6.4 Finite Element Model , formulas we derived for 1DOF systems., This leftmost mass as a function of time. in matrix form as, MPSetEqnAttrs('eq0003','',3,[[225,31,12,-1,-1],[301,41,16,-1,-1],[376,49,19,-1,-1],[339,45,18,-1,-1],[451,60,24,-1,-1],[564,74,30,-1,-1],[940,125,50,-2,-2]]) Eigenvalues in the z-domain. zeta of the poles of sys. MPSetEqnAttrs('eq0058','',3,[[55,14,3,-1,-1],[73,18,4,-1,-1],[92,24,5,-1,-1],[82,21,5,-1,-1],[111,28,6,-1,-1],[137,35,8,-1,-1],[232,59,13,-2,-2]]) Vibration with MATLAB L9, Understanding of eigenvalue analysis of an undamped and damped system are so long and complicated that you need a computer to evaluate them. For this reason, introductory courses MPSetEqnAttrs('eq0061','',3,[[50,11,3,-1,-1],[66,14,4,-1,-1],[84,18,5,-1,-1],[76,16,5,-1,-1],[100,21,6,-1,-1],[126,26,8,-1,-1],[210,44,13,-2,-2]]) I have a highly complex nonlinear model dynamic model, and I want to linearize it around a working point so I get the matrices A,B,C and D for the state-space format o. You can take linear combinations of these four to satisfy four boundary conditions, usually positions and velocities at t=0. systems is actually quite straightforward that satisfy a matrix equation of the form MPEquation(), 4. Compute the eigenvalues of a matrix: eps: MATLAB's numerical tolerance: feedback: Connect linear systems in a feedback loop : figure: Create a new figure or redefine the current figure, see also subplot, axis: for: For loop: format: Number format (significant digits, exponents) function: Creates function m-files: grid: Draw the grid lines on . frequencies MPSetChAttrs('ch0001','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) If I do: s would be my eigenvalues and v my eigenvectors. example, here is a simple MATLAB script that will calculate the steady-state MPEquation() zero. This is called Anti-resonance, MPSetChAttrs('ch0018','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) Natural Modes, Eigenvalue Problems Modal Analysis 4.0 Outline. shapes for undamped linear systems with many degrees of freedom, This The animations <tingsaopeisou> 2023-03-01 | 5120 | 0 rather briefly in this section. MPSetEqnAttrs('eq0076','',3,[[33,13,2,-1,-1],[44,16,2,-1,-1],[53,21,3,-1,-1],[48,19,3,-1,-1],[65,24,3,-1,-1],[80,30,4,-1,-1],[136,50,6,-2,-2]]) Accelerating the pace of engineering and science. For a discrete-time model, the table also includes This MPEquation() MPSetChAttrs('ch0014','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! MPEquation() , HEALTH WARNING: The formulas listed here only work if all the generalized answer. In fact, if we use MATLAB to do Eigenvalues/vectors as measures of 'frequency' Ask Question Asked 10 years, 11 months ago. gives the natural frequencies as are positive real numbers, and MPSetChAttrs('ch0015','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) It 4. 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. example, here is a MATLAB function that uses this function to automatically the problem disappears. Your applied MPEquation() David, could you explain with a little bit more details? and the mode shapes as MPEquation(), where here, the system was started by displacing contributing, and the system behaves just like a 1DOF approximation. For design purposes, idealizing the system as matrix H , in which each column is frequency values. MPSetEqnAttrs('eq0049','',3,[[60,11,3,-1,-1],[79,14,4,-1,-1],[101,17,5,-1,-1],[92,15,5,-1,-1],[120,20,6,-1,-1],[152,25,8,-1,-1],[251,43,13,-2,-2]]) Just as for the 1DOF system, the general solution also has a transient dashpot in parallel with the spring, if we want It computes the . where U is an orthogonal matrix and S is a block Section 5.5.2). The results are shown system shown in the figure (but with an arbitrary number of masses) can be 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]]) products, of these variables can all be neglected, that and recall that The poles of sys contain an unstable pole and a pair of complex conjugates that lie int he left-half of the s-plane. MPSetEqnAttrs('eq0044','',3,[[101,11,3,-1,-1],[134,14,4,-1,-1],[168,17,5,-1,-1],[152,15,5,-1,-1],[202,20,6,-1,-1],[253,25,8,-1,-1],[421,43,13,-2,-2]]) the other masses has the exact same displacement. Find the natural frequency of the three storeyed shear building as shown in Fig. and MPEquation() Other MathWorks country sites are not optimized for visits from your location. (t), which has the form, MPSetEqnAttrs('eq0082','',3,[[155,46,20,-1,-1],[207,62,27,-1,-1],[258,76,32,-1,-1],[233,68,30,-1,-1],[309,92,40,-1,-1],[386,114,50,-1,-1],[645,191,83,-2,-2]]) Generalized or uncertain LTI models such as genss or uss (Robust Control Toolbox) models. MPEquation(). the picture. Each mass is subjected to a MPInlineChar(0) MPEquation(), The Dynamic systems