Modal Comparison Of Thin Carbon Fiber Beams

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Term paper for my advanced vibrations class at Cal Poly SLO, performing modal analysis on a carbon fiber beam both experimentally and analytically.

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Modal Comparison Of Thin Carbon Fiber Beams

  1. 1. December 5, 2008, San Luis Obispo, CA, USA MODAL COMPARISON OF THIN CARBON FIBER BEAMS Caleb A. Bartels California Polytechnic State University-San Luis Obispo Mechanical Engineering Department San Luis Obispo, CA, USA ABSTRACT The following paper will consider the relationship of two thin The carbon fiber beams have both undergone the same carbon fiber test pieces with different geometries. Both manufacturing process. The carbon fiber-epoxy layup from carbon fiber beams undergo modal analysis and compare the AS4/3510-6 material comprised 8 layers of carbon with resulting mode shapes and natural frequencies. The laminate angles of [45/-45/45/-45/-45/45/-45/45]. The rectangular beam is analyzed using the method of assumed engineering properties for each laminate are listed below in modes, also known as the Ritz series method, to solve for the Table 1. mode shapes to validate experimental data. Analysis for the trapezoidal beam was beyond the scope of the project due to Table 1. Engineering properties for a carbon fiber lamina. the varying geometry and plate-like behavior. Thickness( in) Elastic Modulus (psi) h E11 E22 INTRODUCTION .0052 200x106 1.4x106 The application of composite materials continues to broaden Shear Modulus (psi) Poisson’s ratio Density (lbm/in3) in engineering design. Due to carbon fiber’s high strength to weight and stiffness to weight ratio, the material is a preferred G12 v12 ρ choice in many engineering applications requiring lightweight, 0.93x10 6 0.3 0.0502 strong structures. Increasing applications of composite materials has also spurred interest in understanding the For the analysis, both beams will be fixed at each end as if material structure and vibration characteristics. clamped to a wall. These boundary conditions will allow more accurate experimental data seen later in the paper, as the The following paper analyzes the vibration response of two beam will be stiffer and will trigger the source easier. slender carbon fiber composite beams. Two carbon fiber test pieces are considered with differing geometry: a rectangular NOMENCLATURE cut-out and a trapezoidal beam. See Figure 1 below for a The following list of terms will be used throughout the schematic top view representation of the two specimens. following paper: A Area (in2) E Young’s Elastic Modulus (psi) f Frequency (Hz) G Shear Modulus (psi) J Moment of Inertia (in4) j Matrix Row Location (-) K Spring Stiffness (lbf/in) L Length (inches) M Mass (pounds-mass) n Matrix Column Location (-) q Generalized Coordinates (-) h Thickness (inches) u Assumed Modes (-) v Poisson’s Ratio (-) Figure 1. Top view displaying geometry of both carbon fiber beams. w Width (inches) Ψ Basis Function (-) For any given beam, there are specific frequencies in which π Pi (-) the structure vibrates in an organized manner that can be ω Natural Frequency (rad/s) described through equations. Each frequency is labeled a φ Modal Coordinates (-) natural frequency, with the corresponding shape of the beam Φ Normalized Modal Coordinates (-) during vibration called the mode shape. This paper compares Ψ Mode Shape (-) the mode shapes between two similar beams and evaluates ρ Density (lbm/in3) validates the experimental data through an analytic study of the rectangular beam. 1
