in. Generator Voltage (V) Generated Power P  V 2 / R (W) Conversion Efficiency (%) Activity 3. Efficiency of the Module 1 fan used as a turbine A simple wind turbine can be built by mounting a set of turbine blades on a shaft of an electric generator. The kinetic energy of moving air is now being converted to the mechanical work of the rotating shaft (in the turbine) and then to electrical energy (in the generator). The entire system is presented in Figure 6. Box Fan DC motor from Module 1 used as DC generator Electrical Load (variable resistor) Wind turbine blades Figure 6 Block diagram of the system used in Module 2 The source of ‘wind’ for this module is a 20” diameter window fan, which generates a stream of air moving at the velocity of up to about 4.5 m/s (high setting). The fan is driven by a 110V AC motor and consumes about 85 watts of electrical power when run on high setting. The DC motor from Module 1 now is used as a DC generator. Just like the DC motor from Module 1 can also be used as a DC generator, the fans built in Module 1 can be used as a turbine. Thus, only two steps are needed to convert the Module 1 system (Figure 3) into the Module 2 system (Figure 6): 1. Replace the dry cell batteries with a 500 ohm variable resistor (similar to Figure 5) 2. Position the generator/turbine unit into the wind generated by the window fan The PASCO ET-8783 AC Generator is no longer used. The following procedure can now be used to determine the power generated by the system: 1. 2. 3. 4. 5. 6. 7. Disconnect one wire from the variable resistor Using the multi-meter, set the variable resistor to R = 100 ohm Connect the resistor to the generator Start the fan at high setting Using the multi-meter, measure the voltage drop V across the resistor Calculate the generated power using equation P  V 2 / R Calculate the generated power using equation   P / Pfan  100% (where Pfan is the nominal power consumption of the box fan) Record all measured and calculated values in Table 5 below. Table 5 First turbine results 100 Resistance R () Voltage V (V) Power P  V 2 / R (W) Conversion Efficiency (%) Wind Turbine and Tower Design Competition Rules The goal of the competition is to design and build a model wind turbine, including the supporting tower, which will deliver the maximum electric power and minimize the tower weight. The general configuration is shown in Figure 8. Only materials and tools listed in Table 6 can be used. Figure 8 The general configuration for the wind turbine competition The following constraints must be obeyed: L = 2 ft (required) x 1) The ‘wind’ is generated by a 20” window fan, which creates a stream of air moving at the velocity of up to about 4.5 m/s (high setting). The fan is elevated so that its axis is about 2 feet above the work bench. 2) The electricity is generated by a small DC generator connected to a 500 ohm variable resistor used as electrical load. 3) The wind turbine of your design (mounted on the generator shaft) is not to exceed 18 inch in diameter and its most protruding part must be at least 12 inch away from the plane of the fan. 4) The turbine must be mounted on a 2”x2” platform and supported by a 24-inch tall truss tower of your design (see Figure 7). The tower must have either 3 or 4 legs, with bottom ends no more than 8 inches apart. If using 3 legs, triangular platform at the top is allowed. 5) The tower (see Figure 7) must withstand the turbine weight of Q = 6 N (1.35 lbs) 6 N (1.35 lbs) and the maximum horizontal (axial) force of 10 N H = 10 N (2.25 lbs) representing the (2.25 lbs) effect of strong winds. The 10 2 in. square at top N force is applied directly to the of tower generator along its axis. All (required) tower members must remain intact in order to pass the Tower bracing to loading test. prevent tower leg A A 6) The generator must be buckling mounted securely to the top of (your design.) the tower (to withstand the axial load) Tower base 7) The design must include the separation up to feet attachments to a 12 inch by 10 inch wooden base. 8 in. (your design) 8) The top mount and the feet attachment are the only places where glue can be used. Figure 7 Basic tower dimensions and loading Table 6 presents complete list of materials, equipment and tools that can be used. Table 6 Materials and equipment that can be used for the Competition Materials Quantity Equipment Quantity 1 1 Balsa Wood – /4” x /4” x 36” 4 20 inch fan 1 per 2 teams 1 1 Balsa Wood – /4” x /16" x 36” 1 3/16” shaft collar – to attach metal 2 disc Popsicle Sticks 15 Rubber tubing – to attach metal disc 1 1 1 Coffee Stirrers - /4” x /16" x 7.5” 30 Digital Multimeter 1 Base, Soft Wood – 12” x 10” 1 Potentiometer – 500 ohm 10 turn 1 x½” 14 inch zip ties 4 DC Motor / Generator 1 Plastic rivets 100 Base / Rod Stand 1 Push Pins 100 Cork pad work surface 1 Corrugated Plastic Sheet – 20” x 1 Laboratory scale 1 per class 20”x 2 mm Tools Quantity Scotch Tape 1 Hex Key for Collars 1 Double Stick Tape 1 Hex Key for Couplings 1 Aluminum Disc – 5.5” x .032” 1 Scissors 1 Wire – black, red, green 3’ each Ruler 1 Generator holder with screws 1 Wire stripper 1 Angle brackets with screws 4 Tin snips 1 Gorilla glue some X-Acto knive 1 Hole piercing tools 2 Hand drill with 1/8” DIA drill bit 1 Each group must bring to the competition: 1) Completely assembled tower (base, tower and turbine) 2) A record of the tower weight (in grams), signed by the instructor 3) A record of the resistance value to be used The judges will set the chosen resistance, determine the generated power, and only then perform the tower loading test. The group’s score in the competition will be based on the following formula (with the Generated Power in watts and the Tower Weight in grams): Score  200  Generated Power  (120  Tower Weight ) Penalties:     If the tower experienced minor failure during the loading test, a 30 gram penalty will be added to the Tower Weight If the tower experienced catastrophic failure during the loading test, a 60 gram penalty will be added to the Tower Weight If glue was used in tower construction a 30 gram penalty will be added to the Tower Weight If a stand was used instead of the tower, the Tower Weight will be taken as 180 grams. Three Competitions: Each among all teams meeting for class at the same time (Wednesday 3PM, Wednesday 6PM, or Friday Noon) Prizes: Three best teams in each competition Grand Prize: Best of all teams Module 2: Wind Turbine NOTE: All subsequent Activities are in preparation for the Competition. Most of them can be performed in any order, or even simultaneously. Activity 4. Design the new turbine Using the basic calculation results from HW Assignment 1 and the Reference section as guidance, build a wind turbine, following the Competition Rules listed above. Note that the electric load resistance can be varied. The important parameters that were selected in HW 1 include the turbine diameter and number of blades. You can experiment with various blade angles, or try to curve them. Remember that you do not have to follow the HW1 results, but you must adhere to the Competition Rules. Use your engineering judgment and build the turbine. Make sure that all turbine blades are of the same size/shape and are attached at the same distance from the aluminum disc center. Otherwise, unacceptable rotor vibration will develop. Activity 5. Determine the maximum power of the new turbine Mount the turbine on the available stand in front of the window fan and determine the maximum generated power a similar procedure as in Activity 3 (page 6), but now allowing for variable resistance. 1. 2. 3. 4. 5. 6. 7. Disconnect one wire from the variable resistor Using the multi-meter, set the variable resistor to R = 100 ohm Connect the resistor to the generator Start the fan at high setting Using the multi-meter, measure the voltage drop V across the resistor Calculate the generated power using equation P  V 2 / R Repeat the above steps seven more times, using the remaining resistance values in Table 7. Record all results in Table 7 below. Resistance R () Voltage V (V) Power P  V 2 / R (W) Table 7 Determination of the maximum power 100 20 10 8 6 4 2 1 Record the determined maximum power values and the corresponding value of the conversion efficiency from AC power driving the fan to DC power output from the generator in Table 8 below. Compare your results with the corresponding values determined in Activity 3 (Table 5). Table 8 Maximum power – Results Resistance R () Voltage V (V) Power P  V 2 / R (W) Conversion Efficiency (%) Module 3: Turbine Tower No wind turbine is complete without a supporting tower, and the three most commonly used tower designs are a monopole, lattice (truss) tower (described in the Reference section), and a pole or truss tower with guy wires. The purpose of this