Design, development and use of a mechanism simulator for aeronautical engineering
Audren Guiho
Department of Aeronautical and Vehicle Engineering Royal Institute of Technology
Stockholm Sweden
Artikeln handlar om en strategi som mål bestå av att utveckla en särskild simulator mekanism, för närvarande utformad av Safran Group. Det finns en tidigare version av samma mekanismen men systemskalor är inte i en like-for-like grund. Alltså, fysiska fenomenen som innebärs i mekanismen och deras magnitud kan inte jämföras med tidigare versionen av systemet. Det är varför en ny simulator har skrivits i kod. Artikeln analyserar strategin som adopterades för att utforma ett system som lägger på en bred uppsättning parametrar.
This paper deals with the strategy for developing a very specific mechanism simulator. This mechanism is currently designed by Safran Group. A previous version of the same mechanism does exist but the scale is not on a like-for-like basis. Therefore, physical phenomena involved in this mechanism and their magnitude are not comparable to the previous version of the mechanism and this is why a new simulator has been developed (specified, coded and validated) from scratch. The paper addresses the strategy adopted for modelling a mechanism laying on a wide set of parameters as well as its use.
I. Nomenclature
II. Introduction
The objective of this master thesis is to develop the simulator of a specific mechanism integrated in a wider aeronautical system. A similar system already exists and includes its own simulator for its mechanism too.
The need for a new simulator comes from the need for a new equivalent system. Actually, the system at stake is re-engineered to a new scale. With this new scale comes a different dynamic, different forces with different magnitudes and moreover, different hypothesis. This justifies the obsolescence of the previous simulator. Furthermore, now that feedbacks on the previous mechanism have been gathered, the simulator could be re modeled to take into account those feedbacks and be rebuilt on more accurate hypothesis.
This paper deals with the strategy set to deliver a fully functional simulator for the mechanism of the new system. It first seems important to understand the system and its mechanism at stake very well, as well as its environment. Then, the focus is on how every single technical solution fulfills the technical functions. After understanding the mechanism itself and the need it is meeting, the following step is to define the use of the simulator and the needs it will be required to meet. This step helps designing the architecture of the simulator. Eventually, all theoretical equations are established and implemented. The simulator is now ready to be used under one of the three different uses possible.
L0 = free length of the spring L2 = length at load point 2 Lc = length at joined coil LK = buckling length Lr = length of plasticization m = step of the spring w = index of the spring d = wire diameter
D = mean diameter of the spring
De = external diameter of the spring G = torsion modulus of material E = elastic modulus of material R’ = stiffness of one coil R = stiffness of the spring n = number of coils
ni = added coils due to end fixation tkc = corrected maximal constraint tzul = acceptable maximum shear stress
III. Methodology
A. Previous work
Even though the new aeronautical system is not fully designed yet, a previous version does exist, based on the same principle. The main difference between the new one and its previous version is the scale. To design the new aeronautical system, the idea is to start from the previous design and bring it up to a new scale. Since the new aeronautical system is bigger, it is heavier, therefore it requires more lift. To increase lift, either speed of the flow (or speed of the system in the flow) or aerodynamic surfaces have to be increased. For this aeronautical system, speed cannot be increased. This why, to increase lift at constant speed, parts have to be wider and, as a direct consequence, heavier. The aim of the mechanism is to generate a controlled motion of some parts in a given time. Therefore, higher loads are applied and more energy is required to move those heavier parts to their final position.
This way, one understands that using the same concept is a good starting point: same mechanism with similar mechanical parts might be reused, but all the parameters (such as mass, inertia, geometry, number of elements…) involved in the dynamic will be affected by this change of scale.
Another benefit coming along with the use of the same mechanism is the feedbacks. Since the mechanism has been implemented years ago, different cases of failure have been analyzed and justified, so that same mistakes should not happen again.