  2. 2. ANALYTICAL METHOD Equations (4) and (5) create the 4x4 mass matrix M used in The differing beams are to be compared by finding the first forming the equation of motion. three natural frequencies of each beam. To solve for the natural frequencies, the equation of motion must be derived The stiffness matrix is derived from the potential energy for each beam. Both beams are modeled as being clamped at equation, resulting in each end and having the same length between clamped ends. Deformation is only considered in the vertical direction, or (6) only as bending. From these restraints, the Ritz Series method developed by Ginsberg applies to the specified problem and can be used in determining the mass and stiffness matrices, M Where and K respectively. Figure 2 allows a visual representation for the analytical model developed as a side view of the beam. (7) The Young’s Modulus of Elasticity E is dependent on the carbon fiber ply’s direction. To find the elasticity in the longitude direction, a Matalab code provided by the manufacturer must be utilized. The Matlab code can be seen Figure 2. Condition considered in analyzing beam, with both ends of a thin beam clamped. in Appendix A, where the longitude elasticity calculated was used throughout the analysis. Substituting equation (7) for j The beam has a width w and thickness h. Only the rectangular and n into (6) yields the expanded stiffness matrix for j≠n and beam may be analyzed using this method, as the trapezoidal j=n. From this, the 4x4 stiffness matrix K is created. beam is too wide to be considered a beam. The analyses of each beam are developed independently. The mass and stiffness matrix now are used to solve the eigen value problem The mode shape can be found by solving for the assumed 2 mode shapes, otherwise known as the Ritz series given by the K M 0 (8) unit vector, Where the solving for the eigensolution yields ω2 and solving for the eigenvector yields φ. This eignenvalue problem is very (1) complex for a four rows by four columns matrix, requiring the use of mathematic solver program such as Matlab. The where Ψj is the basis function and qj is the generalized natural frequency is calculated from the vector of ω2. The coordinates. N is the number of mode shapes that will be modal coordinates φ is used to calculate the normalized modal solved for. For this problem, four modes will be calculated, coordinates Φ which leads to the development of the mode with the first three being of interest. Ψj can be found from the shapes based on x using boundary conditions. For a beam clamped at x=0 and x=L, N n jn j ( x) (2) j 1 Only the first the first three mode shapes, Ψ, are desired, so n will range from 1 to 3. Since the method being analyzed is a For the Ritz series method, the effective mass matrix can be four term Ritz series, N will equal four. Three mode plots are derived by the kinetic energy of bending into a function of Ψj, graphed that visualize how the beam responds during yielding excitation at the corresponding frequencies. (3) EXPERIMENTAL METHOD The mode shapes of the carbon beams were found Substituting Ψ into Mjn, the equation expands to experimentally to compare with the analytical work developed on the beam models. To find the mode shapes experimentally, each beam must be subjected to a roving impact test, where an (4) impact hammer with an attached force transducer excites specified locations. An accelerometer attached to a specified When j=n, the mass matrix equation reduces to location records the output motion, which is then analyzed using a spectrum analyzer to yield the magnitude, in dB, of the (5) response over a range of frequencies, in Hz. A uni-directional accelerometer was used to Figure 3 displays an overall view of the set up of the experiment. 2
