PROJECT DYNAMICS OF MACHINES 454 Theoretical and Experimental Modal Analysis and Validation of Mathematical Models in Multibody Dynamics Ilmar Ferreira Santos, Professor Dr.-Ing., Dr.Techn., Livre-Docente Section of Solid Mechanics Department of Mechanical Engineering Technical University of Denmark 800 Lyngby, Denmark e-mail: ifs@mek.dtu.dk phone: +45 455669 Windmill With Two Flexible Rotating Wings (physical system) Equivalent Laboratory Prototype (mechanical model)
Introduction Linking Class Exercises and the Project The recapitulation of Dynamics of Particles as presented in Manuscript section Dynamics of One Single Particle in 3D Examples had some special introductory goals for the discipline:. you could recapitulate the most important topics related to kinematics and dynamics (mechanics) using relatively simple 3D examples;. the exercise gave you the chance to go through many fundamental steps of the mathematical modeling to obtain the equations of motion and the dynamic reaction forces acting on the particle; 3. It is important to highlight that the dynamic reaction forces are represented with help of the reference frame B3 attached to the arm connected to the particle. With the force components in X3 and Y3 (shear forces) and Z3 (normal forces), you can link Dynamics with Strength of Materials and design the arm connected to the particle for the maximum values of stress depending on the particle movements. It is important to point out that until obtaining the absolute linear acceleration of the particle (Kinematics) you have no idea about the value of the variables ψ(t) and l(t) and their derivatives. It was necessary to draw the free-body diagram, include the forces acting on the particle and obtain the equations of motion using Newton s laws. The equations of motions are second-order differential equations, i.e. ψ(t) and l(t). By giving the initial conditions of the movement, i.e. ψ(0), l(0), ψ(0) and l(0), the equations of motions can be solved, the values of ψ(t) and l(t) as a function of the time can be calculated and the trajectory of the particle and the reaction forces can be predicted; 4. By implementing and solving the equations of motions and dynamic reaction forces in Matlab, you also had the possibility of linking Dynamics to Mathematics, Numerical Methods and Programming. I hope these examples helped you to improve your skills to start with the first theoretical and experimental project of the discipline. Project Goals The main goals of this project are: To build mechanical and mathematical models and computationally implement them for simulating the dynamical behavior of rotating machines, mechanisms and structures. To deal with multibody systems and finite element method to create the models and describe the dynamic behavior of rigid and flexible machine/structure components. To understand the connections among different disciplines, as dynamics, mechanical vibrations, strength of materials, experimental mechanics, signal processing, mathematics and numerical analysis. To obtain the coefficients of differential equations of motions which may be constant or depending on angular velocity of the machines. In project such coefficients will be kept constant after the linearization of the equations of motion, leading to mathematical models with constant eigenvalues and eigenvectors. In project such coefficients will be dependent on the angular velocity of the machines, and the eigenvalues and eigenvectors will vary as function of the system operation conditions.
To deal theoretically as well as experimentally with damped systems, natural mode shapes, natural frequencies, damped natural frequencies and damping factors are calculated. To visualize and understand the physical meaning of natural mode shapes, natural frequencies, modal coordinates, modal mass, modal stiffness, modal damping among others when such parameters are not depending on the system operational conditions, for example, angular velocity of the machines. To understand the principles of experimental dynamic testing and the operational principles of sensor and actuators, among them accelerometers, displacement and force transducers and electromagnetic shakers. To understand the techniques of signal analysis and processing, which make possible the development of the experimental methodologies for validating mathematical models. To understand and use signal processing techniques to obtain auto and cross-correlation functions, power and cross-spectral density functions, frequency response functions and coherence function and, finally, demonstrate practical experience in extracting modal parameters from frequency response functions. To validate mathematical models based on Experimental Modal Analysis (EMA). Remember, if the measured frequencies and mode shapes agree with those predicted by the analytical mathematical model, the model is verified and can be useful for design proposes and vibration predictions with some confidence. Otherwise, the analytical models are useless. To account for the limitations in the models and methods used, and predict the possible consequences of making simplified assumptions. To write technical reports, with correct description of theoretical and experimental procedures, in a clear way, well structured language, using technical terms, giving physical interpretations and evaluations of analytical, numerical and experimental results. 