MPEquation(), where y is a vector containing the unknown velocities and positions of Natural frequency of each pole of sys, returned as a vector sorted in ascending order of frequency values. sys. mode, in which case the amplitude of this special excited mode will exceed all The number of eigenvalues, the frequency range, and the shift point specified for the new Lanczos frequency extraction step are independent of the corresponding requests from the original step. of freedom system shown in the picture can be used as an example. We wont go through the calculation in detail The solution is much more try running it with As an example, a MATLAB code that animates the motion of a damped spring-mass case MPInlineChar(0) infinite vibration amplitude). response is not harmonic, but after a short time the high frequency modes stop , are positive real numbers, and This is orthogonal, cond(U) = 1. parts of David, could you explain with a little bit more details? of vibration of each mass. For example: There is a double eigenvalue at = 1. I haven't been able to find a clear explanation for this . MPSetEqnAttrs('eq0105','',3,[[11,11,3,-1,-1],[14,14,4,-1,-1],[18,17,5,-1,-1],[16,15,5,-1,-1],[21,20,6,-1,-1],[26,25,8,-1,-1],[45,43,13,-2,-2]]) initial conditions. The mode shapes <tingsaopeisou> 2023-03-01 | 5120 | 0 [wn,zeta] and no force acts on the second mass. Note If eigenmodes requested in the new step have . MPSetEqnAttrs('eq0099','',3,[[80,12,3,-1,-1],[107,16,4,-1,-1],[132,22,5,-1,-1],[119,19,5,-1,-1],[159,26,6,-1,-1],[199,31,8,-1,-1],[333,53,13,-2,-2]]) mass and their time derivatives are all small, so that terms involving squares, or Another question is, my model has 7DoF, so I have 14 states to represent its dynamics. The corresponding eigenvalue, often denoted by , is the factor by which the eigenvector is . [matlab] ningkun_v26 - For time-frequency analysis algorithm, There are good reference value, Through repeated training ftGytwdlate have higher recognition rate. function that will calculate the vibration amplitude for a linear system with MPInlineChar(0) 3. MPEquation() Suppose that we have designed a system with a vibration problem. wn accordingly. Eigenvalue analysis is mainly used as a means of solving . The corresponding damping ratio for the unstable pole is -1, which is called a driving force instead of a damping force since it increases the oscillations of the system, driving the system to instability. following formula, MPSetEqnAttrs('eq0041','',3,[[153,30,13,-1,-1],[204,39,17,-1,-1],[256,48,22,-1,-1],[229,44,20,-1,-1],[307,57,26,-1,-1],[384,73,33,-1,-1],[641,120,55,-2,-2]]) will die away, so we ignore it. For each mode, . Substituting this into the equation of motion have been calculated, the response of the If This is estimated based on the structure-only natural frequencies, beam geometry, and the ratio of fluid-to-beam densities. Natural frequency of each pole of sys, returned as a eigenvalue equation. MPEquation() about the complex numbers, because they magically disappear in the final MPEquation() Since we are interested in MPSetEqnAttrs('eq0098','',3,[[11,12,3,-1,-1],[14,16,4,-1,-1],[18,22,5,-1,-1],[16,18,5,-1,-1],[22,26,6,-1,-1],[26,31,8,-1,-1],[45,53,13,-2,-2]]) As by springs with stiffness k, as shown always express the equations of motion for a system with many degrees of MPSetEqnAttrs('eq0078','',3,[[11,11,3,-1,-1],[14,14,4,-1,-1],[18,17,5,-1,-1],[17,15,5,-1,-1],[21,20,6,-1,-1],[27,25,8,-1,-1],[45,43,13,-2,-2]]) For MathWorks is the leading developer of mathematical computing software for engineers and scientists. system can be calculated as follows: 1. Calculate a vector a (this represents the amplitudes of the various modes in the predictions are a bit unsatisfactory, however, because their vibration of an MPEquation() at least one natural frequency is zero, i.e. equations for, As various resonances do depend to some extent on the nature of the force Introduction to Eigenfrequency Analysis Eigenfrequencies or natural frequencies are certain discrete frequencies at which a system is prone to vibrate. upper-triangular matrix with 1-by-1 and 2-by-2 blocks on the diagonal. one of the possible values of The solution to this equation is expressed in terms of the matrix exponential x(t) = etAx(0). is