Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Material Type: Paper; Professor: Chandy; Class: Electrical and Computer Engineering Design I; Subject: Electrical & Computer Engineer; University: University of Connecticut; Term: Spring 2009;

Typology: Papers

2009/2010

1 / 53

Download Health Monitoring of Structure with Cable Members under Tension | ECE 4901 and more Papers Electrical and Electronics Engineering in PDF only on Docsity! University of Connecticut HEALTH MONITORING OF STRUCTURES WITH CABLE MEMBERS UNDER TENSION Senior Design Team Christopher Von Kohorn – Mechanical Engineering Lead on cable vibration Jeff Urban – Electrical & Computer Engineering Lead on digital design Eric Snapper - Electrical & Computer Engineering Lead on signal processing Sponsor Dr. Jonathan Russell - Civil Engineering, Coast Guard Academy Advisors Dr. Sung Yeul Park – Electrical Engineering Dr. Kevin Murphy – Mechanical Engineering Dr. John Bennett – Mechanical Engineering Dr. Rich Dino – Business / Management December 2009 ECE 4901 ECE 4901 Table of Contents 1. Abstract…………………………………………………………………………………… …………………3 2. Introduction……………………………………………………………………………… ……………….4 a. Problem Statement………………………………………………………………………….. 4 b. Background………………………………………………………………………… …………….5 3. Theory……………………………………………………………………………………… ………………..7 4. Approach…………………………………………………………………………………… ……………….9 5. Design Concept…………………………………………………………………………………… ……10 a. Power Supply……………………………………………………………………………… …..12 b. Accelerometer…………………………………………………………………… ……………14 c. Band-Pass Filter………………………………………………………………………………16 d. PC Oscilloscope……………………………………………………………………… ……….17 e. Attachment and Installation Tool…………………………………………………..18 6. Concerns…………………………………………………………………………………… ……………..19 7. Data Processing………………………………………………………………………………… …….21 a. Peak Fitting………………………………………………………………………………… …..25 8. Castleman Lab Setup. ……………………………………………………………………………..26 a. Lab Setup Calculations……………………………………………………………………30 9. Broadcasting Tower Testing…………………………………………………………………….32 a. Waterford Tower Testing…………………………………………………………………32 2 ECE 4901 2. Introduction Problem Statement: Instruments are to be designed that measure the cable tension in guy wires supporting broadcasting towers through the application of an algorithm developed by Dr. Jonathan Russell of the Coast Guard Academy. The algorithm will be enhanced by the University of Connecticut Senior Design Team and implemented in a portable laptop based system as well as a self-contained permanently installed system. The technique determines the tension of a cable by identifying the observed resonant frequencies and determining the best match within a matrix of predicted natural frequencies for the guy wire predicted at a given tension and temperature. Dr. Russell’s has provided code to determine the natural frequencies for a cable given its geometry and material properties. The Senior Design Team is enhancing the algorithm by automating the peak extraction and comparison processes, and building electronic systems to identify the tension automatically. 5 ECE 4901 Background: Cable tension structures are those that incorporate metallic cables under tension as structurally significant elements. Such structures include cable stay bridges, broadcasting towers, and high-voltage power lines. The cable tension of guy wires attached to antenna towers (Analog & Digital: AM / FM / TV / Loran) must be monitored periodically to ensure that cables maintain their tensile load. While likelihood of failure is low, costs of failure can be high. Failure can result in loosening of the cable, bending or twisting of the central tower or cable breakage. Such failures can damage the tower, causing loss of operation, possible human injury, and requiring replacement of the tower. Towers are typically 1000 ft tall, ranging up to 2000ft. The central truss is formed as a triangle shaped 10ft on a side, with columns 3-6” wide of 0.5” wall tubing. Cables are strung between points on the tower and the ground in groups of three at each of several heights on the tower. These cables control the movement of the tower, rather than hold it fixed in place, and often hold on the order of 40,000 lbs tension. Full size towers incorporate heavy steel cables, inches thick, which typically sag 50 – 70ft from vertical. The fundamental (base) frequency of each cable ranges from 2Hz down to 0.2Hz (0.5 second to 5 seconds). The current method of performing accurate cable tension measurements uses a slow, labor, personnel and equipment intensive process of attaching a coupling to each cable and physically measuring the cable tension using a hydraulic cylinder to take the load. The method generates results with 5 – 10% accuracy. This method must be repeated for each cable. Less accurate methods are used for small towers and for initial construction of larger towers. On method involves line-sighting using a scope installed parallel to the cable, near the base. Another involves exciting a pulse in the cable and counting the cycle time for the pulse to move between the base and tower. With any method, the center tower will