Design, Construction, and Testi ng of an Electrodynamic
Contents
1. LVDT is fastened to the base plate there is a physical limitation to displacement of the base plate This won t cause too much inconvenience since our position waveforms integrated from velocity is near symmetrical about zero so displacement of the base plate is negligible The holder for the dashpotis shown below Figure 8 Dashpot stand The setscrews on the SmartMotor were carefully removed and preserved as we detached the motor from the wave flume The SmartMotor was mounted onto the frame through two vertical aluminum beams as shown in Fig 11 Figure 9 Mounting the SmartMotor onto Unistrut The frame cables SmartMotor base plate and LVDT stand were completed within the first three to four weeks of the semester The need for a dashpot holder came around the 12th week by which time it was built by J and attached onto the structure Upon completion the complete view of the shake table is shown in Figure 10 Figure 10 Panoramic view of the shake table upon completion 12 3 Theory 3 1 Derivation Transfer function of a Second Order Underdamped System in time and Laplace domain Consider a mass spring damper system with an input of x t Equation of motion will establish the following d r dy c ky x t mo deg T x t Define the damping variable to be c c 20wym Also recall that the relationship for resonant frequency is w E Substituting the above two equations into the equ2. is L 26 Thus we find the spring constant of the lumped mass system BEA lbs D ni We can then find the resonant frequency wg 14 3 3 Procedures for frequency domain analysis Sample code for this procedure is attached in Appendix B FFT fftshift fft baseAcce dt time 2 time 1 f 1 length time baseAcce 2 dt baseAcce plot f abs FFT Performing this analysis onthe El Centro acceleration response we obtain the following frequency amplitude correspondence El Centro Frequency Amplitude Correspondence 35 abs fft baseA n2 n2 co CO en e en o adh ea P a aal 17 SL diia TW ua Ca A ot Ki 4 z 0 2 4 6 8 Freq H2 Figure 12 FFT of El Centro acceleration response 3 4 Theoretical calculation finding the resonant frequencies of a two story stick mass building 15 The building to be analyzed is the simple steel rigid frame shown in Fig The weights of the floors are the same and measured out before assembled It is further assumed that the structural properties are uniform along the length of the building We model the building as a two story shear building which can be represented by the spring mass system shownin Fig eo ei Figure 13 Multimass spring mo del for a two story shear building The weight is weighed in Ibm W W ilb The mass can be calculated m m 0026lbs in Since the girders are assumed to be rigid the stiffness spring constant of each
3. section To find the structure s resonant frequencies we shook it using an impulse Our PCB accelerometer was fixed onto a nut glued on the top floor The acceleration response was plotted and then we applied the FFT procedure to obtain the frequency domain information 38 10 baseA e 300 No 8 100 abs fft baseA 50 40 30 20 10 0 10 20 30 40 50 Freq Hz Figure 38 Time and frequency domain responses to an impulse of a 2 story structure The two resonant frequencies are 6 1 Hz and 19 2 Hz This is a standard procedure for finding the resonant frequencies of the multi story structures We applied the same approach to a three story stick mass structure 250 T T T T T T T T T 200 F 4 OD CH T 1 ce e T A abis Tl baseA kel Wen 50 40 30 20 20 D 10 20 30 40 m Freq Hz 39 Figure 39 Time and frequency domain responses to an impulse of a 3 story structure 40 9 References Engineering 12 Website Erik Cheever Structural Dynamics Theory and Computation Mario Paz MOOG Animatics 2014 SmartMotor User s Guide Class 5 SmartMotor Technology Moog Animatics Unistrut 2013 Unistrut Retrieved January 2014 from Atkore International Inc bon www unistrut us 41 Appendices AppendixA Analog Output oo Analog velocity output to simulate waveforms Bill Wu ENGR 90 oo beep dispi seue specify position pause 3 be
4. the right or left side of an equation respectively The SmartMotor SM23165DT is equipped with 15 I O points which can be physically accessed by a DB 15 D sub Connector The pin numbers and their corresponding functions are listed in the following 20 PIN _5V I O Connector 1 2 3 12 15 V O 0 GP or Enc A or Step Input WO 1 GP or Enc B or Dir Input WO 2 Positive Over Travel or GP MO 3 Negative Over Travel or GP VO 4 GPorRS485 ACom 1 MO 5 GP or RS485 B Com 1 VO 6 G command or GP Phase A Encoder Output Phase B Encoder Output RS 232 Transmit Com 0 RS 232 Receive Com 0 5VDC Out Ground Ground Main Power 12 VDC to 48BVDC Specifications 25mAmp Sink Source 10Bit 0 5VDC A D Sink Source DAR 108it 0 5VDC A D 25mAmp Sink Source 10Bit 0 5VDC A D 25mAmp Sink Source 10Bit 0 5VDC A D 25mAmp Sink Source 10Bit 0 5VDC A D Sink Source 25mAmp 1UBKt U 5bVDC A D 25mAmp Sink Source 10Bit 0 SVDC A D 25mAmp Sink Source 25mAmp Sink Source 50mAmps Max If DE Option Control Power separate from Main Power Notes 1 5MHz max as Enc or Step input 1 5MHz max as Enc Or Dir Input 115 2KBaud Max 115 2KBaud Max Redundant connection on Main Pwr Connector 1152KBaud Max 1152KBaud Max With DE option this becomes separate control power input Figure 18 SmartMotor Connector Pinouts Diagra DB 15 D sub Connector The 5V I O is push pull To use t
