Structural Optimization and Modeling of

Large Dynamic Structures for Controls Simulation

 

Marvin F. “Tim” Campbell, Adel Elsaie

 

Vertex-RSI Special Structures Division,

2600 Technology Drive, Plano, Texas 75074, USA

 

 

                                                                                                                                                                 ABSTRACT 

 

Dynamic Radio and Optical Astronomy instruments are becoming larger and more sophisticated, so they demand better, more precise control to enable the narrow field optics to perform their functions.  Coaxing performance into the sub arc- second region of pointing and tracking requires much more than just reducing friction and getting stronger gearboxes.  It requires the designers to develop a fundamental feel of the structure’s personality and its dynamic properties, the system nonlinearities, and macro- and microscopic aspects of the drive systems, bearings and other motion components of the instrument.

 

Keywords: Optimization, structures, LaGrange, dynamics, time domain, frequency domain, simulation, STARSTRUC

 

1.      Introduction

 

While communications antenna systems are getting smaller due to the higher frequencies being used, radio and optical telescopes are becoming larger and more precise so they can reach deeper into the cosmos than ever before.  Also, advances in technology have allowed improvements in surface accuracy of the reflectors and reflector segments, and precision active alignment of the segments of composite reflectors.  With these advances come much narrower beam widths and tighter fields of view.  To make use of the higher precision optics, the designer must improve repeatability in pointing and tracking to get the more precise control to enable the highly refined, pinpoint optics to perform their functions.  Current generation instruments were developed to have pointing and tracking in the single integer arc-second region, while the coming generations push the performance requirements down into the sub arc-seconds, with repeatability into the hundredths.   This is not easy to do, even for the smaller dome shrouded optical instruments being designed today.   But now, with radio telescopes of 100 meter aperture (Greenbank and Eifflesburg) and larger (305 meter Arecibo), with similar class, multi-mirror optical instruments to follow, how do we coax the relatively more crude large structures and drives to perform to these higher standards? 

 

Pushing the repeatable performance down into the low sub arc-second region is a challenge, and to do so demands much more than just fabricating to precision dimensions and trying to push friction to almost nothing.  To be more effective, we have to depend upon a combination of optimal design with an in-depth understand of the macro and micro performance of the drive systems and support bearings, and the detailed nature of system nonlinearities to circumvent those problems that limit performance.  This paper outlines some of the engineering tasks that must be accomplished to ensure that the instrument as designed will meet its stated performance goals.  In a nutshell, nothing can be left to chance.

 

To be sure, the design of very large, precision pointing structures is a highly evolutionary, experienced-based endeavor that depends upon a skilled and artistic design team that has exhibited success in achieving demanding performance goals in the past.  The primary reason for this is that large projects, particularly those in astronomy, require years to develop the scientific justification, requirements and funding, followed by a long period for design, fabrication, installation and testing.  This is not conducive to the quick training of an inexperienced staff.  A special personality is required of the design team member, one that accepts the very long development cycle before the satisfaction of seeing a finished product is achieved.  Personalities that seek the quick satisfaction of standard quarterly business cycles may not be comfortable with this.  Not every one fits in.

 


 

 

2.      System Design Overview

 

To achieve the tracking performance necessary, the design must begin with an optimally designed structure that achieves the maximum of stiffness with the minimum weight to get the desired best natural frequencies and lowest deflections.  The drive system detail must be included in the structural analysis, because resonant frequencies are a global phenomenon, depending as much on the stiffness of the several stages of the gear reducer as on the largest load bearing members in the structure.  Then, the structure must be dynamically simulated in frequency domain for linear controls analysis, followed by a time domain simulation for inclusion of significant non-linear effects such as gear backlash, static and kinetic friction, instrument and command quantization, time lags, nonlinear spring rates, etc. 

 

3.      Optimum Structural Design

 

To achieve an optimum structural design, one must satisfy the following objectives and constraints:

 

·         Minimize the structural weight and cost while optimizing the natural frequencies of the structure.  For very large structures this is of utmost importance. Lower weight and high natural frequencies equates to higher performance.  Lower cost improves the probability of getting funding for the project.

 

·         The design must meet the minimal deformation criteria necessary to hold the reflector shape and to control deflections that would otherwise degrade pointing and tracking.

 

·         Sufficiently high structural and locked rotor resonances must be maintained to enable the control system bandwidth to overcome lower frequency perturbations that would move the system away from the desired pointing vector.  To accomplish this, it is vital that the designers know and understand the frequency spectrum and magnitude of perturbations that can degrade the optical qualities and tracking of the instrument.

