1

Figure 1. Top view displaying geometry of both carbon fiber beams.

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

carbon fiber test pieces with different geometries. Both

carbon fiber beams undergo modal analysis and compare the

resulting mode shapes and natural frequencies. The

rectangular beam is analyzed using the method of assumed

modes, also known as the Ritz series method, to solve for the

mode shapes to validate experimental data. Analysis for the

trapezoidal beam was beyond the scope of the project due to

the varying geometry and plate-like behavior.

INTRODUCTION The application of composite materials continues to broaden

in engineering design. Due to carbon fiber’s high strength to

weight and stiffness to weight ratio, the material is a preferred

choice in many engineering applications requiring lightweight,

strong structures. Increasing applications of composite

materials has also spurred interest in understanding the

material structure and vibration characteristics.

The following paper analyzes the vibration response of two

slender carbon fiber composite beams. Two carbon fiber test

pieces are considered with differing geometry: a rectangular

cut-out and a trapezoidal beam. See Figure 1 below for a

schematic top view representation of the two specimens.

For any given beam, there are specific frequencies in which

the structure vibrates in an organized manner that can be

described through equations. Each frequency is labeled a

natural frequency, with the corresponding shape of the beam

during vibration called the mode shape. This paper compares

the mode shapes between two similar beams and evaluates

validates the experimental data through an analytic study of

the rectangular beam.

The carbon fiber beams have both undergone the same

manufacturing process. The carbon fiber-epoxy layup from

AS4/3510-6 material comprised 8 layers of carbon with

laminate angles of [45/-45/45/-45/-45/45/-45/45]. The

engineering properties for each laminate are listed below in

Table 1.

Table 1. Engineering properties for a carbon fiber lamina.

Thickness( in) Elastic Modulus (psi)

h E11 E22

.0052 200x106 1.4x106

Shear Modulus (psi) Poisson’s ratio Density (lbm/in3)

G12 v12 ρ

0.93x106 0.3 0.0502

For the analysis, both beams will be fixed at each end as if

clamped to a wall. These boundary conditions will allow

more accurate experimental data seen later in the paper, as the

beam will be stiffer and will trigger the source easier.

NOMENCLATURE The following list of terms will be used throughout the

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 (-)

w Width (inches)

Ψ Basis Function (-)

π Pi (-)

ω Natural Frequency (rad/s)

φ Modal Coordinates (-)

Φ Normalized Modal Coordinates (-)

Ψ Mode Shape (-)

ρ Density (lbm/in3)

2

ANALYTICAL METHOD The differing beams are to be compared by finding the first

three natural frequencies of each beam. To solve for the

natural frequencies, the equation of motion must be derived

for each beam. Both beams are modeled as being clamped at

each end and having the same length between clamped ends.

Deformation is only considered in the vertical direction, or

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

and K respectively. Figure 2 allows a visual representation for

the analytical model developed as a side view of the beam.

Figure 2. Condition considered in analyzing beam, with both ends of a

thin beam clamped.

The beam has a width w and thickness h. Only the rectangular

beam may be analyzed using this method, as the trapezoidal

beam is too wide to be considered a beam. The analyses of

each beam are developed independently.

The mode shape can be found by solving for the assumed

mode shapes, otherwise known as the Ritz series given by the

unit vector,

(1)

where Ψj is the basis function and qj is the generalized

coordinates. N is the number of mode shapes that will be

solved for. For this problem, four modes will be calculated,

with the first three being of interest. Ψj can be found from the

boundary conditions. For a beam clamped at x=0 and x=L,

(2)

For the Ritz series method, the effective mass matrix can be

derived by the kinetic energy of bending into a function of Ψj,

yielding

(3)

Substituting Ψ into Mjn, the equation expands to

(4)

When j=n, the mass matrix equation reduces to

(5)

Equations (4) and (5) create the 4x4 mass matrix M used in

forming the equation of motion.