B. System approach
The first step by dealing with an existing mechanism is to analyze and understand it by conducting a so-called “system approach”. It means not only studying the mechanism in itself, but studying every other outside system it is meant to interact with, whenever during its lifespan. By listing every single interaction with the external field of the mechanism and defining the life cycle of the mechanism, it’s is easier not to forget any function the mechanism is required to fulfill. Those functions linking the mechanism with other external systems are called “constraints functions”. The results of this study are gathered in one diagram this way:
A table then gathers and develops every single constrain function named FCi. Thanks to this study, one can understand why the mechanism does exist and what need(s) it is meeting. Further on, while designing the simulator, one has to remember those constrain functions and tailor the simulator to it.
FP1 Main function of the mechanism
FC1 Be subjected to aerodynamic effects while guarantying the main function FC2 Not breaking under aerodynamic effects
FC3 FC4
FC5 Ensure safety for maintenance or manufacturing staff FC6 Be easy to maneuver, install, uninstall
FC7 FC8
Fig 1: Mechanism’s environment and constraints functions
Fig 2: Table of constraint functions FC1
FC2
FC4 FC3
FC8 FC7 FC5 FC6
Aero flux Outside 3
Outside 1 Maintenance /
manufacturing staff Mechanism
FP1
This study being done, a good understanding of the mechanism and its interaction with other external systems is earned. In order to specify, structure and code the simulator of the mechanism, it is also required to well understand how the mechanism in itself works. This is the purpose of the next study. It consists of naming the main function of the mechanism first. Then the question is how this main function is fulfilled. The answer is a list of technical functions, each of them also fulfilled by some technical sub functions. This goes until the last technical function is basic enough to identify one single mechanical part or a combination of mechanical parts fulfilling this function. This is called the technical solution. The results from this study are gathered in a table this way:
This step offers a better understanding of the mechanism itself. Besides, it helps understanding the interaction between all parts it is composed of and the aim of each of those parts. This will definitely help during the programming process.
From needs met by the mechanism itself, come constraints on the simulator of this mechanism.
Understanding the needs met by the mechanism is a very different thing from understanding the needs met by its simulator. The very end of this system approach is to see the simulator itself as a whole system and define all its functions (and sub functions) and solutions (here equations used or coding tricks). This helps designing the architecture of the simulator before even coding it.
C. Coding the simulator
Now that the mechanism is well understood, it can be implemented. The first step is to code the simulator without even thinking about the optimization of the future mechanism. The goal here is to understand what is happening in the mechanism while it is moving. This step starts by naming all the parameters involved and their relations. The relations between parameters can be found thanks to geometric considerations or dynamic principles on each moving part. Those equations are implemented under Matlab and Simulink. Since the temporal evolution of the mechanism is dynamic, Simulink enable to solve algebraic loops step by step and find absolute solutions to differential equations. Appendix 1 shows the architecture of the simulator.
Under Simulink, there are two kinds of solver [1]. The first one is “linear search”. To find the minimum of a function and solve differential equations, it checks the area around a starting point in each direction. The second one is “trust region”. This one compares a fit model with the function in a specific region. The fit is evaluated by comparing the ratio of expected improvement from the model approximation with the actual improvement observed. Both of those solvers gives very close results but the solving strategy is not the same. The simulator is designed to choose the estimated best solver for each simulation. But if, for whatever reason, the solver fails to handle the simulation, it automatically switch to the other one and try again.
Technical solution
Technical solution To integrate with the
sysem Technical sub function 1
encastrement avec le
Technical function 3 Technical sub function 1 activateur
Technical sub function 2 To store
energy
Store at several points Main
function Be supplied with energy
To restore energy
Disable the release
Technical solution Technical solution
Spring
Technical solution
Fig 3: Functional description of the mechanism
What tends to make the simulator really hard to understand is that relations linking the different parameters are not linear. Figure 4 shows a representation of the simulator where X is an input parameter vector including all parameters involved (lengths, weight, elasticity of parts, or temperature…) and Y is an output result vector that describes the evolution of interesting values such as the position, speed and acceleration of parts as a function of time.
D. Validation
Once relations between parameters are well encoded, the simulator has to be approved. To do so, specific sets of parameters are chosen in order to try every functions of the simulator. Those specific parameters are chosen in a range including the operating range of the mechanism. With specific parameters, specific output results are expected. For instance, setting every single dissipative phenomenon (friction…) to zero makes results more predictable. For instance if a mass falls in the void and dissipative phenomena are ignored, the expected result is the mass to fall with its acceleration to be exactly 1g. This recipe is conducted for each single part of the simulator as well as for the whole simulator. This in-depth study of each function of the simulator and their association gives rise to validation paper gathering every piece of proof showing the simulator can be trusted.
