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DYNAMICS OF MACHINERY 41514
PROJECT 1
Digital Twins (Mathematical Models) in Multibody
Dynamics and Their Validation via Experimental
Modal Analysis
Holistic Overview of the Project Steps & Their Conceptual Links
Ilmar Ferreira Santos, Professor, Dr.-Ing., Dr.Techn.Technical University of Denmark
Department of Mechanical EngineeringSection of Solid Mechanics, 404 – 006
e-mail: [email protected]
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1 Introduction – Linking Class Exercises and the Project
The recapitulation of Dynamics of Particles as presented in the manuscript — section ”Dynamics ofOne Single Particle in 3D – Examples” — had some special introductory goals for the discipline:
1. You could recapitulate the most important topics related to kinematics and dynamics (mechan-ics) using relatively simple 3D examples;
2. The exercise gave you the chance to go through many fundamental steps of the mathematicalmodeling to obtain the equations of motion and the dynamic reaction forces acting on theparticle;
3. It is important to highlight that the dynamic reaction forces are represented with help of thereference frame B3 attached to the arm connected to the particle. With the force componentsin X3 and Y3 (shear forces) and Z3 (normal forces), you can link Dynamics with Strengthof Materials and design the arm connected to the particle for the maximum values of stressdepending on the particle movements. It is important to point out that until obtaining theabsolute linear acceleration of the particle (Kinematics) you have no idea about the value ofthe variables ψ(t) and l(t) or 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 motion are second-order differential equations as they include 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 equationsof motion can be solved for the values of ψ(t) and l(t) as functions of time and the trajectoryof the particle and the reaction forces can be predicted;
4. By implementing and solving the equations of motion and dynamic reaction forces in Mat-lab, you also had the possibility of linking Dynamics to Mathematics, Numerical Methods andProgramming.
I hope these examples helped you to improve your skills and prepare for the first theoretical andexperimental project in the discipline.
2 Project Goals
The main goals of this project are:
• To build mechanical and mathematical models and computationally implement them forsimulating the dynamical behavior of rotating machines, mechanisms and structures.
• To deal with multibody systems and the finite element method to create the modelsand describe the dynamic behavior of rigid and flexible machine/structure components.
• To understand the connections between different disciplines, as dynamics, mechanicalvibrations, strength of materials, experimental mechanics, signal processing, mathematics andnumerical analysis.
• To obtain the coefficients of differential equations of motions which may be constant or de-pending on angular velocity of the machines. In project 1, such coefficients will be keptconstant after the linearization of the equations of motion, leading to math-ematical models with constant eigenvalues and eigenvectors. In project 2 suchcoefficients will be dependent on the angular velocity of the machines, and the eigenvalues andeigenvectors will vary as function of the system operation conditions.
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• 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, among others, natural mode shapes,natural frequencies, modal coordinates, modal mass, modal stiffness and modaldamping when such parameters are not depending on the system operational conditions suchas the angular velocity.
• To understand the principles of experimental dynamic testing and the operationalprinciples of sensor and actuators, among them accelerometers, displacement and forcetransducers and electromagnetic shakers.
• To understand the techniques of signal analysis and processing, which make possiblethe development of the experimental methodologies for validating mathematical models.
• To understand and use signal processing techniques to obtain auto and cross-correlationfunctions, power and cross-spectral density functions, frequency response functions and coher-ence functions and, finally, to demonstrate practical experience in extracting modal parametersfrom frequency response functions.
• To validate mathematical models based on Experimental Modal Analysis(EMA). Remember, if the measured frequencies and mode shapes agree with those predictedby the analytical mathematical model, the model is verified and can be useful for designproposes and vibration predictions with some confidence. Otherwise, the analyticalmodels are useless.
• To account for the limitations in the models and methods used, and predict the pos-sible consequences of making simplifying assumptions.
• To write technical reports, including correct descriptions of theoretical as well as experimentalprocedures, in a clear and well-structured language, using proper technical terms to providephysical interpretation and evaluation of analytical, numerical and experimental results.