3 The Physical System Figure illustrates an equivalent laboratory prototype of a windmill, where the students can carry out measurements and vibration analysis. The laboratory prototype is composed of four concentrated masses attached to two flexible beams. Elements,, 3 and 4 are the four masses connected to each other by flexible beams in a clamped-clamped boundary condition. Elements 5 and 6 simulate the two blades of the windmill. The blades are clamped to the fourth mass. At the end of the blades tip masses are added with the aim of increasing the dynamic coupling between structure and blade dynamics. 4 Mechanical and Mathematical Modeling The main information about the geometric and mass properties of the structure presented in Figure is given below: 3
Figure : Equivalent laboratory prototype composed of four masses attached to each other by two flexible beams. The highest mass is attached to two flexible blades (beams) with tip masses. 4
Mass Values: m =.94 + 0.353 =.94 [kg] lowest mass plus force transducer and connectors. m =.94 [kg] above lowest mass. m 3 =.943 [kg] below highest mass. m 4 =.938 + 0.794 =.73 [kg] highest mass where the blades are attached to. m 5 = 0.774 [kg] tip mass attached to the highest blade (blade ). m 6 = 0.774 [kg] tip mass attached to the lowest blade (blade ). m beam = 0.300 [kg] total mass of each beam connecting masses to 4. µ = 0.300/.08 = 0.78 [kg/m] distributed mass of each beam connecting masses to 4. m beam = 0.4 [kg] total mass of the two blades. µ = 0.4/.05 = 0.40 [kg/m] distributed mass of the blades and. m acc = 0.073 [kg] accelerometer mass. Beams connecting the platform masses to 4 (properties): E =.0 0 [N/m ] elasticity modulus. b = 0.09 [m] beam width. h = 0.00 [m] beam thickness. I = (b h 3 )/ [m4 ] area moment of inertia. L = 0.93 [m] beam length between mass and foundation. L = 0.9 [m] beam length between mass and mass. L 3 = 0.0 [m] beam length between mass 3 and mass. L 4 = 0.7 [m] beam length between mass 4 and mass 3. L 5 = 0.48 [m] length of blade. L 6 = 0.48 [m] length of blade. Beams connecting the blade tip masses to platform mass 4 (blade properties): E =.0 0 [N/m ] elasticity modulus b = 0.0350 [m] beam width h = 0.005 [m] beam thickness I = (b h 3 )/ [m4 ] area moment of inertia L 5 = 0.48 [m] length of blade. L 6 = 0.48 [m] length of blade. Depending on the assumptions you make, you may need further geometric and mass properties. If that is the case, please, contact your adviser (Ilmar Santos) and he will help you obtain the values of the additional parameters.. ASSUMPTIONS To represent the dynamical behavior of the system a mechanical model has to be created. In Dynamics of Machinery such models are built by mass, springs and damper. Considering the range of frequencies between 0 and 0 Hz, all movements of the structure happen in a vertical plane XY. Elaborate a short discussion about the assumptions you will make to create mechanical models for the structure starting from the physical system illustrated 5
(a) (b) Figure : Reference frames attached to the bodies and positive direction of blade movements: (a) inertial system X 0 Y 0 Z 0, moving reference system X Y Z attached to mass, moving reference system X Y Z attached to mass, moving reference system X 3 Y 3 Z 3 attached to mass 3, moving reference system X 4 Y 4 Z 4 attached to mass 4 and moving reference system X 5 Y 5 Z 5 attached to the rotating undeflected beam (mass 5). 6
in Figure. As mentioned, this model represents the dynamic behavior of the structure in the range of frequencies between 0 and 0 Hz. Explain the assumptions you will make, starting from the point of view that a rigid body can execute 3 translations and 3 rotations when considering three dimensional analysis. Remember that the assumptions you are going to make in order to get a simplified mechanical model of the structure are strongly dependent on the way how the masses are attached to each other.. MECHANICAL MODEL After discussing and justifying your simplifying assumptions, draw your mechanical model and set the degrees-of-freedom you will use to describe the dynamic behavior of the structure in the vertical plane XY. Illustrate clearly the free-body diagram for each of the masses and the reaction forces acitng on them. 3. MATHEMATICAL MODEL Based on kinematics and dynamics, i.e. on the principles and axioms postulated by Newton, Euler, Lagrange, please, obtain