require its own visual inspection to check straightness, non-twistedness, and to check bolted connections and bracket attachments. When the tower is first constructed, with the cable tensions being adjusted for the first time, this process can take weeks to accomplish. This is because the tensions of the cables must be adjusted simultaneously until the equilibrium position of the tower is attained. Vibration techniques to determine tension in bridges are mature technologies that have been widely studied and applied. These techniques begin to fail on broadcast tower guy cables as they grow larger and incorporate significant sag and significant lumped masses at points along their length. Tower guy cables present a different problem than bridge risers, which are comparatively massive, stiff and taut. Bridge analysis techniques are accurate only for the 1st, 2nd and 3rd frequencies, not afterwards. On guy cables, these lower frequencies may have much lower amplitudes than higher frequencies, which can be used 6 ECE 4901 up to approximately the 20th harmonic. Furthermore, the bridge techniques do not allow for the inclusion of lumped masses along the length of the cable. The presence of significant point masses on a cable has been shown to cause banding of the natural frequencies, a significant problem to overcome when attempting vibration analysis of a system. Dr. Russell approached the Tech-Knowledge Portal Program (TKP, an EDA University Center within the Office of Technology Commercialization) with a product / business concept that should solve these problems, allowing vibrational measurement techniques to be applied to large guy cables on towers. The technique uses frequency spectrum data up to the 20th harmonic of the fundamental frequency to accurately predict natural frequencies, despite the presence of lumped masses, significant sag, and inclined cable spans. After discussion with Dr. Russell, he has expressed interest in funding this as an Entrepreneurial Senior Design project. He would be interested in participating in a commercial startup business should the project be successful. Figure 1: Tension tool for small cables Figure 2: Line-sighting using a scope 7 ECE 4901 Approach By implementing Dr. Russell’s cable tension measuring algorithm, a smaller and more efficient portable device will be designed to monitor the cable tension. The completed device should be far more cost effective than the current process, meeting a need within the niche business of cable installation and inspection. The permanently installed version will offer continuous monitoring, not presently available in the marketplace, and may serve as a platform for additional monitoring applications. The portable type may be temporarily attached to the cable to take a quick reading, and the collected data compared to baseline data from the original installation. This would be a fast and reliable method, and could conceivably eliminate a majority of the work associated with the currently used procedures, while delivering significant cost savings. The device would be usable by a minimal service team, ideally a single technician. The permanent type could be permanently attached to a cable, performing measurements on a periodic or semi-continuous basis. Data communication could be wireless or hard-wired, and by a schedule or on-demand. The device would need to be durable under weather conditions, wear and corrosive effects, and would require a power source, which may be hard-wired, or harvest vibrational or solar energy. The upgrade from a portable to a permanent system will allow the team to reduce costs, allow as-needed maintenance inspections, and may offer a platform for additional applications. The team will pursue both versions, beginning with the portable type to verify and refine our technique. This will be used as a test fixture to better understand the problem, refine minimum specifications and justify the design of the permanent, self-contained system. After creating an initial, working portable system, the team will swap in prototype components and build additional functionality into the working system to transition to the permanent self-contained system. 10 ECE 4901 4.Design Concept The team’s initial design goals are to create a handheld device that can gather and store data to be easily accessed for analyzing. To fulfill these needs, the team is initially using a digital PC oscilloscope that allows the team to gather data from our accelerometer and easily transfer the results to MATLAB. This setup is convenient because the data can be stored for repeated experimentation and interpretation as the analysis software is developed. The digital oscilloscope package that the team has obtained is equipped with an automatic FFT function, to convert our vibrational responses to the frequency spectrum. The team has opted to collect samples in the time domain, using the FFT generated frequency spectrum only for secondary verification – implementing an FFT through code to allow for an easier transition to further design stages. The digital oscilloscope package is intended for initial testing to verify the technique and determine minimum specifications for the digital circuit. Switching to either a microcontroller of FPGA meeting the minimum specifications will reduce the overall cost and size of the device. From here, it can either begin processing the data internally to output a concluded tension, or it can transmit the data to an offsite location where the analysis can be performed. This microcontroller/FPGA approach, however, is merely conceptual at this point in time, as the team would like to improve the coding aspect itself before moving forward with making our device more practical and marketable. Figure 4: Portable prototype block diagram 11 ECE 4901