3 The Physical System
Figure 1(a) shows a windmill composed of 2 rotating wings and figure 1(b) illustrates an equivalentlaboratory prototype of a windmill for the students to carry out measurements and vibration analysis.The laboratory prototype is composed of four concentrated masses attached to two flexible beams.Elements 1, 2, 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 bladesare clamped to the fourth mass. At the end of the blades, tip masses are added with the aim ofincreasing the dynamic coupling between structure and blade dynamics.
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(a) (b)
Figure 1: (a) Windmill composed of 2 rotating wings; (b) Equivalent laboratory prototype com-posed of four masses attached to each other by two flexible beams. The highest mass is attachedto two flexible blades (beams) with tip masses.
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4 Mechanical and Mathematical Modeling – Toward a DigitalTwin
The main information about the geometric and mass properties of the structure presented in Figure1 is given below:
• Mass Values:
m1 = 1.941 kg + 0.353 kg = 2.294 kg – lowest mass plus force transducer and connectors.
m2 = 1.941 kg – above lowest mass.
m3 = 1.943 kg – below highest mass.
m4 = 1.938 kg + 0.794 kg = 2.732 kg – highest mass where the blades are attached to.
m5 = 0.774 kg – tip mass attached to the highest blade (blade 1).
m6 = 0.774 kg – tip mass attached to the lowest blade (blade 2).
mbeam1= 0.300 kg – total mass of each beam connecting masses 1 to 4.
µ1 = 0.300 kg/1.08m = 0.278 kg/m – distributed mass of each beam connecting masses 1–4.
mbeam2= 0.421 kg – total mass of the two blades.
µ2 = 0.421 kg/1.05m = 0.401 kg/m – distributed mass of the blades 1 and 2.
macc = 0.073 kg – accelerometer mass.
• Beams connecting the platform masses 1 to 4 (properties):
E = 2.0 · 1011N/m2 – elasticity modulus.
b1 = 0.0291m – beam width.
h1 = 0.0011m – beam thickness.
I1 = (b1h31)/12
[
m4]
– area moment of inertia.
L1 = 0.193m – beam length between mass 1 and foundation.
L2 = 0.292m – beam length between mass 2 and mass 1.
L3 = 0.220m – beam length between mass 3 and mass 2.
L4 = 0.271m – beam length between mass 4 and mass 3.
L5 = 0.428m – length of blade 1.
L6 = 0.428m – length of blade 2.
• Beams connecting the blade tip masses to platform mass 4 (blade properties):
E = 2.0 · 1011N/m2 – elasticity modulus
b2 = 0.0350m – beam width
h2 = 0.0015m – beam thickness
I2 = (b2h32)/12
[
m4]
– area moment of inertia
L5 = 0.428m – length of blade 1.
L6 = 0.428m – length of blade 2.
• 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 thevalues of the additional parameters.
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(a) (b)
Figure 2: Reference frames attached to the bodies and positive direction of blade movements: (a)inertial system X0−Y0−Z0, moving reference system X1−Y1−Z1 attached to mass 1, movingreference system X2−Y2−Z2 attached to mass 2, moving reference system X3−Y3−Z3 attachedto mass 3, moving reference system X4−Y4−Z4 attached to mass 4 and moving reference systemX5 − Y5 − Z5 attached to the rotating undeflected beam (mass 5).
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1. ASSUMPTIONS – To represent the dynamical behavior of the system a mechanical model hasto be created. In Dynamics of Machinery such models are built by mass, springs and damper.Considering the range of frequencies 0 to 10Hz, all movements of the structure happen in avertical plane XY. Elaborate a short discussion about the assumptions you will make to createmechanical models for the structure starting from the physical system illustrated in Figure 1.As mentioned, this model represents the dynamic behavior of the structure in the range offrequencies 0 to 10Hz. Explain the assumptions you will make, starting from the point ofview that a rigid body can execute three translations and three rotations in a three-dimensionalanalysis. Remember that the assumptions you are going to make in order to get a simplifiedmechanical model of the structure are strongly dependent on how the masses are attached toeach other.