The stiffness matrix is derived from the potential energy

equation, resulting in

(6)

Where

(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

in Appendix A, where the longitude elasticity calculated was

used throughout the analysis. Substituting equation (7) for j

and n into (6) yields the expanded stiffness matrix for j≠n and

j=n. From this, the 4x4 stiffness matrix K is created.

The mass and stiffness matrix now are used to solve the eigen

value problem

02MK (8)

Where the solving for the eigensolution yields ω2 and solving

for the eigenvector yields φ. This eignenvalue problem is very

complex for a four rows by four columns matrix, requiring the

use of mathematic solver program such as Matlab. The

natural frequency is calculated from the vector of ω2. The

modal coordinates φ is used to calculate the normalized modal

coordinates Φ which leads to the development of the mode

shapes based on x using

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

four term Ritz series, N will equal four. Three mode plots are

graphed that visualize how the beam responds during

excitation at the corresponding frequencies.

EXPERIMENTAL METHOD The mode shapes of the carbon beams were found

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

impact hammer with an attached force transducer excites

specified locations. An accelerometer attached to a specified

location records the output motion, which is then analyzed

using a spectrum analyzer to yield the magnitude, in dB, of the

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.

)(1

xj

N

j

jnn

3

Figure 3. Full set-up of roving impact experiment.

The experiments were performed on the Cal Poly- San Luis

Obispo campus in the Vibrations Laboratory located in

Building 13-101. The carbon fiber material was provided by

Aaron Williams, a student who is a member of Cal Poly’s

Human Powered Vehicle team.

To satisfy clamped boundary conditions, two medal plates

sandwiched each end of the carbon beams. A C-clamp was

then applied to the metal plates to fully ensure that the

clamping force was applied evenly throughout the width of the

beam’s boundary conditions. The metal plates were clamped

to the top of cinder blocks to avoid interference of the carbon

fiber beam’s vibration response with the table. The clamping

of the beam is shown in Figure 4 below.

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 5. View of impact points to be analyzed.

The LDS spectral analyzer was, in turn run, by the RT Focus

pro program on the laptop, recording each experimental data

taken. Each point averaged five recordings of impact

response, where the output was saved to import into MEscope

for further analysis. The data acquisition was triggered by the

input force transducer, where the program reported the

accelerometer’s output, the magnitude plot for the frequencies

excited, and the coherence of the averages taken per data

point. The recorded data spanned a frequency up to 2000 Hz

for the trapezoidal beam.

Certain points close to the accelerometer were harder to obtain

accurate data than others, as the accelerometer response plot

needs careful watching to ensure the output signal is not

overloaded. Other points further away from accelerometer

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 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.

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 7. Frequency response function for the rectangular beam.

Figure 8. Frequency response function for the trapezoidal beam model.

The mode shapes were simulated for each natural frequency

selected from the frequency response plot. The following

figure depicts the first three mode shapes resulting from the

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.

Figure 9. Mode shapes for trapezoidal and rectangular geometries from

experimental data.

Table 2. Comparison of modes based on geometry and method.

Mode

Rectangular

f (Hz)

Trapezoidal

f(Hz)

1st 20.5 96.9

2nd 172 192

3rd 350 932

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.

The mode shapes plotted in Matlab from the corresponding

natural frequencies were noticeably wrong. By changing

parameters discussed in the following section, the mode

shapes were plotted as follows in Figure 10, matching the

mode shapes from the experimental method.

Mode

Experimental

f (Hz)

Theoretical

f(Hz)

Percent Error

(%)

1st 20.5 21.5 4.65

2nd 172 171 0.58

3rd 350 367 4.63

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-15

-10

-5

0

5

10

15

Distance along beam (x/L)

PS

I j(x)

First Mode

Second Mode

Third Mode

Figure 10. Numerical plot of first three mode shapes for the rectangular

beam.

DISCUSSION The mode shapes of the rectangular beam found through

experimentation follow the expected results of what the first

three shapes should look like visually. All points on the first

mode shape vibrate in complete unison. The second mode has

one node, where the beam is oscillating 90 degrees out of

phase. From the mode shape plotted in MEscope, one peak

seems much more pronounced than the other peak, which is

smaller in amplitude yet oscillating 90 degrees out of phase.