5. time and structure acceleration and use Matlab s curve fitting toolbox baseAhs vs timehs Structure Transfer Function baseAhs 2 4 5 8 10 12 14 16 18 20 timehs Figure 31 Curve fit for H tmcture The curve fitting results are as follows 1 2 w 6 6 0 0036 42s R 0 87 oF Notice the time constant is much bigger than that of Ha This give us a valid reason to ignore Hy 33 8 5 Discussion onthe significance of H plate The facttsructure Tpit gives us one reason to ignore Hy Also recall that its time constant corresponds to a cutoff frequency of 5 Hz Plotting a vertical line of 5 Hz onthe frequency amplitude of El Centro El Centro Frequency Amplitude Correspondence 35 abs fft bss eA n2 n2 co c en e en o Freq Hz Figure 32 Content of El Centro captured by Hy Observe the majority of the earthquake s amplitude are in the low frequency region and are captured by the base plate s transfer function This points to another valid reason to ignore it Based on the fact that the base plate s transfer function has a significantly smaller time constant compared to that of the structure s and that its cutoff frequency is higher enough to capture the majority content of the El Centro in fact most major earthquake behave similar to El Centro in that the majority of their content lies in the low frequency region we have decided to ignore the base plate s transfer funct
6. Design Construction and Testing of an Electrodynamic Uniaxial Bench Scale Shake Table Bill Wu Advisors Professor Erik Cheever and Professor Faruq Siddiqui Department of Engineering Swarthmore College May 8th 2015 Abstract A bench scale shake table was designed and constructed for the Swarthmore College Engineering Department The frame of the shake table was made of Unistrut It s powered bya motor with a complete built in servo control system that takes analog output as the velocity through an A D converter Two transfer functions were developed one that connects the motor to the base plate and the other connecting the plate to a simple test structure lumped mass model Sensors LVDT and accelerometers were hooked onto the base plate and structure to measure acceleration and displacement data to be sent back to Matlab for further analysis and plotting We used Matlab s linear simulator sim to calculate the theoretical output for a known input waveform and compared the results to the actual measured output by the accelerometer The results matched well for impulse functions step functions and arbitrary waveforms such as the El Centro earthquake Two multi story stick mass systems were shaken at their respective resonant frequencies observe the effects of resonance Acknowledgements I would like to thank Professor Cheever for his patience and wisdom along every single step of this project I also want to thank Professor Si
7. LSM1 xM cable The motor can be powered using a 24 48 VDC power supply An Agilent N5746A power supply was used which has a DC output voltage rating of 40 V 19 A current and 760 W power A schematic of the communication setup between a PC and the motor is shown in Figure 15 KITUSB232485 CBLSM1 xM Erni e PS24V8AG 110 g or PS42V6AG 110 f Figure 15 RS 232 Through USB for D Style Motors MOOG Animatics 2014 SmartMotor User s Guide 4 2 Programming Notes Due to the direct connection of the motor to a serial port the motor can be controlled using the MOOG Animatics SmartMotor interface SMI Software or Matlab SMI enables easy communication through the terminal program which provides immediate response to a given command Additionally the SMI software contains debugging options and can obtain information from the motor including current position velocity acceleration and voltage Because of these characteristics the SMI software is ideal for troubleshooting In addition to the SMI software Matlab can be used to communicate with the motor by creating a serial port and then sending text files through the serial port to the SmartMotor using the fprintf command A screenshot ofthe SMI programming interface is shown in Figure 16 18 File Edit View Communication Compile Tools Window Help Bag BiG ib WF wen gt s OnE le Find Motors Com3 Ethemet USB CAN Channel O E D I Detected Con