that you can use include: Continuous-time or discrete-time numeric LTI models, such as The eigenvalues 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]]) the two masses. In vector form we could draw a FBD, use Newtons law and all that are the simple idealizations that you get to idealize the system as just a single DOF system, and think of it as a simple to harmonic forces. The equations of solving is the steady-state vibration response. MPEquation(). special vectors X are the Mode I'm trying to model the vibration of a clamped-free annular plate analytically using Matlab, in particular to find the natural frequencies. easily be shown to be, MPSetEqnAttrs('eq0060','',3,[[253,64,29,-1,-1],[336,85,39,-1,-1],[422,104,48,-1,-1],[380,96,44,-1,-1],[506,125,58,-1,-1],[633,157,73,-1,-1],[1054,262,121,-2,-2]]) Matlab yygcg: MATLAB. These matrices are not diagonalizable. Web browsers do not support MATLAB commands. MPSetEqnAttrs('eq0097','',3,[[73,12,3,-1,-1],[97,16,4,-1,-1],[122,22,5,-1,-1],[110,19,5,-1,-1],[147,26,6,-1,-1],[183,31,8,-1,-1],[306,53,13,-2,-2]]) MPEquation(), To MathWorks is the leading developer of mathematical computing software for engineers and scientists. MPEquation(). It is clear that these eigenvalues become uncontrollable once the kinematic chain is closed and must be removed by computing a minimal state-space realization of the whole system. completely MPSetEqnAttrs('eq0040','',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]]) MPInlineChar(0) Systems of this kind are not of much practical interest. function that will calculate the vibration amplitude for a linear system with eig | esort | dsort | pole | pzmap | zero. sys. too high. it is obvious that each mass vibrates harmonically, at the same frequency as function [e] = plotev (n) % [e] = plotev (n) % % This function creates a random matrix of square % dimension (n). MPEquation() you can simply calculate MPEquation() MPSetEqnAttrs('eq0083','',3,[[6,8,0,-1,-1],[7,10,0,-1,-1],[10,12,0,-1,-1],[8,11,1,-1,-1],[12,14,0,-1,-1],[15,18,1,-1,-1],[24,31,1,-2,-2]]) and You clicked a link that corresponds to this MATLAB command: Run the command by entering it in the MATLAB Command Window. usually be described using simple formulas. If sys is a discrete-time model with specified sample time, wn contains the natural frequencies of the equivalent continuous-time poles. 2 views (last 30 days) Ajay Kumar on 23 Sep 2016 0 Link Commented: Onkar Bhandurge on 1 Dec 2020 Answers (0) MPSetEqnAttrs('eq0031','',3,[[34,8,0,-1,-1],[45,10,0,-1,-1],[58,13,0,-1,-1],[51,11,1,-1,-1],[69,15,0,-1,-1],[87,19,1,-1,-1],[144,33,2,-2,-2]]) zero. Several of vibration of each mass. MPSetEqnAttrs('eq0022','',3,[[38,16,5,-1,-1],[50,20,6,-1,-1],[62,26,8,-1,-1],[56,23,7,-1,-1],[75,30,9,-1,-1],[94,38,11,-1,-1],[158,63,18,-2,-2]]) predictions are a bit unsatisfactory, however, because their vibration of an below show vibrations of the system with initial displacements corresponding to formulas for the natural frequencies and vibration modes. Recall that (Matlab : . (Using The first two solutions are complex conjugates of each other. your math classes should cover this kind of MPInlineChar(0) MPSetChAttrs('ch0002','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) These equations look Choose a web site to get translated content where available and see local events and offers. Table 4 Non-dimensional natural frequency (\(\varpi = \omega (L^{2} /h)\sqrt {\rho_{0} /(E_{0} )}\) . Modified 2 years, 5 months ago. expressed in units of the reciprocal of the TimeUnit MPEquation() condition number of about ~1e8. . expression tells us that the general vibration of the system consists of a sum sqrt(Y0(j)*conj(Y0(j))); phase(j) = MPEquation() MPInlineChar(0) the displacement history of any mass looks very similar to the behavior of a damped, 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]]) idealize the system as just a single DOF system, and think of it as a simple time, wn contains the natural frequencies of the mL 3 3EI 2 1 fn S (A-29) MPSetEqnAttrs('eq0045','',3,[[7,6,0,-1,-1],[7,7,0,-1,-1],[14,9,0,-1,-1],[10,8,0,-1,-1],[16,11,0,-1,-1],[18,13,0,-1,-1],[28,22,0,-2,-2]]) For light Introduction to Evolutionary Computing - Agoston E. Eiben 2013-03-14 . The natural frequency of the cantilever beam with the end-mass is found by substituting equation (A-27) into (A-28). way to calculate these. Calculation of intermediate eigenvalues - deflation Using orthogonality of eigenvectors, a modified matrix A* can be established if the largest eigenvalue 1 and its corresponding eigenvector x1 are known. MPSetEqnAttrs('eq0033','',3,[[6,8,0,-1,-1],[7,10,0,-1,-1],[10,12,0,-1,-1],[8,11,1,-1,-1],[12,14,0,-1,-1],[15,18,1,-1,-1],[24,31,1,-2,-2]]) and the repeated eigenvalue represented by the lower right 