2. MECHANICAL MODEL – After discussing and justifying your simplifying assumptions, drawyour mechanical model and state the degrees of freedom you will use to describe the dynamicbehavior of the structure in the vertical plane XY. Illustrate clearly the free-body diagram foreach of the masses and the reaction forces acting on them.
3. MATHEMATICAL MODEL – Based on kinematics and dynamics, i.e. on the principles andaxioms postulated by Newton, Euler and Lagrange, please, obtain the mathematical model(equations of motion and dynamic reaction forces) for representing the structure dynamics inthe frequency range 0 to 10Hz. Describe in detail (free-body diagrams) how to obtain suchequations of motion and dynamic reaction forces. Consider that the two flexible bladesattached to the top mass can rotate around the axis Z with constant angular velocity θ, i.e. suchthat θ(t) varies linearly in time. Write the equations of motion in the form x(t) = F(x(t),x(t))knowing that
x(t) = { x1(t) x
2(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 linearspring forces based on the constitutive equation of a beam. The equations of motion shouldbe written as a function of the masses mi, structure properties, i.e. Li, Ii, E, constant angularvelocity of the rotor θ and the position angle θ(t). Do not use numerical values here, onlysymbols.
It is important to recall the student’s attention to the fact that x1(t) is the linear movement of
the lowest mass relative to the inertial coordinate system. Nevertheless, the linear displacementof the additional masses, namely x
2(t), x
3(t), x
4(t), x
5(t), and x
6(t) are not defined relative to
the inertial system. They are defined relative to each other, i.e. the linear movement x2(t) of
the second lowest mass is defined relative to the lowest mass movement x1(t), x
3(t) is defined
relative to x2(t), x
4(t) is defined relative to x
3(t), and x
5(t) and x
6(t) relative to x
4(t). Do
not use other definitions of the degrees of freedoms than the definition specified here!
4. LINEARIZATION – Imagine now that the angular velocity of the blades is θ0 = 0 and θ(0) = 0,as illustrated in figure 2(b). Simplify your set of equations of motion and write them in matrixform, specifying the mass M and stiffness K matrices and write their coefficients as a functionof the structure properties, i.e. mi, Li, Ii, E, etc. Do not use numerical values here, onlysymbols.
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 trying to create a mathemat-ical model representing structural and/or machine dynamics, is to define the damping matrix
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D and to obtain each element of this. Before you continue with the project, try to think aboutit. How could you get the damping matrix D? Remember that when you do not know how toexactly 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 theexperimental stage of the project.
In this phase of the project we are going to make a very strong assumption that the dampingmatrix D can be written as a linear combination of the matrices M and K that could be moreeasily obtained. In other words, we will assume ”proportional damping”. This proportionallygives three possible cases:
(a) D1 = αM
(b) D2 = βK
(c) D3 = αM+ βK
Explain the physical meaning of such three assumptions, and how the vibration energy willbe dissipated depending on the movements of the structure? Could you give some physicalexamples?
8. DAMPING FACTOR – The structure of the damping matrix D was defined with the assumptionof proportional damping, but the question now is: How do you obtain the proportionality con-stants 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 workwith. If you remember from the experimental classes, a realistic approximation of the structuraldamping factor ξ is somewhere between 0.001 and 0.005, according to the experimental resultsobtained using the equation:
ξ =
12πN ln
(
y0yN
)
√
1 +[
12πN ln
(
y0yN
)]2.
