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ASSIGNMENT 4 Problem 1 Find displacement and stresses in the crankshaft when engine runs at the first natural frequency of the crankshaft ASSIGNMENT 4 PROBLEM 2 35% Support hd_head.SLDPRT A Hard drive head is subjected to a random base excitation in y direction. The frequency range is 0-2500Hz The acceleration time history is in rvib1.txt. Use poweri.exe to generate acceleration PSD Find RMS and PSD displacement of tip A. What is the probability that displacement exceeds the percentage of RMS displacement given in the next slide? Deliverables: SW model with response plot and study ready to run. ASSIGNMENT 4 PROBLEM 2 35% Jon Richard 110% 90% ASSIGNMENT 4 PROBLEM 2 35% 2.0 1.5 1.0 0.5 0.0 0 0.5 1 -0.5 1.5 2 time [s] -1.0 -1.5 -2.0 Acceleration time history in r_vib1.txt This acceleration time history contains 7680 data obtained during 1.5 s with 5120 samples per second. Overall G RMS = 0.44. The amount of data makes it impractical to run a dynamic time analysis. Assuming that this is a stationary random process we will use this acceleration time history to calculate the Acceleration Power Spectral Density curve. Acceleration time history is stored in r_vib1.txt ASSIGNMENT 4 PROBLEM 2 35% Required input: Enter input filename: r_vib1.txt Specify Input file type: acceleration Select Input Unit G Select Output Band Type constant spectral bandwidth Select the number of samples per segment 64 Mean removal yes Window type Hanning (see below for explanation) 2.0 1.5 1.0 0.5 0.0 0 0.5 1 1.5 2 -0.5 -1.0 -1.5 -2.0 Rectangular windows works well with nonstationary data, ideally with quiet periods before and after the main events Hanning window is recommended for stationary data Input to program poweri.exe required to generate Acceleration Power Spectral Density curve from acceleration time history. See also the next slide. ASSIGNMENT 2 PROBLEM 4 35% Input to program poweri.exe required to generate acceleration power spectral density curve from acceleration time history. ASSIGNMENT 4 PROBLEM 2 35% Output files from program poweri.exe. File A_G2Hz.psd must now be processed in Excel. Once parsed into columns, paste it into SW ASSIGNMENT 4 PROBLEM 2 35% 30000 25000 20000 15000 10000 5000 0 0 500 1000 1500 2000 2500 frequency [Hz] Acceleration PSD curve we’ll use as input to FEA. Plot has been created in Excel using “smooth curve” option. ASSIGNMENT 4 PROBLEM 2 35% G RMS Hz 2 0.0016 Δf = 2.5Hz 0.0014 0.0012 Number of samples per segment: 2048 0.0010 0.0008 0.0006 0.0004 frequency [Hz] 0.0002 0.0000 G RMS Hz 2.0 1.5 2 0.0016 0 500 1000 1500 0.0012 0.5 0.0008 0.5 1 -0.5 1.5 0.0006 2 time [s] 0.0004 -1.0 0.0002 -1.5 0.0000 frequency [Hz] 0 -2.0 Time history G RMS Hz 2 500 1000 1500 0.0016 2000 2500 3000 Δf = 40Hz 0.0014 0.0012 Number of samples per segment: 128 0.0010 Acceleration PSD curves produced using different numbers of bins (frequency ranges). The area under the curve is approximately the same for all curves. 3000 Number of samples per segment: 512 0.0010 0 2500 Δf = 10Hz 0.0014 1.0 0.0 2000 0.0008 0.0006 0.0004 0.0002 frequency [Hz] 0.0000 0 500 1000 1500 2000 2500 3000 ASSIGNMENT 4 PROBLEM 2 35% Acceleration time PSD generator history data poweri.exe Excel SW Random Vibration 2.0 1.5 1.0 0.5 0.0 0 0.5 1 1.5 2 -0.5 -1.0 -1.5 -2.0 Acceleration PSD r_vib1.txt Time domain Frequency domain You may download PSD generator from: http://www.designgenerator.com/transfer/Ariadne.zip Summary of PSD curve generation ASSIGNMENT 4 PROBLEM 2 35% Mode 1 390Hz Mode 3 1272Hz Mode 2 1182Hz Mode 4 2005Hz Results of modal analysis ASSIGNMENT 4 PROBLEM 2 35% RMS displacement result RMS displacement value investigated displacement value probability that absolute displacement will exceed that displacement is probability that absolute displacement will be less than that displacement 8.80E-02 1.00E-01 26% 74% RESPONSE SPECTRUM METHOD 14 SEISMIC RECORDS The ground displacements recorded by a seismometer located directly above a fault that ruptured during the 1985 Mw = 8.1, Michaocan, Mexico earthquake. Ground motion histories (North South directions) for El Centro (1940) “How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja 15 SEISMIC RECORDS CALIFORNIA ZONE 2 G The earthquake is a non-stationary process and the methods of Random Vibrations can not be used. Here, each graph contains 6145 data points Dynamic Time analysis is impractical. time [s] Special analysis method called Response Spectrum Method has been developed to analyze long duration non stationary processes like the CALIFORNIA ZONE 4 earthquake or pyrotechnic shock events. G time [s] 16 RESPONSE SPECTRA Each data point on the response spectrum curve represents the peak response from a time history analysis of the earthquake applied to 1-DOF oscillator system. The ordinate defines the natural period of the oscillator, and the abscissa the peak response. Repeating the procedure for a great many frequencies defines a continuous curve for an assumed level of damping. The figure shows the response of 2% damped system tuned to 1Hz and 2Hz and the transfer of the peak calculated responses to the response spectrum graph. “How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja 17 RESPONSE SPECTRA Response spectra are typically presented for a damping ratio of 5% that is considered to be typical for buildings. This does not mean that a damping ration of 5% is appropriate for any given analysis. When response spectra are used as an input, a single smoothed spectrum derived from several events or several response spectra from different events are used. The use of smoothed response spectra implies the use of several earthquake records. A given shock response spectrum does not have a unique corresponding time history. “How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja 18 RESPONSE SPECTRA There are a number of approximate methods for scaling 5% damped spectra for other damping levels. Given two consistent spectral of different damping values, the spectrum for an intermediate damping values can be calculated based on interpolation between spectral amplitude and natural logarithm of damping ratio. “How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja 19 CONCEPT OF THE RESPONSE SPECTRUM The concept of the Response Spectrum Method is based on the following observations: 1 A system vibrating in resonance can be described as a single degree of freedom harmonic oscillator system with certain equivalent mass, equivalent stiffness and damping. 2 A response of a system with more that one resonant frequency can be represented as a combination of responses of harmonic oscillators, each harmonic oscillator corresponding to particular resonant frequency. This is the basis of the modal superposition method. 3 If the excitation frequency is equal to one of structure resonance frequencies, then the system response to that excitation is controlled only by system damping. Mass does not matter, stiffness does not matter, they have no impact on system’s response. The only thing controlling your system response is its damping! Imagine two vastly different systems: A harmonic oscillator and a bridge. Those two systems have two important things in common: the natural frequency and damping. Now imagine that both the harmonic oscillator and the bridge are excited by the same excitation that happens to have the same frequency as the resonance frequency of both the harmonic oscillator and the bridge. The response (e.g. the maximum displacement) will be the same for the harmonic oscillator and the bridge! 4 Say you agree that structure dynamic response can be adequately described by using the modal superposition method based on four modes. Then rather than studying the response of the actual structure, you can study the response of four harmonic oscillators with natural frequencies corresponding to those four modes and damping the same as damping of the structure you study. In particular, if you study the structure response to seismic excitation, then rather than testing the actual structure, you can just subject those four oscillators to the earthquake acceleration time history and record e.g. the maximum acceleration of each oscillator as a function of oscillator’s frequency. This way you will build the response spectrum curve. 