8. T and dashpot were later constructed and stabilized onto Unistrut pieces AUnistrut metal framing system was deemed optimal for the shake table s frame as it can be used to supporta large load while also be taken aparteasily for storage The Unistrut system contains specially configured lipped channels made of carbon steel that can be connected in a variety of configurations using fabricated Unistrut fittings Solidworks a3D CAD modeling software was used to design the flume The various Unistrut channels and fittings were obtained online from the Unistrut CAD Library Different configurations of the channels were then considered to be built around the dimensions of the flume The final design of the frame is shown in Figure ITEM NO ary DESCRIPTION LENGTH 1 2 Unistrut P1000 35 20in 2 4 Unistrut P1000 46 37 in 3 4 Unistrut P1000 49 63 in 4 10 Fittings shapes wing 5 2 Unistrut P1000 24 00 in s 6 2 Unistrut P1000 17 21 in LV E Ee GES Jee ze rer scair RAW mum pee MAU Shake SURE ACT rub mers Toireaecrs Figure 6 Unistrut design and dimension of the shake table The holder for the LVDT is made of a hollow section aluminum bar as shown in Figure 7 10 Figure 7 LVDT stand The easiness to make additions on Unistrut is manifested The rod of the LVDT is fastened to the base plate through a tapped hole Also note that once the rod of the
9. ab script file FFT Procedure List of Figures Figure 1 Animatics SmartMotor SM23165DT Figure 2 National Instrument PCI 6221 Figure 3 A D Convertor Box Figure 4 Linear Variable Differential Transformer LVDT Figure 5 PCB 352B70 Accelerometer Figure 6 Unistrut design and dimension of the shake table Figure 7 LVDT stand Figure 8 Dashpot stand Figure 9 Mounting the SmartMotor onto Unistrut Figure 10 Panoramic view of the shake table upon completion Figure 11 Idealized Single DOF lumped mass structure used for testing Figure 12 FFT of El Centro acceleration response Figure 13 Multimass spring model for a two story shear building Figure 14 Frequency domain response to an impulse of a 2 story structure Figure 15 RS 232 Through USB for D Style Motors MOOG Animatics 2014 SmartMotor User s Guide Figure 16 SmartMotor Interface example screenshot Figure 17 Linear and sinusoidal motion Figure 18 SmartMotor Connector Pinouts Figure 19 Setup ofthe DAQ system Figure 20 Test Run for calibration Figure 21 Lumped mass structure used for testing Figure 22 Two story Stick Mass System Figure 23 Connections details ofthe two story stick mass system Figure 24 The Big Picture Figure 25 Command input and recorded input for an impulse Figure 26 Command inputand recorded input for a sinusoid Figure 27 Command input and recorded input for the El Centro Figure 28 Zoom in of Figure 27 Figure 29 cftoolto obta
10. an configure data acquisition hardware and read data into MATLAB and Simulink for immediate analysis We can also send out data over analog and digital output channels provided by data acquisition hardware In Matlab we create a session object that we can configure to perform operations using a CompactDAQ device w daq createSesion ni where ni is the name of the vendor To collect data from our LVDT and accelerometer we added two analog input channels to Matlab s DAQ session To store data we created one analog output 23 channel After the desired waveform is generated inside a vertical array we transfer it into the DAQ through the following command in Matlab queueOutputData w motion To collect the data that the sensors collect during motion we used the following command data w startForeground Data will consist of two columns acceleration and displacement both in terms of voltage Using the calibration ratios given in the Introduction section we can obtain the acceleration and speed in terms of SI units a data 1 98 1 m s x data 2 01 m 24 6 Calibration While we transfer data from Matlab to the SmartMotor through the DAQ system the array is seen as voltages not velocity in SI units Calibration is needed so we can input waveforms of velocity in terms of SI units m s We took a sample test run Fig depicts time and position voltage on the two ax
11. and measured output for a sinusoid For the El Centro the 1 5im result is in blue while the measured acceleration by the accelerometer is in green El Centro 1 d 1 L 2 4 6 8 10 12 14 Figure 35 Theoretical and measured output for El Centro We see that the 1sim results for impulse and sinusoid input are good in in terms of amplitude and frequency They are both for relatively short durations For the El Centro output we can see the basic shape of the predicted result matches that of the measured acceleration However there is a noticeable shift throughout the entire waveform starting before the first second 8 7 Addition ofa dashpotto enhance the El Centro simulation results We then speculated the cause of the shift for the 1sim results for the El Centro Recall that our structure system is very lightly damped with a damping coefficient of near zero 0036 and the time constant of over 30 seconds Such light damping could cause the delay in response which is then amplified by the long duration ofthe El Centro waveform 36 To verify our speculation we added a dashpot stand onto our frame again made easy by choice of our framing material Unistrut The rod of the dashpot is fastened onto the structure by gluing a No 5 nut onto its front surface There is a even great limit to the amount of possible displacement as the dashpot s rod is even shorter than that of the LVDT Due to this limitation we co