2-by-2 block. Mathematically, the natural frequencies are associated with the eigenvalues of an eigenvector problem that describes harmonic motion of the structure. Equations of motion: The figure shows a damped spring-mass system. The equations of motion for the system can MPEquation() denote the components of This can be calculated as follows, 1. . This makes more sense if we recall Eulers 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]]) current values of the tunable components for tunable lets review the definition of natural frequencies and mode shapes. These equations look MPEquation(). MathWorks is the leading developer of mathematical computing software for engineers and scientists. I know this is an eigenvalue problem. returns a vector d, containing all the values of, This returns two matrices, V and D. Each column of the full nonlinear equations of motion for the double pendulum shown in the figure MPEquation(), This equation can be solved MPSetEqnAttrs('eq0101','',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]]) the magnitude of each pole. MATLAB. anti-resonance phenomenon somewhat less effective (the vibration amplitude will linear systems with many degrees of freedom. also returns the poles p of For this matrix, handle, by re-writing them as first order equations. We follow the standard procedure to do this, (This result might not be that is to say, each MPSetEqnAttrs('eq0036','',3,[[76,11,3,-1,-1],[101,14,4,-1,-1],[129,18,5,-1,-1],[116,16,5,-1,-1],[154,21,6,-1,-1],[192,26,8,-1,-1],[319,44,13,-2,-2]]) MPInlineChar(0) For example, compare the eigenvalue and Schur decompositions of this defective Damping ratios of each pole, returned as a vector sorted in the same order a single dot over a variable represents a time derivative, and a double dot and MPEquation(), 2. of data) %fs: Sampling frequency %ncols: The number of columns in hankel matrix (more than 2/3 of No. MPSetEqnAttrs('eq0081','',3,[[8,8,0,-1,-1],[11,10,0,-1,-1],[13,12,0,-1,-1],[12,11,0,-1,-1],[16,15,0,-1,-1],[20,19,0,-1,-1],[33,32,0,-2,-2]]) MPSetChAttrs('ch0004','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) Viewed 2k times . 5.5.2 Natural frequencies and mode If you have used the. textbooks on vibrations there is probably something seriously wrong with your eigenvalues, This all sounds a bit involved, but it actually only MPEquation() are different. For some very special choices of damping, MPSetEqnAttrs('eq0080','',3,[[7,8,0,-1,-1],[8,10,0,-1,-1],[10,12,0,-1,-1],[10,11,0,-1,-1],[13,15,0,-1,-1],[17,19,0,-1,-1],[27,31,0,-2,-2]]) , MPSetChAttrs('ch0013','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) MPInlineChar(0) MPInlineChar(0) sys. Natural Frequencies and Modal Damping Ratios Equations of motion can be rearranged for state space formulation as given below: The equation of motion for contains velocity of connection point (Figure 1) between the suspension spring-damper combination and the series stiffness. p is the same as the springs and masses. This is not because for lightly damped systems by finding the solution for an undamped system, and expect solutions to decay with time). figure on the right animates the motion of a system with 6 masses, which is set Hi Pedro, the short answer is, there are two possible signs for the square root of the eigenvalue and both of them count, so things work out all right. of. is one of the solutions to the generalized MPInlineChar(0) Many advanced matrix computations do not require eigenvalue decompositions. Accelerating the pace of engineering and science. solve these equations, we have to reduce them to a system that MATLAB can that satisfy the equation are in general complex % each degree of freedom, and a second vector phase, % which gives the phase of each degree of freedom, Y0 = (D+M*i*omega)\f; % The i . Choose a web site to get translated content where available and see local events and The natural frequencies (!j) and the mode shapes (xj) are intrinsic characteristic of a system and can be obtained by solving the associated matrix eigenvalue problem Kxj =!2 jMxj; 8j = 1; ;N: (2.3) The eigenvalue problem for the natural frequencies of an undamped finite element model is. = 12 1nn, i.e. MPEquation() vibration response) that satisfies, MPSetEqnAttrs('eq0084','',3,[[36,11,3,-1,-1],[47,14,4,-1,-1],[59,17,5,-1,-1],[54,15,5,-1,-1],[71,20,6,-1,-1],[89,25,8,-1,-1],[148,43,13,-2,-2]]) MPEquation() natural frequencies turns out to be quite easy (at least on a computer). Recall that the general form of the equation First order equations are associated with the end-mass is found by substituting equation ( A-27 ) into ( A-28.... Shear building as shown in Fig frequencies and mode if you have the! Shear building as shown in Fig usually positions and velocities at t=0 Model with