This approval study analyzes magnitude of results (as seen in previous part) but also direction of evolution (increasing or decreasing) of results. Knowing the results from a simulation of a specific set of parameter, the results from another simulation having few differences in its parameter vector has to go in the right direction. For instance, a first simulation is made with a set of parameters. Then a second is done but with one single difference: inertia of a part is doubled. Since the inertia doubled, it is harder to move this part, so the simulation should foresee a bigger time of arrival. Another test is to remove every single dissymmetric parameters from the vector. This way every similar parts of the mechanism has to react the same way if exited by the same solicitation. Those verifications do not analyze the strength of phenomena but their evolution when one dimension of the parameter vector is shifted a bit.
Since the simulator is supposed to predict performances of both the former and the new mechanism, the idea is to perform a final test including output results from the former mechanism. Since the former mechanism already exists, experiments have been conducted in specific conditions (input parameter vector known for the simulator to be able to compute former mechanism performances) and results can be found in archives. To do so, the parameter vector is tailored to coincide with parameters from the former version of the mechanism and for which real experiments has already been made. If after many comparisons, output results from the simulator are the same as measured values from real experiment (within an error margin), then the simulator is approved and delivers the results it is expected to.
Here is what is obtained by running the simulator on the former version of the mechanism in standard conditions:
Fig 5: Acceleration of 4 different parts of the mechanism, results from experiments Fig 6: Velocity of 4 different parts of the mechanism, results from experiments Fig 4: Schematic representation of the simulator
Mechanism
REK 207 - Wing angular velocity after cycling at -55 °C
-5 -2.5 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30
2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.6 2.61 2.62
Time (s)
Angular velocity (rad/s)
Wing 1 Wing 2 Wing 3 Wing 4 Loom switch Start
Y1
Y2
Yi
Ym
0 0 Yi
0 IV. Using the simulator
Now that the simulator is trust-worthy, it can be used. There are three different ways to use the simulator. The first way to use the simulator is as a designing tool. By studying the impact of every input parameter on the performances of the mechanism, the simulator can help designing the mechanism and improve its performances through a sensitivity analysis. Besides, to ensure meeting specifications over a percentage of success, a statistical study is conducted thanks to the simulator. Eventually, the last available use is to tailor the input parameter vector, including all input parameters involved in the dynamics of the mechanism, to characterize a specific experiment made in a lab. The aim is to compare the results of the experiment on the real mechanism with what is foreseen by the simulator.
Understanding the difference between the real test and the simulation might eventually help readjusting the model.
A. Helping designing the new mechanism.
To do so, the strategy is to consider a nominal configuration for the mechanism with a specific input parameter vector. Then many simulations with small differences around the nominal input parameter vector are launched. Eventually, the performances of the aeronautical system equipped with different instances of the mechanism are compared. By launching all those simulations corresponding to different parameters, one of them will show better performances and will be a better nominal configuration for the mechanism.
What is important now, is how to choose the input parameters among all existing possibilities and what specific output result is legit to be compared (total duration of the simulation, remaining energy in the mechanism at the end …) from a simulation to another, to acknowledge the prevalence of one set of input parameters on another. This specific output result of importance is given by looking at the corresponding component of the output result vector. To get this output result of interest, a projection on the specific dimension of the vector (the dimension of interest) is necessary. The simulation and the following projection are together a scalar function of the input parameter vector and the aim is to minimize this function. Let’s call this function the “cost function”. Since the cost function is not linear, the matlab function “fmincon” is used to find its minimum.