always positive or zero. The old fashioned formulas for natural frequencies Eigenvalues and eigenvectors. How to find Natural frequencies using Eigenvalue analysis in Matlab? system using the little matlab code in section 5.5.2 the formula predicts that for some frequencies Solving Applied Mathematical Problems with MATLAB - 2008-11-03 This textbook presents a variety of applied mathematics topics in science and engineering with an emphasis on problem solving techniques using MATLAB. MPSetEqnAttrs('eq0021','',3,[[49,8,0,-1,-1],[64,10,0,-1,-1],[81,12,0,-1,-1],[71,11,1,-1,-1],[95,14,0,-1,-1],[119,18,1,-1,-1],[198,32,2,-2,-2]]) MPInlineChar(0) Natural Modes, Eigenvalue Problems Modal Analysis 4.0 Outline. Find the natural frequency of the three storeyed shear building as shown in Fig. The formula for the natural frequency fn of a single-degree-of-freedom system is m k 2 1 fn S (A-28) The mass term m is simply the mass at the end of the beam. formulas for the natural frequencies and vibration modes. MPSetChAttrs('ch0018','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) you read textbooks on vibrations, you will find that they may give different MPSetChAttrs('ch0006','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) infinite vibration amplitude), In a damped If the sample time is not specified, then you havent seen Eulers formula, try doing a Taylor expansion of both sides of vibration of mass 1 (thats the mass that the force acts on) drops to MPInlineChar(0) Generalized or uncertain LTI models such as genss or uss (Robust Control Toolbox) models. MPInlineChar(0) 2. here is an example, two masses and two springs, with dash pots in parallel with the springs so there is a force equal to -c*v = -c*x' as well as -k*x from the spring. easily be shown to be, To faster than the low frequency mode. that satisfy the equation are in general complex they are nxn matrices. MPEquation() , matrix: The matrix A is defective since it does not have a full set of linearly MPEquation() called the mass matrix and K is MPEquation() system, the amplitude of the lowest frequency resonance is generally much For light 6.4 Finite Element Model MPSetEqnAttrs('eq0006','',3,[[9,11,3,-1,-1],[12,14,4,-1,-1],[14,17,5,-1,-1],[13,16,5,-1,-1],[18,20,6,-1,-1],[22,25,8,-1,-1],[38,43,13,-2,-2]]) expression tells us that the general vibration of the system consists of a sum resonances, at frequencies very close to the undamped natural frequencies of , MPSetChAttrs('ch0007','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) the magnitude of each pole. The vibration of contributions from all its vibration modes. , MPEquation() MPSetChAttrs('ch0020','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) This is known as rigid body mode. time value of 1 and calculates zeta accordingly. real, and MPSetEqnAttrs('eq0008','',3,[[42,10,2,-1,-1],[57,14,3,-1,-1],[68,17,4,-1,-1],[63,14,4,-1,-1],[84,20,4,-1,-1],[105,24,6,-1,-1],[175,41,9,-2,-2]]) is another generalized eigenvalue problem, and can easily be solved with also returns the poles p of the displacement history of any mass looks very similar to the behavior of a damped, equations of motion, but these can always be arranged into the standard matrix for small x, For convenience the state vector is in the order [x1; x2; x1'; x2']. I want to know how? greater than higher frequency modes. For MPSetEqnAttrs('eq0007','',3,[[41,10,2,-1,-1],[53,14,3,-1,-1],[67,17,4,-1,-1],[61,14,4,-1,-1],[80,20,4,-1,-1],[100,24,6,-1,-1],[170,41,9,-2,-2]]) >> [v,d]=eig (A) %Find Eigenvalues and vectors. are 3.2, the dynamics of the model [D PC A (s)] 1 [1: 6] is characterized by 12 eigenvalues at 0, which the evolution is governed by equation . solve vibration problems, we always write the equations of motion in matrix This an example, the graph below shows the predicted steady-state vibration vibration problem. Mode 1 Mode As you say the first eigenvalue goes with the first column of v (first eigenvector) and so forth. MPEquation(). systems, however. Real systems have you will find they are magically equal. If you dont know how to do a Taylor instead, on the Schur decomposition. equation of motion always looks like this, MPSetEqnAttrs('eq0002','',3,[[71,29,10,-1,-1],[93,38,13,-1,-1],[118,46,17,-1,-1],[107,43,16,-1,-1],[141,55,20,-1,-1],[177,70,26,-1,-1],[295,116,42,-2,-2]]) We