The third natural frequency has two nodes and three peaks.

Again the front two peaks are much more pronounced than the

rear peak on the image.

The first natural frequency was originally omitted during the

initial testing, as no noticeable peak was clearly seen in the

frequency response graph. After studying the data closer, it

was determined that the first natural frequency occurred below

the first spike at 172 Hz. The beam experiences first modal

oscillation at a frequency of 20.5 Hz, indicating the first

natural frequency.

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

the mode shapes matched the experimental mode shapes

graphed and resemble the expected shapes of a clamped-

clamped beam, the natural frequencies changed dramatically,

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.

The mode shapes plotted of the trapezoidal plate did not align

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

thin plate undergoing vibration with clamped-free-clamped-

free vibration. The modes resemble a plates mode shapes, yet

are distorted due to the complex geometry. Analysis on the

plate would be an extensive project that could produce

interesting results. Those results would give a clearer

representation of carbon fibers vibration quality, as the effect

of the layer angles would be more pronounced in the results.

REFERENCES 1. Zhang, Shaohui. "Modeling and vibration analysis of

a composite supporter for aerospace applications."

Advanced Composite Mater 14(2005) 199-210. 25

Nov 2008

2. Wei, Z.. "Delamination Assessment of Multilayer

Composite Plates Using Model-Based Neural

Networks ." Journal of Vibration and Control

11(2005) 607-625. 25 Nov 2008

3. Jianxin, Gao. "Vibration and Damping Analysis of a

Composite Plate with Active and Passive Damping

Layer." Applied Mathematics and Mechanics 20Oct.

1999 1075-1086. 25 Nov 2008

4. Barkanov, E.. "TRANSIENT RESPONSE OF

SANDWICH VISCOELASTIC BEAMS,."

Mechanics of Composite Materials 36March 2000

367-368. 25 Nov 2008

5. Numayr, K.S.. "INVESTIGATION OF FREE

VIBRATIONS OF COMPOSITE." Mechanics of

Composite Materials 42(2006) 331-346. 25 Nov 2008

6

APPENDIX A- CARBON FIBER LAYUP MATLAB CODE

REFERENCE: AARON WILLIAMS

Cal Poly-San Luis Obispo Senior Mechanical Engineer Student

12/5/08 6:44 AM F:\ME 517\Term Project\CLT2.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 allclose allclc %set up a diary filediary 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 defaultformatl=[ 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

12/5/08 6:44 AM F:\ME 517\Term Project\CLT2.m 2 of 4

%Temperature change input

%service temp %DT=-280 DT=-280 % find the total thicknesstotal = 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 wellNT = 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 strainR = [1 0 0; 0 1 0; 0 0 2]; % locate the bottom of the first plyh(1) = -thick/2.;imax = n + 1; %loop for rest of the ply distances from midsurf

12/5/08 6:44 AM F:\ME 517\Term Project\CLT2.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 matricesfor 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

12/5/08 6:44 AM F:\ME 517\Term Project\CLT2.m 4 of 4

disp('_____________________________________________________________________________________')disp('Stiffness matrix elements:')disp(' ')%change the display format for compliance matrixformat short eQA = 1.0*AB = .5*BD = (1/3)*D K = [A, B; B, D]; disp('_____________________________________________________________________________________')disp ('Plate Compliance Matrix:') disp(' ')C = inv(K);disp(C)

7

APPENDIX B- RITZ SERIES DEVELOPMENT CALCULATIONS

Maple and Matlab code

(1)

Continuation of Rectangular Beam Ritz Series Calculations

L d 4.3 :w d 1.5 :t d 0.04 :A d w$t :

J d w$t3

12:

macc d .000838 :xm d 2.15 :

ψj dxL

$ 1K xL

$sinj$π$x

L:

ψn := xL

$ 1K xL

$sinn$π$x

L:

ψjm :=xm

L$ 1K

xm

L$sin

j$π$xm

L:

ψnm :=xm

L$ 1K

xm

L$sin

n$π$xm

L:

Mjsn d

0

LxL

2$ 1K x

L

2$sin

j$π$xL

$sinn$π$x

L$ρ $ A dxCmacc$ψjm$ψnm :

Mj = n d

0

LxL

2$ 1K x

L

2$sin

j$π$xL

2

$ρ$A dxCmacc$ψjm2

:

Kjsn d

0

L

E $ J $j$πL2 $ 1K 2$x

L$cos

j$π$xL

K2L2 C

j$πL

2

$xL

Kx2

L2

$sinj$π$x

L$

n$πL2 $ 1K 2$x

L$cos

n$π$xL

K2L2 C

n$πL

2

$xL

Kx2

L2

$sinn$π$x

Ldx :

Kj = n d

0

L

E $ J $j$πL2 $ 1K 2$x

L$cos

j$π$xL

K2L2 C

j$πL

2

$xL

Kx2

L2

$sinj$π$x

L$

j$πL2 $ 1K 2$x

L$cos

j$π$xL

K2L2 C

j$πL

2

$xL

Kx2

L2

$sinj$π$x

Ldx :

M11 := eval Mj = n, j = 1, n = 1 :

M12 := eval Mjsn, j = 1, n = 2 :

(2)

M13 := eval Mjsn, j = 1, n = 3 :M14 := eval Mjsn, j = 1, n = 4 :M21 := eval Mjsn, j = 2, n = 1 :M22 := eval Mj = n, j = 2, n = 2 :M23 := eval Mjsn, j = 2, n = 3 :M24 := eval Mjsn, j = 2, n = 4 :M31 := eval Mjsn, j = 3, n = 1 :M32 := eval Mjsn, j = 3, n = 2 :M33 := eval Mj = n, j = 3, n = 3 :M34 := eval Mjsn, j = 3, n = 4 :M41 := eval Mjsn, j = 4, n = 1 :M42 := eval Mjsn, j = 4, n = 2 :M43 := eval Mjsn, j = 4, n = 3 :M44 := eval Mj = n, j = 4, n = 4 :Md linalg matrix 4, 4, M11, M12, M13, M14, M21, M22, M23, M24, M31, M32, M33, M34, M41, M42, M43,

M44

0.006286467569 ρC0.00005237500000, K1.488869734 10-12 ρ, K0.001862313345 ρ

K0.00005237500000, K4.903826696 10-13 ρ ,

K2.356239086 10-12 ρ, 0.004424154224 ρ, K1.185129071 10-11 ρ, K0.001961943275 ρ ,

K0.001862313345 ρK0.00005237500000, K8.478958577 10-12 ρ, 0.004324524294 ρ

C0.00005237500000, 1.371063856 10-12 ρ ,

K5.159180235 10-13 ρ, K0.001961943275 ρ, 1.371064063 10-12 ρ, 0.004307759642 ρK11 := eval Kj = n, j = 1, n = 1 :K12 := eval Kjsn, j = 1, n = 2 :K13 := eval Kjsn, j = 1, n = 3 :K14 := eval Kjsn, j = 1, n = 4 :K21 := eval Kjsn, j = 2, n = 1 :K22 := eval Kj = n, j = 2, n = 2 :K23 := eval Kjsn, j = 2, n = 3 :K24 := eval Kjsn, j = 2, n = 4 :K31 := eval Kjsn, j = 3, n = 1 :K32 := eval Kjsn, j = 3, n = 2 :K33 := eval Kj = n, j = 3, n = 3 :K34 := eval Kjsn, j = 3, n = 4 :K41 := eval Kjsn, j = 4, n = 1 :K42 := eval Kjsn, j = 4, n = 2 :K43 := eval Kjsn, 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,

(3)

K44

7.856703985 10-7 E, 9.908137777 10-16 E, K0.000001202724288 E, K5.975875614 10-16 E ,

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

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%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.clcclear allclose 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)];

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%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 PHIPHI=V; %The mode function PSI, a function of the modal matrix and the basis function, can now be calculated and plotted:L=4.3 ;% inchesx=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%shapesplot(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')