5 If the natural frequencies of the examined structure fall “in-between” frequencies of your oscillators, the structure response can be interpolated. So if you know structure natural frequencies and you also know the acceleration response spectrum curve you have just constructed by examining the response on harmonic oscillators, you can find out by interpolation, the maximum acceleration of the structure corresponding to each mode. Double integration will give you the maximum displacement. 6 Notice that the information of interactions between modes has been lost in the above process. Therefore, it must be re-built using the SRSS or the Absolute Sum method, as described further. 20 GENERATING RESPONSE SPECTRUM The Response Spectrum model is a series of Single Degree of Freedom (SDOF) harmonic oscillators with different natural frequencies subjected to a particular earthquake ground motion. Each SDOF system has a unique time history response to a given base input. The shock response spectrum is the peak absolute acceleration of each SDOF oscillator to the history base input. The SDOF oscillators are arranged in order of ascending natural frequency. The oscillator on the far left is the most compliant one; and the oscillator on the far right is the most stiff system. We subjected the oscillators to a half sine shock. Note that: 1. The most compliant system is "isolated" from the shock pulse. 2. The stiffer systems almost follow the input pulse exactly. They have very little reverberation. 3. Some of the middle systems actually reach the highest amplitude due to a transient resonance effect. 21 Base excitation GENERATING RESPONSE SPECTRUM 1 0 0 50 100 150 200 250 300 350 400 time Response of SDOF oscillators to half-sine pulse. Courtesy of Vibrationdata 22 2.0 vertegi4 yg GENERATING RESPONSE SPECTRUM 1.5 G [ m / s2 ] 1.0 Base input 0.5 0.0 yg -0.5 -1.0 -1.5 -2.0 0 10 20 time [ s ] 30 40 time Step 1 A synthesized waveform; typical acceleration time history Step 2 A series of SDOF harmonic oscillators is subjected to the excitation as defined by the waveform from step 1. 10 Maximum acceleration 1 0.1 0.01 0.001 0.1 1 10 100 Frequency [Hz] Step 3 Response spectrum curve ready to be used as input in the Dynamic Shock analysis. It is usually plotted in log scale. 23 USING RESPONSE SPECTRUM FOR FEA Maximum acceleration 10 1 0.1 0.01 0.001 0.1 1 10 100 Frequency [Hz] Response spectrum methods is ubiquitous in earthquake engineering, with very widespread support in general purpose FE systems, and almost universal application. It is important to remember that it is a response spectrum: i.e. once you know the natural frequencies of the structure, in simple cases domminated by a single mode, the respoinse can be simply read from a graph. FE implementation of the procedure merely automate the interpolation, and add post-processing to combine effects of mutliple modes and shaking directions. 24 USING RESPONSE SPECTRUM FOR FEA Step 1 Modal analysis of the analyzed structure determines modes, frequencies and mass participation factors. Structure is thus represented by a series of SDOF oscillators. Step 2 The acceleration response for each mode (for each SDOF oscillator) is interpolated using the response spectrum curve (at the resonant frequency of corresponding mode). Step 3 The complete response of each mode is found based on the acceleration response found in step 2 Step 8 An assumption is made to find the combined response of all SDOF oscillators because the time at which oscillator (each mode) reached its peak value was discarded when the response was calculated. There is no precise way of combining the modes to find the total response. 25 USING RESPONSE SPECTRUM FOR FEA Full model shows 40.6% total mass participation factor based on 20 modes. Truncated model shows 56.5% total mass participation factor based on the same 20 modes. Note on mass participation: Most analysis standards call for the total mass participation factor of 80% or more. However, this requirement is often difficult to satisfy especially if, due to the nature of supports, only a small portion of structure participates in vibrations. 