Mechanical Engineering - Mechanical Systems Dynamics

Laboratory notes - Report 2

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MECHANICAL SYSTEM DYNAMICS Finite Elements Method (FEM) in structural dynamics: software implementation in Matlab environment – part 1 I. La Paglia 1. Mesh generation [USER INPUT: *. inp ] 2. Definition of the global and local reference systems [USER INPUT: *. inp ] 3. Removal of external constraints and introduction of corresponding constraint forces [USER INPUT: *. inp ] 4. Energy functions formulation in the local nodal coordinates of each element [FEM PROGRAM: loadstructure ()] 5. Coordinate transformation from the local to the global reference system [FEM PROGRAM: assem (), calling el_tra ()] 6. Matrix assembling, for the entire structure [FEM PROGRAM: assem ()] Procedure for building a FE model of a structure 2 Main.m % structure data m = … % check max element length Lmax = … % build *. inp file (mesh, constraints) loadstructure () % nodes definition % elements definition % draw structure dis_stru () % build and assemble matricies assem () el_tra () % build M and K local ref. % assemble total M and K I. La Paglia Mechanical Systems Dynamics • Element length The element must work in quasi -static region! The element first natural frequency is � � :5;should be sufficiently grater than the maximum frequency of interest 3 ��� Typically, we can produce a mesh such that  � :-; � ��� R s
�w thus 3 Must consider • Discontinuities I 5 I 6 Property change Geometry change I. La Paglia Mechanical Systems Dynamics Mesh generation .��� L � 6 s
�w 3 ��� ', I ��� Example - Properties of the system 4 A simply supported aluminum beam with rectangular constant cross -section I. La Paglia Mechanical Systems Dynamics 3 ��� L srr t� \ .��� L � 6 s
�w 3 ��� ', I ��� L uvz II 5 We must define : ▪ Nodes : o Constraints (x, y, θ) o Coordinates (x, y) ▪ Elements : o Connected nodes o Properties (m, EA, EJ) 4 beam elements (300 mm) and 5 nodes *NODES n – constr.( x,y,theta ) – coor . x,y 1 1 1 0 0 0 2 0 0 0 0.3 0 3 .. *BEAMS n - input node - output node - property 1 1 2 1 2 2 3 1 3 .. *PROPERTIES 1 0.864 2.176e7 1.1605e2 Node number Constraint on x: 1 true , 0 false coordinates Element number Connected nodes Element property number m, EA, EJ Property number I. La Paglia Mechanical Systems Dynamics Example - Meshing 6 ! FEM(1) ! 1st Exercise ! --------------------------------------------- ! list of nodes : *NODES ! n. of node - constraint code ( x,y,theta ) - x coordinate - y coordinate. 1 1 1 0 0.0 0.0 2 0 0 0 0.3 0.0 3 0 0 0 0.6 0.0 4 0 0 0 0.9 0.0 5 1 1 0 1.2 0.0 ! end card *NODES *ENDNODES ! --------------------------------------------- ! list of elements : *BEAMS ! n. of elem. - n. of input node - n. of output node – n. of prop. 1 1 2 1 2 2 3 1 3 3 4 1 4 4 5 1 *ENDBEAMS ! --------------------------------------------- ! List of properties : *PROPERTIES ! N. of prop. – m – EA - EJ 1 0.864 2.176e7 1.1605e2 *ENDPROPERTIES File extension Start comments with «!», compiler ignores following characters NO white lines admitted *NODES start definition of nodes *ENDNODES end definition of nodes *BEAMS start definition of beam finite elements *ENDBEAMS end definition of beam finite elements *PROPERTIES start definition of element properties *ENDPROPERTIES end definition of beam finite elements properties I. La Paglia Mechanical Systems Dynamics Input file: MSD_FEM_01.inp • loadstructure () Processes the *. inp file and returns some useful variables • assem () Taking info from previous function outputs, assembles M and K matrices The final matrices M and K are of the whole system (free and constrained), as we’ll see in the following. 7 [file_i,xy,nnod,sizew,idb,ndof,incid,l,gamma,m,EA,EJ,posit,nbeam,pr]= loadstructure ; [M,K]=assem(incid,l,m,EA,EJ,gamma,idb); I. La Paglia Mechanical Systems Dynamics Input file processing loadstructure () function Call Outputs file_i name of the *. inp file analysed xy N × 2 matrix containing the coordinates of the nodes nnod total number of nodes of the structure sizew maximum dimension of the structure idb N × 3 matrix , numbering each degree of freedom (free and constrained) with different progressive numbers. N is the number of nodes, 3 are the degrees of freedom for each node (1, 2, 3) = (x, y, θ) ndof number of total degrees of freedom incid incidence