module is to design and build a wood framed truss tower supporting your wind turbine. Activity 6. Build the turbine tower Using the calculation results from HW Assignment 2 as the guidance, design and build a truss tower which can support the defined vertical and horizontal loads. The dimensions of the tower and loading conditions shall conform to those shown in Figure 7 of the Competition Rules. The two competing goals here are that the tower is strong enough and lightweight at the same time. The towers will be tested during Week 4 to ensure that they can withstand the design loads. Figure 9 Truss towers Tower legs shall be constructed using ¼ in. square balsa wood. Bracing members (connecting the legs) may be constructed from thinner balsa wood, Popsicle sticks and/or coffee stirrers. Connections between the tower legs and bracing elements can be made using push pins (thumb tacks) or plastic rivets. Table 6 in the Competition Rules lists all available materials. You can use either 4 legs (easier) or 3 legs (lighter). Material properties of the wood are provided in the Reference section. Bracing shall be sufficient to brace the tower legs against a buckling failure. You may choose bracing that corresponds with your Euler buckling calculations or select different bracing, as you choose. Consider the tower analysis you completed as your construct your tower. Several different types of bracing are shown in Figure 9 and Figure 10. Activity 7. Build the turbine mount One of the design challenges is how to attach the wind turbine to the truss tower. Design and build a sturdy turbine mount to be attached to the top of the tower. The mount must be strong enough to support the forces described in the Competition Rules. One possible design is shown in Figure 11, but you can also use the provided custom mount. Determine the weight of the tower with no turbine mount or the turbine. The weight determination must be supervised and recorded by the Projects Instructor. Figure 10 Simple plane trusses Activity 8. Attach tower feet to the base Attach the tower feet to the tower base in such a way that the tower can withstand the forces due to the wind generated by the fan. It is best to use the provided angle brackets, but you can also use Gorilla glue (at the end of one class period) to strengthen the feet attachment. Figure 11 Turbine mount REFERENCE Energy and Power Quoting excerpts from “Energy” entry from “… the most common definition [of energy] is that it is the capacity of a system to perform work. The definition of work in physics is the movement of a force through a distance, and energy is measured in the same units as work. If a person pushes an object x meters against an opposing force of F newtons, F∙x joules (newton-meters) of work has been done on the object; the person's body has lost F∙x joules of energy, and the object has gained F∙x joules of energy.” Power is the rate of energy use or transfer. It is expressed in units of energy per unit of time, most notably watts (one watt equals one joule per second). So a rated power of a device (like a refrigerator or a TV set) indicates at what rate the device consumes energy. If you use a 1000-watt microwave oven for 6 minutes (0.1 hours), then the device will consume 100 watt-hour (or 0.1 kWh) of electricity. Note that kWh is a unit of energy, not power. How much will this electricity cost? About 1 cent! Microwave ovens are very efficient way to heat up food. Let’s introduce the forms of energy and power most relevant to engineering.  Kinetic energy is the energy of motion. If an object of mass m is moving with velocity v, the kinetic 2 energy is KE  12 mv  Potential energy (of gravity) is the energy an object gains when it is lifted in a gravitational field. If an object of mass m is raised by h, it gains the potential energy of PE  mgh , where g is gravitational acceleration. Electrical energy is the energy of moving electrons that can be converted to heat (in a resistor) or work (in an electric motor). If voltage V applied to a device results in current flow I for a period of time t, the electrical energy consumed is E  (VI )  t  P  t . Note that P  VI is power. Chemical energy is the energy absorbed or released during a chemical reaction. For example, fuels (gasoline, natural gas, etc.) when burnt have their chemical energy converted to heat. Food also has chemical energy which is essential to functioning of our bodies. Nuclear energy is created when mass of atoms is converted to energy in a nuclear reaction. If the 2 mass is reduced by m, the amount of energy released is E  mc , where c is the speed of light (about 3 108 m/s, or 300,000,000 m/s, or 186,200 miles per second!).    