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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

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ASSIGNMENT 4 PROBLEM 2 35% Support A Hard drive head is subjected to a random base excitation in y direction. The frequency range is Hz 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. hd_head.SLDPRT

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Jon 110% Richard 90% ASSIGNMENT 4 PROBLEM 2 35%

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time [s] acceleration [G] 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 = 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 235%

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Required input: Enter input filename:r_vib1.txt Specify Input file type:acceleration Select Input UnitG Select Output Band Typeconstant spectral bandwidth Select the number of samples per segment64 Mean removal yes Window typeHanning (see below for explanation) Input to program poweri.exe required to generate Acceleration Power Spectral Density curve from acceleration time history. See also the next slide. Rectangular windows works well with non- stationary data, ideally with quiet periods before and after the main events Hanning window is recommended for stationary data ASSIGNMENT 4 PROBLEM 235%

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Input to program poweri.exe required to generate acceleration power spectral density curve from acceleration time history. ASSIGNMENT 2 PROBLEM 4 35%

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. 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%

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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 235%

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time [s] frequency [Hz] Number of samples per segment: 2048 Number of samples per segment: 512 Number of samples per segment: 128 Time history Acceleration PSD curves produced using different numbers of bins (frequency ranges). The area under the curve is approximately the same for all curves. ASSIGNMENT 4 PROBLEM 2 35%

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Acceleration time history data PSD generator poweri.exe Excel r_vib1.txt Acceleration PSD SW Random Vibration Summary of PSD curve generation Time domainFrequency domain You may download PSD generator from: ASSIGNMENT 4 PROBLEM 2 35%

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Mode 1390Hz Mode 21182Hz Mode 31272Hz Results of modal analysis Mode 42005Hz ASSIGNMENT 4 PROBLEM 2 35%

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RMS displacement result ASSIGNMENT 4 PROBLEM 2 35%

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14 RESPONSE SPECTRUM METHOD

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15 Ground motion histories (North South directions) for El Centro (1940) “How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja 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.

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16 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. Special analysis method called Response Spectrum Method has been developed to analyze long duration non stationary processes like the earthquake or pyrotechnic shock events. CALIFORNIA ZONE 2 CALIFORNIA ZONE 4 G G time [s] SEISMIC RECORDS

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17 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. RESPONSE SPECTRA “How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja

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18 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. RESPONSE SPECTRA “How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja

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19 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

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20 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. CONCEPT OF THE RESPONSE SPECTRUM

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21 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. GENERATING RESPONSE SPECTRUM

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22 GENERATING RESPONSE SPECTRUM Response of SDOF oscillators to half-sine pulse. Courtesy of Vibrationdata time Base excitation

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23 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. Step 3 Response spectrum curve ready to be used as input in the Dynamic Shock analysis. It is usually plotted in log scale. acceleration Frequency [Hz] Maximum acceleration Base input GENERATING RESPONSE SPECTRUM

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24 USING RESPONSE SPECTRUM FOR FEA 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. Frequency [Hz] Maximum acceleration

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25 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. USING RESPONSE SPECTRUM FOR FEA

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26 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. USING RESPONSE SPECTRUM FOR FEA

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27 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. SRSSThe 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: Absolute Sum The sum of the absolute values of the contributions of each mode Tends to overestimate the response and can be very conservative. USING RESPONSE SPECTRUM FOR FEA

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28 GENERATING RESPONSE SPECTRUM CURVE Acceleration time history has 6145 data points This is input to Shock Response curve generator. vertegi4.txt

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29 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. 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. 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. The amplification factor relates to using the formula below. Is % of critical damping: Response spectrum generator; Courtesy of Vibrationdata Input file in ASCII format containing acceleration time history. The amplification factor 25 corresponds to = 2 % The amplification factor 10 corresponds to = 5 % vertegi4 mm.txt GENERATING RESPONSE SPECTRUM CURVE

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30 secondsm/s 2 Input Time history of an earthquake Maximum positive and maximum negative acceleration (6145 data points) as function of time Output from SRS generator (after processing in Excel) Max. absolute acceleration (49 data points) as function of frequency for 5% damping frequencymm/s 2 m/s 2 GENERATING RESPONSE SPECTRUM CURVE

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31 frequency Maximum absolute acceleration mm/s 2 Generated Shock Response Spectrum curve GENERATING RESPONSE SPECTRUM CURVE

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32 Acceleration time history data SRS generator SRS.exe Excel vertegi4.txt SRS.txt SolidWorks Summary of SRS curve generation Time domain Frequency domain GENERATING RESPONSE SPECTRUM CURVE

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33 SEISMIC ANALYSIS Example of a model

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