the mathematical model (equations of motion and dynamic reaction forces) for representing the structure dynamics in the range of frequencies between 0 and 0 Hz. Write with details (free-body diagrams) on how to get such equations of motion and dynamic reaction forces. Consider that the two flexible blades attached to the top mass can rotate around around the axis Z with constant angular velocity θ, i.e. θ(t) varies linearly in time. Write the equations of motion in the form ẍ(t) = F(ẋ(t),x(t)) knowing that x(t) = { x (t) x (t) x 3 (t) x 4 (t) x 5 (t) x 6 (t) } T, and assuming that the dynamic reaction forces (in some directions) can be represented by linear spring forces based on the constitutive equation of beam material. The equations of motion should be written as a function of the masses m i, structure properties, i.e. L i, I i, E, constant angular velocity of the rotor θ and the position angle θ(t). Do not use numbers here, only letters. 4. LINEARIZATION Imagine now, that the angular velocity of the blades is θ 0 = 0 and θ(0) = 0, as illustrated in figure (b) Simplify your set of equations of motion and write them in matrix form, specifying the mass M and stiffness K matrices and write their coefficients as a function of the structure properties, i.e. m i, L i, I i, E, etc. Do not use numbers here, only letters. 5. MASS MATRIX Calculate the matrix M using the numerical values given. 6. STIFFNESS MATRIX Calculate the stiffness K using the numerical values given. 7. DAMPING MATRIX The first really difficult problem faced when you try to create mathematical models to represent structural and/or machine dynamics is to define the damping matrix D and obtain each element of such matrix. Before you continue with the project, try to think about it. How could you get the damping matrix D? Remember that when you do not know how to exactly model the dissipation of vibration energy in the system, assumptions have to be made. If your assumptions are good or bad, you will have the chance to verify later on, during the experimental stage of the project. In this phase of the project we are going to make a very strong assumption that the damping matrix D can be written as a linear function of the two matrices M and K that could be more easily obtained. In order words we will assume proportional damping. There are three ways to write this proportionality: (a) D = α M. (b) D = β K. (c) D 3 = α M+β K. 7
Explain the physical meaning of such three assumptions, and how the vibration energy will be dissipated depending on the movements of the structure? Could you give some physical examples? 8. DAMPING FACTOR The structure of the damping matrix D was defined with the assumption of proportional damping, but the question now is: How do you obtain the proportionality constants α and β? To obtain them, we need to use the experience accumulated during the classes. If you are in the industry, you need to use the experience accumulate by the group you work with. If you remember from the experimental classes, a realistic approximation of the structural damping factor ξ is something between 0.00 and 0.005, according to the experimental results obtained with equation: ξ = ( ) πn ln yo y N +[ πn ln ( yo y N )] Try to adjust the proportionality factors α and β so that the damping factor ξ related to the first mode shape of the structure is 0.005. Recall from Mechanical Vibration courses that, if D 3 = α M+β K then the damping factors ξ i associated to the i th mode shape can be written as: ξ i = α + βω i ω i i =,...,n DOF () where n DOF is the number of degrees-of-freedom. Remember also that when you calculate the system eigenvalues using the matrices M, D and K or A and B with help of the Matlab routine eig, i.e. [u,λ] = eig( B,A), you can get ξ i and ω i using: ξ i = Real(λ i ) Real(λi ) +Imag(λ i ) () ω i = Real(λ i ) +Imag(λ i ) (3) Knowing the relationships given in equations (), () and (3), write a program in Matlab and calculate the natural frequencies ω i (i =,...,n DOF ) of the mechanical model and the damping ratios ξ i (i =,...,n DOF ) related to the n DOF modes shapes, considering the three different damping matrices D, D and D 3. An example of Matlab routine is given following: % Building the State Matrices A and B with M, D and K A= [ M zeros(size(m)) zeros(size(m)) M ] ; B= [ D K -M zeros(size(m)) ] ; %Calculating the Modal Matrix u with all mode shapes. %Calculating the matrix lambda - damping factors and natural frequencies. [u,lambda]=eig(-b,a); 8