  3. 3. Figure 5. View of impact points to be analyzed. The LDS spectral analyzer was, in turn run, by the RT Focus Figure 3. Full set-up of roving impact experiment. pro program on the laptop, recording each experimental data The experiments were performed on the Cal Poly- San Luis taken. Each point averaged five recordings of impact Obispo campus in the Vibrations Laboratory located in response, where the output was saved to import into MEscope Building 13-101. The carbon fiber material was provided by for further analysis. The data acquisition was triggered by the Aaron Williams, a student who is a member of Cal Poly’s input force transducer, where the program reported the Human Powered Vehicle team. accelerometer’s output, the magnitude plot for the frequencies excited, and the coherence of the averages taken per data To satisfy clamped boundary conditions, two medal plates point. The recorded data spanned a frequency up to 2000 Hz sandwiched each end of the carbon beams. A C-clamp was for the trapezoidal beam. then applied to the metal plates to fully ensure that the clamping force was applied evenly throughout the width of the Certain points close to the accelerometer were harder to obtain beam’s boundary conditions. The metal plates were clamped accurate data than others, as the accelerometer response plot to the top of cinder blocks to avoid interference of the carbon needs careful watching to ensure the output signal is not fiber beam’s vibration response with the table. The clamping overloaded. Other points further away from accelerometer of the beam is shown in Figure 4 below. were difficult to obtain consistent excitation responses and good coherence in the averaging of the data. The rectangular carbon beam was analyzed in similar fashion to the trapezoidal beam, where the ends were clamped by the steel plates and held in place by the C-Clamp. The beam was broken up into 15 nodes. The nodes were placed every inch in the x direction and every half for the width. The accelerometer was placed in the middle of the beam at point 8 to ensure the accelerometer would not be placed on a node of motion. The set up for the rectangular beam can be seen in Figure 8. Figure 4. Carbon beam clamped between two metal plates. The trapezoidal beam was divided up into 14 distinct points to perform the impact test on. The points were evenly spaced every inch throughout the face of the beam, as seen in Figure 5. The accelerometer was attached to point 6 using wax. The accelerometer and input hammer are connected to a source amplifier by microdot cables and relayed to the LDS analyzer. Figure 6. Roving impact experimental setup for rectangular beam. After the both roving impact tests are finished for all points, the data can be saved in RT pro and exported to MEscope to perform a modal analysis on the beam with the exported data. 3
  4. 4. A surface model was created and points were assigned to match the points sketched out on the physical beams. RESULTS After data collection was complete, the experimental natural frequencies were found through analyzing the peaks of the imaginary value of the magnitude frequency response graph. This plot was created by over-laying the frequency response for all points excited with the input hammer. The resulting frequency response plots are displayed below, and the first three frequencies are noted with a red bar. Figure 9. Mode shapes for trapezoidal and rectangular geometries from experimental data. Table 2. Comparison of modes based on geometry and method. Mode Rectangular Trapezoidal f (Hz) f(Hz) 1st 20.5 96.9 2nd 172 192 3rd 350 932 Figure 7. Frequency response function for the rectangular beam. The integration required to solve the analytic method was developed in the symbolic solver Maple. The corresponding mass and stiffness matrices were solved in the numerical solver Matlab to produce the first three natural frequencies of the beam, seen in Table 3. The code can be reviewed in Appendix B. Table 3. Validation of experimental test compared to analytic method for the rectangular beam. Mode Experimental Theoretical Percent Error f (Hz) f(Hz) (%) 1st 20.5 21.5 