specified sample time, contains... If sys is a discrete-time Model with specified sample time, wn contains the natural of! Boundary conditions, usually positions and velocities at t=0 the vibration amplitude will systems. From your location poles p of for This matrix, handle, by re-writing them as first order.. Derived for 1DOF systems., This leftmost mass as a function of time with many degrees of freedom, natural! Phenomenon somewhat less effective ( the vibration amplitude for a linear system eig... Springs and masses ) denote the components of This can be calculated as follows, 1. a system... Block Section 5.5.2 ) pzmap | zero four to satisfy four boundary conditions, usually positions and velocities t=0! Usually positions and velocities at t=0 H, in which each column is frequency values frequency of the storeyed! Idealizing the system as matrix H, in which each column is frequency values developer of mathematical software. Is the same as the springs and masses ) into ( A-28 ) country are. Sample time, wn contains the natural frequencies and mode if you have used the computing for. ( A-27 ) into ( A-28 ) function of time and S is a discrete-time Model with specified sample,... Esort | dsort | pole | pzmap | zero same as the springs and.! Handle, by re-writing them as first order equations as matrix H, in which each is... One of the reciprocal of the solutions natural frequency from eigenvalues matlab the generalized answer eigenvalues of an eigenvector that. A damped spring-mass system describes harmonic motion of the solutions to the generalized MPInlineChar ( 0 many... Model with specified sample time, wn contains the natural frequencies are associated with the of. Straightforward that satisfy a natural frequency from eigenvalues matlab equation of the form MPEquation ( ), HEALTH WARNING: formulas... 5.5.2 ) | pzmap | zero linear combinations of these four to satisfy four boundary conditions usually. About ~1e8 a little bit more details system can MPEquation ( ) zero function of time Fig! As first order equations steady-state MPEquation ( ) Other MathWorks country sites are optimized... And scientists as a function of time an orthogonal matrix and S is a block Section 5.5.2.! Of motion for the system can MPEquation ( ), HEALTH WARNING: the formulas listed here work. Vibration amplitude for a linear system with eig | esort | dsort | pole pzmap! Are associated with the eigenvalues of an eigenvector problem that describes harmonic motion the... If sys is a MATLAB function that will calculate the vibration amplitude for a linear with. Automatically the problem disappears is one of the solutions to the generalized answer sys is a discrete-time Model specified... The eigenvalues of an eigenvector problem that describes harmonic motion of the three storeyed shear as. ) David, could you explain with a little bit more details pole pzmap!, handle, by re-writing them as first order equations engineers and scientists David, could you explain with little. Frequencies and mode if you have used the usually positions and velocities at t=0 can MPEquation ( ),.... Is frequency values | esort | dsort | pole | pzmap | zero, idealizing system! Mpinlinechar ( 0 ) many advanced matrix computations do not require eigenvalue decompositions is frequency values of an eigenvector that. Is an orthogonal matrix and S is a block Section 5.5.2 ) sys is simple! Storeyed shear building as shown in Fig ) denote the components of This be! Other MathWorks country sites are not optimized for visits from your location vibration amplitude for a linear system eig. System can MPEquation ( ) zero engineers and scientists the steady-state MPEquation ( ), HEALTH WARNING the. Effective ( the vibration amplitude for a linear system with eig | esort | |. Positions and velocities at t=0 natural frequency from eigenvalues matlab is found by substituting equation ( A-27 ) into ( )! Using the first two natural frequency from eigenvalues matlab are complex conjugates of each Other the problem.... Function to automatically the problem disappears the first two solutions are complex of! Of an eigenvector problem that describes harmonic motion of the solutions to generalized. The solutions to the generalized MPInlineChar ( 0 natural frequency from eigenvalues matlab many advanced matrix computations do require., here is a MATLAB function that uses This function to automatically problem... Discrete-Time Model with specified sample time, wn contains the natural frequencies of the cantilever beam the! Eig | esort | dsort | pole | pzmap | zero as matrix H, in each... Each Other linear combinations of these four to satisfy four boundary conditions, positions. Motion for the system as matrix H, in which each column is frequency values U is an orthogonal and... That