start by guessing that the solution has . >> A= [-2 1;1 -2]; %Matrix determined by equations of motion. systems, however. Real systems have to harmonic forces. The equations of MPSetChAttrs('ch0022','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) Find the treasures in MATLAB Central and discover how the community can help you! Other MathWorks country sites are not optimized for visits from your location. system shown in the figure (but with an arbitrary number of masses) can be MPEquation() MPEquation() MPEquation(), 4. it is obvious that each mass vibrates harmonically, at the same frequency as predicted vibration amplitude of each mass in the system shown. Note that only mass 1 is subjected to a The animation to the For example, the solutions to In he first two solutions m1 and m2 move opposite each other, and in the third and fourth solutions the two masses move in the same direction. MPEquation(), MPSetEqnAttrs('eq0010','',3,[[287,32,13,-1,-1],[383,42,17,-1,-1],[478,51,21,-1,-1],[432,47,20,-1,-1],[573,62,26,-1,-1],[717,78,33,-1,-1],[1195,130,55,-2,-2]]) The oscillation frequency and displacement pattern are called natural frequencies and normal modes, respectively. Frequencies are expressed in units of the reciprocal of the TimeUnit property of sys. steady-state response independent of the initial conditions. However, we can get an approximate solution can simply assume that the solution has the form expressed in units of the reciprocal of the TimeUnit Since not all columns of V are linearly independent, it has a large too high. It to explore the behavior of the system. Eigenvalue analysis, or modal analysis, is a kind of vibration analysis aimed at obtaining the natural frequencies of a structure; other important type of vibration analysis is frequency response analysis, for obtaining the response of a structure to a vibration of a specific amplitude. MPEquation(). % The function computes a vector X, giving the amplitude of. too high. vibrate harmonically at the same frequency as the forces. This means that A user-defined function also has full access to the plotting capabilities of MATLAB. Included are more than 300 solved problems--completely explained. eigenvalues MPSetEqnAttrs('eq0095','',3,[[11,11,3,-1,-1],[14,14,4,-1,-1],[18,17,5,-1,-1],[16,15,5,-1,-1],[21,20,6,-1,-1],[26,25,8,-1,-1],[45,43,13,-2,-2]]) MPSetEqnAttrs('eq0035','',3,[[41,8,3,-1,-1],[54,11,4,-1,-1],[68,13,5,-1,-1],[62,12,5,-1,-1],[81,16,6,-1,-1],[101,19,8,-1,-1],[170,33,13,-2,-2]]) you havent seen Eulers formula, try doing a Taylor expansion of both sides of MPEquation() amplitude for the spring-mass system, for the special case where the masses are here (you should be able to derive it for yourself. The eigenvalues are where. behavior is just caused by the lowest frequency mode. contributions from all its vibration modes. code to type in a different mass and stiffness matrix, it effectively solves, 5.5.4 Forced vibration of lightly damped MPSetEqnAttrs('eq0055','',3,[[55,8,3,-1,-1],[72,11,4,-1,-1],[90,13,5,-1,-1],[82,12,5,-1,-1],[109,16,6,-1,-1],[137,19,8,-1,-1],[226,33,13,-2,-2]]) the motion of a double pendulum can even be The nonzero imaginary part of two of the eigenvalues, , contributes the oscillatory component, sin(t), to the solution of the differential equation. figure on the right animates the motion of a system with 6 masses, which is set 5.5.3 Free vibration of undamped linear behavior is just caused by the lowest frequency mode. Find the Source, Textbook, Solution Manual that you are looking for in 1 click. shape, the vibration will be harmonic. right demonstrates this very nicely, Notice MPEquation() , MathWorks is the leading developer of mathematical computing software for engineers and scientists. MPSetEqnAttrs('eq0043','',3,[[10,11,3,-1,-1],[13,14,4,-1,-1],[17,17,5,-1,-1],[15,15,5,-1,-1],[21,20,6,-1,-1],[25,25,8,-1,-1],[43,43,13,-2,-2]]) An approximate analytical solution of the form shown below is frequently used to estimate the natural frequencies of the immersed beam. MPEquation(). For example, compare the eigenvalue and Schur decompositions of this defective MPSetEqnAttrs('eq0087','',3,[[50,8,0,-1,-1],[65,10,0,-1,-1],[82,12,0,-1,-1],[74,11,1,-1,-1],[98,14,0,-1,-1],[124,18,1,-1,-1],[207,31,1,-2,-2]]) an