4.65 2nd 172 171 0.58 3rd 350 367 4.63 Figure 8. Frequency response function for the trapezoidal beam model. The mode shapes plotted in Matlab from the corresponding natural frequencies were noticeably wrong. By changing The mode shapes were simulated for each natural frequency parameters discussed in the following section, the mode selected from the frequency response plot. The following shapes were plotted as follows in Figure 10, matching the figure depicts the first three mode shapes resulting from the mode shapes from the experimental method. experimental studies and data collected. The mode shapes were created using the program MEscope, where a surface was defined representing the given beam. Points were numbered and meshed in the surface to correlate with the measured data. Table 2 summarizes both experimental mode shape frequencies found using this process. 4
  5. 5. the mode shapes matched the experimental mode shapes 15 graphed and resemble the expected shapes of a clamped- clamped beam, the natural frequencies changed dramatically, 10 yielding 70 percent error in the second and third natural frequency. The change in natural frequency magnitude was not expected, as such a small change was made to the code. 5 The mode shapes plotted of the trapezoidal plate did not align PSIj(x) 0 to the shapes or frequencies of the rectangular beam. After further thought, the results confirm that the trapezoidal plate could not be modeled as a beam, but needs to be treated as a -5 thin plate undergoing vibration with clamped-free-clamped- free vibration. The modes resemble a plates mode shapes, yet First Mode are distorted due to the complex geometry. Analysis on the -10 Second Mode Third Mode plate would be an extensive project that could produce interesting results. Those results would give a clearer -15 representation of carbon fibers vibration quality, as the effect 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Distance along beam (x/L) of the layer angles would be more pronounced in the results. Figure 10. Numerical plot of first three mode shapes for the rectangular beam. DISCUSSION REFERENCES The mode shapes of the rectangular beam found through 1. Zhang, Shaohui. "Modeling and vibration analysis of experimentation follow the expected results of what the first a composite supporter for aerospace applications." three shapes should look like visually. All points on the first Advanced Composite Mater 14(2005) 199-210. 25 mode shape vibrate in complete unison. The second mode has Nov 2008 one node, where the beam is oscillating 90 degrees out of 2. Wei, Z.. "Delamination Assessment of Multilayer phase. From the mode shape plotted in MEscope, one peak Composite Plates Using Model-Based Neural seems much more pronounced than the other peak, which is Networks ." Journal of Vibration and Control smaller in amplitude yet oscillating 90 degrees out of phase. 11(2005) 607-625. 25 Nov 2008 The third natural frequency has two nodes and three peaks. 3. Jianxin, Gao. "Vibration and Damping Analysis of a Again the front two peaks are much more pronounced than the Composite Plate with Active and Passive Damping rear peak on the image. Layer." Applied Mathematics and Mechanics 20Oct. 1999 1075-1086. 25 Nov 2008 The first natural frequency was originally omitted during the 4. Barkanov, E.. "TRANSIENT RESPONSE OF initial testing, as no noticeable peak was clearly seen in the SANDWICH VISCOELASTIC BEAMS,." frequency response graph. After studying the data closer, it Mechanics of Composite Materials 36March 2000 was determined that the first natural frequency occurred below 367-368. 25 Nov 2008 the first spike at 172 Hz. The beam experiences first modal 5. Numayr, K.S.. "INVESTIGATION OF FREE oscillation at a frequency of 20.5 Hz, indicating the first VIBRATIONS OF COMPOSITE." Mechanics of natural frequency. Composite Materials 42(2006) 331-346. 