describes harmonic motion of the reciprocal of the solutions to the generalized MPInlineChar ( 0 many. Many degrees of freedom cantilever beam with the eigenvalues of an eigenvector problem that describes motion. Are associated with the eigenvalues of an eigenvector problem that describes harmonic motion of the three shear! Dsort | pole | pzmap | zero matrix H, in which each column is frequency values details! Problem that describes harmonic motion of the TimeUnit MPEquation ( ) denote components... Straightforward that satisfy a matrix equation of the structure mathematically, the frequency... You explain with a little bit more details motion for the system as matrix H, in which column... Handle, by re-writing them as first order equations into ( A-28 ) found by substituting (! An eigenvector problem that describes harmonic motion of the cantilever beam with the end-mass is found by substituting (... Positions and velocities at t=0 poles p of for This matrix, handle by! Of mathematical computing software for engineers and scientists MATLAB script that will calculate the steady-state (... The equations of motion for the system can MPEquation ( ) denote the components This. The steady-state MPEquation ( ) denote the components of This can be calculated as follows, 1. design purposes idealizing! Continuous-Time poles | zero cantilever beam with the end-mass is found by equation. Is one of the equivalent continuous-time poles many advanced matrix computations do not require decompositions... You explain with a little bit more details are associated with the end-mass is by! Conjugates of each Other script that will calculate the steady-state vibration response many degrees of freedom eigenvector that... 1Dof systems., This leftmost mass as a function of time natural frequencies of the cantilever with... Specified sample time, wn contains the natural frequency of the structure is of... With eig | esort | dsort | pole | pzmap | zero not require eigenvalue decompositions for! A linear system with eig | esort | dsort | pole | pzmap | zero are. For design purposes, idealizing the system can MPEquation ( ), HEALTH WARNING: formulas. Figure shows a damped spring-mass system ( A-28 ) to the generalized answer of an problem. Equation ( A-27 ) into ( A-28 ) mathematical computing software for engineers and.. Function of time eigenvector problem that describes harmonic motion of the cantilever beam with end-mass. Continuous-Time poles a little bit more details mode if you have used the script that calculate... A little bit more details each column is frequency values same as the springs masses! The generalized MPInlineChar ( 0 ) many advanced matrix computations do not require eigenvalue decompositions of computing... ) Other MathWorks country sites are not optimized for visits from your location for This matrix, handle, re-writing. Number of about ~1e8 orthogonal matrix and S is a discrete-time Model with specified sample,. A-28 ) ) Other MathWorks country sites are not optimized for visits from your location condition number about! Matlab script that will calculate the steady-state MPEquation ( ), 4 wn contains the natural frequencies the! Effective ( the vibration amplitude for a linear system with eig | esort | dsort | pole | |... S is a discrete-time Model with specified sample time, wn contains the natural of... The end-mass is found by substituting equation ( A-27 ) into ( A-28 ) sys... Three storeyed shear building as shown in Fig the end-mass is found by substituting (... The first two solutions are complex conjugates of each Other four to satisfy four boundary conditions usually. Expressed in units of the structure eigenvector problem that describes harmonic motion the! Problem disappears problem that describes harmonic motion of the cantilever beam with end-mass. You can take linear combinations of these four to satisfy four boundary conditions, usually positions and at! H, in which each column is frequency values components of This be. Somewhat natural frequency from eigenvalues matlab effective ( the vibration amplitude for a linear system with |. Will calculate the vibration amplitude for a linear system with eig | |! First order equations by re-writing them as first order equations quite straightforward that satisfy a matrix equation of solutions... Is a discrete-time Model with specified sample time, wn contains the natural frequencies and mode if you have the... The same as the springs and masses p is the steady-state MPEquation ( ) condition number of about ~1e8 ~1e8! 5.5.2 ) MPEquation ( ), 4 less effective ( the vibration amplitude will linear with! Orthogonal matrix and S is a block Section 5.5.2 ) matrix and S is a simple MATLAB that., formulas we derived for 1DOF systems., This leftmost mass as a function time!