 

·         Stresses in the structure must be sufficiently low to prevent extraordinary static stresses (from acts of God) or fatigue failure, thus leading to an efficient and safe design with good, long term performance.

 

·         Applicable codes such as AISC and/or other standard engineering design codes must be satisfied.

 

·         The buckling properties of the structure must be included in the analysis, to ensure the desired performance of the optimized lightweight design, and the natural frequency calculations must include the degrading effects of geometric stiffness.

 

For small structures these things can be performed manually with reasonable effectiveness, but on very large structures with thousands of elements of beams, shells and solids, it becomes a daunting task.  Thus the only practical way is to use automated structural optimization.  In a practical and viable optimizer, the above six competing constraints should be handled simultaneously rather than separately, since an over simplification from sequential consideration can be counter-productive in a real and complex optimization problem.  For example, in a structure with 10,000 elements, manually optimizing the most critical 50 elements for all load cases is a significant challenge, but how does one optimize the remaining 9950 members that contain the bulk of the weight within a reasonable time and at a reasonable cost?  Vertex-RSI Special Structures utilizes a finite-element, structural optimization program known as STARSTRUC.  Developed over 25 years by Dr. Adel Elsaie, STARSTRUC has been used successfully to optimize aircraft components, offshore platforms, antennas and other large structures.  Using it typically results in weight savings of about 30% over manually designed structures while improving structural performance.

 

The optimization procedure in STARSTRUC is applied to linear elastic structures with fixed geometry and material properties.  The layout of the structure is not changed during the design procedure.  In other words, the art and geometry of the structure is left to the designer, while component member selection is left to the optimizer.  One must merely start with a crude guess of the member sizes, and STARSTRUC will optimize to the same result whether the initial estimate is over or under the optimized weight of the system.

 


 

 

1.       STARSTRUC Optimization

 

Structural optimization in STARSTRUC deals with minimum weight sizing of a structural system under multiple constraints when the structure is subject to:

 

1.        Behavior constraints: the upper limits on stresses, upper and lower limits on displacements, limits on critical buckling loads and minimum natural frequencies of vibration.

 

2.        Side constraints: the minimum and maximum allowable element sizes, size proportion constraints, and fixing the size of some parts of the structural model.

 

The design variables in the structure may be divided into two groups - active and passive.  Active design variables are those values that may be changed to achieve an optimized design subject to design codes, displacement, buckling, and vibration constraints.  Passive design variables remain unchanged.  A passive design variable is usually governed by the minimum component size constraints, usually established by the designer for practical reasons. An active design variable is determined by its role in satisfying the displacement, buckling, and vibration constraints, and is generally larger than the minimum size constraints.  Discussion of the automated optimization process is too lengthy to be presented here, so references are presented at the end of this paper.

 

Because radio and optical telescope instruments are so dependent upon maintaining high natural frequencies, it is of particular importance that the optimization process includes the effects of geometric stiffness.  All of us have seen analyses that have been optimistic with regard to lowest natural frequencies.  Poorly meshed analysis models tend to be optimistic anyway, but many FEM programs tend to analyze for “free vibration” mode shapes and frequencies, in which the analysis is conducted in a mathematically weightless environment.  Study of even a simple cantilever beam indicates that its lowest natural frequency can vary substantially with an axial load, down if in compression, and up if in tension.  Since most structures are biased into compression by weight, It should be expected that the actual natural frequencies of the structure be lower than those observed in a weightless environment, i.e. those predicted by a “free vibration” FEM analysis.  Therefore, if geometric stiffness is not included in an optimization sequence, or any FEM analysis for that matter, it is likely that one of the primary goals of the optimization will not be met!

 

To use the STARSTRUC optimizer, one must first develop a suitable geometry for the structure, first assuming or estimating the sizes of the various members.  Then, one must establish the appropriate optimization goals besides just low weight.  These could include minimum frequencies of mode shapes, constraints on types of members in particular places, minimum and maximum member sizes for reasons of practicality, members that should be identical for balance and symmetry, etc.  Once the constraints are set, the optimizer can perform its function and size all members in the structure while meeting all the constraints.  STARSTRUC does this quickly, in 4 to 6 automatic optimization passes.