Power Converting Devices For each conversion process (see Figure 12), the conversion efficiency can be defined as useful power output Pout divided by the required power input Pin :   Pout / Pin and power losses Ploss as: Power Input Power Converting Device Useful Power Figure 12 Power conversion Ploss  Pin  Pout It can be shown that the conversion efficiency of a sequence of n conversion processes is equal to the product of the individual conversion efficiencies:  system  1  2   n There are countless examples of power converting devices. In this project, you will be using batteries, electric motors, electric generator, resistors, fans and wind turbines. Wind Turbines Wind turbines convert the kinetic energy of blowing wind into work of a rotating shaft. Based on the rotor configuration, they can be classified as either horizontal axis or vertical axis turbines (Figure 13). Most commonly used rotor designs are shown in Figure 14 and include: two- or three-bladed propeller, Darrieus, multi-bladed (a.k.a. U.S. Farm Windmill) and Savonius. The first two operate on so called lift principle (same as airplane wing) and require high speed of rotation, while the last two operate on so called drag principle and work best at low speed of rotation (see also Figure 15 discussed below). The total kinetic power carried by wind impacting a wind turbine is 3 Pwind  12 AVwind where Vwind is the wind speed,  is the density of air and A is the area swept by the turbine rotor. The actual power P generated by a wind turbine is obviously smaller than Pwind : 3 P  CP Pwind  12 CP AVwind ( 0  CP  1) where CP is a so called power coefficient, The power produced of the air blowing over the generator blades is therefore proportional to the cube of the wind speed and to the surface area swept by the fan blades. It can be shown that regardless of how perfect the turbine is, the power coefficient can never exceed 8 27 , or about 59% (the Betz limit). The power coefficient of actual wind turbines is even smaller than this, as shown in Figure 15. The graph presents the power coefficient CP of several turbine types as a function of the so called advance ratio (or tip speed ratio)  defined as: V r   tip  Vwind Vwind where r is the rotor radius and  is its angular velocity*. Note that the multi-blade turbine has * Note that  (rad / s )  N (rev/min )  Figure 13 Horizontal vs. vertical axis wind turbines Horizontal Axis High-Speed (Propeller) Vertical Axis Darrieus Lift Effect (High Speed) Savonius (top view) Drag Effect (Low Speed) Figure 14 Propeller, Darrieus, Multi-Bladed and Savonius wind turbines Figure 15 Power coefficient as a function of advance ratio 2 (rad/rev) 2N  (rad / s ) . 60 (s/min) 60 the maximum power coefficient CP of about 15% when tip speed is about 90% of the wind speed. For high-speed turbines CP can exceed 45%. Design of Truss Tower All structural systems are composed of a number of basic structural elements – beams (bending members), columns (compression members), hangers (tension members), and so forth. These elements can be combined to form larger structures such as buildings, bridges, towers, and other structures. In this project, you will design and construct a tower to support your windmill. The legs of the tower serve as tension and compression members to resist the downward gravity loads and the overturning effects due to wind. Applied Loads To determine the internal forces in the tower legs along the height of the tower, the tower can be idealized as a cantilever beam turned on its side with a point load, H, acting at the end of the cantilever and equivalent to the wind force pushing on the tower and windmill, as well as an additional axial load, Q, equal to the weight of the tower and the windmill pushing downward in the direction of gravity. The base of the tower is a rigid support that resists lateral loads (wind loads), axial loads (gravity loads), and bending moments caused by overturning forces. Q = 6 N (1.35 lbs) H = 10 N (2.25 lbs) L = 2 ft (required) x 2 in. square at top of tower (required) A A Tower bracing to prevent tower leg buckling (your design.) Tower base width between 2 in. and 6 in. (your design) Figure 16 - Forces acting on windmill. At any one section (A-A), the tower must resist the axial load, Q, a shearing force equal to the lateral force, H, and an overturning moment (bending moment) caused by the lateral load from the wind, H, acting over a distance “x” from the top of the tower (Figure 16). Note that the axial force and lateral shear force in the tower remain constant over the height of the tower. However, the bending moment, M = Hx, increases from a value of M = 0 lb-ft at the top of the tower where x = 0 ft to a maximum value of M = HL at the base of the tower where x = L ft (Figure 17). Q=6N (1.35 lbs) Bending Moment, M = Hx, vs. x Shear Force, H vs. x Axial Force, Q, vs. x QL Qx L = 2 ft x H Q 0 H = 10 N (2.25 lbs) Figure 17 - Shear, axial, and bending moment vs. height of tower. Equilibrium An element is in equilibrium provided it is at rest if originally at rest. To maintain equilibrium, it is necessary to satisfy Newtown’s first law of motion, which requires the resultant force acting on a particle to be equal to zero. Eq. 1. ∑F=0 Newtown’s second law of motion, Eq. 2. F = ma, provides that if F = 0, the acceleration of the object must equal 0, since the object will have mass. If the tower is at rest – termed a state of static equilibrium – both the linear acceleration and angular acceleration equal zero and the following equations of static equilibrium must be satisfied: Eq. 3. Eq. 4. Eq. 5. ∑ Fy = 0 ∑ Fx = 0 ∑M=0 The sum of the forces in the y-direction, sum of the forces in the x-direction, and sum of the moments about any point equal 0. If these equations are not satisfied, the object is not at rest, but rather is accelerating or moving, which in structures is related to instability and failure. The static equilibrium equations can be used to calculate compression forces and tension forces in the tower legs due to the axial load, Q, and the overturning moment, Hx, at any point along the height of the tower as shown in Figure 18. Note that the distance x is defined from the top, rather than the bottom of the tower. Q = 6 N (1.35 lbs) H = 10 N (2.25 lbs) Top of tower x Arbitrary point, x, from top of tower A F1 W F2 Figure 18 - Forces acting on Tower Section A moment is simply the multiplication of a force by its perpendicular distance from the force vector to the point in question. Selecting an arbitrary point A on the tower as shown above and summing moments about Point A, provides the following equation: Eq. 6. ∑ 𝑀𝐴 = 0 = 𝐻𝑥 + 𝑄𝐿 2 − 𝐹2 𝑊 which can be solved for F2. Note that selecting Point A in line with the tower leg force, F1, eliminates F1 from the moment equation, such that the only unknown is force, F2. Now the only unknown force is, F1. Summing forces in the y-direction provides the following equation: Eq. 7. ∑ 𝐹𝑦 = 0 = 𝐹2 − 𝑄 + 𝐹1 which can be solved for F1. Negative values for force indicate a downward force (tension in the tower leg; whereas, positive values of force indicate an upward force (compression in the tower leg). F1 and F2 are the internal axial forces in the tower legs required to resist the applied loads, H and Q. Typical towers are 3-leg or 4-leg in plan view (Figure 19). For a 4-leg tower, the forces F1 and F2 are simply divided in half to determine the force in each tower leg. For a 3-leg tower, the force F1 is divided in half, but the force F2 must be resisted by one leg of the tower alone. Or, F2 is divided in half, and F1 is resisted by one leg. ½ F1 ½ F2 ½ F1 ½ F2 ½ F1 ½ F1 F2 Figure 19 - Forces acting on Tower Legs Internal Member Stresses The tower resists the shear force, through internal shear stresses in the wood. When all tower legs are equal in size, each leg resists an equal proportion of the shear force, H. Therefore, each tower leg in a 3-leg tower resists 1/3 H; whereas, each leg in a 4-leg tower resists ¼ H. Once the internal forces in the tower legs are calculated, the forces are used to calculate axial stresses in the tower legs through the relationship: Eq. 8. 𝜎= 𝑃 𝐴 where: σ = axial tensile or compressive stress (psi) P = axial load in tower leg = F1 or F2 (lbs) A = cross sectional area of tower leg (in2) Using an allowable stress design approach for the tower design, the actual stresses in the tower legs should be no more than the strength of the material in compression, fc, or tension, ft, divided by a factor of safety. For your project, the factor of safety, F.S. = 3 such that: Eq. 9. where: Eq. 10. Eq. 11. 