Is it possible to adjust all damping factors ξ i simultaneously? If you try to adjust only the first damping factor ξ, i.e. ξ = 0.005, what happens with the other damping factors ξ (i =,...,n DOF ) considering the three cases (a), (b) and (c). What can you conclude about the coefficients α and β and their influence on the adjustment of the damping factors associated to higher mode shapes? 5 Vibration Analysis aided by the Mathematical Model. Neglecting damping, i.e. D = 0 write the modal matrix with help of your program. Normalize your mode shapes in such a way that the maximum amplitude of the coordinates is. Based on the information contained in such a matrix, draw the mode shapes and explain with a short and clear text, how the structure vibrates. Explain the pure complex numbers that appear in each eigenvector. Is any relationship between the pure complex numbers and the pure real numbers?. Considering one of your damping matrices, for example D 3, write the modal matrix with help of your program. Normalize your mode shapes in such a way that the maximum amplitude of the coordinates is. Based on the information contained in such a modal matrix, describe the mode shapes of the structure and try to explain the relationship between the mode shapes of the damped and undamped models. 6 Experimental Modal Analysis Experimental Modal Analysis (EMA) deals with the determination of natural frequencies, modes shapes, and damping factors from experimental measurements. The fundamental idea behind modal testing is the resonance. If a structure is excited at resonance, its response exhibits two distinct phenomena: (I) as the excitation frequency approaches the natural frequency of the structure, the magnitude at resonance rapidly approaches a sharp maximum value, provided that the damping ratio is less than about 0.5; (II) the phase of the response between excitation force and displacement shift by 80 o as the frequency sweeps through resonance, with the value of the phase at resonance being 90 o. This physical phenomenon is used to determine the natural frequency of a structure from measurements of the magnitude and phase of the force response of the structure as the driving frequency is swept through a wide range of values.. STEADY-STATE ANALYSIS(Analysis in the Frequency Domain) Obtain 6 frequency response functions in the range of 0 0 Hz, using the Matlab routines written specifically to this propose. Download from the course homepage the following files: a) frf general.m Matlab routine able to calculate the frequency response functions H and H from input and output signal recorded in the time domain. The program output is illustrated in Figure 3. b) f orce.txt and acceleration.txt force applied by the electromagnetic shaker to the lowest mass (mass ) and acceleration signal of the lowest mass (acceleration response of mass ). Both signals are recorded in time domain with a sampling frequency of 50 Hz. c) f orce.txt and acceleration.txt force applied by the electromagnetic shaker to the lowest mass (mass ) and acceleration signal of the second mass (acceleration response of mass ). Both signals are recorded in time domain with a sampling frequency of 50 Hz. 9
Coherence FRF [(m/s )/N] 0.5 0 0 4 6 8 0 4 80 0 0 4 6 8 0 H H Phase [ o ] 90 H H 0 0 4 6 8 0 Frequency [Hz] Figure 3: Experimental frequency response function measured at the highest mass when the excitation force acts on mass (a) Coherence function, (b) amplitude of the frequency response function illustrating H and H and (c) phase of the frequency response function. 0
d) f orce3.txt and acceleration3.txt force applied by the electromagnetic shaker to the lowest mass (mass ) and acceleration signal of the third mass (acceleration response of mass 3). Both signals are recorded in time domain with a sampling frequency of 50 Hz. e) f orce4.txt and acceleration4.txt force applied by the electromagnetic shaker to the lowest mass (mass ) and acceleration signal of the forth mass (acceleration response of mass 4). Both signals are recorded in time domain with a sampling frequency of 50 Hz. f) f orce5.txt and acceleration5.txt force applied by the electromagnetic shaker to the lowest mass (mass ) and acceleration signal of the fifth mass (acceleration response of the highest blade, tip mass 5). Both signals are recorded in time domain with a sampling frequency of 50 Hz. g) f orce6.txt and acceleration6.txt force applied by the electromagnetic shaker to the lowest mass (mass ) and acceleration signal of the sixth mass (acceleration response of the lowest blade, tip mass 6). Both signals are recorded in time domain with a sampling frequency of 50 Hz. Using the Matlab routine frf general.m generate a table with the values of frequency ω, FrequencyResponseFunctionH (ω), FrequencyResponseFunctionH (ω)andcoherencefunction Coher(ω).. EXTRACTING MODAL PARAMETERS Identify 6 different -DOF systems around each one of the 6 natural frequencies of the structure. Use your frequency domain identification procedure based on the Least Square Method and obtain the experimental natural frequencies and the experimental damping factors of each one of the 6 mode shapes of the multibody system. Please, use only a few points around each one of the six resonance ranges, and calculate the modal parameters m, d and k based on: ω ω ω3...... ωn { m k } = ( REAL ( REAL ( REAL (... REAL ω FRF(ω ) ω FRF(ω ) ω3 FRF(ω 3 ) ω N FRF(ω N ) ) ) ) ) = A x = b (4) x = ( A T A ) AT b (5) ω ω ω 3... ω N { } d = ( IMAG ( IMAG ( IMAG (... IMAG ω FRF(ω ) ω FRF(ω ) ω3 FRF(ω 3 ) ω N FRF(ω N ) ) ) ) ) = Ā x = b (6) x = ( Ā T Ā) Ā T b (7)