Data Acquisition
Digital Circuit
(Microcontroller or FRGA)
Data Preprocessing
Data Storage or
Communication
Figure 5: Self-contained prototype block diagram
eee eee eee
12
ECE 4901 5b. Accelerometer: The core of the vibrational measurement system is an accelerometer to convert cable motions into electrical signals for analysis. Ideally, the accelerometer should utilize communications that are EMI resistant when considering frequencies within the broadcast range(s) emitted by antennas installed on the tower, sensitive for the appropriate range of g-forces experienced by the cable, and ideally be sensitive across the complete range of relevant frequencies, 0.1Hz – 40Hz. EMI interference from tower broadcasts occurs at much higher frequencies (100kHz +) than the relevant cable vibrations. Digital communications may occur at these rates. The long leads for the portable system, and possibly for the permanent system, may have induced signals at the broadcast frequencies. Therefore, accelerometers utilizing digital communications should be avoided. Alternatively, an accelerometer with an analog (DC) voltage will be used. For this type of accelerometer, the DC voltage output differs from a center reference point (half of the reference voltage) by an amount proportional to the measured g-force, positive or negative. The range of g-forces experienced by the cable should remain within 0.0g – 2.0g (ambient gravity +- 1.0g) under expected cable conditions. These values may not hold under extenuating circumstances, such as icing induced galloping conditions. The low frequency response characteristics limited choices of accelerometer technologies to those that can measure DC or near-DC signals. Piezo-resistive and variable capacitance devices have this quality and were strongly considered. A leading contender piezo-electric accelerometer, and an excellent match for all specifications is Measurement Specialties’ 4000A model, available in several sensitivities. This accelerometer is designed for measuring long duration transients, with rated frequency response down to DC. This choice was very expensive at $335. Besides lower specified frequency handling capability, other advantages of this model such as temperature compensation are not necessary for our project. It is also very accurate, however since the team is measuring only frequencies, and not the magnitude of forces, the accuracy of the output is inconsequential. The accelerometer chosen for the project was Analog Devices ADXL32X line. This is a close match to the ideal specifications, only missing the very low end of the rated frequency response, falling off below 0.5Hz. It is designed for sensing motion and low frequency vibrations. It incorporates a capacitance micro-sensor with a pulsed square wave applied to a silicon capacitor incorporating a suspended plate. The amplitude of the resulting signal is dependent on the relative position of the plates, and low pass filtered to 15 ECE 4901 produce a consistent signal. This technology is both inexpensive ($30 installed on a board) and durable, handling high-g shocks without internal damage. The bandwidth is adjusted to limit high frequency response using capacitors installed outside of the chip itself, on the breakout board. The chip includes two axis capability, however only one axis is necessary for our application. The other output is left disconnected. Our board was sourced from SparkFun Electronics and had pre-installed 1uF capacitors, which correspond to a 50Hz cut-off. The filtering will be discussed in more detail in the following section. The initial accelerometer for the project (Analog Devices ADXL321) was sourced before fully understanding the cable behavior. A conservative choice was made to select a range of +-18g, in case vibrations exceeded +-1g from ambient. This resulted in lower sensitivity, and smaller amplitude components may have been lost below the noise floor. The accelerometer will be replaced with a more sensitive version within the same model line, the ADXL322, with a range of +-2.0g. Figure 7: ADXL321 accelerometer Figure 8: ADXL321 internal schematic Figure 9: Accelerometer breakout board schematic 16 ECE 4901 5c. Band-Pass Filter: On the accelerometer signal output, a band pass filter is installed to reduce the effect of unwanted noise and allow the maximum resolution on the scope. The filter is composed of a low pass filter built into the accelerometer breakout board and a high pass filter at the point of the oscilloscope input. 