Fig 7: Acceleration of 4 different parts of the mechanism, results from simulation
Fig 9: Schematic representation of the simulator and projection
Specific output result of importance, used to compare a simulation to another
Simulation
Projection
Mechanism
Time Time
Fig 8: Velocity of 4 different parts of the mechanism, results from simulation
To choose those input parameter vectors, constraints are applied on each component of the vector. For instance, due to the room available in the system, some mechanical parts cannot exceed a certain length, or, due to the maximum acceptable mass of the system, the mass of each mechanical part has to be restricted between two values. This step is equivalent to draw a hypercube of n dimensions that delineates the range of each one of the n input parameters. This cube is then regular- grid-meshed and each point of this mesh is to be passed through the cost function (simulation + projection), giving the specific result to be compared to other results from each other points of the mesh.
Here is a projection on two dimensions of the input parameter and its constraints:
( 𝑋1𝑚𝑖𝑛 𝑋2𝑚𝑖𝑛 𝑋𝑖𝑚𝑖𝑛… 𝑋𝑗𝑚𝑖𝑛
… 𝑋𝑛𝑚𝑖𝑛)
<
( 𝑋1𝑋2
… 𝑋𝑖 𝑋𝑗… 𝑋𝑛)
<
( 𝑋1𝑚𝑎𝑥 𝑋2𝑚𝑎𝑥 𝑋𝑖𝑚𝑎𝑥… 𝑋𝑗𝑚𝑎𝑥
… 𝑋𝑛𝑚𝑎𝑥)
The conceptual result of this parametric study is a n+1 dimension-nappe constrained in space which is minimized around a certain point of the mesh. This point is the n dimensional input parameter vector that makes mechanism the most efficient in the selected direction of the output results, Yi.
It is really important to decrease as much as possible the step of the mesh in each direction so that probability to find the absolute minimum of the cost function is high. Here is the impact of a poorly- chosen mesh grid in a one dimension case:
Cost function Yi
Fig 11: Coordinates in the input parameter vector of the minimum of the cost function Absolute minimum
of cost function
Fig 10: Coordinates in the input parameter vector of the minimum of the cost function Xj Xi
Xi constraints Xj constraints
Coordinates of absolute minimum
Meshed hypercube (here dimension 2)
Ximin
Ximax Xjmin
Xjmax
Nominal input parameter vector
The fmincon function needs a starting point to process minimization. From this starting point, the optimization algorithm finds a way to fall down. On the figure 12, each starting point (blue points) can only fall in pit A or C. This way, the absolute minimum of the function is never found. On figure 13 though, thanks to a smaller step, the probability for a starting point to be already at the edge of the minimum pit is higher. So it is more likely to end up in the absolute minimum pit.
Once the cost function has been minimized, the set of input parameters giving the most efficient mechanism is theoretically known. Every single parameter is chosen to match the set of input parameters hereby defined. But minimizing the cost function is not the only requirement. It is also required to check that the solution selected now meet specifications. If not, this solution needs to be abandoned and the second minimum of the cost function needs to be considered. After this step, the result is a set of parameters which is optimal in term of performances and meets the required specifications.
The following section is a design example of one part in the mechanism: the springs. The section details the strategy applied to find a good nominal point for some coordinates (the coordinates in relation with the spring (mass, length diameter, stiffness…) in the parameter vector.
The mechanism at stake features springs as a source of energy. The mechanism is supposed to meet the specifications during its lifespan (whatever happens to the mechanism). Therefore, this is really important to understand the process underneath the spring’s aging for two reasons. First of all, to model this process in the simulator. Secondly, to design springs giving the right amount of energy even at the end of the life span. To guaranty this, the spring is designed in a way to minimize the aging process.
A
B
C
A
B
C
Fig 12: Poorly meshed hypercube
Fig 13: Well meshed hypercube
Thanks to the simulator, the initial needed force (in order for the mechanism to fulfill performances) from the spring is known, F2. Usually, the step of the spring m is given by 0.3*D where D is the mean diameter of the spring.
m = 0.3*D
For one coil of the spring, the maximum displacement is:
Δ = m - d = 0.3*D - d The spring index, in order for the spring to be manufactured, is between 4 and 20 and given by:
w = D/d
At maximal displacement, the force has to be F2. F2 = R'*Δ Where R’ is the stiffness of one coil given by
R' = G*d4 / (8*D3)
This, combined with (3) yields to:
F2 = G*d2*(0.3*w -1) / (8 w3)
By stating, w (usually 10) and G (depends on material used), the wire diameter of the sprig, d, is found.