12. ation of motion d y dy Nia M 2 ts 2 de wo dt wy Kw x t Performing Laplace transform on both sides of the equation we get Y s s 2 wos w Kw X s Rearrange to get X s Kwg Y s s2 2 wys wg H s which is the transfer function for a second order underdamped system when the input and output are both in terms of velocity Similarly the transfer function for velocity in acceleration out can be derived 2 Kwys Hs s s 2 wss wg To obtain these transfer functions in the time domain we can setup all the variables using Matlab s symbolic toolbox and perform inverse Laplace through Matlab s iLaplace command The time domain transfer function for velocity in velocity out is 13 1 wt ai H t 1 Tiare ft sin Land t acos Q The time domain transfer function for velocity in acceleration out is sinh ot 1 g H t wie nf cosh ot 1 E m m 3 2 Theoretical calculation findingthe resonant frequency of a lumped mass system The lumped mass system used and the cross section ofthe cantilever beam is shown in Figure E FH yum 4 125 TT Figure 11 Idealized Single DOF lumped mass structure used for testing The second moment of area is in 12 6144 The elastic modulus of aluminum is E 10 105 psi the weight of the aluminum block with the attached SparkFun accelerometer and Arduino is W 1 905 lbs the length ofthe cantilever beam
13. city acceleration and position are given in encoder units A conversion is therefore necessary to specify the position in inches velocity in inches per second and acceleration in inches per second squared The Animatics SM23165DT used in this project rotates with 4000 encoder counts per revolution Additionally the HLD60 Actuator with Internal Rollers has a 12 5 mm displacement per revolution Using simple programs we can achieve linear and sinusoidal motion as depicted by Figure 17 est Run Linear Motion terw s Test Run Sinusoidal Motion Test Run Linear Motion 2 1em s Figure 17 Linear and sinusoidal motion However SMI programs are only good for achieving relatively simple motion A comprehensive list of desired position and or velocity needs to be hardcoded into the program This is not practical when we re ultimately trying to implement an arbitrary waveform For example if the waveform of velocity data is spaced at 20ms and has a duration of 30s there are 1 500 velocity data points to write into the SMI program which is not realistic or desirable 4 4 I O Ports Upon further study of the User Manual we discovered the SmartMotor s I O Functions which are extremely flexible and provide a variety of digital and analog input and output capability Each I O point has a corresponding pre assigned variable name within the programming environment and can be read from or written to by placing it on
14. ddiqui for his guidance on implementing testing structures Professor Everbach for helping me understand the previous codes developed for the motor at the start of the semester Grant Smith and James Johnson for their patience guidance and support in the machine shop a place I now no longer dread Table of Contents Abstract Acknowledgement Table of Contents 1 Introduction 1 10bjective and Motivation 1 2 Key Components ofthe Shake Table 2 Construction 3 Theory 3 1 Derivation Transfer function of a Second Order Underdamped System in time and Laplace domain 3 2 Theoretical calculation finding the resonant frequency of a lumped mass system 3 3 Procedures for frequency domain analysis 3 4 Theoretical calculation finding the resonant frequencies of a two story stick mass building 4 Animatics SmartMotor 4 1 Communication 4 2 Programming Notes 4 3 Motion Commands 4 4 I O Ports 4 5 Analog Velocity 5 Data Acquisition DAQ 6 Calibration 7 Testing Structures 7 1 Lumped Mass System 7 2 Two story Stick Mass Model 8 Results 8 1 The Big Picture 8 2 Comparing command input with measure velocity output 8 3 Obtaining Hplate 8 4 Obtaining Hstructure 8 5 Discussion on the significance of Hplate 8 6 Using Matlab s LSIM 8 7 Addition of a dashpotto enhance the El Centro simulation results 8 8 FFT analysis of two multistoried stick mass structure 9 References Appendices Appendix A Matlab script file Analog Velocity Appendix B Matl