 

 

4.      Dynamic Simulation

 

While the structural analysis proceeds, it is prudent to develop the equations of motion and dynamic models of the system for a classic linear, frequency domain analysis leading to the preliminary control system design, followed by a time domain analysis to allow for non-linear effects evaluation.  While many standard finite element structural analysis programs can be used to develop transfer functions for various segments of a structural model, we choose to revert to a more classic approach of developing lumped spring-mass models (LMM) of the structure based on its mode shapes.   A model is created for each axis of rotation.   In doing this, we take advantage of the fact that resonant frequencies (eigenvalues) and mode shapes (eigenvectors) are global phenomena, i.e. properties of the structure taken as a whole, unlike stresses which are local phenomena.  Further, the development of automatic controls to manage the pointing and tracking is centered about the lowest 2-4 frequencies, so the use of lumped mass-spring models that replicate the desired mode shapes is entirely appropriate and adequate.  The last 40 years of antenna development has verified this.  Once the LMM is developed, the controls designer may analyze it with both linear and nonlinear analysis tools.  For the nonlinear analysis, he is free to insert appropriate nonlinear features into it to complete the simulation.  Unlike a model created from a grouping of possible transfer functions, in the LMM there is an obvious one-to-one correspondence between the nonlinear feature (usually a component property) and a definable structural entity.  Examples of nonlinarities that may be included are:

 

·         Gear backlash

·         Coulomb friction from bearings, brake and seal drag, motor brush drag, etc.

·         Nonlinear spring rates of bearings, gears and gearboxes

·         Quantization of digital commands

·         Encoder and tachometer quantization

·         Time delays, usually associated with sample-and-hold data.

·         Operational discontinuities such as step commands

·         Brake application and buffer impacts

 

1.       LaGrange's Equations and Model Subsets

 

LaGrange’s Equations can be found in almost any college dynamics textbook.  However with the treatment they are usually given, they appear to be difficult to learn and use, but this is not really the case.  Although they are derived from a rather rigorous use of Hamilton’s Principle and calculus of variations, they constitute a methodical, almost “cook book” way to develop a dynamic system’s differential equations of motion.  They are based on easy to write expressions for Kinetic energy, T, Potential energy, V, and if viscous friction is involved, dissipation energy, F.  After that, rather straight forward partial and time derivatives are used to produce the equations of motion.  For an antenna or telescope designer, they are extremely useful for allowing one to take a lumped mass and spring representation of a structure, and determine its representative equations of motion.  A set of Frequency and Time Domain simulations can then be developed to ascertain the system’s performance before it is actually built. Vertex-RSI staff members have used this method to develop equations of motion for a number of tracking antennas, including shipboard, airborne and ground based.  These include antennas for communications, radar and astronomy, and for size ranges from 1 meter through 100 meters in diameter. 

 

 

2.       LaGrange's Equations

 

The forces of a generalized system of n independent coordinates are defined as follows:

 

 

   where i = 1 to n                                                            (1)

 

 

where     L = T – V,  the LaGrangian                                                                                                                             (2)

and         T = the system Kinetic energy in terms of the

                V = the system Potential energy in terms of the

 = the ith generalized coordinate, such as the displacement of a mass or a rotation angle for a moment of inertia.

 

                F = the Rayleigh Dissipation Function.

 

                 = the ith externally applied force (normally applied to the ith mass).

 

The Raleigh Dissipation Function is present when the frictional forces are linear and proportional to the velocities squared, .  Then

                                                                                                                                      (3)

 

where     = the ith viscous friction coefficient.

 

To illustrate the methodology, the following example of a gearbox driven load, driven by a motor on top of a pedestal mass is presented.  See Figure 1 for the system configuration.

 

Text Box: Figure 1:  Elevation Drive System

 

 

 

To illustrate the use of LaGrange’s Equations, a representative set of equations of motion for an dish antenna’s elevation axis gear drive system was developed.  Upon developing an understanding of the methodology presented here, a designer should be able to develop a reasonably good set of motion equations for almost any antenna or telescope drive system.  That having been done, the roles of the structures and controls designers can work along separate but parallel analysis paths.  The structure designer may proceed along the path of refining the details of his design, protected by the global nature of eigenproblems in that there is little he can or will do to change the mode shapes of the design.  Meanwhile the controls designer can vary the springs within the LMM to move the frequencies both below and above the specification value in order to bound his design space for the potential variability of the real, installed structure.