Try to adjust the proportionality factors α and β such that the damping factor ξ1 related tothe first mode shape of the structure is 0.005. Recall from your Mechanical Vibration coursesthat, if D3 = αM+ βK then the damping factor ξi associated to the i-th mode shape can bewritten as:
ξi =α
2ωi
+βωi
2, i = 1, ..., n
DOF(1)
where nDOF
is the number of degrees of freedom. Remember also that when you calculate thesystem eigenvalues using the matrices M, D and K or A and B with help of the Matlab routineeig, i.e. [u, λ] = eig(−B,A), you can get ξi and ωi using:
ξi =−Real(λi)
√
Real(λi)2 + Imag(λi)2and ωi =
√
Real(λi)2 + Imag(λi)2 (2)
Knowing the relationships given in equations (1) and (2), write a program in Matlab and calculatethe natural frequencies ωi (i = 1, ..., n
DOF) of the mechanical model and the damping ratios
ξi (i = 1, ..., nDOF
) related to the nDOF
modes shapes, considering the three different dampingmatrices D1, D2 and D3. An example of Matlab routine is given following:
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% 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);
Is it possible to adjust all damping factors ξi simultaneously? If you try to adjust onlythe first damping factor ξ1, i.e. ξ1 = 0.005, what happens with the other damping factors ξ1(i = 2, ..., 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 tohigher mode shapes?
5 Vibration Analysis aided by the Mathematical Model
1. Neglecting damping, i.e. D = 0, write the modal matrix with help of your program. Normalizeyour mode shapes in such a way that the maximum amplitude of the coordinates is 1. Basedon the information contained in such a matrix, draw the mode shapes and explain with a shortand clear text, how the structure vibrates. Explain the pure complex numbers that appear ineach eigenvector. Is there any relationship between the pure complex numbers and the pure realnumbers?
2. Considering one of your damping matrices, for example D3, write the modal matrix with helpof your program. Normalize your mode shapes in such a way that the maximum amplitude ofthe coordinates is 1. Based on the information contained in such a modal matrix, describe themode shapes of the structure and try to explain the relationship between the mode shapes ofthe damped and undamped models.
6 Experimental Modal Analysis
Experimental Modal Analysis (EMA) deals with the determination of natural frequen-cies, modes shapes, and damping factors from experimental measurements. The fundamentalidea behind modal testing is the resonance. If a structure is excited at resonance, its response ex-hibits two distinct phenomena:
1. as the excitation frequency approaches the natural frequency of the structure, the magnitude atresonance rapidly approaches a sharp maximum value, provided that the damping ratio is lessthan about 0.5;
2. the phase of the response between excitation force and displacement shifts by 180◦ as thefrequency sweeps through resonance, with the value of the phase at resonance being 90◦.
These physical phenomena are used to determine the natural frequency of a structure from mea-surements of the magnitude and phase of the force response of the structure as the driving frequencyis swept through a wide range of values.
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1. STEADY-STATE ANALYSIS (Analysis in the Frequency Domain) – Obtain 6 frequency responsefunctions in the range 0 to 10Hz, 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 H1
and H2 from the input and output signals recorded in the time domain. The programoutput is illustrated in Figure 3.
0 2 4 6 8 10 120
0.5
1
Coh
eren
ce
0 2 4 6 8 10 120
2
4
FR
F [(
m/s
2 )/N
]
H1H2
0 2 4 6 8 10 120
90
180
Frequency [Hz]
Pha
se [o ]
H1H2
Figure 3: Experimental frequency response function measured at the highest mass when theexcitation force acts on mass 1 – (a) Coherence function, (b) amplitude of the frequency responsefunction illustrating H1 and H2 and (c) phase of the frequency response function.
(b) force1.txt and acceleration1.txt – force applied by the electromagnetic shaker tothe lowest mass (mass 1) and acceleration signal of the lowest mass (acceleration responseof mass 1). Both signals are recorded in time domain with a sampling frequency of 50Hz.
(c) force2.txt and acceleration2.txt – force applied by the electromagnetic shaker tothe lowest mass (mass 1) and acceleration signal of the second mass (acceleration responseof mass 2). Both signals are recorded in time domain with a sampling frequency of 50Hz.