15. ep pause 0 1 disp yyy elt clear clear global dag createSession ni Rate 100 No duration in seconds required with analog output ow zz addAnalogOutputChannel w Dev3 ao0 Voltage addAnalogInputChannel w Dev3 ai0 Voltage addAnalogInputChannel w Dev3 ail Voltage de w Channels 1 Range 10 10 w Channels 2 Range 10 10 load telcentrino mat a el a Ott 2 40 IS v el cumsum a el mean a el dt x el cumsum v el mean v el dt oo Impulse S ones 10 1 0 ss ones 10 1 0 092 r ones 400 1 0 v s 22 r oo dp oe oo 9 Sinusoid time 0 dt 10 ssl 04 sin l time ss2 0d siar 2 ime 42 ss3 04 sin 3 time Ss4 104 g11 5 time ssh B3XxO4 sim Y time z v s ssi ss2 Ss3 ss4 Ssb r 3 Gis 0 04 1 1 sqrt 1 z 2 exp z w x sin w sqrt 1 z y kees G w 2 exp x w z cosh x w z 2 Dxt14213 oe z einh eew z 2 1j 01 2 j z 2 ly 1723 El Centres y SE GITL TOQU oe oe Pasemetion 2 47 v 0 0385 queueOutputData w Basemotion S serial COM3 creates serial port object for motor eet all the properties of the port s BaudRate 9600 s DataBits 8 s Terminator SREP RT S RequestToSend MONET s FlowControl software s DataTerminalReady off s parity none open e connect serial port object to the moter disp Connected to m
16. es By calculating the slope through extracting two data points we found the actual velocity to be 2 1 cm s The voltage fed into the DAQ was 3 04V Taking into account the 2 5V offset in our analog velocity code we calculated the conversion ratio from SI units to SmartMotor voltages 1 Jd calibration rato Test Run Linear Motion 2 1cm s T T T T position voltage 0 1 1 fi fi 1 L 1 0 5 1 1 5 2 25 3 3 5 time E Figure 20 Test Run for calibration Thus in our Matlab code we added the following line for unit conversion motion 2 5 DH 0885 where v is the vertical array of analog velocity of the waveform to be queued into the DAQ system 25 7 Testing Structures 7 1 Lumped Mass System The lumped mass system used is shown in Figure 21 Figure 21 Lumped mass structure used for testing The second moment of area is I bh lt int The elastic modulus of aluminum is E 10 10 psi the weight of the aluminum block with the attached SparkFun accelerometer and Arduino is W 1 905 lbs the length of the cantilever beam is L 26 The spring constant of the lumped mass system is k 278 Theresonant frequency is oy E 1 19 Hz 7 2 Two story Stick Mass Model The modelis shown in Figure 22 26 Figure 22 Two story Stick Mass System The columns are aluminum all threads with a diameter of 0 186 The two stories are made from 1 8 plywood and we
17. figuration IV Open alMotors _ Com39600AS232Ch08NT CG MN Coma RS232 9600 bps MP PT 0 VT 60000 ADT 100 G END Send G Motort Com3 5 0 4 7 gf Ethernet usa W CAN Channel 0 125000 bps Figure 16 SmartMotor Interface example screenshot In configuration the connection between the motor to the PC com port is established through Com3 The terminal window acts like Matlab s command window Commands can be given to control the motor B EJ allows us to upload a script file onto SmartMotor s hard drive Only one program can be stored at a time With each upload the previous version is erased and replaced Prior to moving the motor the over travel limits and fault bits must be cleared To Clear the over travel limits the EIGN command Enable Inputs as General Use is used The historical fault bits are cleared by using the ZS command The following command lines must therefore be included in all code or typed directly into the SMI terminal before the motor can move EIGN 2 EIGN 3 ZS After these commands are set the LED light on the SmartMotor will change from solid red to flashing green indicating that the motor is ready and on standby to move 4 3 Motion Commands The Animatics SmartMotor can move using different modes including a torque mode velocity mode absolute position mode and relative position mode A basic absolute move example is shown below MP VT 100000 ADT 1000 PT 20000 G 19 The velo
18. he I O functions in our code we must first deactivate default on board I O functions To take inputs the following command is useful INA V1 exp where exp is the I O Bit Number or Status Word Number This can be passed in from a variable 4 5 Analog Velocity From the User Manual we found the following sample code based upon which we were able to achieve motion based on an analog output of velocity 21 EIGN W 0 Disable hardware limits KP 3020 Increase stiffness from default KD 10010 Increase damping from default F Activate new tuning parameters ADT 100 Set maximum acceleration MV Set to Mode Velocity d 10 Analog dead band 5000 full scale 0 2500 Offset to allow negative swings m 40 Multiplier for speed w 10 Time delay between reads b 0 Seed b c10 Label to create infinite loop a INA V1 3 o x a b Take analog 5Volt FS reading Set x to determine change in input IF x gt d Check if change beyond deadband VT b m Multipiier tor appropriate speed G Initiate new velocity ELSEIF x d Check if change beyond deadband VT b m Multiplier for appropriate speed G Initiate new velocity ENDIF End If statement b a Update b for prevention of hunting WAIT w Pause before next read GOTO10 Loop back to label END Obligatory END never reached A deadband is an interval of a signal domain or band where no action occurs the system is dead For our purpose however the