 

 

3.       La Grange Application Example – Elevation Drive System

 

Figure 1 represents the lumped mass model of the motor armature inertia, Jm, the gearbox and pedestal inertia, Jp, and the load inertia, JL.  Also included are their associated springs, Ki and viscous frictions, Bi.  Note that the viscous drive friction, Bm, is lumped at the motor shaft, and the drive compliance KG is lumped at the output shaft of the gearbox.  The rotation angles of the various moments of inertia and the gearbox output are represented by the Greek letter .  The corresponding angular velocities are denoted by , and the angular accelerations are denoted by .  The gearbox is a speed reducer of ratio N:1, where N can be either positive or negative (negative when a clockwise motor rotation causes a counterclockwise load rotation).  The motor torque, Tm, is applied against the motor inertia, Jm, and reacts against the gear case and pedestal inertia, Jp

 

Because there are only three masses, Jm, Jp and JL, so there are only three independent variables, denoted ,  and .  The fourth coordinate, , is dependent and can be defined by an equation that relates the relative output shaft rotation of the gearbox to the relative input shaft rotation:

 

                                                                                                                                                              (4)

or,                          

                                                                                                                                                     (5)

 

To form the LaGrangian, the Kinetic Energy, T, is needed and is given by:

 

                                                                                                                                           (6)

 

and the Potential Energy, V, is given by:

 

                                                                                                                                             (7)

 

Substituting equation (5) into (7) gets the result:

 

                                                                                                            (8)

 

Upon combining the Kinetic and Potential energies, equations (6) and (8), the LaGrangian is formed:

 

                                                     (9)

 

The Rayleigh Dissipation Function, F, is given by:

 

                                                                                                            (10)

 

From LaGrange’s Equations for qm (= q1),

 

                                                                                                                                                             (11a)

 

                                                                                                                          (11b)

 

                                                                                                                                                              (11c)

 

And the forcing function Q1 is:

 

                                                                                                                                                                                      (11d)

 

Upon combining equations (11) with (2), the dynamic time domain equation of motion for the Motor mass is formed:

 

                                                                                     (12)

 

Using a similar process for qL (= q2) and qp (= q3), one can develop the remaining two equations of motion as follows:

 

For qL:

                                                                                            (13)

and finally for qp:

    (14)

 

Equations (12), (13) and (14) represent the simultaneous linear, time-domain dynamic equations of motion for the system defined by Figure 1.  Note that the external forcing function Tm is independent because its value varies with the electric current applied to the motor. For a full system of equations for an antenna, another set must be developed for the Azimuth axis.  Upon solving these equations for the accelerations of, and , one can create the simulation block diagram presented in Figure 2, and then develop a simulation model on a program such as Simulink. 

 

Why create a lumped mass-spring model like this in lieu of creating a linear simulation model on a finite element program? There are several reasons:

 

·         Mode shapes and frequencies of a structure are global properties, while stresses and deflections are local properties.  This means that fairly significant localized variations in the structure have little effect on the global properties other than slight changes in frequency.  This allows the controls development and structural refinement efforts to be independent.

 

·         Once the LMM is created, it can be quickly tuned to a range of natural frequencies just by changing the springs.  Thus, the controls designer can bound his design problem by examining a wide range of potential structural frequencies, should the FEM analysis fail to produce an accurate estimate of the real system frequencies.  To do this within the finite element model requires a reconfiguration of the many components of the structure.

 

·         In the LMM, the placement of nonlinearities such as backlash, command and sensor quantization, time delays, non-linear springs, friction models, etc. can be readily accomplished because there is direct physical one-to-one correspondence between the masses and springs in the system and the real features of the system.


 

 

Figure 2:  Simulation Block Diagram – Elevation Drive System


 

5.      Macroscopic vs. Microscopic Effects

 

To achieve the finest resolution of control, it is important to understand motion components such as bearings, gears, wheels etc., in response to the way they act under the types of loads they are carry.  Also, it is important to utilize the control system to provide stiffness where there is little or none otherwise.  This, of course, is the secondary purpose of the control system following pointing and tracking.  Unfortunately, there are too many servo and drive designs on otherwise excellent astronomy instruments that fail to establish this capability effectively, so they fail to meet their tight, 1-2 arc-second specifications.  The controls designer must realize that loaded rolling elements and gears do not have linear spring rates, and have non-trivial amounts of deflection when compared to an arc-second or two.  Further, while these devices get stiffer with load, conversely, they approach zero stiffness as their instantaneous loads pass through zero.  Since the drive system stiffness is one of the multipliers in the closed loop control system gain, this amounts to a region of unstable, low gain operation that exists around any near zero external load condition.  Unfortunately, this is where most instruments spend most of their operational life!

