Rolling Radar Conceptpage 1

Rolling Radar ConceptPage 1

Project Number: 05426

Rolling Radar ConceptPage 1

Rolling radar concept

Brian McManus- Mechanical Engineering Rochester Institute of Technology / Aaron Halterman- Mechanical Engineering Rochester Institute of Technology
Caleb Herry- Mechanical Engineering Rochester Institute of Technology / Gabor Ferencz- Electrical Engineering Rochester Institute of Technology / Gabriel Chan- Electrical Engineering Rochester Institute of Technology

Rolling Radar ConceptPage 1

Abstract

In search for a better rotating radar, Lockheed Martin (LMCO)has created a rolling radar concept in which a large circular array rotates along a concentric rail system and is propelled by a magnetic drive. This concept has been only partially developed by LMCO and is lacking critical analyses of reliability, the internal gravity drive, control system, electromagnetic track configuration, array position indicator, and tribology of the wheels and axle. The scope of this Senior Design project was to help LMCO assess the feasibility of the rolling radar concept by completing the aforementioned analyses. Once accomplished, LMCO will be able to utilize the analyses to help acquire additional resources for advancing research and design.

A number of parametric analyses were done in MS Excel as part of the Senior Design feasibility assessment: weight estimation, start/stop time, shear/moment stresses, and size determination of the radar array, support wheel, and axle. A user-friendly interface was created to allow LMCO to continue the parametric analyses as parameters become solidified. An electromagnetic track configuration was chosen to aid in development of a servo control. This servo control, along with the dynamics of motion and wind disturbances, was used to create a radar control system that was modeled with Matlab to simulate the motion of this radar. The design of the array position indicator was an initial goal of the design team but had to be dropped because the design of the radar was deemed too preliminary to accomplish this task.

background

Lockheed Martin MS2, located in Syracuse, NY designsfixed and transportable radar systems. Reliability is a great concern for the fixed radar systems, as physically rotating massive structures creates a variety of load bearing and other mechanical reliability problems. An alternate design was therefore conceptualized to attempt to remedy these reliability concerns.

The current rolling radar concept consists of a large array fixed to an axle and supported at the other end by a smaller support wheel. This system sits on a concentric inner and outer track where power will be transferred electromagnetically through the outer track allowing the array to rotate and scan 360 degrees. The system is propelled by a magnetic carriage, or drive, that rides on the inside circumference of the array by an electromagnetic rail.

nomenclature

α / Angle of wind velocity relative to direction of motion (rad)
β / Degree of gravity drive rotation (deg)
μ0 / Permeability of free space (N/A2)
Φ1 / Degree of rotation of array (Rad)
Φ2 / Degree of rotation of support wheel (Rad)
θ / Array tilt angle (deg)
B / Magnetic field (T)
Cd / Drag coefficient
Fd / Force of drag, N
g / Standard gravity (m/s2)
I / Current (A)
J / Rotational Inertia
K / Stiffness of Axle, (N/m2)
l / Distance between bottom of gravity drive and electromagnetic track (m)
ma / Mass of array (kg)
mg / Mass of gravity drive (kg)
msw / Mass of support wheel (kg)
mw / Mass of axle (kg)
N / Number of coils
ρ / Density of air, kg/m3
r / Resistance of coil (Ω)
ra / Radius of array (m)
rsw / Radius of support wheel (m)
T / Torque on Array (Nm)
t / Time (sec)
V / Velocity of gravity drive carriage (m/s)
ω / Angular velocity (rad/sec)
Wm / Width of magnetic material on gravity drive carriage (m)
Vw / Wind velocity (m/s)
x / Displacement of gravity drive carriage (m)

parametric analyses

The goal of the parametric study is to evaluate the feasibility and practicality of particular parameter sets. To accomplish this, the analysis focuses on parts of the system under heavy loading, high stresses, operational requirements, and other perceived problem areas. Parts of the system that are of particular interest include the axle, tracks, gravity drive mass, overall weight distribution, and start-up time.

The size of the gravity drive needed to initiate rotation for such a large radar system, which will be approximately six meters in diameter and fifty tons, is largely unknown until analysis has been completed. Determining this weight is in integral part of the overall feasibility of the concept.

Assumptions need to be made in order to simplify the analysis of the gravity drive. To support its own weight, the outer ring of the array will be made of a high-density composite which, along with the electromagnets, will be the primary sources of weight in the array. With this in mind, a hollow ring assumption for the array can be made. Additional assumptions are as follows:

Assumptions:

No Friction

Axle - Rigid Body

Axle - Constant Cross-section

Axle – Linear Radius Variance with Array Size

Axle – Linear Wall Thickness Variance with Radius

Array - Solid Ring

Array – Constant Thickness

Support – Solid Steel Disk

Support – Linear Thickness Variance with Radius

These assumptions set up a foundation which provides a means of evaluating system characteristics. For example, since we assumed there to be no friction, the mass of the gravity drive was calculated based on a required startup time (time required to accelerate the radar up to operational speed) and the desired rotational speed (operational speed) with the Eq. (1).

/ (1)

Microsoft Excel and Visual Basic were used for evaluation and examination of parameters like size, mass, internal and external forces, stresses, and dimensions. The total system and each individual component can be evaluated by this method. Furthermore, results were plotted across parameter values for one or more variables. For example, Figure 1was generated from Eq. (1) with the gravity drive mass as the dependent parameter, start up time as the primary independent parameter, and array radius as a secondary independent parameter.

Figure 1: Gravity Drive Mass Variance with Start-up Time

Examining any parametric often reveals an optimum or simplified representation of a governing equation or parameter relationship which can be automated inside the program. This results in better parametric generation and representation. Ultimately, a streamlined effect-based analytical study of the systems feasibility and practicality is desired.