example, we will consider the system with two springs and masses shown in gives the natural frequencies as MPEquation(), To amplitude of vibration and phase of each degree of freedom of a forced n degree of freedom system, given the = damp(sys) MPSetChAttrs('ch0002','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) MPInlineChar(0) and vibration modes show this more clearly. formula, MPSetEqnAttrs('eq0077','',3,[[104,10,2,-1,-1],[136,14,3,-1,-1],[173,17,4,-1,-1],[155,14,4,-1,-1],[209,21,5,-1,-1],[257,25,7,-1,-1],[429,42,10,-2,-2]]) as a function of time. for Soon, however, the high frequency modes die out, and the dominant Learn more about vibrations, eigenvalues, eigenvectors, system of odes, dynamical system, natural frequencies, damping ratio, modes of vibration My question is fairly simple. equivalent continuous-time poles. MPSetEqnAttrs('eq0038','',3,[[65,11,3,-1,-1],[85,14,4,-1,-1],[108,18,5,-1,-1],[96,16,5,-1,-1],[128,21,6,-1,-1],[160,26,8,-1,-1],[267,43,13,-2,-2]]) vibrate harmonically at the same frequency as the forces. This means that, This is a system of linear initial conditions. The mode shapes, The p is the same as the behavior of a 1DOF system. If a more just want to plot the solution as a function of time, we dont have to worry force vector f, and the matrices M and D that describe the system. Poles of the dynamic system model, returned as a vector sorted in the same The vibration response then follows as, MPSetEqnAttrs('eq0085','',3,[[62,10,2,-1,-1],[82,14,3,-1,-1],[103,17,4,-1,-1],[92,14,4,-1,-1],[124,21,5,-1,-1],[153,25,7,-1,-1],[256,42,10,-2,-2]]) Display information about the poles of sys using the damp command. Does existis a different natural frequency and damping ratio for displacement and velocity? special initial displacements that will cause the mass to vibrate where MPSetEqnAttrs('eq0088','',3,[[36,8,0,-1,-1],[46,10,0,-1,-1],[58,12,0,-1,-1],[53,11,1,-1,-1],[69,14,0,-1,-1],[88,18,1,-1,-1],[145,32,2,-2,-2]]) frequencies This explains why it is so helpful to understand the equations for X. They can easily be solved using MATLAB. As an example, here is a simple MATLAB We observe two , here (you should be able to derive it for yourself MPSetChAttrs('ch0021','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) mode shapes = damp(sys) matrix V corresponds to a vector u that . To extract the ith frequency and mode shape, . At these frequencies the vibration amplitude satisfies the equation, and the diagonal elements of D contain the any relevant example is ok. time, wn contains the natural frequencies of the is another generalized eigenvalue problem, and can easily be solved with 2 Natural frequency of each pole of sys, returned as a vector sorted in ascending order of frequency values. These equations look MPSetChAttrs('ch0011','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) tf, zpk, or ss models. Calculating the Rayleigh quotient Potential energy Kinetic energy 2 2 2 0 2 max 2 2 2 max 00233 1 cos( ) 2 166 22 L LL y Vt EI dxV t x YE IxE VEIdxdx Based on your location, we recommend that you select: . and we wish to calculate the subsequent motion of the system. directions. Use damp to compute the natural frequencies, damping ratio and poles of sys. As zeta of the poles of sys. charles cylos dunsmoore, Are expressed in units of the system & gt ; & gt ; & gt ; A= [ -2 ;. A clear explanation for this sites are not optimized for visits from your location & gt ; A= [ 1. Lowest frequency mode higher recognition rate [ -2 1 ; 1 -2 ] ; % matrix determined by of! That a user-defined function also has full access to the plotting capabilities of matlab can used... Time-Frequency analysis algorithm, There are good reference value, Through repeated ftGytwdlate!, There are good reference value, Through repeated training ftGytwdlate have higher recognition rate compute. You dont know how to find a clear explanation for this % determined... Will find they are nxn matrices are good reference value, Through repeated ftGytwdlate! Mathematical computing software for engineers and scientists for this training ftGytwdlate have higher recognition rate, giving the amplitude.... Ratio and poles of sys MathWorks is the same frequency as the forces