50Hz low pass filters are included on both the X and Y axis accelerometer outputs, using on-chip 32kOhm resistors and 0.1uF capacitors surface-mounted to the breakout board. This is close enough to the design specification of 40Hz. The ADXL32X accelerometer is rated down to 0.5Hz, however it is sensitive to DC components (ambient gravity), so the rating can be considered more of an acceptable tolerance than a hard cut-off. To allow the highest resolution measurements possible using the scope, the accelerometer output was normalized to 0V average output, through the use of a 0.15Hz high pass filter. This should be close enough to the design value of 0.1Hz. This was formed by a 1uF capacitor in line with the signal ouput wire from the accelerometer to the scope, and the scope’s own 1Mohm input resistance. 17 ECE 4901 6. Concerns Structural details and weather conditions can be problematic for a vibrational measurement system, and must be considered in our design process. Structural: Cables vary with decreasing relative stiffness as cables attached at higher points are considered. An extra top cable made from multiple materials is often installed to adjust the electromagnetic properties of the structure. The electrical properties of the cable materials and their geometry influence the transmission characteristics of the antenna structure. One material extends from the top of the tower, an insulator is installed along the length of the cable, and the balance of the cable is made of another material. If the team attempts to incorporate the measurement of tension for these cables into the final device, it will prove problematic. However, no solution for addressing this problem is being considered at this moment as these cables are structurally insignificant. Some towers incorporate paired cables that can be challenging for diagnosis. These are fairly rare and the team’s investigation will not concern such designs. It may be possible to use vibration measurement techniques on paired cables, however this is currently beyond the scope of the project. Components attached to the cable may affect its vibration characteristics. Installed components may include anti-galloping cables and rattling cowbells. The tower itself, if it is excited to vibrate, may drive cable vibrations. Accurate drawings are necessary for accuracy of Dr. Russell’s method as posited. It is currently unknown what the degree of accuracy necessary for the device will be as the team is currently testing the algorithm and has been unable to conduct any meaningful tests as of yet. However, it is posited that a possible solution to minimizing the effects of initial inaccuracies may be possible via measuring calculated tension change in the cable due to temperature change (possible with permanent installed device). By measuring the calculated change in tension with respect to the temperature, it may be possible to more accurately determine the unstrained length of the cable over time. However, while the team would like to pursue this possibility further, this option will only be pursued upon the successful completion of a final prototype design with a fully functional algorithm. 20 ECE 4901 Weather: Weather conditions are problematic, especially when considering a permanently installed system that is expected to take measurements continuously or on demand. Wind can cause resonant standing waves. Ice can weigh down a cable, affecting it’s structural properties such that the predictions will no longer be accurate. Ice can also accumulate asymmetrically along the cable, creating an airfoil shape that may induce large amplitude galloping motions. Lighting will regularly impact the tower and may contact the cable. While the tower and cables are constructed of conducting steel, induced voltages will need to be considered, especially near the surface of the cable. A permanent system will need to be protected from high voltage damage to the circuit through measures that may include positioning electronics away from the cable surface and shielding of the electronic component case. 21 ECE 4901 7. Data Processing Data processing will be performed initially on the laptop using MATLAB, however the intent of the project is to implement as much data processing capability as feasible within the digital circuit. For the permanent system, all processing will ideally take place within the circuit, which will output a tension value. At minimum the team seeks to accomplish preprocessing of the raw data within the circuit to reduce data volume for transmission to the analysis point. Each cable of each structure has unique geometry and design properties and will require its own matrix for frequency comparison. This requires a level of customization that will be most easily achieved by a software interface on a laptop for the portable version, and a customized table for a specific cable programmed into the permanent device. Accuracy of the test is dependent on the length of the test. Longer tests will allow greater precision of the FFT