All lengths in meters. E = = 7800 kg/m 3

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Charity Sara Quinn
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1 Poblem desciption In this poblem, we apply the component mode synthesis (CMS) technique to a simple beam model All lengths in metes. E = N/m 2 = 7800 kg/m 3 The beam is a fee-fee beam. Fo simplicity, we conside only motions of the beam model in the xy plane. Theefoe each beam node has thee degees of feedom: the x tanslation, the y tanslation and the z otation. Because the beam model that we use has 6 nodes, the total numbe of degees of feedom n is 18. In the CMS method, the degees of feedom ae divided into bounday degees of feedom and inteio degees of feedom. In this poblem, we will assign the degees of feedom at the two end-nodes of the beam model to be bounday degees of feedom. Since thee ae 6 degees of feedom total at the two end-nodes, the numbe of static constaint modes k is 6. We will also set the numbe of fixed inteface dynamic vibation modes q to 3. The esulting size of the educed matices K, M is ( k q) ( k q) 9 9. As pat of the CMS solution, we will also obtain appoximations fo the fist 8 natual fequencies and mode shapes. Fo a eview of the CMS technique, see Section of the ADINA Stuctues Theoy and Modeling guide. In this poblem solution, we demonstate the following topics that have not been pesented in pevious poblems: Setting up a component mode synthesis analysis Befoe you begin Please efe to the Icon Locato Tables chapte of the Pime fo the locations of all of the AUI icons. Please efe to the Hints chapte of the Pime fo useful hints. ADINA R & D, Inc. 60-1

2 This poblem can be solved with the 900 nodes vesion of the ADINA System. Much of the input fo this poblem is stoed in the following file: pob60_1.in. You need to copy this file fom the folde samples\pime into a woking diectoy o folde befoe beginning this analysis. Invoking the AUI and choosing the finite element pogam Invoke the AUI and set the Pogam Module dop-down list to ADINA Stuctues. Defining the model Model definition We have pepaed a batch file (pob60_1.in) that defines the following items: Poblem heading Selection of maste degees of feedom, so that the beam vibates in the xy plane. Geomety points and lines Coss-section Element goup 1, which is a Hemitian beam element goup. The element goup contains 5 equally spaced beam elements. Choose File Open Batch, navigate to the woking diectoy o folde, select the file pob60_1.in and click Open. The gaphics window should look something like this: TIME Y Z X U 1 U 2 3 No bounday conditions assigned 60-2 ADINA Pime

3 Specifying the analysis options fo the CMS method Poblem 60: Component mode synthesis applied to a beam model Set the Analysis Type dop-down list to Fequencies/Modes and click the Analysis Options icon. Set the Solution Method to Component Mode Synthesis (Subspace), set the Numbe of Fequencies/Mode Shapes to 8, set the Numbe of Mode Shapes to be Pinted to 8, check the Allow Rigid Body Mode button and set the # of Fixed Inteface Modes to 3. Then click the Bounday DOFs... button. In the Bounday DOFs fo Component Mode Synthesis dialog box, fill in the table as follows, then click OK twice to close both dialog boxes. Node # Deg. of Feedom 1 X-Tanslation 1 Y-Tanslation 1 Z-Rotation 6 X-Tanslation 6 Y-Tanslation 6 Z-Rotation Geneating the data file, unning ADINA Stuctues Click the Save icon and save the database to file pob60. Click the Data File/Solution icon, set the file name to pob60, make sue that the Run Solution button is checked and click Save. When ADINA Stuctues is finished, the Message Window should look something like this: Initializing... Stage 1 Allocating 32.0 MB of memoy... Initializing... Stage 2 Initial ADINA memoy allocation mb (o 4.0 mw) Stating Solution Pocess... Input phase... Assemblage of linea matices. Calculate and stoe the load vecto Memoy used by the in-coe spase solve.. : 0.2 mb (o 0.0 mw) Total memoy used by the pogam : 32.2 mb (o 4.0 mw) Factoization completed. Stating calculation of constaint modes... Step numbe = 1 step size = E+00 time = E+00 Step numbe = 2 step size = E+00 time = E+00 Step numbe = 3 step size = E+00 time = E+00 Step numbe = 4 step size = E+00 time = E+00 Step numbe = 5 step size = E+00 time = E+00 Step numbe = 6 step size = E+00 time = E+00 Allocated memoy without spase solve.... : 32.0 mb ( 4.0 mw) Memoy used without spase solve : 0.1 mb ( 0.0 mw) Stating calculation of fixed inteface vibation modes... Iteation numbe = 1 Numbe of initially conveged eigenvalues = 0 Iteation numbe = 2 Numbe of initially conveged eigenvalues = 0 ADINA R & D, Inc. 60-3