 

For example, a single drive gear and/or axis bearing may go from zero to full load with about 0.010 inch (0.25 mm) deflection, with a nominal of about 0.005 inch (0.12 mm) for the normal drive load.  This is due to highly nonlinear properties and low stiffness of the components near the zero load point.  If the load direction is reversed, the deformation must recover and move the opposite direction before the bearing (or gear) will generate a full opposing reaction force.  As an result, gear mesh or bearing might shift about .010 inch (0.25 mm) or so during a load reversal.  If the drive radius is 60 inches (1.5 m), then this represents a potential low stiffness region of over 200 arc-seconds over and above the gear backlash.  Thus, with the use of a single drive or multiple drives without counter-torque, even the use of zero-backlash gearboxes does not eliminate a substantial range of lost motion due to low stiffness. Wind turbulence will excite and increase the motion.  The result is a slow, low frequency servo oscillation and lack of wind turbulence control because of a low gain instability about the null load point.  The effect is often described as “hunting”.  Regardless of the terms used, it is still an instability, and it generally means that the control system will never satisfy its pointing and tracking goals.  The reality of such a system is that it is usually more stable in the presence of a steady wind, because the load biases the nonlinear drive components into a more linear, higher stiffness (and thus higher gain) region. 

 

Because the magnitude of instability is reduced by lowering bearing friction, designers tend to assume that friction near zero is the answer that will cure all problems.  While very low friction is good, it has no effect on externally applied loads such as wind buffeting.  To control this problem, only sufficient drive system stiffness and control loop gain will suffice.  To achieve this, it is necessary to bias multiple drives against each other to reshape the fundamental system stiffness curves of the drive system.  In this way, the inherent null in stiffness (zero slope of the load-deflection curve) can be eliminated. Figure 3 represents the stiffness curve of a single, zero backlash drive system.  Figure 4 represents the much superior stiffness profile of a pair of countertorqued drives.  This curve is created when dual drives are preloaded against each other at the preload points indicated in Figure 3.

 

6.      Conclusions

 

Large radio and optical telescopes require more than just creating a structure and pushing it around with a cheap drive system.  The large weight and inertial loads must be minimized and structurally optimized to the extent possible to make the system more economically feasible and easier to control.  To accomplish this, structural optimization  both reduces weight and places structural stiffness where it will do the most good.  Once this is accomplished, the high performance system must be modeled and simulated to allow design perfection of the control system.  Finally, it is important to realize the impact of nonlinear springs in the drive system components on the ability of the controls to provide tight, precision pointing and tracking.  Multiple, countertorqued drives are essential to accomplish this.

 

 

 

 


 

 

        Figure 3:  Unbiased Component Stiffness        Figure 4:  Counter-loaded Component Stiffness

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

7.      References

 

1.        Marvin F. Campbell, numerous proprietary documents on large structures, controls and simulation.

2.        Elsaie, A.M., Tabarrok, B., and Fenton, R.G., “STARSTRUC: Structural Optimization Software and its Applications,” Seventh Symposium on Engineering Application of Mechanics, University of Toronto, Toronto, Canada (June 1984)

3.        Elsaie, A.M., “STARSTRUC: Structural Optimization Program for Large Systems,” Finite Element Method, Modeling, and New App., CED. Vol. 101, ASME Conference, Chicago (July 1986)

4.        Elsaie, A.M., “Optimizing Structural Design,” CAE (Oct. 1986)

5.        Elsaie, A.M., Klindra, B.E., Love, M.H., and Rogers, W.A., “Optimization of Aircraft Structure Using STARSTRUC,” SAE General Aviation Aircraft Meeting, Wichita, Kansas (Apr. 1987)

6.        Elsaie, A.M., and Santillan, R. Jr., “Structural Optimization of Landing Gears Using STARSTRUC,” SAE General Aviation Aircraft Meeting, Wichita, Kansas (Apr. 1987)

7.        Elsaie, “STARSTRUC User’s Guide,” 1989

 

 

 

                                                                               

Correspondence:  Email:  Tcampbell@vertexdallas.com;  Telephone 972-424-1557;  FAX  972-424-8285;

Vertex-RSI Special Structures Division, 2600 Technology Drive, Suite 500, Plano, TX  75074, USA.