function, however the tension is more likely to shift during longer test intervals, resulting in blurred frequency spectrum, which may be less likely to result in a conclusive match. Per Dr. Russell’s experience using manual inspection of traces, optimal performance can be obtained with measurements on the order of 30 seconds in length. To conclusively determine tension, 10 of 15 consecutive measurements should contain a peak frequency for it to be considered for comparison to the calculated natural frequency matrices. At any given point on the cable, tension and environmental conditions, some natural frequencies will be represented, and others not. Dr. Russell’s existing method of frequency matching is based on human pattern recognition and intuition in determining a match. The team will need to identify additional criteria to determine the threshold to positively identify a match. Dr. Russell’s claimed accuracy is within 1%, which is superior to measuring requirements of 5% tolerance. The team will need to conclude its own level of accuracy once a method is developed. 22 ECE 4901 ten times that of the Waterford tower runs will give a much better defined frequency spectrum. Figure 12: The vibrational response, as well as the resulting frequency spectrum from the use of an FFT on a data sample from the Waterford tower Figure 13: the vibrational response, as well as the resulting frequency spectrum from the use of a FFT on a data sample from a small cable fixed to a door handle. The results can be seen to be both closer together and better defined when compared to the previous figure, which had a sampling time of about 1/10 of the small cable response results 25 ECE 4901 7a. Peak Fitting Once the frequency spectrum has been generated, a peak fitting algorithm will be applied to identify peaks that exceed a minimum relevant threshold. The algorithm will smooth the spectrum using least squares curve fitting. Downward zero-crossings in the first derivative of this smoothed spectrum will be identified as the peaks. This technique is useful primarily for signals that have several data points in each peak, not for spikes that have only one or two points. If adjustments to the smoothing coefficients are not effective in reducing incorrect identification of peaks, further limitations may need to be introduced. Figure 14: An example of how the peak extraction technique is going to work. Note that these results are not actual results from the team and are simply an theoretical example of how the technique will work 26 ECE 4901 8. Castleman Lab Setup The current lab setup was constructed in the Castleman lab at the University of Connecticut. There were several conditions that needed to be met by this setup for. The base of the structure needed to be movable in order to simulate different horizontal spans between the base of the test cable and its top. The exact unstrained length of the cable needed to be measurable to approximately 1/16th of an inch for Dr. Russell’s ‘known unstrained length’ method. The tension of the cable had to be converted into a force applied directly in series with the force transducer. These conditions were met using the following lab setup. For the top of the test cable. The test end of the cable is wound around the end of the c-clamp, which is attached to a beam, with the final portion taped. The tape allows for exact knowledge of where the test length of the cable begins off of the c-clamp. For the base of the cable, an aluminum structure was made. The baseplate of this structure was ½” thick aluminum, over a foot in length, and about 5” in width. The additional area of the base plate was necessary in order to be able to place extra masses on the base plate to weight it down. This allowed the structure to be movable, yet create a rigid test setup when needed by relying on the static friction between the cement floor and the aluminum baseplate. Three additional ½” thick, 1 foot in length, and 2” wide bars were welded together to form the base structure shown in figure 4. In the center of the base plate, the plate was tapped to allow the force transducer to thread into it rigidly. Directly above the tapped hole, an 1/8” diameter hole was drilled in the top of the base structure. The test cable was bent through the 1/8” hole in the top of the structure. This allowed the tension of the cable to be converted into a vertical force applied through a connecting block (described further below) onto the force transducer screwed into the base plate. It is theorized that the frictional effects of the test cable being bent through the hole in the top of the base structure will be negated by giving the test cable a large excitation around this point (allowing for the test cable to slip around the bend as necessary to create equal tension on both sides of the bend). This effect has yet been untested. 27 ECE 4901 A final purpose is to determine how well the algorithm handles temperature change in the test cable. The team intends to compare results from daytime (normal temperature) testing of the cable to results from nighttime (cold temperature) testing of the cable. This drop in temperature will shorten the unstrained length of the cable, and thus will increase the tension. 