matrix N × 6, same idea as for idb , but with N number of elements and 6 the degrees of freedom of each element: (1, 2, 3, 4, 5, 6) = ( x1, y 1, θ1, x2, y 2, θ2) 8 [file_i,xy,nnod,sizew,idb,ndof,incid,l,gamma,m,EA,EJ,posit,nbeam,pr]= loadstructure ; I. La Paglia Mechanical Systems Dynamics Input file processing: loadstructure () loadstructure () function Call Outputs l vector containing the length of each element gamma vector containing the rotation angle of each element with respect to the global reference system m vector containing the mass per unit length of each element EA, EJ vectors containing EA and EJ for each elements posit N × 2 matrix containing the xy positions of the elements nbeam number of elements pr vector containing the properties of each element 9 I. La Paglia Mechanical Systems Dynamics [file_i,xy,nnod,sizew,idb,ndof,incid,l,gamma,m,EA,EJ,posit,nbeam,pr]= loadstructure ; Input file processing: loadstructure () ... what are idb and incidence matrix incid for? Considering the simple example 10 If we want to know which is the index of the row, in the assembled matrices, corresponding to the θ DoF of the second node (mid span), we use idb matrix as : index = idb (2,3) = 4 Whereas if we want to know which is the index of the row corresponding to the y DoF of the second node of the second element (II), we would type : index = incid (2,3+2) = 9 I. La Paglia Mechanical Systems Dynamics Input file processing: loadstructure () Example: main code 11 Going back to the example clear all ; close all ; clc L = 1.2; % [m] E = 68e9; % [Pa] b = 40e -3; % [m] h = 8e -3; % [m] r = 2700; % [kg/m^3] m = r*b*h; % [kg/m] J = 1/12*b*h^3; % [m^4] A = b*h; % [m^2] EA = E*A; % [N] EJ = E*J; % [Nm^2] Omax = 100*2* pi ; a = 1.5; Lmax = sqrt (pi^2/a/ Omax * sqrt (EJ/m)); % load the input file and assemble the structure [file_i,xy,nnod,sizew,idb,ndof,incid,l,gamma,m,EA,EJ,posit,nbeam,pr ]=loadstructure; % draw the structure dis_stru (posit,l,gamma,xy,pr,idb,ndof); % assemble mass and stiffness matrices [M,K]= assem (incid,l,m,EA,EJ,gamma,idb ); I. La Paglia Mechanical Systems Dynamics Useful to build the input file 12 I. La Paglia Mechanical Systems Dynamics M FF M FC M CF M CC The mass matrix M is symmetric. According to the boundary conditions, it can be partitioned as ➢ Similarly, also the stiffness matrix K is symmetric and is partitioned accordingly. Mass matrix M Once the M and K matrices have been assembled, natural frequencies and mode shapes of the system can be finally computed. Remembering that 13 FF free dofs inertia/stiffness seen by free dofs FC constrained dofs inertia/stiffness seen by free dofs CF free dofs inertia/stiffness seen by constrained dofs CC constrained dofs inertia/stiffness seen by constrained dofs Natural frequencies and mode shapes are obtained by solving the following eigenvalue problem [modes omega2] = eig (inv(MFF)*KFF); omega = diag (sqrt(omega2)); % Sort frequencies in ascending order [omega_sorted omega_sorted_indices ] = sort(omega); % Sort mode shapes in ascending order modes_sorted = modes(:, omega_sorted_indices ); eig matlab function can be used to this aim (NB frequencies are not in increasing order!) I. La Paglia Mechanical Systems Dynamics eig returns a diagonal matrix « omega2» of eigenvalues and a matrix « modes» whose columns are the corresponding eigenvectors Natural frequencies and mode shapes For the simple case of the pin -pin aluminium beam, the analytical solution can be considered as a reference for computing the natural frequencies of the system Natural frequencies with different value of element length 14 L Mode Analytical Freq [Hz] FEM L 0.4m Freq [Hz] FEM L 0.3m Freq [Hz] FEM L 0.2m Freq [Hz] FEM L 0.1m Freq [Hz] 1° 12.64 12.65 12.65 12.64 12.64 2° 50.57 51.17 50.77 50.61 50.57 3° 113.78 126.29 115.86 114.23 113.82 4° 202.28 234.82 224.51 204.67 202.51 5° 316.06 420.17 356.86 324.45 316.92 I. La Paglia Mechanical Systems Dynamics Natural frequencies and mode shapes 2 15 Mode shapes drawing I. La Paglia Mechanical Systems Dynamics scale_factor = 1.5; mode = modes_sorted (:,ii); figure(); diseg2( mode,scale_factor,incid,l,gamma, posit,idb,xy ); Undeformed structure drawing dis_stru (posit,l,gamma,xy,pr,idb,ndof ); Natural frequencies and mode shapes 1° mode 2° mode 3° mode scale factor for the visualization of the mode shape Assuming that damping