𝜎 ≤ 𝜎𝑎𝑙𝑙𝑜𝑤 𝑓 𝑐 (𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛) 𝜎𝑎𝑙𝑙𝑜𝑤 = 𝐹.𝑆.=3 𝑓 𝑡 (𝑡𝑒𝑛𝑠𝑖𝑜𝑛) 𝜎𝑎𝑙𝑙𝑜𝑤 = 𝐹.𝑆.=3 The following materials are provided for the tower structure: Table 9. Material Properties Material Cross-Sectional Dimensions Balsa Wood ¼ in. x ¼ in. x 3 ft Medium Grade Popsicle Sticks 3/32 in. x 3/8 in. x 4 ½ in. long Birch Wood 0.06 in. x ¼ in. x 7 ½ in. long (Coffee Stirrers) Buckling of Compression Members Strength and Stiffness Properties Compressive Strength, fc = 1750 psi Tensile Strength, ft = 2890 psi Modulus of Elasticity Comp., E= 66,700 psi Unknown Compressive Strength, fc = 750 psi Tensile Strength, ft = 600 psi Shear Strength, fv = 195 psi Modulus of Elasticity, E = 1,500,000 psi For many compression members, instability of the member occurs prior to reaching the compressive strength of the material. This instability is known as buckling, which is an out of plane displacement of the column. out of plane displacement (instability) Figure 20 - Buckling In 1757, Leonhard Euler, a Swiss mathematician, presented mathematically described the critical buckling load, P: Eq. 12. 𝑃= 𝜋2 𝐸𝐼 𝐿2 where: E = Modulus of Elasticity of Material (see material table) I = Moment of Inertia, 𝐼= Eq. 13. 𝑏ℎ 3 12 The Euler buckling load is related to the stiffness, EI, of the compression element and its effective length, L. For a column pinned (allowed to rotate) at its supports, L equals the full length of the column. Stability of Built-up Structures In built-up structures with members that are pinned together, triangles form stable shapes. They cannot be deformed without deforming the individual members. Squares and rectangles can easily (with minute force) be pushed over such that they collapse. Pinned connections →UNSTABLE Pinned connections → STABLE Pinned connections → STABLE Figure 21 - Stability of structures with pin-connected elements In order to prevent the tower legs from buckling, bracing points can conservatively be designed using the Euler buckling equation and a factor of safety F.S. = 3. Such that the unbraced length of the tower leg, Lb, is calculated using: Eq. 14. 𝜋2 𝐸𝐼 3𝐶 𝐿𝑏 = √ where: P = the compressive force in the tower leg Electrical Components DC motor: As the diagram below indicates, current flows from a DC voltage supply through the brushes into a split ring commutator. The commutator is mounted on a movable armature. For simplicity, this is shown as one loop of wire. The brushes provide continuous electrical contact from the battery to the armature by sliding over the surface of the commutator. As the armature and commutator assembly rotates, the current carrying wire moves between two stationary permanent magnets. The magnets, referred to as the stator, interact with the wire loop through a magnetic field. The current carrying wire produces a magnetic field in a direction perpendicular to the plane of the loop. Together the magnetic fields generate a net force on the wire (known as the electromagnetic force or EMF). At a critical point in its rotation, when the poles from the wire loop and the permanent magnets align, the current direction is reversed in the wire due to the action of the split ring commutator. The momentum of the armature forces the rotational motion to continue. (This illustration is an example of a two-pole motor. It illustrates the basic principles involved in the DC motor. Commercial versions of DC motors are much more sophisticated.) A DC motor can also be used as a DC generator. In this case the available mechanical power (for example wind power) is used to rotate the commutator assembly, resulting in a DC voltage generated between the electrical terminals. AC Generator: In many ways the AC generator is similar to the DC motor. Of course in this instance, mechanical energy is introduced into the system and electrical energy is what results, which is the reverse of a motor. As the diagram below indicates, an armature, stator and brushes are present, but compare the structure of the commutator to the DC motor commutator. Instead of a split ring as in the case of the DC motor, there are two separate rings. As the armature rotates, a continuous electrical current is produced. Instead of changes in current direction as in the wire loops, a smooth, sinusoidal waveform