30 ECE 4901 8a. Lab Setup Calculations: There are several calculations that will need to be done to confirm the veracity of the test setup. These are detailed below. One of the first tests that needs to be performed is that of determining the static friction coefficient (k) between the concrete floor of the test lab and the aluminum base plate of the base structure. This will be accomplished via the following method: The force transducer shall be attached to the side of the base structure. The force it takes to drag the base structure will be measured by the force transducer. This will be repeated for varying amounts of weight placed on the base structure, making sure that the weights do not slip. Thus, the coefficient of friction will be calculated via the following formula: Wfk / (1) Where f is the measured frictional force and W is the total weight of the base structure. Calculating this coefficient will allow the team to know how much weight needs to be placed on the base structure to ensure a no slip condition. The tension at the bottom of the cable (Tb) can be measured via the following equation: AFTb / (2) Where F is the cable force measured by the force transducer and A is the unstrained area of the cable. The tension at the top of the cable (Tt) can be calculated via the following equation: D*A*L Tb Tt (3) Where L is the unstrained length of the cable and D is the weight density of the cable. Another factor that must be considered is the deflection of the aluminum support beams in the base structure due to the bending moment of the tension. This must be examined to determine if there will be a significant impact on the horizontal span between the two points of the cable. In this case, only one of the beams needs to be considered as the dimensions and properties for both beams are the same, as would be the bending moment due to the tension of the cable. The first thing to calculate is the bending moment. inchesFM 12*)cos(* (4) 31 ECE 4901 Where theta is the angle made between the bottom test end of the cable and the base structure. The beam is modeled as a cantilevered beam with a moment load, which yields the following equation I)*E*(2 / inches)^2 (12*M y (5) Plugging in equations 4 and 2 yields the following I)*E*(2 / inches)^3 (12*)cos(*(F/A) y (6) This equation will allow the team to calculate the deflection along the horizontal span due to bending. Using a theoretical maximum force of 50 lbf and a maximum case angle theta of 15 degrees, and a modulus of elasticity of 10^7 psi for the aluminum bars, a theoretical maximum case deflection was calculated to be y = 0.013852 inches. Thus, the bending deflection is considered insignificant. 32 ECE 4901 10. Timeline Stage Oct Nov Dec Jan Feb March April Create Prototype Measuring Device Create setup in Test Lab Conduct Field Tests Verify/Modify Statics Equations Translate Readings to Frequency Spectrum Curve Fit // Peak Extraction Build Mounting Fixture Test & Revise Prototype Design Develop Business Plan Figure 20: Gantt chart. Boxes in green have been completed. Boxes in red represent planned work in that area Currently, the team is working on implementing the code given to us by Dr. Russell as well as constructing code for peak extraction. The team is also working on building a new mounting fixture for the prototype device. 35 ECE 4901 11. Budget To-date expenditures of the team: Item Amount Analog Devices ADXL321 Accelerometer $29.95 BNC breakout cable $1.95 PicoScope 2203 PC oscilloscope $245.56 Shipping $3.22 Total: $280.68 Planned expenditures of the team: None at this time. 36 ECE 4901 References 1. Russell, Jonathan C. Statics and Dynamics of Tall Guyed Radio Navigation Towers. May 1995. 2. Irvine, Max H. Cable Structures. Cambridge, Mass: MIT Press, 1981. 3. Kitchin, Charles, Mike Shuster, and Bob Briano. "Reducing the Average Power Consumption of Accelerometers." Analog Devices. 08 December 2009 <http://www.analog.com/ static/imported-files/application_notes/320665656385 11AN-378.pdf> 4. Budynas, Richard G., and J. K. Nisbett. Shigley's Mechanical Engineering Design. 8th ed. New York, NY: McGraw-Hill, 2008. 5. Russell, Jonathan C., and T. J. Lardner. "Experimental Determination of Frequencies and Tension for Elastic Cables." Journal of Engineering Mechanics. (October 1998): 1067-1072. 6. Starossek, U. "Cable Dynamics - A Review." Structural Engineering International. 171-176. 7. Ren, Wei-Xin, Hao-Liang Liu, and Gang Chen. "Determination of Cable Tensions Based on Frequency Differences." Engineering Computations. 25.2 (October 2007): 172-189. 09 December 2009 <www.emeraldinsight.com/10.1108/02644400810855977>. 