4 Final numbe of fixed vibation inteface modes = 3 Allocated memoy without spase solve.... : 32.0 mb ( 4.0 mw) Memoy used without spase solve : 0.1 mb ( 0.0 mw) Stating calculation of eigenvalues fo educed system matices Iteation numbe = 1 Numbe of initially conveged eigenvalues = 0 Iteation numbe = 1 Numbe of initially conveged eigenvalues = 0 Iteation numbe = 2 Numbe of initially conveged eigenvalues = 0 Iteation numbe = 3 Numbe of initially conveged eigenvalues = 4 Final numbe of conveged eigenvalues = 9 ADINA Stuctues has pefomed seveal opeations: 1) Computed 6 static steps. Each static step coesponds to one static constaint mode (column in Φ ). Thee ae 6 constaint modes, because thee ae 6 bounday degees of feedom. c 2) Computed the natual fequencies and mode shapes of the fixed inteface dynamic vibation modes (columns in Φ ). Thee ae 3 fixed inteface dynamic vibation modes. 3) Computed the educed matices K, n M. These ae 9 9 matices. 4) Computed appoximations to the fist 8 natual fequencies and mode shapes of the oiginal analysis model. Examining the educed matices The educed matices ae witten to file pob60.out4. Use a text edito to open this file. The fist few lines of the file should look something like the pintout at the top of the next page. These ae the nonzeo enties of matix K. The emaining lines of the file should look something like the pintout at the bottom of the next page. These ae the nonzeo enties of matix M. File pob60.out4, beginning: KXX 1P,5E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E ADINA Pime

5 E E E E E E E E E E E+01 File pob60.out4, end: MXX 1P,5E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+01 Examining the appoximate natual fequencies and mode shapes of the oiginal analysis model Now close all open dialog boxes, set the Pogam Module dop-down list to Post-Pocessing (you can discad all changes), click the Open icon and open pothole file pob60. ADINA R & D, Inc. 60-5

6 Click the Next Solution icon thee times to display mode 4, then click the Modify Mesh Plot icon, click the Element Depiction... button, check the Display Beam Coss Section field, click the Advanced tab, set the # Segments fo Neutal Axis to 4 and click OK twice to close both dialog boxes. When you click the Node Symbols icon and the Show Oiginal Mesh icon, the gaphics window should look something like this: MODE 4, F TIME MODE MAG Y Z X Click the Next Solution and Pevious Solution icons to examine the othe modes. The natual fequencies should be as given in the second column of the following table: Mode numbe Appoximate fequency using CMS method (Hz) Fequency using subspace method (Hz) 1 ~0 ~0 2 ~0 ~0 3 ~0 ~ ADINA Pime

7 Fo compaison, we have computed the fist nine natual fequencies and mode shapes of the oiginal beam model (using the subspace method), and the esults ae in the thid column of the above table. It is seen that modes 1-6 ae in good ageement, but that modes 7-8 ae not in good ageement. The accuacy of the natual fequencies and mode shapes computed using the CMS method can be impoved by using moe fixed inteface dynamic vibation modes. Recall that in the above analysis, we ae using only 3 fixed inteface dynamic vibation modes. Exiting the AUI: Choose File Exit to exit the AUI (you can discad all changes). ADINA R & D, Inc. 60-7

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300 N All lengths in meters. Step load applied at time 0.0.

300 N All lengths in meters. Step load applied at time 0.0. Problem description In this problem, we subject the beam structure of problem 1 to an impact load as shown. 300 N 0.02 0.02 1 All lengths in meters. Step load applied at time 0.0. E = 2.07 10 11 N/m 2