The, the external diameter of the spring is computed:
De = D + d
This value is really important because it characterizes the total room needed to put this spring in the mechanism.
The previous computations assume that coils are getting joined to get F2. But under the force F2, the spring needs to respects some rules. Among them, one specifies that the minimal space between two coils is 0.15*d. The step of the spring is then modified to be:
m = Δ + 1.15*d
For this mechanism, the total deformation f = L0 - L2 is known. The number of coils is given by:
n = f*R' / F2
Eventually the free length of the spring is computed:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10) Fig 14: Nomenclature of a spring [2]
Fig 15: Step of the spring [2]
L0 = n*(delta + 1.15*d) + ni*d
Where ni describes the end of the spring. For a better distribution of stresses, the ends of the spring are chosen to be closed and ground which yields to ni = 1.5 [3].
The total stiffness of the spring is:
R = G*d4 / (8*n*D3)
Now that the spring is designed, some checks have to be done in order to avoid buckling and plasticization. Buckling might change the direction of the force given by the spring and there might not be enough force along the spring axis to meet specifications. Plasticization might speed up aging and loss in maximal force given by the spring.
- First, the spring shall not buckle. To avoid buckling, L2 should not be less than the critical buckling length LK. LK < L2 is required.
If (L0/D) < π*[(2*m + 1) / (m + 2)]1/2 then LK= 0 (no risk of buckling)
Else μ (depends on material used) μ = E / (2G) - 1
ν (depends on end fixation used)
- Secondly, the maximum allowed stress should not be met, anywhere in the spring. Moreover, an irregularity of the distribution of the stresses in the section of the wire can be observed. The highest stress is on the internal envelope of the coils of the spring.
The coefficient of correction k is thus defined according to the curvature rate. Accord to DIN standards, k = (w + 0.5) / (w - 0.75)
The corrected maximal constraint is then:
tkc = 8*D*R*k*(L0-Lc)/(pi*d3)
This value should not exceed tzul, the acceptable maximum shear stress of the material: tkc < tzul. Let’s define the compression length where the corrected constraint reach tzul:
Lr = L0 - (pi*d3*tzul)/(8*D*R*k)
Same as for buckling, to avoid plasticization, L2 > Lr is required.
If the spring is designed according to this standards, with the chosen material, it is supposed to have a small relaxation equivalent to a loss of 1.25% under the force given at L2 at the end of the life span. Experimental studies give the following result:
(11)
(12)
(13)
(14)
(15) Fig 16: Constraints in the wire [2]
A margin of security is taken so that 5% of relaxation is considered. The simulator is then used to estimate performances of the mechanism when F2 is in fact 95%F2. This strategy enables to prove that, even at the end of the life span (and a 5% relaxation tops), the mechanism will meet specifications.
This study gives a rather good nominal point in the parameter vector (L0, n, R…) as well as constraints (Lr, LK, room allowed in the system for the mechanism…)
B. Conducting a statistical study.
The aim of this step is to take into account differences between the theoretical model and the reality by giving a statistical behavior to unknown phenomena. Since the mechanism is not theoretical but real, and will be industrialized, there will be some differences between theoretical model and what will be built. This mainly comes from manufacturing tolerances or aging of some parts. In the previous section, a nominal configuration has been found. But for instance, length of any parts are always given with a quantifiable uncertainty. This is why, the mechanism is required to guarantee meeting specifications even if its configuration is slightly shifted from the optimal parameter vector.
First of all, for each parameter of the input parameter vector, the maximal difference between the nominal value and a statistical possible value is estimated thanks to data sent by the supplier. Each parameter is now framed between a maximal and a minimal value (due to tolerances [5]) around the nominal value, as can be seen in the next graph where Xi, Xj and Xk represent three different dimensions (randomly chosen among every dimension of the problem for the example) of the input parameter vector.
Nominal value Maximal value Minimal value
Xj
Xk
Fig 17: Relaxation of the spring [4]
Fig 18: Statistical dispersion around nominal value for 3 parameters 1.25
After this step, random points in this hypercube are chosen, and simulations are run to establish whether the slightly shifted input parameter vector leads to output results meeting specifications or not.