ratios �� have been identified by means of proper modal parameters identification techniques, the coefficients α and β necessary to compute the damping matrix can be identified 16 B = [0.01 0.015 0.0098 0.012]'; A = zeros(4,2); for ii=1:4 A(ii,:) = [1/(2*w(ii)) w(ii)/2]; end ab = (A'*A) \A'*B; ab = A \B; CFF = ab(1)*MFF + ab(2)*KFF; Pseudoinverse matrix \ least square error Suppose to know the damping ratios of the first 4 modes I. La Paglia Mechanical Systems Dynamics Damping matrix where %�� L � / �� E � >- �� ? We want now to compute the frequency response function due to a concentrated force applied in the middle of the beam Pay attention to: - have a node where the force is applied - the frequency range of the input force 17 ! Node 7 36 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 38 39 35 Node 7 Y displacement ➢ 18 ° row I. La Paglia Mechanical Systems Dynamics Frequency Response Function 3 ��� L wrr  t� � � 5 3 ��� R s
�w \ .��� L sww II 36 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 38 39 35 Frequency response function can thus be computed 18 F0 = zeros( ndof ,1); F0(18) = 1; om = (0:1:500)*2*pi; for ii=1:length(om) A = -om(ii)^2*MFF + 1i*om(ii)*CFF + KFF; X(:,ii) = A \F0; end 9 → y of node 4 10 → θ of node 4 18 → y of node 7 (colocated ) We get as many FRF as the number of dofs of the system . I. La Paglia Mechanical Systems Dynamics Frequency Response Function F 3 6 / �� E F3 %�� E - �� : A��� L (4A��� 18 ° row being (4 uw xs As an example: MECHANICAL SYSTEM DYNAMICS Finite Elements Method (FEM) in structural dynamics: software implementation in Matlab environment – part 2 I. La Paglia G� G� 2 Another example I. La Paglia Mechanical Systems Dynamics 1. Verify that an element length of 0.5 m is consistent with a dynamic analysis up to 100 Hz 2. Write the *. inp file for the structure 3. Compute the structure’s natural frequencies and vibration modes up to the 3 rd one 4. Compute the damping matrix according to the experimental damping ratios provided 5. Compute the frequency response function of y displacement of node 5 [ [ ] ] A 3 Concentrated elements I. La Paglia Mechanical Systems Dynamics ➢ How to deal with concentrated elements? G� G� 4 Concentrated elements I. La Paglia Mechanical Systems Dynamics Procedure 1. Build the *. inp of the structure neglecting concentrated elements 2. Compute the mass matrix M and the stiffness matrix K 3. Add concentrated elements contributions directly into the corresponding matrices ❑ M asses ❑ Springs ❑ Dampers (if any, after computing the damping matrix C) 4. Partition the matrices to get M FF and K FF 5. Compute the damping matrix C FF 5 Concentrated elements *NODES 1 1 1 1 -1.0 1.0 2 0 0 0 -0.5 1.0 3 0 0 0 0.0 1.0 4 0 0 0 0.5 1.0 5 0 0 0 1.0 1.0 6 0 0 0 0.0 0.5 7 1 1 0 0.0 0.0 *ENDNODES *BEAMS 1 1 2 1 2 2 3 1 3 3 4 1 4 4 5 1 5 3 6 1 6 6 7 1 *ENDBEAMS *PROPERTIES 1 9.75 2.57e7 1.34e4 *ENDPROPERTIES I. La Paglia Mechanical Systems Dynamics 1. Build the *. inp of the structure neglecting concentrated elements G� G� 6 Concentrated elements [file_i,xy,nnod,sizew,idb,ndof,incid,l,gamma,m,EA,EJ,posit,nbeam,pr]=loadstructure; % draw the structure dis_stru(posit,l,gamma,xy ,pr,idb,ndof ); Number structure nodes 7 Number of structure d.o.f . 16 Number of properties 1 Number of beam FE 6 I. La Paglia Mechanical Systems Dynamics +5 constrained [M,K]=assem(incid,l,m,EA,EJ,gamma,idb); ➢ Symmetric sparse matrix ➢ Similar results for stiffness matrix 1. Build the *. inp of the structure neglecting concentrated elements 2. Compute the mass matrix M and the stiffness matrix K G� G� The mass m lof the concentrated element is related to the node 5 displacement along x and y directions, while the inertia Jl is associated to the rotation of node 5. According to idb matrix 7 Concentrated mass I. La Paglia Mechanical Systems Dynamics Node 5 3. Add concentrated elements contributions directly into the corresponding matrices G� G� 8 Concentrated spring I. La Paglia Mechanical Systems Dynamics Concentrated spring kx elongation is given by x displacements of node 5 while spring ky elongation is given by y displacements of node 5. ➢ According to the same procedure, also G� spring contribution can be accounted for Node 5 According to idb matrix 3. Add concentrated elements contributions directly into the corresponding matrices 4. Partition the matrices