26 USING RESPONSE SPECTRUM FOR FEA Modal combination methods An assumption has to be made to find the resultant structural response since the time at which the mode reached its peak value was discarded when the response was calculated. There is no precise way of combining the modes to find the total response. SRSS The Square Root of the Sum of the Squares Tends to underestimate the response if modes are spaced out closely. Closely spaced modes are typical for structures with high torsional stiffness and in structures with identical members. Modes are closely spaced if, for example, there are repetitive elements in the structure. Modes are considered closely spaced if: CF fi fi 1 0.1 fi 1 Absolute Sum The sum of the absolute values of the contributions of each mode Tends to overestimate the response and can be very conservative. 27 GENERATING RESPONSE SPECTRUM CURVE 2 1.5 1 0.5 0 0 5 10 15 20 25 30 35 -0.5 -1 -1.5 -2 vertegi4.txt Acceleration time history has 6145 data points This is input to Shock Response curve generator. 28 GENERATING RESPONSE SPECTRUM CURVE vertegi4 mm.txt Input file in ASCII format containing acceleration time history. The starting frequency may be defined by a testing standard or some other convention. Consideration should also be given to the natural frequency of the structure. The starting frequency defines the frequency of the lowest frequency oscillator. The amplification factor relates to using the formula below. Is % of critical damping: Q 1 2 ζ The amplification factor 25 corresponds to = 2 % The amplification factor 10 corresponds to = 5 % Octave spacing is similar to the frequency spacing on a piano keyboard. Each successive key is 1/12 octave higher than the previous key. The frequency spacing is sometimes chosen by an established convention, such as some testing standard or by experience. The frequency spacing effectively defines the number of oscillators. Primary shock is the response that occurs during the shock pulse. Residual shock pulse is the response that occurs after the shock pulse. These terms mainly apply to a well-defined shock pulse such as a half-sine pulse. The earthquake shock, however, may taper off in a very gradual manner. Thus a somewhat arbitrary point must be chosen as the "end" of the shock. Response spectrum generator; Courtesy of Vibrationdata 29 GENERATING RESPONSE SPECTRUM CURVE seconds 16.93 16.935 16.94 16.945 16.95 16.955 16.96 16.965 16.97 16.975 16.98 16.985 16.99 16.995 17 17.005 17.01 17.015 17.02 17.025 17.03 17.035 17.04 17.045 17.05 17.055 17.06 17.065 17.07 17.075 m/s2 -825.69 -530.89 -248.48 -1.18 244.61 477.66 613.73 775.75 980.39 1151.4 1334.03 1536.91 1740.5 1970.92 2206.33 2355.73 2508.93 2662.86 2734.67 2756.32 2739.77 2688.63 2619.15 2552.68 2540.24 2545.58 2595.85 2786.82 3067.25 3432.76 m/s2 -0.08257 -0.05309 -0.02485 -0.00012 0.024461 0.047766 0.061373 0.077575 0.098039 0.11514 0.133403 0.153691 0.17405 0.197092 0.220633 0.235573 0.250893 0.266286 0.273467 0.275632 0.273977 0.268863 0.261915 0.255268 0.254024 0.254558 0.259585 0.278682 0.306725 0.343276 Input Time history of an earthquake Maximum positive and maximum negative acceleration (6145 data points) as function of time frequency mm/s2 0.1000 94.52 0.1122 111.40 0.1260 131.80 0.1414 156.80 0.1587 187.50 0.1782 225.90 0.2000 273.90 0.2245 334.70 0.2520 411.20 0.2828 506.90 0.3175 630.80 0.3564 799.00 0.4000 1013.00 0.4490 1294.00 0.5040 1713.00 0.5657 2244.00 0.6350 3149.00 0.7127 4458.00 0.8000 7013.00 0.8980 13120.00 1.0080 20710.00 1.1310 22560.00 Output from SRS generator (after processing in Excel) Max. absolute acceleration (49 data points) as function of frequency for 5% damping 30 GENERATING RESPONSE SPECTRUM CURVE 60000 Maximum absolute acceleration mm/s2 50000 40000 30000 20000 10000 0 0 1 10 100 frequency Generated Shock Response Spectrum curve 31 GENERATING RESPONSE SPECTRUM CURVE Acceleration time SRS generator history data SRS.exe Excel SolidWorks 2 1.5 1 0.5 0 0 5 10 15 20 25 30 35 -0.5 -1 -1.5 -2 SRS.txt vertegi4.txt Frequency domain Time domain Summary of SRS curve generation 32 SEISMIC ANALYSIS Example of a model 33