19. igh 1 lb They are equally spaced along the column with each story 8 in height At the support the aluminum square plate had 14 tapped holes which was bigger than the diameter of the selected all thread A special nut was made on the lathe to fit both the tapped hole on the aluminum square plate as well as the all thread 10 32 nuts were used to stabilize the two stories so they wouldn t move along the column even during high frequency motion Details of the special nut and the 10 32 s are shown in Figure 23 Figure 23 Connections details of the two story stick mass system As shown in the Theory section the resonant frequencies are w 8 69Hz w 20 59 Hz The three story stick mass system is identical in nature and therefore is notshown in this report 27 8 Result 8 1 The Big Picture The system outline of the shake is shown below in Figure 24 Transfer Functions Construction d Matlab E Motor elo rs Figure 24 The Big Picture 8 2 Comparing command input with measure velocity output After we calibrated the SmartMotor s analog velocity program we experimented with a variety of input We queued an impulse sinusoid and the El Centro into the A D converter and recorded the base plate s position using the LVDT We then plotted both the differentiated LVDT data to obtain velocity with the input velocity against time Comparing command input with measured velocity For an impulse the Matlab com
20. in curve fit for Hplate Figure 30 Response to Impulse Figure 31 Curve fit for Hstructure Figure 32 Content of El Centro captured by Hplate Figure 33 Theoretical and measured output for an impulse Figure 34 Theoretical and measured output for a sinusoid Figure 35 Theoretical and measured output for El Centro Figure 36 Response to an impulse for more heavily damped system Figure 37 Theoretical and measured output for El Centro for heavily damped system Figure 38 Time and frequency domain responses to an impulse of a 2 story structure Figure 39 Time and frequency domain responses to an impulse of a 3 story structure 1 Introduction 1 1 Objectives and motivation A bench scale shake table can be a meaningful to study a model s response to a variety of dynamic loading including earthquakes which are arbitrary waveforms The study of earthquakes is important because earthquakes cause billions of dollars in damage every year around the world and tens of thousands of deaths and injuries In the United States alone around 1 200 deaths have been recorded since 1900 Many more fatalities occurred in earthquakes elsewhere Buildings are normally adequately designed for gravity and vertical loads Thus lateral movements provided by the uniaxial shake table can introduce bending to the columns and torsion if the center of mass and center of resistance are offset Testing of models of actual buildings and building prototypes is one me
21. input are waveforms that could be considered digital signal is constantly changing and thus setting the deadband to 10 is detrimental to obtaining consistency We changed d to 0 To achieve a maximum speed of around0 1 m s we experimented with the multiplier value and changed it from 40 to 95 The wait time is decreased to 5 w 5 toaccommodate higher sampling rates The most important line of this sample code is the following a INV V 1 3 o INV V1 3 means we are retrieving input from I O point 3 or pin number 4 Voltage is also scaled in millivolts where 3456 would be 3 456V Setting the offset to 2500 o 2500 means the 2 5V is the zero velocity voltage This code causes the SmartMotor s velocity totrack an analog input 22 5 Data Acquisition System Data acquisition plays an important role in connecting Matlab and the SmartMotor It queues the desired waveform sent from Matlab in an array format While the SmartMotor is activated its code retrieves the line of data to use as the velocity input The basic setup is shown in Figure 19 Figure 19 Setup of the DAQ system As shown in Fig the analog output portion of the A D converter is then connected to Pin 4 and ground ofthe SmartMotor s I O port On the analog input section of the convertor two connections go the accelerometer and LVDT s power source respectively It is largely realized through Matlab s data acquisition toolbox With the toolbox we c
22. ion Ey B 6 Using Matlab s LSIM 34 1s im simulates the time response of continuous or discrete linear systems to arbitrary inputs lsim sys u t producesa plot of the time response ofthe dynamic system model sys to the input history t u The vector t specifies the time samples for the and consists of regularly spaced time samples The input u is an array having as many rows astime samples and as many columns as system inputs For instance ifsys isaSISO system then u is a t by 1 vector If sys has three inputs then u is at by 3 array The model sys can be continuous or discrete SISO or MIMO Since our system is a discrete SISO both u and t have only one column After obtaining the structure s transfer function Ay yctre we obtained the theoretical acceleration output for threetypes of command input impulse sinusoid and El Centro to see how well they compare For the Impulse input the 1 5imresultis in blue while the measured acceleration by the accelerometer is in green Impulse nek h E D r Dir f D r al 0 44 4 ei y 1 1 f 1 1 1 1 L L 4 0 1 2 3 4 5 p 7 Figure 33 Theoretical and measured output for an impulse 3M For the Sinusoidal input the 1sim result is in blue while the measured acceleration bythe accelerometer is in green 35 I cl 05 1 15 2 25 3 35 4 45 5 S Figure 34 Theoretical