to get M FF and K FF 5. Compute the damping matrix C FF 9 Concentrated elements MFF = Mtot(1:ndof,1:ndof); KFF = Ktot(1:ndof,1:ndof); B = [0.01 0.015]'; A = zeros(2,2); for ii=1:2 A(ii,:) = [1/(2*w(ii)) w(ii)/2]; end ab = A \B; CFF = ab(1)*MFF + ab(2)*KFF; I. La Paglia Mechanical Systems Dynamics ➢ Symmetric sparse matrix ➢ Similar results for stiffness matrix 10 Natural frequencies and mode shapes Ml_h = diag([ml ml Jl]); E_ml = zeros(3,21); E_ml(:,10:12) = eye(3); Ml = E_ml’*Ml_h*E_ml; 21x21 Mtot = M + Ml; 21x21 MFFtot = Mtot(1:ndof,1:ndof); 16x16 E_kx = zeros(1,21); E_kx(10) = 1; Kx = E_kx’*kx*E_kx; 21x21 E_ky = zeros(1,21); E_ky(11) = 1; Ky = E_ky’*ky*E_ky; 21x21 Ktot = K + Kx + Ky; 21x21 KFFtot = Ktot(1:ndof,1:ndof); 16x16 [x0,w0] = eig (MFFtot \KFFtot ); w = sqrt( diag (w0)); [w,ind ] = sort(w); f = w/(2*pi); mode = x0(:, ind ); -- undeformed -- mode 1 -- mode 2 -- mode 3 I. La Paglia Mechanical Systems Dynamics G� G� 11 Another example I. La Paglia Mechanical Systems Dynamics 1. Verify that an element length of 0.5 m is consistent with a dynamic analysis up to 100 Hz 2. Write the *. inp file for the structure 3. Compute the structure’s natural frequencies and vibration modes up to the 3 rd one 4. Compute the damping matrix according to the experimental damping ratios provided 5. Compute the frequency response function of y displacement of node 5 NOW IT’S YOUR TURN [ [ ] ] A G� G� 12 Concentrated elements ➢ How to deal with this concentrated element? I. La Paglia Mechanical Systems Dynamics 13 Concentrated elements If the finite element approach is considered The spring elastic potential energy reads like I. La Paglia Mechanical Systems Dynamics ➢ >- � �? stiffness matrix of the G spring in the local reference frame 14 Concentrated elements Projection onto the global reference frame with thus I. La Paglia Mechanical Systems Dynamics Rotation matrix ➢ >- � �? stiffness matrix of the G spring in the global reference frame 15 Concentrated elements Assembly into the global reference frame thus I. La Paglia Mechanical Systems Dynamics According to idb matrix ➢ >- �? contribution of the G spring to the overall stiffness matrix ➢ So that finally ' � v � x
�su � sw L s r r r s r r r s ' � s � u
�s � u L s r r r s r r r s 16 Concentrated elements K_k_L = [1 0 0 −1 0 0]’*k*[1 0 0 −1 0 0]; g = 7/4*pi; lambda_k = [cos(g) sin(g) 0 −sin(g) cos(g) 0 0 0 1]; Lambda_k = [lambda_k zeros(3,3) zeros(3,3) lambda_k ]; K_k_G = Lambda_k’* K_k_L * Lambda_k; idofn2 = idb(2,:); idofn6 = idb(6,:); idofk = [idofn2 idofn6]; K(idofk, idofk) = K(idofk, idofk) + K_k_G; I. La Paglia Mechanical Systems Dynamics 17 Concentrated elements ➢ How to deal with this concentrated element? I. La Paglia Mechanical Systems Dynamics G� G� 8 1. We need to place a node in correspondence of the concentrated mass, with the proper constrains 2. We need to properly modify the mass and stiffness matrices to account for the concentrated elements MECHANICAL SYSTEM DYNAMICS Finite Elements Method (FEM) in structural dynamics: software implementation in Matlab environment – part 3 I. La Paglia Suppose you want to compute the structure response considering the first 3 modes only. The modal superposition approach can be used. 2 Modal superposition approach N x3 - Matrix collecting the modal vectors of modes 1, 2 and 3 - Mass matrix - Stiffness matrix - Damping matrix - Generalized force vector I. La Paglia Mechanical Systems Dynamics 3 Modal superposition approach % Modal matrices ii = 1:3 % first 3 mode shapes Phi = modes(:,ii); Mmod = Phi'*MFF*Phi; Kmod = Phi'*KFF*Phi; Cmod = Phi'*CFF*Phi; Fmod = Phi'*F0; % FRF in modal superposition approach for ii = 1:length(Om) xx_mod (:,ii) = ( -Om(ii)^2*Mmod+1i*Om(ii)* Cmod+Kmod )\Fmod ; end xx_2m = Phi * xx_mod ; % 35x3 3x501 FRF_mod = xx_2m( idb (4,2),:); Input node : 7 y Output node : 4 y I. La Paglia Mechanical Systems Dynamics T4A�� � 4 Distributed load Work -equivalent nodal forces/moments Example: a constant distributed load applied on element II (between nodes 2 and 3) I. La Paglia Mechanical Systems Dynamics � � II The distributed load on a beam can be replaced by the equivalent forces at the nodes . At first, these can be computed solving the constraint forces of a fixed -fixed beam (3 times hyperstatic), subjected to the same load . ➢ Constraint bending moments are responsible for the beam