Specifications of the mechanism also state how many errors are tolerated. If only 1 mistake over 106 is tolerated, then 8x106 simulations are run. If only 8 mistakes (performances does not meet specifications) occur, then the mechanism is guaranteed to behave as expected (even if slightly shifted from nominal configuration) for almost all scenarii.
Since there is here a statistical approach, it might be interesting to study again one of the 8x106 simulations to understand precisely why specifications are not reached. This is why, during iterations, output results are written in a data base with all input parameters involved and all output results. But those are only initial and final states of the simulation. The reason why specifications are not fulfilled might happen in the middle of the simulation. It might be a lack of energy in the mechanism as well as a solver problem. It is really important in such statistical approach to understand why the simulation failed in order not to count a solver issue as a wrongly defined input parameter vector. In order to investigate this kind of problems, a function in the simulator is created to reach any line of the database and re play the simulation.
C. Comparing simulator’s output results with a specific experiment
The idea here is to use the simulator to compare results from real life and a forecast. The results from real experiments are not yet conducted on the new mechanism since it does not exists yet. Once they will be carried out, they will be compared with the forecast. If there is no differences (or relative small ones) then it appears that the simulator handle pretty well every phenomena at stake. If there is some differences, then those experiments might help to reconsider the magnitude of some phenomena.
This is called readjusting the model.
V) Summary
Programming the simulator of this specific mechanism is not exactly like starting from scratch.
Indeed, since a former version of the mechanism exists, it enables having an experimental approach that helps modeling phenomena and using feedbacks on difficulties that might have been encountered.
Furthermore, the “system approach” at the very beginning of the thesis enabled to define, with accuracy, the boundaries of the mechanism. Moreover, investigating interactions with other elements of its environment outside of the boundaries, helps to define the needs and constraints the mechanism has to meet.
After this theoretical study, it becomes easier to take as many phenomena as possible into account while programming and define a complete and efficient strategy for the coding and approval part.
Once approved, the simulator can be used for different applications: designing the mechanism thanks to the simulator, trying to optimize the performances of the mechanism when it is designed, or conducting a statistical study to guaranty specifications will be met even if parts show small quantifiable defects. Later on, when the mechanism will be manufactured, comparing specific experiments with the simulator’s results will help readjust the simulator.
VI) Conclusion
The simulator developed during this master thesis represents a major tool for design, dimensioning the mechanism. It is being used by engineers at this very moment: the design of the mechanism is being engineered with the help of the simulator and mechanism’s predicted performances increase at each step of iteration. The System Approach strategy set to achieve the goal of coding a mechanism simulator in a limited time can be used for other projects. In a relatively close future, the model used in the simulator will be enhanced by experiments and feedbacks.
VI. References
[1] https://fr.mathworks.com/help/simulink/ug/typesofsolvers.html;jsessionid=f0962863b1adfdb4cd0d5324a3bf [2] http://www.meca.insa-toulouse.fr/~paredes/Springs2K/index.php?men=com&ide=Cnomenclature
[3] http://www.meca.insa-toulouse.fr/~paredes/Springs2K/
[4] Results from Spring calculator from IST
[5] Guide des sciences et technologies industrielles, Jean-Louis Fanchon, Nathan, March 2011
VII. Appendix
Appendix 1: Architecture of matlab the simulator used to run one simulation
The main script calls every other scripts in the order specified on the left.
Since some results depends on others, in it really important to control the order in which each script is called.
Initialization: In this script, every input parameter is given the nominal value.
Specific case instantiation: When the simulation is a bit shifted from the nominal parameters, this script re-write on every variable that differs from the nominal configuration, and affects its new value to it.
Then many computations scripts are called. Their objective is to create new variables, depending on previous variables from initialization and the case specification.
Then the simulation on Simulink happens. It computes step by step the dynamic evolution of the mechanism, solving every algebraic loops.
Once the simulation over, Results are plotted and the data base is filled.
Initialization
Specific case instantiation Computation 1
Computation 2
Computation 3
Geometric computations Aerodynamics computations
Dynamic computations
Simulation
Ploting / Data base writting
MAIN