is created. Dry Cell Batteries: This voltage source converts chemical to electrical energy. An ideal DC voltage source produces a constant voltage output irrespective of the circuit in which it is connected. As a practical matter, this is impossible to achieve for all possible loads. Consider a short circuit placed directly across a DC voltage source, i.e. a wire lead connected from one terminal to the other. Since the short has negligible resistance, assuming a short length of wire is used, Ohm’s law requires that the current draw must be infinite. No real DC source can ever satisfy this requirement. There are six principal types of commercial batteries. You may have heard of some of these used in some everyday applications. (1) Zinc Batteries. These are commonly referred to as “flashlight” batteries. There are two types, the carbon-zinc “regular-duty” and the zinc-chloride “heavy-duty” batteries. The latter type can last from 25 to 50 percent longer than the former. (2) Alkaline Batteries. These batteries have an alkaline manganese dioxide chemical makeup that can last five to eight times longer than the carbon-zinc types. They also come in a rechargeable version, although they require a special low-current, recharger for this purpose. (3) Nickel-Cadmium Batteries. These batteries were specifically designed to last for over 500 recharges. “Ni-Cads” do not hold a charge nearly as long as zinc or alkaline batteries and must be frequently charged. They also suffer from a “memory” effect, whereby the useful capacity of the battery is reduced if the Ni-Cad cells are not fully discharged before being recharged. Another disadvantage of this battery type is that the polarity can change, the positive terminal can become negative and the negative terminal can change to positive, under certain circumstances such as when the battery is left discharged for too long or if it is discharged below 75%–80% capacity. (4) Nickel Metal Hydride Batteries: “NiMH” batteries can be recharged over 400 times and do not have the problem of “memory” as the Ni-Cads have. They have a low internal resistance and can supply large amounts of current, for example in running motors and speakers. They can develop large amounts of heat in supplying current and must therefore be isolated from temperature sensitive circuit components such as integrated circuits. In addition, they can lose their charge by simply being placed in storage. (5) Lithium and Lithium-Ion Batteries: These batteries are largely used in laptop computers. They have the highest energy density, i.e. stored electrical energy per weight, and can retain their charge for months. They can be recharged, but require special recharging circuitry. (6) Lead Acid Batteries: Lead plates that are immersed in an acid-based electrolyte form the internal construction of the battery. The cells usually are sealed (SLA), but have pores that allow gases to escape during recharging. Lead-acid batteries are reasonably heavy. A single 6.0 V battery may weigh up to four or five pounds. They are used in automotive applications to provide an energy source to start gasoline engines. Gelled type batteries use a special thickened electrolyte and are the most common form of SLA battery. “Lemon” (Galvanic) Cell: In a lemon cell, two nails are inserted through the skin of the lemon to the inside of the fruit. One nail is galvanized, meaning it is coated in zinc. The other nail is copper coated. The zinc atoms dissolve in the acid of the lemon juice. The acid serves as the electrolyte, providing hydrogen ions, H+. The electrolyte produces zinc ions, Zn2++, while leaving behind two electrons, 2e-. This reaction is termed oxidation. As the zinc is enters the electrolyte, two positively charged hydrogen ions, H+ from the electrolyte combine with two electrodes on the surface of the copper terminal and form a hydrogen gas molecule, H2. This reaction is termed reduction. For the reactions to occur, electrons flow in the wire from zinc to the copper terminal. These electrons combine with the hydrogen ions already in the solution, forming an electrical closed circuit. The voltage produced across the terminals of the cell is about 0.4 V with a small current of about 1.0 mA (=1x10-3A). By using Ohm’s Law, V=IR, we can calculate the resistance associated with the lemon cell to be about 400 Ω.
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