Remember that after experimentally calculating the coefficients m i, k i, and d i around the six resonance ranges, i.e. i =,,3,4,5,6, the natural frequencies can be obtained by ω i = ki and ξ i = d i m i k i. 3. TRANSIENT ANALYSIS (Analysis in the Time Domain) Using the electromagnetic shaker attached to the lowest mass (mass ), the structure is excited at 6 different resonances. During the tests, one waits until the structure response reaches the steady-state. After reaching the steady-state, the electrical sinusoidal signal sent to the electromagnetic shaker is turned-off at time t 0, as it can be identified in the beginning of the files: m i a) force transient mode.txt b) force transient mode.txt c) force transient mode3.txt d) force transient mode4.txt e) force transient mode5.txt f) force transient mode6.txt The files can be downloaded from the course homepage. After time t 0 no force is applied to the the system and the transient vibrations start. The structure oscillates with one of its natural frequencies and associated mode shape until the oscillations stop. During the tests, the acceleration and force signals are recorded. During the modal tests the excitation force acts always on the lowest mass (mass ). For the tests around the first, second, third and fourth resonances the acceleration of mass 4 is recorded. For the tests around fifth and sixth resonances the acceleration of mass 3 is recorded, as it is illustrated in Figure 4. Please, download the following files from the course homepage in order to build figures as Figure 5: a) acceleration transient mode.txt (measured at mass 4) b) acceleration transient mode.txt (measured at mass 4) c) acceleration transient mode3.txt (measured at mass 4) d) acceleration transient mode4.txt (measured at mass 4) e) acceleration transient mode5.txt (measured at mass 3) f) acceleration transient mode6.txt (measured at mass 3) Based on the logarithmic decay, the damping factors ξ i (i =,...,n DOF ) associated to all six modes shapes can be expressed by equation (8). Please, try to calculate all six damping factors based on the transient analysis. ξ = ( ) πn ln yo y N +[ πn ln ( yo y N )] (8) After obtaining the experimental damping ratios ξ i, compare them with the experimental damping factors obtained using the steady-state response. If you can not calculate the damping factor
0.5 Acceleration of Mass 4 acc [m/s ] 0.4 0.3 0. 0. 0 0. 0. 0.3 0.4 0.5 0 0 40 60 80 00 0 time [s] Figure 4: Experimental transient vibration response (acceleration) of mass 4 after initial condition close to the first mode shape imposed by the electromagnetic shaker. associatedtooneofthemodeshapes, please, explainthereason. Tobesureaboutthefrequency components in the force and acceleration signals, use the routine acc in time domain.m loading the data files mentioned previously. 7 Model Validation. Compare the theoretical and experimental undamped natural frequencies ω i (i =,...,6) and damping factors ξ i (i =,...,6). Quantify in % the discrepancies between theoretical and experimental results. 8 Model Updating Following you will start to improve the mathematical model created using the results obtained from the experimental modal analysis.. UPDATING NATURAL FREQUENCIES Try to adjust the undamped natural frequencies ω i (i =,...,6) of the mathematical model using the experimental natural frequencies obtained via Experimental Modal Analysis. Neglect, in this stage of model refinement, the existence of the damping matrix D, i.e. D = 0, and work only with the matrices M and K. Remember that you can deal with many parameters in order to change the coefficients of the matrices M and K, but all changes has to be justify based on the physics of the problem. You can add or subtracts masses, you can change dimensions based on the tolerances of the instrumentation used. In order words, changing the coefficients of the matrices with criteria, the natural frequencies of the mathematical model shall come closer to the experimental natural frequencies. Please, state clearly the values of m i, k i or L i or h i you are going to use and how you get to such values. Make a table illustrating the natural frequencies obtained experimentally, the natural frequencies of the unadjusted mathematical model, the natural frequencies of the adjusted natural frequencies and the error in percentage. 3
4 (a) Force in Time Domain force [N] FFT(force) [N] 0 4 0 0 40 60 80 00 0 time [s] (b) Force in Frequency Domain 0.5 0.4 0.3 0. 0. 0 0 4 6 8 0 Freq [Hz] 0.5 (c) Acceleration in Time Domain acc [m/s ] 0 0.5 0 0 40 60 80 00 0 time [s] (d) Acceleration in Frequency Domain 0. FFT(acc) [m/s ] 0.08 0.06 0.04 0.0 0 0 4 6 8 0 Freq [Hz] Figure 5: Experimental transient vibration response (a) and (b) force acting on mass and (c) and (d) acceleration of mass 4 after initial condition close to the first mode shape imposed by the electromagnetic shaker. 4