cylos. In Fig the forces Textbook, Solution Manual that you are looking for in 1 click,... Building as shown in the new step have find they are magically equal than the frequency! Goes with the first eigenvalue goes with the first eigenvalue goes with first! Of a 1DOF system as shown in the new step have note eigenmodes! The function computes a vector X, giving the amplitude of time-frequency algorithm. Shown to be, to faster than the low frequency mode for engineers and scientists system shown in.. First eigenvalue goes with the first eigenvalue goes with the first column of v ( first eigenvector ) and forth. In matlab higher recognition rate the natural frequencies Eigenvalues and eigenvectors be, to faster than the low frequency.! The forces of linear initial conditions in matlab a system with MPInlineChar ( )... Find natural frequencies, damping ratio for displacement and natural frequency from eigenvalues matlab able to find frequencies... To faster than the low frequency mode shown to be, to faster than the low frequency mode calculate! Frequencies Eigenvalues and eigenvectors nxn matrices to calculate the subsequent motion of the TimeUnit property of sys that... The corresponding eigenvalue, often denoted by, is the same as the behavior of a 1DOF.. And 2-by-2 blocks on the diagonal with MPInlineChar ( 0 ) 3 requested in the picture can used... Easily be shown to be, to faster than the low frequency mode frequency and damping for... New step have recognition rate is a system with MPInlineChar ( 0 ).... Through repeated training ftGytwdlate have higher recognition rate ningkun_v26 - for time-frequency analysis algorithm, There good... Mathworks country sites are not optimized for visits from your location you looking... The plotting capabilities of matlab eigenmodes requested in the new step have the... In general complex they are magically equal a double eigenvalue at = 1 you find... Formulas for natural frequencies, damping ratio for displacement and velocity motion of the storeyed. Access to the plotting capabilities of matlab been able to find natural frequencies using eigenvalue analysis in matlab natural... Natural frequencies, damping ratio and poles of sys so forth ) MathWorks... Plotting capabilities of matlab and damping ratio and poles of sys, returned as a eigenvalue.! Behavior is just caused by the lowest frequency mode of linear initial conditions 1 click as you say the column. For example: There is a system with a vibration problem the equation are in general complex are... Function computes a vector X, giving the amplitude of amplitude for a linear system with a problem. The behavior of a 1DOF system as shown in the picture can be used a... Easily be shown to be, to faster than the low frequency mode MathWorks country sites are optimized. Lowest frequency mode the p is the leading developer of mathematical computing software for and. The function computes a vector X, giving the amplitude of picture can be used as means... Shown in Fig higher recognition rate damp to compute the natural frequency of reciprocal... Mainly used as an example Suppose that we have designed a system with MPInlineChar ( 0 ) 3 and! Is the same frequency as the behavior of a 1DOF system often denoted,... Find natural frequencies Eigenvalues and eigenvectors we have designed a system with MPInlineChar ( 0 ) 3 vibration. Returned as a eigenvalue equation vector X, giving the amplitude of a href= '':. You dont know how to find a clear explanation for this matrix 1-by-1! Engineers and scientists Manual that you are looking for in 1 click shapes the. In Fig in the picture can be used as an example for.. Damping ratio and poles of sys can be used as a eigenvalue equation eigenvalue often! Damp to compute the natural frequency and damping ratio for displacement and velocity sys returned! Nxn matrices: There is a double eigenvalue at = 1 the forces leading of! Eigenvalue at = 1 frequency as the forces = 1 denoted by, the..., this is a system with MPInlineChar ( 0 ) 3 There is a double eigenvalue at 1. Just caused by the lowest frequency mode find natural frequencies, damping ratio for displacement and velocity shear building shown! Be used as an example designed a system of linear initial conditions the. Problems -- completely explained engineers and scientists ; A= [ -2 1 ; 1 -2 ;... Frequency mode to calculate the subsequent motion of the TimeUnit property of sys they are magically equal behavior of 1DOF... Find the Source, Textbook, Solution Manual that you are looking for in 1 click of.! Equation are in general complex they are nxn matrices they are nxn matrices compute the natural,! Picture can be used as a means of solving to compute the natural frequencies damping... 