deflection! � � � � x � � � � � � x � � � � � � x � � � � x � � x � II 5 Distributed load Work -equivalent nodal forces/moments As an alternative, let us consider the work -equivalent nodal forces, computed according to beam element shape functions The load must be expressed in the local reference frame I. La Paglia Mechanical Systems Dynamics Pay attention to the element orientation �� 6 Distributed load Work -equivalent nodal forces/moments I. La Paglia Mechanical Systems Dynamics Example: a constant distributed load applied on element II (between nodes 2 and 3) Notice that in the proposed example Thus Same as before! � � II 7 Distributed load Work -equivalent nodal forces/moments Example: a constant distributed load applied on element II (between nodes 2 and 3) Rotation matrix I. La Paglia Mechanical Systems Dynamics Nodal forces of beam element II in local and global reference frame ( γII = 0) Local reference frame Global reference frame � � II 8 Distributed load Work -equivalent nodal forces/moments Example: a constant distributed load applied on element II (between nodes 2 and 3) 6 x (3xNnod) matrix 1 for dofs of nodes 2 and 3 I. La Paglia Mechanical Systems Dynamics Finally, the sum over all the elements according to the expansion matrix ' � is performed � � II 9 Distributed load Work -equivalent nodal forces/moments Example: a constant distributed load applied on element II (between nodes 2 and 3) p0 = 100; %[N/m] p0G = [0 p0]'; FG = zeros (3*nnod,1); for ii = 2 % cicle over elements involved : in this case only element II gammaii = gamma(ii); Lk = l(ii); p0L = [cos(gammaii) sin(gammaii); -sin(gammaii) cos(gammaii)]*p0G; FkL = [Lk/2; 0 ; 0 ; Lk /2; 0 ; 0 ]*p0L(1) + ... [0 ; Lk /2; Lk^2/12; 0 ; Lk /2; -Lk^2/12]*p0L(2); FkG = [cos(gammaii) -sin(gammaii) 0 0 0 0; sin(gammaii) cos(gammaii) 0 0 0 0; 0 0 1 0 0 0; 0 0 0 cos(gammaii) -sin(gammaii) 0; 0 0 0 sin(gammaii) cos(gammaii) 0; 0 0 0 0 0 1]* FkL ; Ek = zeros (6,3* nnod ); for jj=1:6 Ek( jj,incid (ii,jj)) = 1; end FG = FG + Ek'* FkG ; end 0 0 5.0000 0.0833 0 5.0000 -0.0833 0 0 0 ... I. La Paglia Mechanical Systems Dynamics 39 x1 10 Static deflection The static deflection of the structure is computed accounting only for constant forces and potential energy derivatives, i.e. generalized component of constant conservative forces. In static conditions Thus xF = KFF \FG(1:ndof); figure(); diseg2(xF,100,incid,l,gamma,posit,idb,xy) title(['static deflection']); Rely on the «diseg2» function to draw the static deflection I. La Paglia Mechanical Systems Dynamics / �� 7T � E % �� 6T � E - �� T � L ( � � 7T � L 6T � L r T � L - �� ? 5( � � 35 x1 11 Exam solution example I. La Paglia Mechanical Systems Dynamics I >GC I ? '# >0 ? ', >0 I 6? External beams 4 4e07 9e03 Internal beams 0.7 1e05 5e03 12 Exam solution example Damping is defined according to the proportional damping assumption :[C]= α[M]+ β[K] . 1. Define aFE model of the structure and save the image of the undeformed structure in a.fig file, named FNxxx 1.fig (F is the first letter of your family name, N is the first letter of your name, xxx are the last three digits of your matriculation number (e .g.Bruni Stefano 123456 :BS 456 ). 2. Calculate the structure vibration modes .Save the images of the first 4 mode shapes in 4 distinct .fig files (named FNxxx 2.fig ,FNxxx 3.fig ,FNxxx 4.fig, FNxxx 5.fig ), with the indication of the associated natural frequencies . 3. Calculate the structure frequency response function which relates the input force at the node k to the output horizontal displacement j(assume the input force to vary in the 0-10 Hz frequency range and set the frequency resolution to 0.01 Hz) .Plot the magnitude and phase diagrams and save the Matlab figures in two files FNxxx 6.fig .Provide a short comment to the diagrams (in the table at the back of this paper) . 4. Calculate the same frequency response function specified in item 3,by developing a model in modal coordinates, limited to the first two modes .Plot the magnitude and phase diagrams superimposed to those of item 3 and save the Matlab figure name FNxxx 7.fig .Provide a short comment to the diagrams (in the table at the back of this paper) . 5. Compute the modal mass and stiffness of the first two modes and reassign the values of α and β so that they result in the following damping ratios for the first two vibration modes :ξ1=2% ,ξ2=3% . 