(d) force3.txt and acceleration3.txt – force applied by the electromagnetic shaker tothe lowest mass (mass 1) and acceleration signal of the third mass (acceleration responseof mass 3). Both signals are recorded in time domain with a sampling frequency of 50Hz.
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(e) force4.txt and acceleration4.txt – force applied by the electromagnetic shaker tothe lowest mass (mass 1) and acceleration signal of the forth mass (acceleration responseof mass 4). Both signals are recorded in time domain with a sampling frequency of 50Hz.
(f) force5.txt and acceleration5.txt – force applied by the electromagnetic shaker tothe lowest mass (mass 1) and acceleration signal of the fifth mass (acceleration response ofthe highest blade, tip mass 5). Both signals are recorded in time domain with a samplingfrequency of 50Hz.
(g) force6.txt and acceleration6.txt – force applied by the electromagnetic shaker tothe lowest mass (mass 1) and acceleration signal of the sixth mass (acceleration responseof the lowest blade, tip mass 6). Both signals are recorded in time domain with a samplingfrequency of 50Hz.
Using the Matlab routine frf_general.m generate a table with the values of frequency ω, Fre-quency Response FunctionH1(ω), Frequency Response FunctionH2(ω) and Coherence FunctionCoher(ω).
2. EXTRACTING MODAL PARAMETERS – Identify 6 different 1-DOF systems around each oneof the 6 natural frequencies of the structure. Use your frequency domain identification procedurebased on the Least Square Method and obtain the experimental natural frequencies and theexperimental damping factors for each 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 modalparameters m, d and k based on:
−ω21 1
−ω22 1
−ω23 1
......
−ω2N 1
{
mk
}
=
Real(
ω21
FRF(ω1)
)
Real(
ω22
FRF(ω2)
)
Real(
ω23
FRF(ω3)
)
...
Real(
ω2N
FRF(ωN )
)
=⇒ Ax = b (3)
=⇒ x =(
ATA)−1
ATb (4)
ω1
ω2
ω3...ωN
{
d}
=
Imag(
ω21
FRF(ω1)
)
Imag(
ω22
FRF(ω2)
)
Imag(
ω23
FRF(ω3)
)
...
Imag(
ω2N
FRF(ωN )
)
=⇒ Ax = b (5)
=⇒ x =(
AT A)−1
AT b (6)
Remember that after experimentally calculating the coefficients mi, ki, and di around the six
resonance ranges, i.e. i = 1, 2, 3, 4, 5, 6, the natural frequencies can be obtained by ωi =√
kimi
and ξi =di
2√miki
.
3. TRANSIENT ANALYSIS (Analysis in the Time Domain) – Using the electromagnetic shakerattached to the lowest mass (mass 1), the structure is excited at 6 different resonances. Duringthe tests, one waits until the structure response reaches the steady-state. After reaching thesteady-state, the electrical sinusoidal signal sent to the electromagnetic shaker is turned off attime t0, as it can be identified in the beginning of the files:
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(a) force_transient_mode1.txt
(b) force_transient_mode2.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 t0 no force is appliedto the the system and the transient vibrations start. The structure oscillates with one of itsnatural 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 forceacts always on the lowest mass (mass 1). For the tests around the first, second, third andfourth resonances the acceleration of mass 4 is recorded. For the tests around fifth and sixthresonances the acceleration of mass 3 is recorded, as it is illustrated in Figure 4. Please, downloadthe following files from the course homepage in order to build figures as Figure 5:
(a) acceleration_transient_mode1.txt (measured at mass 4)
(b) acceleration_transient_mode2.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)
0 20 40 60 80 100 120−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5Acceleration of Mass 4
acc
[m/s
2 ]
time [s]
Figure 4: Experimental transient vibration response (acceleration) of mass 4 after initial condi-tion close to the first mode shape imposed by the electromagnetic shaker.