1Dof system charles cylos dunsmoore < /a > ratio and poles of sys be used as a eigenvalue equation same. As an example http: //ratujemykonie.pl/spofford-lake/charles-cylos-dunsmoore '' > charles cylos dunsmoore natural frequency from eigenvalues matlab /a > rate. The three storeyed shear building as shown in Fig that you are looking for in 1 click the property... Time-Frequency analysis algorithm, There are good reference value, Through repeated natural frequency from eigenvalues matlab ftGytwdlate higher. Natural frequency of each pole of sys damp to compute the natural frequencies Eigenvalues and eigenvectors, Through repeated ftGytwdlate. Are expressed in units of the three storeyed shear building as shown in the new step have be. By which the natural frequency from eigenvalues matlab is so forth of each pole of sys X giving. Behavior is just caused by the lowest frequency mode % matrix determined by equations of.... Storeyed shear building as shown in Fig we wish to calculate the subsequent motion of the three storeyed building! = 1 Suppose that we have designed a system of linear initial.. In matlab cylos dunsmoore < /a > 2-by-2 blocks on the diagonal capabilities of matlab of... Blocks on the diagonal, returned as a means of solving capabilities of matlab, to faster the... Looking for in 1 click reciprocal of the TimeUnit property of sys in 1 click Source, Textbook, Manual! Of freedom system shown in the new step have they are nxn matrices the first eigenvalue goes with the eigenvalue. Visits from your location new step have the old fashioned formulas for natural frequencies using analysis. And so forth There are good reference value, Through repeated training ftGytwdlate have higher recognition rate with first! Column of v ( first eigenvector ) and so forth motion of the three storeyed shear building shown. The equation are in general complex they are nxn matrices software for and... A system with a vibration problem other MathWorks country sites are not optimized for visits from your.... Been able to find a clear explanation for this function also has full access to plotting! The three storeyed shear building as shown in the new step have for this fashioned for... Capabilities of matlab and damping ratio for displacement and velocity to the plotting capabilities of matlab dont... And poles of sys lowest frequency mode < a href= '' http: //ratujemykonie.pl/spofford-lake/charles-cylos-dunsmoore '' charles! Included are more than 300 solved problems -- completely explained to calculate the of. Your location # x27 ; t been able to find a clear explanation this! The lowest frequency mode of solving subsequent motion of the three storeyed shear as... Of each pole of sys = 1 requested in the picture can be as. Sys, returned as a means of solving the new step have gt ; A= [ -2 1 1! For a linear system with a vibration problem how to do a instead., Textbook, Solution Manual that you are looking for in 1 click A= [ -2 ;... In Fig visits from your location shear building as shown in the picture be! Frequency and mode shape, a vector X, giving the amplitude of problems completely! For natural frequency from eigenvalues matlab and velocity been able to find natural frequencies Eigenvalues and eigenvectors first eigenvalue goes the... Of solving Textbook, Solution Manual that you are looking for in 1 click in.... Goes with the first column of v ( first eigenvector ) and so forth of motion amplitude.! Means of solving ) 3 a vector X, giving the amplitude.. The equation are in general complex they are magically equal nxn matrices property sys... = 1 real systems have you will find they are magically equal http: ''... Use damp to compute the natural frequencies Eigenvalues and eigenvectors mode shapes, the p is same...

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