37 ECE 4901 HHo = To*l/Chord Weighto = Rho * Chord Vo = To*abs(h)/Chord + Weighto DO WHILE ((abs(RhoTemp - Rho).gt.Tol)) RhoTemp = Rho CALL ThD (h,l,A,To,Tol,EA,Rho,HHo,Weighto,Vo, 1 HH,V,Ttop,Lo,Weight,lH,lV,hV) Rho = A * Gamma + Ins / Lo HHo = HH Vo = V Weighto = Weight END DO ELSE Rho = A * Gamma + Ins/Lo Weight = Rho * Lo Ho = Weight Vo = Weight CALL TwD (h,l,A,Tol,EA,HH,V,To,Ttop,Lo,Weight, 1 lH,lV,hV,Ho,Vo) ENDIF * ds is the segment length used in matlab ds=40 WRITE(6,603)l,h,HH,V,To,E,Lo,Weight,A,ds 603 FORMAT(2X,F9.3,2x,F9.3,2x,F9.3,2x,F9.3,2x,F9.3,2x,F10.1,2X,F9.3,2x,F9.3,2x,F5.3,2x,F4.1) 1002 RETURN END SUBROUTINE ThD (h,l,A,To,Tol, EA,Rho,HHo,Weighto,Vo, HH,V,Ttop,Lo,Weight,lH,lV,hV) ************************************************************************************************ **** * SUBROUTINE ThD SOLVES THE EQUATIONS OF THE ELASTIC CATENARY FOR H, V AND Lo ************************************************************************************************ **** IMPLICIT REAL*12 (A-M,O-Z),INTEGER*2 (N) INTEGER*2 Count INTEGER*1 Pr * The non-dimensional constants for the subprogram are the following: eps = To/EA 40 ECE 4901 lhat = Rho*l/To hhat = Rho*h/To * The initial values (guesses) at the non-dimensional subprogram variables are as follows: x = HHo/To z = Vo/To w = Weighto/To * The starting values for the remainders of the three non-dimensional equations are as follows: rlhat = eps * w * x + x * ( log(z/x + SQRT((z/x)**2 + 1)) - log((z-w)/x + SQRT(((z-w)/x)**2 + 1))) - lhat rhhat = eps * w * (z - w/2) + x * ( SQRT((z/x)**2+1) - SQRT(((z-w)/x)**2 + 1)) - hhat rThat = SQRT(x**2 + (z - w)**2) - 1 * Initialize internal subprogram variables to zero count = 0 lhatx = 0 lhatz = 0 lhatw = 0 hhatx = 0 hhatz = 0 hhatw = 0 Thatx = 0 Thatz = 0 Thatw = 0 DET = 0 delx = 0 delz = 0 delw = 0 DO WHILE ((abs(rlhat).gt.Tol).or.(abs(rhhat).gt.Tol).or.(abs(rThat).gt.Tol)) count = count + 1 * This portion calculates the partial derivatives of lhat, hhat and That with respect to x, z, and w lhatx = eps * w + log(z/x + SQRT((z/x)**2 + 1)) - log((z-w)/x + SQRT(((z- w)/x)**2 + 1)) - (z/x)/SQRT((z/x)**2 + 1) + ((z-w)/x)/SQRT(((z-w)/x)**2 + 1) lhatz = 1/SQRT((z/x)**2 + 1) - 1/SQRT(((z- w)/x)**2 + 1) lhatw = eps * x + 1/SQRT(((z-w)/x)**2 + 1) hhatx = lhatz hhatz = eps * w + (z/x)/SQRT((z/x)**2 + 1) - ((z-w)/x)/SQRT(((z-w)/ x)**2 + 1) hhatw = eps * (z-w) + ((z-w)/x)/SQRT(((z-w)/x)**2 + 1) Thatx = 1/SQRT(((z-w)/x)**2 + 1) Thatz = ((z-w)/x)/SQRT(((z-w)/x)**2 + 1) Thatw = - ((z-w)/x)/SQRT(((z-w)/x)**2 + 1) 41 ECE 4901 * This calculates the determinant of the flexibility matrix DET = (-hhatz*lhatw + hhatw*lhatz)*Thatx - (-Thatz*lhatw + Thatw*lhatz)*hhatx + (-Thatz*hhatw + Thatw*hhatz)*lhatx * This portion calculates the increments for x, z and w delx = -1/DET * ( (-Thatz*hhatw + Thatw*hhatz) * rlhat + ( Thatz*lhatw - Thatw*lhatz) * rhhat + (-hhatz*lhatw + hhatw*lhatz) * rThat) delz = -1/DET * ( ( Thatx*hhatw - Thatw*hhatx) * rlhat + (-Thatx*lhatw + Thatw*lhatx) * rhhat + ( hhatx*lhatw - hhatw*lhatx) * rThat) delw = -1/DET * ( ( Thatz*hhatx - Thatx*hhatz) * rlhat + (-Thatz*lhatx + Thatx*lhatz) * rhhat + ( hhatz*lhatx - hhatx*lhatz) * rThat) IF (x+delx.ge.1) THEN x = 0.99999999 ELSE x = x + delx ENDIF z = z + delz w = w + delw * This calculates the remainder of the lhat, hhat and That equations based on the new values of x, z, and w rlhat = eps * w * x + x * ( log(z/x + SQRT((z/x)**2 + 1)) - log((z-w)/x + SQRT(((z-w)/x)**2 + 1))) - lhat rhhat = eps * w * (z - w/2) + x * ( SQRT((z/x)**2+1) - SQRT(((z-w)/x)**2 + 1)) - hhat rThat = SQRT(x**2 + (z - w)**2) - 1 END DO * calculate the actual values of H, V, Weight and Lo HH = x * To V = z * To Weight= w * To Lo = Weight / Rho Ttop = (V**2 + HH**2)**.5 StrTop = Ttop/A StrBot = To/A * calculate the dimensional partial derivatives of H and V, used in spring constant calculations lH = lhatx*Lo/Weight lV = lhatz*Lo/Weight hV = hhatz*Lo/Weight RETURN END SUBROUTINE TwD (h,l,A,Tol,EA,HH,V,To,Ttop,Lo,Weight 1 ,lH,lV,hV,Ho,Vo) ************************************************************************************************ **** 42 ECE 4901 DANNEG.m: % DETERMINANT AND # OF NEGATIVE DIAGONAL TERMS nn=2*(n-1); nneg=0; determinant=0; format short e %row 1 is unchanged if a(1,1)<0 nneg=nneg+1; end determinant=a(1,1); %row 2 a(2,2)=a(2,2)-a(1,2)*a(1,2)/a(1,1); a(2,3)=a(2,3)-a(1,2)*a(1,3)/a(1,1); a(2,4)=a(2,4)-a(1,2)*a(1,4)/a(1,1); if a(2,2)<0 nneg=nneg+1; end determinant=determinant*a(2,2); %rows 3 to nn-2 for i=3:2:nn-3 a(i ,1)=a(i,1)-a(i-2,3)^2/a(i-2,1)-a(i-1,3)^2/a(i-1,2); a(i ,2)=a(i,2)-a(i-2,3)*a(i-2,4)/a(i-2,1)-a(i-1,3)*a(i-1,4)/a(i-1,2); if a(i,1)<0 nneg=nneg+1; end a(i+1,2)=a(i+1,2)-a(i-2,4)^2/a(i-2,1)-a(i-1,4)^2/a(i-1,2)-a(i,2)^2/a(i,1); a(i+1,3)=a(i+1,3)-a(i,2)*a(i,3)/a(i,1); a(i+1,4)=a(i+1,4)-a(i,2)*a(i,4)/a(i,1); if a(i+1,2)<0 nneg=nneg+1; end determinant=determinant*a(i,1)*a(i+1,2); end %row nn-1 a(nn-1,3)=a(nn-1,3)-a(nn-3,3)^2/a(nn-3,1)-a(nn-2,3)^2/a(nn-2,2); a(nn-1,4)=a(nn-1,4)-a(nn-3,3)*a(nn-3,4)/a(nn-3,1)-a(nn-2,3)*a(nn-2,4)/a(nn-2,2); if a(nn-1,3)<0 nneg=nneg+1; end determinant=determinant*a(nn-1,3); %row n a(nn,4)=a(nn,4)-a(nn-3,4)^2/a(nn-3,1)-a(nn-2,4)^2/a(nn-2,2)-a(nn-1,4)^2/a(nn-1,3); if a(nn,4)<0 nneg=nneg+1; end determinant=determinant*a(nn,4); 45 ECE 4901 Guy4.m: h=316.67;E=26400000.0;A=0.00068911;L=504.0;W=5.3004628; nins=6; ins=[72 144 216 288 360 432;0.23498 0.23498 0.23498 0.23498 0.23498 0.23498]; sumins=(1.40988); W=W-sumins; Rho=W/L;Mass=W/386.4; % ins is a 2d