Based on the logarithmic decay, the damping factors ξi (i = 1, ..., nDOF
) associated to all sixmodes shapes can be expressed by equation (7). Please, try to calculate all six damping factors
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based on the transient analysis as
ξ =
12πN ln
(
y0yN
)
√
1 +[
12πN ln
(
y0yN
)]2. (7)
After obtaining the experimental damping ratios ξi, compare them with the experimental damp-ing factors obtained using the steady-state response. If you can not calculate the damping factorassociated to one of the mode shapes, please, explain the reason. To be sure about the frequencycomponents in the force and acceleration signals, use the routine acc_in_time_domain.m load-ing the data files mentioned previously.
7 Model Validation
1. Compare the theoretical and experimental undamped natural frequencies ωi (i = 1, ..., 6) anddamping factors ξi (i = 1, ..., 6). Quantify in percent the discrepancies between theoretical andexperimental results.
8 Model Updating
In the following, you will start to improve the mathematical model created using the results obtainedfrom the experimental modal analysis.
1. UPDATING NATURAL FREQUENCIES – Try to adjust the undamped natural frequenciesωi (i = 1, ..., 6) of the mathematical model using the experimental natural frequencies obtainedvia Experimental Modal Analysis. Neglect, in this stage of model refinement, the existence ofthe damping matrix D, i.e. D = 0, and work only with the matrices M and K. Remember thatyou can deal with many parameters in order to change the coefficients of the matrices M and K,but all changes have to be justified based on the physics of the problem. You can, for example,add or subtracts masses or change dimensions based on the tolerances of the instrumentationused. In order words, changing the coefficients of the matrices in a qualified manner, the naturalfrequencies of the mathematical model shall come closer to the experimental natural frequencies.Please, state clearly the values of ∆mi, ∆ki or ∆Li or ∆hi you are going to use and how youget to such values. Make a table illustrating the natural frequencies obtained experimentally,the natural frequencies of the unadjusted mathematical model, the natural frequencies of theadjusted natural frequencies and the error in percentage.
2. ADJUSTING α AND β COEFFICIENTS – After adjustment of the natural frequencies anddealing with the matrices M and K, try to adjust the damping factors/ratios ξi (i = 1, ..., 6)of the analytical model using the experimental damping ratios ξiexp (i = 1, ..., 6) obtainedvia Experimental Modal Analysis. Remember that α and β are the two parameters of themathematical model to be adjusted in order to achieve a better match of the six damping factors.Once you have more experimental data (six experimental damping factors) than coefficients (twocoefficients α and β), you can, if you want, decide for a least-square fitting. Re-writing equation
13
0 20 40 60 80 100 120−4
−2
0
2
4(a) Force in Time Domain
forc
e [N
]
time [s]
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5(b) Force in Frequency Domain
FF
T(f
orce
) [N
]
Freq [Hz]
0 20 40 60 80 100 120−0.5
0
0.5(c) Acceleration in Time Domain
acc
[m/s
2 ]
time [s]
0 2 4 6 8 100
0.02
0.04
0.06
0.08
0.1(d) Acceleration in Frequency Domain
FF
T(a
cc)
[m/s
2 ]
Freq [Hz]
Figure 5: Experimental transient vibration response – (a) and (b) force acting on mass 1 and(c) and (d) acceleration of mass 4 after initial condition close to the first mode shape imposedby the electromagnetic shaker.
14
(1) in a matrix form, one gets:
ξ1expξ2expξ3expξ4expξ5expξ6exp
=
12ω1exp
ω1exp
21
2ω2exp
ω2exp
21
2ω3exp
ω3exp
21
2ω4exp
ω4exp
21
2ω5exp
ω5exp
21
2ω6exp
ω6exp
2
{
αβ
}
=⇒ Ax = b =⇒ x =(
AT A)−1
AT b (8)
3. UPDATIONG DAMPING FACTORS – After adjusting all coefficients of the matrices M, K andD, please, re-calculate with help of your matlab program the new values of the six theoreticaldamping factors ξi, i = 1, ..., 6. Make a table illustrating all six theoretical damping factorand also the six experimental damping factors, obtained using the steady-state analysis and thetransient analysis. Make an error analysis and present the deviations in percent.