25 Nov 2008 Calculation of the natural frequencies for the rectangular beam followed the method developed earlier in this paper. The resulting frequencies matched closely with the experimental data after determining the location of the first natural frequency. A problem occurred while attempting to plot the mode shapes for the corresponding frequencies. The mode shape plot outputted was correct for the first natural frequency, yet resembled a third mode for the actual second mode shape and a fourth mode for the actual third mode shape. The Matlab and Maple code was reviewed extensively to find what was changing the mode shapes while plotting. The problem occurred in the M and K matrices. Both M and K matrices calculated had several very small terms. The M matrix had 7 terms of the magnitude 10-13 lbm or smaller, while the K matrix had 7 terms a magnitude 10-7lbf/in or smaller. The values of these terms were almost zero compared to the other magnitudes inside the respective matrices. These small terms were set to zero and the mode shapes were recalculated, resulting in the correct mode shapes seen in Figure 10. While 5
  6. 6. APPENDIX A- CARBON FIBER LAYUP MATLAB CODE REFERENCE: AARON WILLIAMS Cal Poly-San Luis Obispo Senior Mechanical Engineer Student 6
  7. 7. 12/5/08 6:44 AM F:ME 517Term ProjectCLT2.m 1 of 4 % Simple CLT File % This one includes hygrothermal % from Dr. Joseph Mello: Professor of Mechanical Engineering % California Polytechnic State University, San Luis Obispo % Basic program provided for students % Hygrothermal details completed by Aaron Williams as a class exercise % Display cleaned up by Aaron Williams % % has plots and pauses (hit return) % play with scaling factors?? % this is a total hack visualization attempt % clear all close all clc %set up a diary file diary CLT.dat %units are US customary (lb, in, E in psi) % total laminate definition in matrix below % [ply angles, thicknesses, matl. #] %Set up for two materials % Data in there now is %1-carbon %2-Eglass % Laminate is defined in this matrix little "L" or l (it looks like a one in default font) disp ('_____________________________________________________________________________________' ) disp('Laminate:') disp(' ') disp(' angle thick matl #') %to change format of l output to default format l=[ 45 .0052 1; -45 .0052 1; 45 .0052 1; -45 .0052 1; -45 .0052 1; 45 .0052 1; -45 .0052 1; 45 .0052 1]; disp(l) % this is the total laminate % cut, paste, edit above to study your laminate of choice
  8. 8. 12/5/08 6:44 AM F:ME 517Term ProjectCLT2.m 2 of 4 %Temperature change input %service temp %DT=-280 DT=-280 % find the total thickness total = sum(l,1); thick = total(1,2); disp('thickness ply count') disp (total(2:3)) % size command to get number of plies n = size(l,1) ; % Lamina Properties % matrix for engineering constants % E1 E2 v12 G12 a11 a22' E = [20.0e6 1.4e6 .30 .93e6 -.5e-6 15e-6; %AS4/3501-6 5.84e6 .9e6 .2 .3e6 0.0e-6 0.0e-6]; %E-Glass/Epoxy disp ('_____________________________________________________________________________________' ) disp('Lamina properties:') disp(' ') disp(' E1 E2 v12 G12 a11 a22') format short e disp (E) %intialize the ply distance and ABD matrices % and not the ermal loads as well NT = zeros(3,1); MT = zeros(3,1); h = zeros(n+1,1); A = zeros(3); B = zeros(3); D = zeros(3); % Form R matrix which relates engineering to tensor strain R = [1 0 0; 0 1 0; 0 0 2]; % locate the bottom of the first ply h(1) = -thick/2.; imax = n + 1; %loop for rest of the ply distances from midsurf