23. mand is ones 10 1 0 092 Since the sample rate of the created DAQ session is 100 Hz our input impulse has a duration of 10 100 0 1s at 092 m s At other times the input velocity is zero The described waveform is depicted precisely by blue in Figure 28 Impulse base motion Na HU LAT Ag M M M al MUT d Vir FEN TA II OI OU II WII D D 5 1 1 5 2 time Figure 25 Command input and recorded input for an impulse The more noisy green data is the position data measured by the LVDT obtained from our DAQ s second analog input channel the first input channel is reserved for the acceleration data which is mentioned later Note that there is an approximately 70 ms delay at the base plate s first sign of acceleration At this point we can not evaluate the significance of this time lag without quantifying the transfer function between the motor and the base plate We will now refer to that transfer function as Hye For a sinusoid the Matlab command is ssl 062 sin 18 time Like with the impulse we plotted the input waveform along with the differentiated LVDT data to obtain Figure 29 Sinusoid 0 1 0 08 0 06 F 0 04 F base motion time Figure 26 Command input and recorded input for a sinusoid There is an apparent time lag but otherwise the two match perfectly in terms of peakamplitude and period For the El Centro we obtained the El Centro north south acceleration data and then perfor
24. med Euler integration to get velocity sample code attached in Appendix B The Matlab command to create the input waveform is v el 1 500 We then plotted the input waveform along with the differentiated LVDT data to obtain Figure 30 base motion 0 5 1 1 5 2 25 3 3 5 4 45 5 5 5 time Figure 27 Command input and recorded input for the El Centro 8 3 Obtaining Hye To gain a close look we zoom in on the first 100 ms of the impulse input LVDT result Figure Di 0 08 S z 0 06 E 2 s 004 0 02 Adi m L 1 L D 0 1 0 2 0 3 0 4 0 5 time Figure 28 Zoom in of Figure 27 31 Create arrays for the time and base plate position and use Matlab s curve fitting toolbox T 01r 008 005 H E 0 04 baseVhp vs timehp si Plate Transfer Function 0 02 1 D 1 1 L L 1 L 0 02 0 04 0 06 0 08 0 1 0 12 timehp Figure 29 cftool to obtain curve fit for Hplate The curve fitting results are as follows 1 wp 52 7 0 60 T 0 03 R 0 81 Wo Note that Hate 8 time constant c is only 0 03 seconds which also translates to a cutoff frequency of around 5 Hz 8 4 Obtaining H structure To obtain the transfer function of the lumped mass structure we applied the same impulse and obtained the following acceleration output results 32 structure acceleration o 0 5 time Figure 30 Response to Impulse Create arrays for the
25. otor remove hardware limits set velocity and acceleration initialize EIGN 2 EIGN 3 Z28 fprintf s initialize go RUN tpristf ie ge Use the startForeground function to start the analog Grau s operation and block MATLAB execution until all data is generated data w startForeground 9 After excution of analog output tell motor to rest command MV VT 0 ADT 200 G END 43 fprintf s command felose cs sp close serial port cleanly hold off pause 1 time l size data l 1 w Rate baseA data 1 98 1 baseP data 2 0 01 baseV diff baseP diff time baseV endt 1 baseV end baseV smooth baseV plot time v time baseV xlabel time ylabel base motion legend Command Output o subplot 2 1 1 plot time v xlabel time ylabel Command SSUBbBLOTIZ 142 plot time baseA xlabel time ylabel Output figure plot time baseA figure plot time data 1 xlabel time ylabel aceel voltage beep giel Losst 44 Appendix B FFT code load two story Subp Lot Zell plot time baseA xlabel Time ylabel baseA subplot 2 1 2 bFET fftshift fft baseA dt time 2 time 1 f 1 length time plot f abs bFFT xlabel Freq Hz ylabel abs fft baseA length baseA 2 dt length baseA 45
