From 0b28de6b2422a087bbe02df8d749f2142cb1d78a Mon Sep 17 00:00:00 2001 From: Nicolas D Mower Date: Sun, 30 Apr 2017 23:06:11 -0400 Subject: [PATCH] first half --- CDM.m | 59 +++++++++++++++++++++++++++++++++++++++++++++++++ Plot1.PNG | Bin 0 -> 23700 bytes Plot2.PNG | Bin 0 -> 33065 bytes defl.asv | 23 +++++++++++++++++++ defl.m | 24 ++++++++++++++++++++ egv.m | 61 +++++++++++++++++++++++++++++++++++++++++++++++++++ montecarlo.m | 30 +++++++++++++++++++++++++ 7 files changed, 197 insertions(+) create mode 100644 CDM.m create mode 100644 Plot1.PNG create mode 100644 Plot2.PNG create mode 100644 defl.asv create mode 100644 defl.m create mode 100644 egv.m create mode 100644 montecarlo.m diff --git a/CDM.m b/CDM.m new file mode 100644 index 0000000..58a1c6e --- /dev/null +++ b/CDM.m @@ -0,0 +1,59 @@ +function [w] = CDM(q,N,P) +%Given an applied distributed load (q N/m), N segments, and P newtons +%transverse load, uses Central Difference Method to solve for deflection at +%segments + +%Initialize Constants +E = 70*10^9; %Pa +b = 0.1; %m +h = 0.01; %m +l = 1; %m + +%Derived variables +I = (b*(h^3))/12; % m^4 + +%Segments +segl = l/N; %m - length of segment + +%Diagonal values +diagonal = ((P*2)/((N^2)*E*I)) + 6; %main diagonal +offDiag = (-P/((N^2)*E*I)) - 4; %off-diagonal + +%Deflection differential +dw = ((segl^4)*q)/(E*I); + +%Loop through all values of A to create values +A = zeros(N-1,N-1); +for row = 1:(N-1) + for column = 1:(N-1) + %diagonal values + if row == column + A(row,column) = diagonal; + end + %off diagonal values + if column == row - 1 + A(row,column) = offDiag; + end + if column == row + 1 + A(row,column) = offDiag; + end + %"off-off" diagonal values + if column == row - 2 + A(row,column) = 1; + end + if column == row + 2 + A(row,column) = 1; + end + end +end + +%Sets corner values to one less than main diagonal values +A(1,1) = ( (2*P) / ((N^2)*(E*I)) ) + 5; +A(N-1,N-1) = ( (2*P) / ((N^2)*(E*I)) ) + 5; + +%Generate B Matrix +B = dw*ones((N-1),1); + +% Calculate deflection +w = A\B; +end diff --git a/Plot1.PNG b/Plot1.PNG new file mode 100644 index 0000000000000000000000000000000000000000..8881490f608fab0d5bdae0c63ea308428bd49450 GIT binary patch literal 23700 zcmeFZc{r49|35y}-KM1FE=6e7g6zsNh)PN!`%bcN$(CVISt>D=$dWCxjIj9A6vo(n&ugf=`#Fy1_N4HeI3X1+!=G7*Ll6y*ZcK;pVuR;>uQ`F zJRC3>j8pycg&Q!~1}qG=4z*?!!qI*%J_DzLofgWtg~f7zYCb{+;RL~UEN z+6aE%di%1G2Mo5O7W&`1dbg}wFqr8n^$X{9@0v3b10Vj~a}y^T{B)Kj;$Z}ALo&5I3%%Kn&%c%=N7hSuW4_AcABs)`QvNxlXEEqEmFI&nLzdZJ~11}>k;@@&c;&x@zW=P9;q;F9w~Bg(HEuwLO?Q1j-0 zfxUqrEeQhatH8=Rl?&`2;izDvwSOi4|NED4VoMS{JkPAcc9|J{^a%L5xo|c1L;E=i ziQO(#oXGq#tJ=bSYN#ogvVJKua6WS+e`P*jXnrQ{boJt7W5d*ZSm10}XyfEeL$pwI z|C^PC8FxM&p2ydd70uk8orx6|LJ2dk+$VSJ*r8AFMSO2RK0!F^MtfB-LCP_xT9D~D z(jR8inu{tG2$;3<9P2K8PxdULZyO*G2qUb(6;|3{<4ZLYylVu%qJMXL{w^}tk@;i9 zn-2b?x%k(4d3jf?=elxD+Fr?eXrx3;gwHKhuPjuzi7wWNUg%$-u}&|4Ra#E->Ce_W z->R5THuuXND5^BWyP6I9d|x3TO_>|-^;s;ptSq8@s`X_UvR}U~QLC4z=+R5db#9BH 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zk?$`kK2_u5@?#wA5pG4V1tz7uF7`Mi4WutWZc=+U%NpA6nkRV}Y+`g>j!nF^zkZeN zwqQ{FnUh|5#%a_AJ5sh+jwc+}MO%M1xjL#;LhP%~8gl36Q8U@z%kLWQX|T%^QkrHC zk8NNL4$gy7JE6-LgYR59AQ(0E;21Ua>0EjeQ{LeI&+Z>AV$R3eTy-&hl=1e{1Ln>} zzL@fTglpQ7Qws4%t)|-H%(5OI-D6##aK9@*UoWLjaolOPsK_FrA%61#i^3{`Wl&F& zR!pPC=#_7}oiS&;t%a;9PVr04lV^wSj?U#>K*Mlfi5rh-DD^dF)A;7Py)u8nNj-`7 zxsQqd`}gJPGHu;AsO>7`U+V4TFWAblBVBZM&so{t=6q^Vh|sD?zCaZ)!9H5`0kAdX zbg)jBZ%1lu?4z<#sUxSB)$n&uJ6VsgZ{ecVx()03#2CTF6z=`^=mKj^;Q3W%8-&6am{n_W^4kc3k+ z^Y3?*$Kevw6lw_R<6DG_f|9dxgf4&|hW6KRV~2|5Yae!6So9dQeUSQ?np|p0DH?yr zaV3aq@xUftb6j!%XufVql8s4|Yucq>D$yp9ptJnvM_jY;oliz;IUyh&>;M3(MvtTdMP`?