  9. 9. 12/5/08 6:44 AM F:ME 517Term ProjectCLT2.m 3 of 4 for i = 2 : imax h(i) = h(i-1) + l(i-1,2); end %loop over each ply to integrate the ABD matrices for i = 1:n %ply material ID mi=l(i,3); v21 = E(mi,2)*E(mi,3)/E(mi,1); d = 1 - E(mi,3)*v21; %Q12 matrix Q = [E(mi,1)/d v21*E(mi,1)/d 0; E(mi,3)*E(mi,2)/d E(mi,2)/d 0; 0 0 E(mi,4)]; %ply angle in radians a1=l(i,1)*pi/180; %Form transformation matrices T1 for ply T1 = [(cos(a1))^2 (sin(a1))^2 2*sin(a1)*cos(a1); (sin(a1))^2 (cos(a1))^2 -2*sin(a1)*cos(a1); -sin(a1)*cos(a1) sin(a1)*cos(a1) (cos(a1))^2-(sin(a1))^2 ]; %Form transformation matrix T2 T2 = [(cos(a1))^2 (sin(a1))^2 sin(a1)*cos(a1); (sin(a1))^2 (cos(a1))^2 -sin(a1)*cos(a1); -2*sin(a1)*cos(a1) 2*sin(a1)*cos(a1) (cos(a1))^2-(sin(a1))^2 ]; %Form Qxy Qxy = inv(T1)*Q*R*T1*inv(R); % build up the laminate stiffness matrices A = A + Qxy*(h(i+1)-h(i)); B = B + Qxy*(h(i+1)^2 - h(i)^2); D = D + Qxy*(h(i+1)^3 - h(i)^3); %load alphs into and array a=[E(mi,5); E(mi,6); 0.0]; %transform cte's axy = inv(T2)*a; %mult by DT to get thermal strain exy exy = DT*axy; %build up thermal load NT = NT + DT*Qxy*axy*(h(i+1)-h(i)); MT = MT + DT*Qxy*axy*(h(i+1)^2 - h(i)^2); %end of stiffness loop end
  10. 10. 12/5/08 6:44 AM F:ME 517Term ProjectCLT2.m 4 of 4 disp ('_____________________________________________________________________________________' ) disp('Stiffness matrix elements:') disp(' ') %change the display format for compliance matrix format short e Q A = 1.0*A B = .5*B D = (1/3)*D K = [A, B; B, D]; disp ('_____________________________________________________________________________________' ) disp ('Plate Compliance Matrix:') disp(' ') C = inv(K); disp(C)
  11. 11. APPENDIX B- RITZ SERIES DEVELOPMENT CALCULATIONS Maple and Matlab code 7
  12. 12. Continuation of Rectangular Beam Ritz Series Calculations L d 4.3 : w d 1.5 : t d 0.04 : A d w$t : w$t3 Jd : 12 macc d .000838 : xm d 2.15 : x x j$π$x ψj d $ 1K $sin : L L L x x n$π$x ψn := $ 1K $sin : L L L xm xm j$π$xm ψjm := $ 1K $sin : L L L xm xm n$π$xm ψnm := $ 1K $sin : L L L L 2 2 x x j$π$x n$π$x Mj s n d $ 1K $sin $sin $ρ $ A dx Cmacc$ψjm$ψnm : L L L L 0 L 2 2 2 x x j$π$x 2 Mj = n d $ 1K $sin $ρ$A dx Cmacc$ψjm : L L L 0 L 2 2 j$π 2$x j$π$x 2 j$π x x Kj s n d E$J$ 2 $ 1K $cos K 2 C $ K 2 L L L L L L L 0 2 j$π$x n$π 2$x n$π$x 2 n$π x x2 $sin $ $ 1K $cos K C $ K 2 L L2 L L L2 L L L n$π$x $sin dx : L L 2 j$π 2$x j$π$x 2 j$π x x2 Kj = n d E$J$ $ 1K $cos K C $ K 2 L2 L L L2 L L L 0 2 2 j$π$x j$π 2$x j$π$x 2 j$π x x $sin $ 2 $ 1K $cos K 2 C $ K 2 L L L L L L L L j$π$x $sin dx : L M11 := eval Mj = n, j = 1, n = 1 : (1) M12 := eval Mj s n, j = 1, n = 2 :
  13. 13. M13 := eval Mj s n, j = 1, n = 3 : M14 := eval Mj s n, j = 1, n = 4 : M21 := eval Mj s n, j = 2, n = 1 : M22 := eval Mj = n, j = 2, n = 2 : M23 := eval Mj s n, j = 2, n = 3 : M24 := eval Mj s n, j = 2, n = 4 : M31 := eval Mj s n, j = 3, n = 1 : M32 := eval Mj s n, j = 3, n = 2 : M33 := eval Mj = n, j = 3, n = 3 : M34 := eval Mj s n, j = 3, n = 4 : M41 := eval Mj s n, j = 4, n = 1 : M42 := eval Mj s n, j = 4, n = 2 : M43 := eval Mj s n, j = 4, n = 3 : M44 := eval Mj = n, j = 4, n = 4 : M d linalg matrix 4, 4, M11, M12, M13, M14, M21, M22, M23, M24, M31, M32, M33, M34, M41, M42, M43, M44 0.006286467569 ρ C0.00005237500000, K1.488869734 10-12 ρ, K0.001862313345 ρ (2) -13 K0.00005237500000, K4.903826696 10 ρ, K2.356239086 10-12 ρ, 0.004424154224 ρ, K1.185129071 10-11 ρ, K0.001961943275 ρ , K0.001862313345 ρ K0.00005237500000, K8.478958577 10-12 ρ, 0.004324524294 ρ C0.00005237500000, 1.371063856 10-12 ρ , K 5.159180235 10-13 ρ, K 0.001961943275 ρ, 1.371064063 10-12 ρ, 0.004307759642 ρ K11 := eval Kj = n, j = 1, n = 1 : K12 := eval Kj s n, j = 1, n = 2 : K13 := eval Kj s n, j = 1, n = 3 : K14 := eval Kj s n, j = 1, n = 4 : K21 := eval Kj s n, j = 2, n = 1 : K22 := eval Kj = n, j = 2, n = 2 : K23 := eval Kj s n, j = 2, n = 3 : K24 := eval Kj s n, j = 2, n = 4 : K31 := eval Kj s n, j = 3, n = 1 : K32 := eval Kj s n, j = 3, n = 2 : K33 := eval Kj = n, j = 3, n = 3 : K34 := eval Kj s n, j = 3, n = 4 : K41 := eval Kj s n, j = 4, n = 1 : K42 := eval Kj s n, j = 4, n = 2 : K43 := eval Kj s n, j = 4, n = 3 : K44 := eval Kj = n, j = 4, n = 4 : K d linalg matrix 4, 4, K11, K12, K13, K14, K21, K22, K23, K24, K31, K32, K33, K34, K41, K42, K43,