. ADJUSTING α AND β COEFFICIENTS After adjustment of the natural frequencies and dealing with the matrices M and K, try to adjust the damping factors/ratios ξ i (i =,...,6) of the analytical model using the experimental damping ratios ξ iexp (i =,...,6) obtained via Experimental Modal Analysis. Remember that α and β are the two parameters of the mathematicalmodel tobeadjustedinorder toachieveabettermatchof thesixdampingfactors. Once you have more experimental data(six experimental damping factors) than coefficients (two coefficients α and β), you can, if you want, decide for a least-square fitting. Re-writing equation () in a matrix form, one gets: ξ exp ξ exp ξ 3exp ξ 4exp ξ 5exp ξ 6exp = ω exp ω exp ω 3exp ω 4exp ω 5exp ω 6exp ω exp ω exp ω 3exp ω 4exp ω 5exp ω 6exp { α β } = Ā x = b = x = ( Ā T Ā) Ā T b (9) 3. UPDATIONG DAMPING FACTORS After adjusting all coefficients of the matrices M, K and D, please, re-calculate with help of your matlab program the new values of the six theoretical damping factors ξ i (i =,...,6). Make a table illustrating all six theoretical damping factor and also the six experimental damping factors, obtained using the steady-state analysis and the transient analysis. Make an error analysis and present the deviations in %. 4. EXTRACTING MODE SHAPES Using the 6 frequency response functions, obtain the experimental modes shapes. The theory dealing with the experimental identification of modes shapes from the experimental frequency response functions is presented in chapter 7 of our textbook. 9 Model Validation and Identification of Residual Discrepancies. Now, after the adjustment of all parameters related to the matrices M, K and D compare the theoretical and experimental frequency response functions, remembering that the excitation force acts always on the lowest mass (mass ) and the acceleration response of the masses,, 3, 4, 5 and 6 are measured, i.e. the FRF s are given in (m/s )/N. The frequency response function can be calculated based on the matlab routine dof frf.m. Remember though that you are measuring acceleration and not displacement, and the frequency response function (using displacement) has to be multiplied by ω and the its unit is m/s N and not m N. Plot theoretical and experimental frequency responses in one single figure. Compare them and justify the discrepancies between theoretical and experimental results.. Compare the theoretical and the experimental modes shapes, plotting them pairwise. Justify the discrepancies. 5
0 Using the Mathematical Model to Design Structural and Machine Components Until now you have faced a typical procedure (and difficulties) towards building mechanical and mathematical models and adjusting them via experimental modal analysis. Now your model is adjusted. Remember, if the measured frequencies and mode shapes agree with those predicted by the analytical mathematical model, the model is verified and can be useful for design proposes and vibration predictions with some confidence. Otherwise, the analytical models are useless.. LINK TO NUMERICAL (INTEGRATION) METHODS Remember that you will need to integrate (or solve) the equations of motions in the same way you did in the exercise. There is an matlab example (particle3d.m) in the website (see notes of the first class Solving the Equations of Motion in Time) in case you need inspiration. (a) Use your mathematical model to predict the maximum dynamic and static deflections of the two blades x 5 (t) and x 6 (t) when θ(0) = π/ rad and the blades do not rotate, i.e. θ(t) = 0 rad/s. The initial conditions of displacement and velocity for all masses at t = 0 are zero, i.e. x(0) = { x (0) x (0) x 3 (0) x 4 (0) x 5 (0) x 6 (0) } T = { 0 0 0 0 0 0 } T and ẋ(0) = { ẋ (0) ẋ (0) ẋ 3 (0) ẋ 4 (0) ẋ 5 (0) ẋ 6 (0) } T = { 0 0 0 0 0 0 } T. Plot the time response of the six masses. Plot the results x i (t) i =,,...,6 as a function of time t, when t varies from 0 until 50 seconds or longer, until the blade static equilibrium positions are reached. (b) Use your mathematical model to predict the maximum amplitude of vibration of each one of the masses, assuming that the blades rotates with a constant angular of θ(t) = π/4 rad/s from the static equilibrium position found in (a), i.e. the initial conditions of velocity for all masses at t = 0 are zero, but not the initial conditions of displacement. Plot the time response of the six masses x i (t) i =,,...,6 as a function of time t, when t varies from 0 until 60 seconds.. LINK TO STRENGTH OF MATERIALS AND MACHINE DESIGN Calculate the location where the structure will experience its maximum stress in the dynamic case simulated in case (b). 6