vector with spacing and wt info for each insulator % n is the number of cable segments and nn is the number of DOF % n is set as the cable length in feet; % numLLs is the number of Eigenvalues to be calculated numLLs=16; %call Kandm subroutine Kandm; %GUESS AT LLAMBDA AND CALCULATE (NORMALIZED) K-LL*M Pendulumsquared=pi^2/l^2*(H*L/(Mass*sqrt(1+(h/l)^2))); %starting point is LL. To start the sequence at other than %the first eigenvalue, set the LL parameter before the desired % eigenvalue, and adjust nneg and nnegt equal to the number of %eigenvalues before the value of LL. Also, adjust the value in %"while" statement of stepping function to include the starting %number of eigenvalues. LL=Pendulumsquared; iteration=1; nneg=0; nnegt1=0; LLt1=LL; dett1=0; data=zeros(2*numLLs,3); %call subroutine a=k-LL*m kLLxm; %call subroutine to calc determinant and # of neg diagonal numbers danneg; LLt1 =LL; dett1 =determinant; %inc is the step value in units of (rad^2/s^2) inc=.8*Pendulumsquared; inc1=inc; %step along determinant function and bound eigenvalues while nneg<(4+numLLs+1) if nnegt1==nneg; dett1=determinant; LL=LL+inc1; kLLxm; 46 ECE 4901 danneg; inc1=inc; elseif nneg-nnegt1==1 data(iteration,1)=LL-inc; data(iteration,2)=dett1; data(iteration,3)=nnegt1; iteration=iteration+1; data(iteration,1)=LL; data(iteration,2)=determinant; data(iteration,3)=nneg; nnegt1=nneg; iteration=iteration+1; else nnegt1-nneg>1; LL=LL-inc1/2; inc1=inc1/2; danneg; end end data %zoom to eigenvalues by factor of 4 with bisection % so as to ensure convergence w/secant search for iii=1:numLLs LL1=data(2*iii-1,1); LL2=data(2*iii,1); d1 =data(2*iii-1,2); d2 =data(2*iii,2); LL=(LL1+LL2)/2; kLLxm; danneg; if nneg==data(2*iii-1,3) data(2*iii-1,1)=LL; data(2*iii-1,2)=determinant; LL=(LL+data(2*iii,1))/2; kLLxm; danneg; if nneg==data(2*iii-1,3) data(2*iii-1,1)=LL; data(2*iii-1,2)=determinant; else data(2*iii,1)=LL; data(2*iii,2)=determinant; end else data(2*iii,1)=LL; data(2*iii,2)=determinant; LL=(LL+data(2*iii-1,1))/2; kLLxm; danneg; if nneg==data(2*iii,3) data(2*iii,1)=LL; data(2*iii,2)=determinant; else 47 ECE 4901 m(2*ins(4,1))=m(2*ins(4,1))+ins(4,2)/386.4; m(2*ins(5,1))=m(2*ins(5,1))+ins(5,2)/386.4; m(2*ins(6,1))=m(2*ins(6,1))+ins(6,2)/386.4; 50 ECE 4901 13. Appendix B – Dr. Russell’s Patent Info 51 ECE 4901 14. Appendix C - Key Terms & Phrases: - Accelerometer – An instrument for measuring acceleration - Anti-Galloping Cable – A cable attached to possible Galloping Cable to restrain it - ‘As Built Print’ – A structural print of the broadcasting tower with the measurements of how the tower was built as opposed to how it was planned - Curve Fitting - To construct a curve with the best fit to a series of data points - Fast Fourier Transform (FFT) Chip – A computer chip that performs signal processing to convert signals from the time domain to a frequency spectrum - Field-programmable Gate Array (FPGA) - An integrated circuit used to implement logical functions - Frequency Spectrum - shows the frequencies occurring from a time domain analysis, as well as the amplitudes, symbolizing the predominance of each frequency with respect to all others - Galloping Cable – A cable that can be excited via wind to structurally dangerous levels of force - Graphical User Interface (GUI) – Digital menus the user can interact with - Microcontroller - An integrated circuit, run by simple functions to perform desired tasks - Natural Frequency – The frequency at which a body tends to vibrate - Oscilloscope - an electronic test instrument that allows voltages to be viewed versus time - Peak Extraction - Determining the locations in which peaks occur in a set of data points - Radio Frequency Interference (RFI) - A disturbance that affects an electrical circuit due to either electromagnetic conduction or electromagnetic radiation emitted from an external source. The disturbance may interrupt, obstruct, or otherwise degrade or limit the effective performance of the circuit - Rattling Cowbell - A device used to dissipate excess energy via making noise attached to the cable - Standing Wave – A wave characterized by a lack of vibration at certain points - Stiffness – The resistance of a body to deformation - Tension – The force per unit area exerted on a cable - A - The unstrained cross-sectional area of the cable - D -The weight density of the cable - E - The modulus of elasticity of the cable - F – The cable force measured by the force transducer - f – The static frictional force - g – The gravitational acceleration constant (32.2 ft/sec^2) - H - The Vertical component of tension of the cable - h - The Vertical span between the base of the cable and the structure it's attached to - I – The Moment of Inertia - i – The Horizontal span between the base of the cable and the structure it’s attached to - k – The static coefficient of friction - L – The unstrained length of the cable - M – Bending Moment - Tb – The tension calculated at the base of the cable - Tt – The tension calculated at the top of the cable - V – The Horizontal component of tension in the cable - W – The weight of the lab setup base structure 52