  14. 14. K44 7.856703985 10-7 E, 9.908137777 10-16 E, K0.000001202724288 E, K5.975875614 10-16 E , (3) 2.910481114 10-16 E, 0.000004725618260 E, K3.464631794 10-15 E, K0.000006111737654 E , K0.000001202724289 E, K4.282843971 10-15 E, 0.00001782640447 E, 5.457926642 10-14 E , K6.227549933 10-16 E, K0.000006111737654 E, 5.484481475 10-14 E, 0.00004988933870 E
  15. 15. 12/5/08 7:02 AM F:ME 517Term ProjectRectangle_Eigen_Mass_Revised.m 1 of 2 %ME 517 Term Project: Modal Analysis of Carbon Fiber Beams %Caleb Bartels %Continuation of analysis of Rectangle Beam using results from Maple %M and K matrix found using Maple, creating two 4x4 matrices without %analyzing E and rho. Now omega^2, or lamda, will be found to determine the %natural frequencies of the beam. clc clear all close all %For Carbon fiber-Epoxy resin Layup of AS4/3501-6 and laminate angles: %[45/-45/45/-45/45/-45/45] %Youngs Modulus of Elasticity found for the carbon fiber material: E=20.12*10^6; %psi (currently an approximate value) %Actual Youngs Modulus Matrix of Lamina: %E=[200*10^6 1.4e6; % 5.84e6, .9e6];%psi %Density of the carbon fiber material: rho=0.0503;%lbm/in^3 (currently an approximate value, reference Free Vibes of Composites paper) %Mass and Stiffness matrix: M=[0.006286467569*rho+0.0005237500000,-1.9450707*10^(-12)*rho,-0.001862313345*rho- 0.0005237500000,-4.474498*10^(-13)*rho; 2.327596815*10^(-12)*rho,0.004424154224*rho,-8.894298648*10^(-12)*rho,-0.001961943275 *rho; -0.001862313345*rho-0.0005237500000,-9.869956020*10^(-12)*rho,0.004324524294*rho+0. 0005237500000,3.339408638*10^(-12)*rho; -5.137652506*10^(-13)*rho,-0.001961943275*rho,-2.085206921*10^(-14)*rho,0.004307759642 *rho]; %M=[0.006286467569*rho+0.0005237500000, 0, -0.001862313345*rho-0.0005237500000, 0; % 0, 0.004424154224*rho, 0, -0.001961943275*rho; %-0.001862313345*rho-0.0005237500000, 0,0.004324524294*rho+0.0005237500000, 0; %0, -0.001961943275*rho, 0, %0.004307759642*rho]; K=E.*[7.856703983*10^(-7),3.924961909*10^(-16),-0.000001202724288,-6.167534369*10^(-16); 8.424189424*10^(-16),0.000004725618260,-2.027393690*10^(-14),-0.000006111737654; -0.000001202724288,-1.555548484*10^(-14),0.00001782640447,6.145278204*10^(-14); -5.747157491*10^(-16),-0.000006111737654,5.324735371*10^(-14),0.00004988933870]; %K=E.*[7.856703983*10^(-7), 0, -0.000001202724288, 0; % 0, 0.000004725618260, 0, -0.000006111737654; % -0.000001202724288, 0, 0.00001782640447, 0; % 0, -0.000006111737654, 0, 0.00004988933870]; %Now the eigen value problem [K-lamda*M=0] will be solved to find lamda: [V,C]=eig(K,M); lamda=[C(1,1);C(2,2);C(3,3);C(4,4)];
  16. 16. 12/5/08 7:02 AM F:ME 517Term ProjectRectangle_Eigen_Mass_Revised.m 2 of 2 %From lamda we can determine the natural frequencies for the first three %modes: display('Natural Frequencies:') omega_1=sqrt(lamda(1,1))/(2*pi) omega_2=sqrt(lamda(2,1))/(2*pi) omega_3=sqrt(lamda(3,1))/(2*pi) %The eigenvector V is the normalized modal matrix PHI PHI=V; %The mode function PSI, a function of the modal matrix and the basis function, can now be calculated and plotted: L=4.3 ;% inches x=linspace(0,L,1000); psi_1=x.*(1-x./L).*sin(1*pi.*x./L)./L; psi_2=x.*(1-x./L).*sin(2*pi.*x./L)./L; psi_3=x.*(1-x./L).*sin(3*pi.*x./L)./L; psi_4=x.*(1-x./L).*sin(4*pi.*x./L)./L; PSI_1=PHI(1,1)*psi_1+PHI(2,1)*psi_2+PHI(3,1)*psi_3+PHI(4,1)*psi_4; PSI_2=PHI(1,2)*psi_1+PHI(2,2)*psi_2+PHI(3,2)*psi_3+PHI(4,2)*psi_4; PSI_3=PHI(1,3)*psi_1+PHI(2,3)*psi_2+PHI(3,3)*psi_3+PHI(4,3)*psi_4; %Plot the mode functions as a function of distance x/L to view the mode %shapes plot(x/L,PSI_1,x/L,PSI_2, x/L, PSI_3) xlabel('Distance along beam (x/L)') ylabel('PSI_j(x)') legend('First Mode', 'Second Mode', 'Third Mode')

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