4. EXTRACTING MODE SHAPES – Using the 6 frequency response functions, obtain the experi-mental modes shapes. The theory dealing with the experimental identification of modes shapesfrom the experimental frequency response functions is presented in chapter 7 of our textbook.
9 Model Validation and Identification of Residual Discrepancies
1. Now, after the adjustment of all parameters related to the matrices M, K and D, comparethe theoretical and experimental frequency response functions, remembering that the excitationforce acts always on the lowest mass (mass 1) and the acceleration response, i.e. the FRFs, ofthe masses 1, 2, 3, 4, 5 and 6 are given in m/(s2N).
The frequency response function can be calculated based on the matlab routine dof2_frf.m.Remember though that you are measuring accelerations and not displacements. A frequencyresponse function relating forces to displacements (a receptance function) has units m/s andshould be multiplied by −ω2 to obtain the corresponding frequency response function relatingforces and accelerations (an inertance function). The latter will hence be given in m/(s2N).Plot theoretical and experimental frequency responses in one single figure. Compare them andjustify the discrepancies between theoretical and experimental results.
2. Compare the theoretical and the experimental modes shapes, plotting them pairwise. Justifythe discrepancies.
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10 Using the Digital Twin (Mathematical Model) to Aid theDesign of Structural and Machine Components
Until now, you have faced a typical procedure (and associated difficulties) applied when buildingmechanical and mathematical models and adjusting them via Experimental Modal Analysis. Nowyour model is adjusted. Remember, if the measured frequencies and mode shapes agree with thosepredicted by the analytical mathematical model, the model is verified and can be useful for designpurposes and vibration predictions with some confidence. Otherwise, the analytical models areuseless.
Here, as an initial approach, you have two choices: i) you can use the proportional damping matrixD previously obtained around the equilibrium when the two blades are in the vertical direction, or ii)you can just neglect the damping. Remember to clearly state your choice in your final report.
1. LINK TO NUMERICAL (INTEGRATION) METHODS – Remember that you will need to inte-grate (or solve) the equations of motions in the same way you did in the exercise 1. There isan matlab example (particle3D.m) in the website (see notes of the first class – Solving theEquations of Motion in Time) in case you need inspiration.
(a) Use your mathematical model to predict the maximum dynamic and static deflections ofthe two blades x
5(t) and x
6(t) when θ(0) = π/2 rad and the blades do not rotate, i.e. θ(t) =
0 rad/s. The initial conditions for displacement and velocity for all masses at t = 0 are zero, i.e.
x(0) = { x1(0) x
2(0) x
3(0) x
4(0) x
5(0) x
6(0) }T = { 0 0 0 0 0 0 }T
and
x(0) = { x1(0) x
2(0) x
3(0) x
4(0) x
5(0) x
6(0) }T = { 0 0 0 0 0 0 }T .
Plot the time response of the six masses. Plot the results xi(t), i = 1, 2, ..., 6 as functions
of time t, when t varies from 0 until 60 seconds or longer, until the blade static equilibriumpositions are reached.
(b) Use your mathematical model to predict the maximum amplitude of vibration of each one ofthe masses, assuming that the blades rotate with a constant angular of θ(t) = π/24 rad/s fromthe static equilibrium position found in (a), i.e. the initial conditions of velocity for all massesat t = 0 are zero, but not the initial conditions of displacement. Plot the time response of thesix masses x
i(t), i = 1, 2, ..., 6 as functions of time t, when t varies from 0 until 200 seconds.
2. LINK TO STRENGTH OF MATERIALS AND MACHINE DESIGN – Calculate the locationwhere the structure will experience its maximum stress in the dynamic case simulated in case(b). Assuming the yield strength of steel (ASTM A36 steel) σ = 250MPa, what can youconclude?
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