26. storyis given by 12 amp 4 EC L 163 Ib in and the individual values forthe sIFBireua indica are thus k 2k 2kz 1632 The equations of motionforthe system are obtained my ky Kay y O my t ky y 0 In the usual manner these equations of motion are solved for free vibration by substituting Y G sin wut c Me Ba ee Cue MIER E ordi otii a Jj 8 w sin wt a The expansion of the determinant of the matrix formation of the equations of motion givesa quadratic equationin w namely mi Smkw K 0 Substitution of the numeric values just obtained mi S3mkw kK 0 Therefore the natural frequencies ofthe structure are 16 w 869Hz w 20 59 Hz Recall the FFT analysis of the data obtained via an impulse on the two story building 300 nN ce e abs fft baseA XU 40 a wm 10 8 0 a a a 55 ren Hz Figure 14 Frequency domain response to an impulse of a 2 story structure The actual natural frequencies are Ww 6 1Hz w 19 2Hz 17 4 Animatics SmartMotor 4 1 Communication The Animatics SmartMotor is electrically driven which enables the creation of arbitrary waveforms as opposed to only hardcoding waveforms using discrete data points of velocity position etc The motor communicates through a PC serial port via a RS232 port A single connection to the 7 Pin Combo D Sub Power amp 1 0 provides power and communication to the motor using a CB
27. thod that is useful in understanding the forces at work Models built of plywood and aluminum all thread allow us to see the seismic behavior of a structure and to understand how the period of an earthquake if it is resonant with a building period will cause the most damage or even collapse The shake table is a device that simulates a dynamic loading or a seismic event It can also be used to create fictional worst case scenarios or resonant frequencies In computer controlled shake tables a computer program generates a signal and a digital signal is sent to a digital analog converter which sends a voltage to the amplifier The amplifier amplifies the voltage and sends it to the shaker platform to which the model is attached The constructed shake table is a one degree of motion shake table meaning that it will move only in one lateral direction A model on a shake table with the same stiffness or resonant frequency as the prototype building will act in a way similar to that of the actual building 1 2 Key components ofthe shake table Motor Animatics SmartMotor SM23165DT The SM23165DT motor with HLD60 Internal Rollers and a stroke length of 400 mm was chosen The carriage shown in Figure 1 moves through a belt driven system classified as a harmonic linear drive POWER amp DATA CONNECTION DIMENSIONS ARE IN INCHES Figure 1 Animatics SmartMotor SM23165DT Data Acquisition DAQ system For our DAQ we used a National Instr
28. uld only run a five second waveform ofthe El Centro First we found the transfer function for our more heavily damped system by applying an impulse and measuring the acceleration response using the accelerometer 0 2 1 structure acceleration 2r 4 4r E Det 4 W I 1 15 2 2 5 3 35 4 45 time Figure 36 Response to an impulse for more heavily damped system We then imported this data into Matlab s curve fitting toolbox and found the following results 7 267 0 08 1 1 7 R 0 38 0 Note the increased damping coefficient and the significantly smaller time constant We then coded this transfer function into Matlab s workspace and performed 1sim on just five seconds of El Centro 37 SS 1 15 3 25 3 35 45 5 Figure 37 Theoretical and measured output for El Centro for heavily damped system Note that shifting only started after the third second mark which shows an improvement from the previous system 8 8 FFT analysis oftwo multistoried stick mass structure We constructed a twos story stick mass model as shown in the Theory section Each floor was sawed from 1 8 plywood at the woodshop then drilled with a No 12 drill at the four corners which are spaced to match the four aluminum square plates on the four corners of the shake table 32 thread inch aluminum all threads with a 186 diameter was used as the columns Connection details are also shown in the Theory
29. ument PCI 6221 shown below Figures 2 and 3 NI PCI 6221 and its placement inside an A D convertor We like it for its Correlated DIO which enables digital and analog functions to be synchronized with hardware timed precision and reduced the need to write an internal timer inside Matlab LVDT Linear Variable Differential Transformer The following figure depicts our LVDT Figure 4 Linear Variable Differential Transformer LVDT Linear variable differential transformer LVDT is a type of electrical transformer used for measuring linear displacement LVDT s are inherently frictionless and converts a position or linear displacement from a mechanical reference into a proportional electrical signal containing phase and amplitude through induction current The calibration ratio is 1V 10mm Accelerometer The following figure depicts out accelerometer Figure 5 PCB 352B70 Accelerometer The accelerometer is a device that measures proper acceleration g force The 98 1m ge calibration ratio for our specific modelis 1V 2 Construction Construction of the shake table consisted of several key components and a few pieces of add ons to support future additions to the shake table The main frame was first erected using Unistrut pieces Then four pieces of cable hung from the top frame pieces to support the base plate which is a piece of plywood The base plate is then connected to the motor Holders for the LVD
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