`k1g*ccuyHj3=BKPGysV zY|iBmlU(($jx^~sKRo;37n$lEJGu`4|M>eTruY$CQ0339$=en-aottqRX`b9rSod% J0P21BzW@WRbq@dl literal 0 HcmV?d00001 diff --git a/defl.asv b/defl.asv new file mode 100644 index 0000000..365d0c6 --- /dev/null +++ b/defl.asv @@ -0,0 +1,23 @@ +function dwdt = defl(x,w) +%For ODE45 script - recursively defines fourth order ODE for deflection +%given q=0 and P=0 + +%Initialize Constants +E = 70*10^9; %Pa +dens = 2700; %kg/m^3 +b = 0.1; %m +h = 0.01; %m +l = 1; %m + +%Placeholder frequenc +%freq = 150; + +%Derived variables +I = (b*(h^3))/12; % m^4 +area = b*h; %m^2 + +%Define constant K in place +k = (dens*area*(freq^2))/(E*I); +dwdt = [w(2);w(3);w(4);k*w(1)]; +end + diff --git a/defl.m b/defl.m new file mode 100644 index 0000000..a73df33 --- /dev/null +++ b/defl.m @@ -0,0 +1,24 @@ +function dwdt = defl(x,w) +%For ODE45 script - recursively defines fourth order ODE for deflection +%given q=0 and P=0, x and w used by ODE45 to solve for deflection over +%position + +%Initialize Constants +E = 70*10^9; %Pa +dens = 2700; %kg/m^3 +b = 0.1; %m +h = 0.01; %m +l = 1; %m + +%Placeholder frequency +%freq = 150; + +%Derived variables +I = (b*(h^3))/12; % m^4 +area = b*h; %m^2 + +%Define constant K to simplify recursive functions for ODE +k = (dens*area*(freq^2))/(E*I); +dwdt = [w(2);w(3);w(4);k*w(1)]; +end + diff --git a/egv.m b/egv.m new file mode 100644 index 0000000..c432cbc --- /dev/null +++ b/egv.m @@ -0,0 +1,61 @@ +function [f1,f2,f3] = egv(N,P) +%Function solves for the natural frequency of a the beam in problem 2 and 3 +%given N segments and transverse load P + +%Initialize Constants +E = 70*10^9; %Pa +dens = 2700; %kg/m^3 +b = 0.1; %m +h = 0.01; %m +l = 1; %m + +%Derived variables +I = (b*(h^3))/12; % m^4 +area = b*h; %m^2 + +%Segments +segl = l/N; %m - length of segment + +%Diagonal values +diagonal = ((P*2)/((N^2)*E*I)) + 6; %main diagonal +offDiag = (-P/((N^2)*E*I)) - 4; %off-diagonal + +%Loop through all values of A to create values +A = zeros(N-1,N-1); +for row = 1:(N-1) + for column = 1:(N-1) + %diagonal values + if row == column + A(row,column) = diagonal; + end + %off diagonal values + if column == row - 1 + A(row,column) = offDiag; + end + if column == row + 1 + A(row,column) = offDiag; + end + %"off-off" diagonal values + if column == row - 2 + A(row,column) = 1; + end + if column == row + 2 + A(row,column) = 1; + end + end +end + +%Sets corner values to one less than main diagonal values +A(1,1) = ( (2*P) / ((N^2)*(E*I)) ) + 5; +A(N-1,N-1) = ( (2*P) / ((N^2)*(E*I)) ) + 5; + +%Built in eigenvalue MATLAB function finds eigenvalues of the matrix +%generated for the beam +egv = eig(A); + +%Plug into function for natural frequencies to solve (for first three - +%more can be generated with following values of egv vector) +f1 = sqrt((egv(1)*E*I)/dens/area/(segl^4)); +f2 = sqrt((egv(2)*E*I)/dens/area/(segl^4)); +f3 = sqrt((egv(3)*E*I)/dens/area/(segl^4)); +end diff --git a/montecarlo.m b/montecarlo.m new file mode 100644 index 0000000..2e1bfe6 --- /dev/null +++ b/montecarlo.m @@ -0,0 +1,30 @@ +function [avg,stdev]=montecarlo(num) +%Generates random values for base and height values +%of the beam, finds according mean and standard deviation +%of the outputted deflections. +%Applied distributed load given as 50 N/m. +%Normal random number generation using 0.1% standard deviation. +%Input 'num' designates number of random samples. + +%Given physical parameters +q = 50; %Input loading 50 N/m +E = 70*10^9; %Pa +l = 1; %m +x = 0.5; %m + +%Loop generates random numbers and places calculated deflections in matrix +for i=1:num + %Generates random values for I + b = normrnd(0.1,0.01); %random base length + h = normrnd(0.01,0.001); %random height value + I = (b*h^3)/12; %calculates random Moment of Inertia (m^4) + + %Calculated corresponding deflections and place in matrix w + w(i) = -(((q*l*(x^3))/(12*E*I)))-(((q*(x^4))/(24*E*I)))-((q*(l^3)*x)/(24*E*I)); +end + +%Find descriptive statistics for set of delfections +avg = mean(w); +stdev = std(w); +end +