6. Calculate the structure static response to the distributed load p0=1e2N/m and provide the vertical displacement of the node i. I. La Paglia Mechanical Systems Dynamics 13 Exam solution example 1 2 3 4 5 6 10 9 8 7 11 12 13 14 15 16 17 18 19 20 21 24 23 22 25 26 I II III IV V VIII VII VI IX X XI XII XIII XIV XV XVI XVII XVIII XIX XX XXI XXII XXIII XXIV XXV XXVI XXVII XXVIII I. La Paglia Mechanical Systems Dynamics I >GC I ? '# >0 ? ', >0 I 6? External beams 4 4e07 9e03 Internal beams 0.7 1e05 5e03 14 Question 1 Define a FE model of the structure and save the image of the undeformed structure in a .fig file, named FNxxx 1.fig (F is the first letter of your family name, N is the first letter of your name, xxx are the last three digits of your matriculation number (e .g.Bruni Stefano 123456 :BS 456 ). [file_i,xy,nnod,sizee,idb,ndof,incid,l,gamma,m,EA,EJ,posiz,nbeam,pr]= loadstructure ; dis_stru (posiz,l,gamma,xy,pr,idb,ndof ) I. La Paglia Mechanical Systems Dynamics 15 Question 2 Calculate the structure vibration modes .Save the images of the first 4 mode shapes in 4 distinct .fig files (named FNxxx 2.fig ,FNxxx 3.fig ,FNxxx 4.fig, FNxxx 5.fig ), with the indication of the associated natural frequencies . I. La Paglia Mechanical Systems Dynamics 16 Question 3 Calculate the structure frequency response function which relates the input force at the node k to the output horizontal displacement j (assume the input force to vary in the 0-10 Hz frequency range and set the frequency resolution to 0.01 Hz) .Plot the magnitude and phase diagrams and save the Matlab figures in two files FNxxx 6.fig .Provide a short comment to the diagrams (in the table at the back of this paper) . I. La Paglia Mechanical Systems Dynamics Damping contribution 17 Question 4 Calculate the same frequency response function specified in item 3, by developing a model in modal coordinates, limited to the first two modes .Plot the magnitude and phase diagrams superimposed to those of item 3 and save the Matlab figure name FNxxx 7.fig .Provide a short comment to the diagrams (in the table at the back of this paper) . I. La Paglia Mechanical Systems Dynamics Modal coordinate transformation 18 Question 5 Compute the modal mass and stiffness of the first two modes and reassign the values of α and β so that they result in the following damping ratios for the first two vibration modes :ξ1=2% ,ξ2=3% . % Damping Matrix om1 = 2* pi *frqord (1); om2 = 2* pi *frqord (2); ab = [1/2/om1 om1/2; 1/2/om2 om2/2]^ -1*[0.02; 0.03]; alpha = ab(1); beta = ab(2); C = alpha*M + beta*K; CFF = C(1:ndof,1:ndof); % Modal matrices for ii = 1:2 % for the first 2 mode shapes mode = modes (:, ordmode (ii)); Phi (:,ii) = mode; end Mmodal = Phi '*MFF* Phi ; Kmodal = Phi '*KFF* Phi ; Cmodal = Phi '*CFF* Phi ; I. La Paglia Mechanical Systems Dynamics 19 Question 6 Calculate the structure static response to the distributed load p0=1e2N/m and provide the vertical displacement of the node i. Static deformation of the structure (scale factor = 10) Δ yi= - 0.0192 m I. La Paglia Mechanical Systems Dynamics 20 Hints for the Assignment I. La Paglia Mechanical Systems Dynamics 7. Compute the static response of the structure due to : a) the deck weight only ; b) the weight of the entire structure . Plot the deformed shape of the structure compared to the undeformed configuration and compute the value of the maximum deflection . ❑ Different strategies can be adopted to introduce the weight of the structure, namely distributed and concentrated loads . ❑ Depending on your choice, properly build the external force vector . MECHANICAL SYSTEM DYNAMICS Finite Elements Method (FEM) in structural dynamics: software implementation in Matlab environment – part 4 I. La Paglia 2 Internal forces I. La Paglia Mechanical Systems Dynamics When dealing with Euler -Bernoulli beam, the internal actions can be computed as follows: 0 L '# �Q �� / L ', � 6S � �6 6 L ', � 7S � �7 According to the FE formulation, the axial and the transverse displacement of a beam section are defined as function of the nodal displacements through the shape functions (which are defined in the local reference frame of each finite element) Q �
�P L = E >� S �
�P L = E >� E ?�6 E @ �7 3 Internal forces Axial force I. La Paglia Mechanical Systems Dynamics � Q � � L > L T � � F T � � . � 0 is constant along the beam element 0 L '# �Q � � � � L �Q � � is constant along the beam element The shape functions for axial deformation are assumed to be linear functions of ξ 4 Internal forces Bending moment and shear force I. La Paglia Mechanical Systems Dynamics The shape functions for bending deformation are assumed to be cubic functions of ξ The bending moment and the shear force are proportional to the second and third partial derivatives of S with respect to ξ / L ', � 6S � �6 � 6S � �6 L t? E x@ � 6 L ', � 7S � �7 � 7S � �7 L x@ 5 Internal forces I. La Paglia Mechanical Systems Dynamics By properly rearranging the partial derivatives according to matrix formulation, the internal forces can be written as the product of the vector collecting the shape functions derivatives and the vector of the nodal coordinates 0 L '# �Q �� �Q �� L > / L ', � 6S � �6 � 6S � �6 L t? E x@ � 6 L ', � 7S � �7 � 7S � �7 L x@ As an example, consider the bending moment M at the specific � axial coordinate along the local reference frame of the finite element ➢ Similar results can be obtained considering the axial and shear force where 6 Internal forces Shear force in section A -A (element XI close to node 12) I. La Paglia Mechanical Systems Dynamics n_el = 11; L_el = l(n_el); idof_i = idb(11,:); idof_j = idb(12,:); lambda = [cos(gamma(n_el)) sin(gamma(n_el)) 0; -sin(gamma(n_el)) cos(gamma(n_el)) 0; 0 0 1]; Xi = lambda*X(idof_i,:); Xj = lambda*X(idof_j,:); d = 2/L_el^3*Xi(2,:) -2/L_el^3*Xj(2,:) ... +1/L_el^2*Xi(3,:) +1/L_el^2*Xj(3,:); T = EJ(n_el)*(6*d); XI A A 7 Motion imposed at the constraints I. La Paglia Mechanical Systems Dynamics Input: node 1 y imposed displacement Output: node 8 rotation U P L U4A�� � θ8 In the case under study: T� L U :P; (� L r (� L r 8 Motion imposed at the constraints I. La Paglia Mechanical Systems Dynamics U P L U4A�� � θ8 being T� an harmonic function of time t T� will be harmonic too with the same frequency Input: node 1 y imposed displacement Output: node 8 rotation :�4 :�4 �< U 9 I. La Paglia Mechanical Systems Dynamics U P L U4A�� � θ8 ncons = nnod*3 -ndof; Y = zeros(ncons,1); Y(idb(1,2) -ndof) = 1; for ii = 1:length(om) X(:,ii) = -(-MFF*om(ii)^2 + 1i*CFF*om(ii) + KFF) \ ... (-MFC*om(ii)^2 + 1i*CFC*om(ii) + KFC)*Y; end FRF = X(idb(8,3),:); Motion imposed at the constraints Input: node 1 y imposed displacement Output: node 8 rotation 10 Time response in steady -state condition I. La Paglia Mechanical Systems Dynamics ncons = nnod*3 -ndof; Y = zeros(ncons,1); Y(idb(1,2) -ndof) = 0.01; %[m] om = 2*pi*60; %[rad/s] xx = -(-MFF*om^2 + 1i*CFF*om + KFF) \ ... (-MFC*om^2 + 1i*CFC*om + KFC)*Y; x = xx(idb(8,3)); U P L U4A�� � θ8 Time response in steady -state condition if U4 L r
�rs I and B L xr *V 1x1 complex vector Input: node 1 y imposed displacement Output: node 8 rotation 11 I. La Paglia Mechanical Systems Dynamics Time response in steady -state condition if U4 L r
�rs I and B L xr *V U P L U4A�� � θ8 dt = 0.0001; t = 0:dt:1; x_t = zeros(length(t),1); for ii = 1:length(t) x_t(ii) = abs (x)*cos(om*t(ii)+angle(x)); end Rely on the IDFT to evaluate the time response Input: node 1 y imposed displacement Output: node 8 rotation Time response in steady -state condition 12 Constraint forces I. La Paglia Mechanical Systems Dynamics In the case under study: T� L U :P; (� L r (� L r Input: node 1 y imposed displacement Output: node 1 y constraint force U P L U4A�� � 4 �- 13 I. La Paglia Mechanical Systems Dynamics U P L U4A�� � 4 �- Compute the force necessary to apply the motion imposed at the constraint with U4 L r
�rs I Y = zeros(ncons,1); Y(idb(1,2) -ndof) = 0.01; for ii = 1:length(om) X(:,ii)= -(-MFF*om(ii)^2+1i*CFF*om(ii)+KFF) \... (-MFC*om(ii)^2+1i*CFC*om(ii)+KFC)*Y; R(:,ii)=( -MCF*om(ii)^2+1i*CCF*om(ii)+KCF)*X(:,ii)+... (-MCC*om(ii)^2+1i*CCC*om(ii)+KCC)*Y; end Ry1 = R(idb(1,2) -ndof,:); Constraint forces Input: node 1 y imposed displacement Output: node 1 y constraint force 14 Assignment I. La Paglia Mechanical Systems Dynamics 15 Hints for the Assignment A tuned mass damper (TMD) is a device consisting of a mass I 6, a spring G6 and a damper ?6. It is mounted on structures in order to reduce the vibration amplitude of a specific resonance. To this end, its natural frequency is “tuned” into a specific structural mode, so that when it is excited by an external force, the damper will resonate out of phase so as to dissipate energy. TMD Structure 6. Design a TMD to be installed on the structure so as to reduce by at least 15 % the amplitude of the vertical vibration of node A in correspondence with the resonance peak of the first mode of vibration, for the same excitation condition of question 4.The increase of the structure mass must be limited to maximum 2% while the damping ratio of the TMD alone must not exceed 30 % .Provide a plot comparing the FRF results corresponding to the structure with and without the TMD . ❑ Tune the TMD on the first natural frequency of the truss bridge ( � 6 L � 5; and investigate the effect of its parameters over the system response. Hint: rely on the modal approach reduced to the first d.o.f. only ❑ Based on the parametric analysis, choose a proper value for the mass I 6 and derive G6 (given � 6); choose the damping ratio �6 and derive ?6 ❑ Install the TMD on the bridge structure. Hint: remember to account for the additional 1 d.o.f. of the TMD both in terms of matrices size and contributions of the lamped parameters ❑ NB: do not change the overall system response! The FRF should change only in the first mode I. La Paglia Mechanical Systems Dynamics 15 Hints for the Assignment A tuned mass damper (TMD) is a device consisting of a mass I 6, a spring G6 and a damper ?6. It is mounted on structures in order to reduce the vibration amplitude of a specific resonance. To this end, its natural frequency is “tuned” into a specific structural mode, so that when it is excited by an external force, the damper will resonate out of phase so as to dissipate energy. TMD Structure 6. Design a TMD to be installed on the structure so as to reduce by at least 15 % the amplitude of the vertical vibration of node A in correspondence with the resonance peak of the first mode of vibration, for the same excitation condition of question 4.The increase of the structure mass must be limited to maximum 2% while the damping ratio of the TMD alone must not exceed 30 % .Provide a plot comparing the FRF results corresponding to the structure with and without the TMD . ❑ Tune the TMD on the first natural frequency of the truss bridge ( � 6 L � 5; and investigate the effect of its parameters over the system response. Hint: rely on the modal approach reduced to the first d.o.f. only ❑ Based on the parametric analysis, choose a proper value for the mass I 6 and derive G6 (given � 6); choose the damping ratio �6 and derive ?6 ❑ Install the TMD on the bridge structure. Hint: remember to account for the additional 1 d.o.f. of the TMD both in terms of matrices size and contributions of the lamped parameters ❑ NB: do not change the overall system response! The FRF should change only in the first mode I. La Paglia Mechanical Systems Dynamics 15 Hints for the Assignment A tuned mass damper (TMD) is a device consisting of a mass I 6, a spring G6 and a damper ?6. It is mounted on structures in order to reduce the vibration amplitude of a specific resonance. To this end, its natural frequency is “tuned” into a specific structural mode, so that when it is excited by an external force, the damper will resonate out of phase so as to dissipate energy. TMD Structure 6. Design a TMD to be installed on the structure so as to reduce by at least 15 % the amplitude of the vertical vibration of node A in correspondence with the resonance peak of the first mode of vibration, for the same excitation condition of question 4.The increase of the structure mass must be limited to maximum 2% while the damping ratio of the TMD alone must not exceed 30 % .Provide a plot comparing the FRF results corresponding to the structure with and without the TMD . ❑ Tune the TMD on the first natural frequency of the truss bridge ( � 6 L � 5; and investigate the effect of its parameters over the system response. Hint: rely on the modal approach reduced to the first d.o.f. only ❑ Based on the parametric analysis, choose a proper value for the mass I 6 and derive G6 (given � 6); choose the damping ratio �6 and derive ?6 ❑ Install the TMD on the bridge structure. Hint: remember to account for the additional 1 d.o.f. of the TMD both in terms of matrices size and contributions of the lamped parameters ❑ NB: do not change the overall system response! The FRF should change only in the first mode I. La Paglia Mechanical Systems Dynamics