Detailed analysis of the axle is necessary since it is the main load-bearing component in the system. The total stress on the axle can be evaluated using mechanics by building on the framework described above. In Figure 2below, stress is the dependent parameter, location on the axle is the primary independent parameter, and four alternative axle radii are displayed for reference (these are 80%, 90%, 110%, and 120% of the current axle radius).

Figure 2: Max Axial Stress Variance with Axle Radius

Based on the data shown in this parametric plot, the axle material and dimensions can be matched to produce desired parameter values such as factor of safety. The stress on a fixed point on the axle fluctuates since the radar is rolling. This fluctuation is evaluated and the stress cycle applied to a fixed point on axle is shown below.

Figure 3: Axial Stress Cycle

Examination of Figure 3yieldsmore optimum relationships. For cyclical stress, zero average stress maximizes life cycles. Additionally, the reaction force ratio between the two wheels can be modified to produce the desired stress cycle.

The following specifications were used for the preliminary feasibility analysis and modeling of the control system. These specifications will change as the design project develops. A user-friendly interface in Excel was created to accommodate future changes. Below is a list of definable variables and relations.

Constant Ratios:

Support radius / array radius.448

Support thickness / support radius..25

Axle radius / array radius..24

Axle wall thickness / axle radius..01

Weight on Support / total weight.75

If a parameter relationship is used, the corresponding constant is calculated automatically.

Constants:

Array Wheel Radius3 m

Array Wheel Thickness1 m

Array Ring Width 1 m

Support Wheel Radius1.35 m

Support Wheel Thickness0.34 m

Axle Radius0.72 m

Axle Wall Thickness7.2 mm

Acceleration due to gravity9.81 m/s2

Tilt Angle30o

Gravity Drive Angle90o

Start-up Time5 min

Array Density1600 kg/m3

Support Wheel Density8000 kg/m3

Axle Density8000 kg/m3

The software evaluating the parameters and parametric relationships includes a user-friendly interface that can be used to further simplify parameter analyses. The interface incorporates all the abilities available from excel in an organized package. It smoothly integrates the use of parameter ratios into more familiar data processing methods. Automatic updates to plots and parameter values allows for easy understanding of results from parameter sets. The interface navigation system also fluidly combines analysis results and parameter set modification. This provides immediate feedback on the improvements and drawbacks of parameter variations.

Electromagnet configuration

The gravity drive system was designed to consist of an electromagnetic track with a permanent magnet drive. When designing such a system, two main assumptions were made. First, the flux in the magnetic circuit remains unchanged during the motion of the gravity drive with respect to the electromagnetic track. Therefore, no electromotive force is inducedin the coils. Second, the motion of the gravity drive with relation to the electromagnetic track takes place slowly, i.e., the acceleration of the gravity drive is close to zero. This allows for the system to be modeled more linearly.

The typical system can be modeled as a combination of mechanical force and electrical power with respect to time producing a field storage and heat. Symbolically, this is given as:

/ (2)

By rearranging the equation above, the following simplified relationship results:

/ (3)

When accounting for the energy stored in the magnetic field, the permeability of the core can be assumed to be so high that all of the stored energy is in the air gap, or the distance between the electromagnetic track and the permanent magnet gravity drive. Also, the effects of fringing are neglected so that the model will have a uniform magnetic energy density.

In order for the model to closely correlate with this system, the force must be related to the current applied to the coils in the electromagnets. The following relationship for the magnetic field can be used to rewrite the force equation to the desired form:

where,
/ (4)

The final, general form of the force equation is given in Eq. (5).

/ (5)

In order to get this form of the equation into a format that allows for an easy implementation in the control system, a linear approximation must be made. The equation is rewritten using the linear approximation method by taking a derivative of the force equation with respect to current. This leads to the transfer function of the electrical system of the gravity drive shown below in Eq. (6).

/ (6)

Because of the assumption that the gravity drive carriage is accelerating slowly enough to consider the velocity as a constant, the following transfer function can be seen in Eq. (7).

/ (7)

This equation is used to describe the relationship between the amount of current that is needed to generate a specific amount of force to propel the gravity drive carriage. The terms on the right-hand side of the equation represent the constants and variables, as noted in the nomenclature table above, that are needed to make the relationship applicable to this particular project.

When used as part of a control system, the transfer function above serves as the regulator of the gravity drive system. The input that is passed into the transfer function will be the current that needs to be applied to the coils of the electromagnets. The output will be the desired force that the gravity drive needs to supply. Based on the various sources of feedback in the control system, the required current applied to the electromagnets and the force generated by the gravity drive can be varied accordingly.

SYSTEM MODEL AND Control

The system dynamics of the radar were modeled and later controlled using the program Matlab. Initially the system was modeled as two rotating cylindrical masses, the array and the support wheel, connected by the axle. The dynamics of the system are captured through the summation of the moments around the central axis; this is shown in Eq. (8), Eq. (9), and Eq. (10).

/ (8)
/ (9)
/ (10)

Where Eq. (8) is the main array moment Eq. (9) is the axel moment equation, and Eq. (10) is the support wheel moment equation. This created a complex 4th order system that was difficult to control. The transfer function is shown below in Eq. (11). This was a result of the twisting affect in the support axle between the two rotating masses.

/ (11)

The input of the function is torque created by the gravitational drive and the output is position.

The complexity of the system dynamics and the difficulty associated with controlling them lead to an alternate representation of the radar. The new model is based on the assumption that there would be negligible twist along the axle. This assumption is based on the fact that the torque is applied through the gravitational drive that is located on the array and has by far the largest inertia of the structure. If the torque was to be applied through the support axle on the array by a motor at the base, the twist could not be considered negligible. With this assumption, the system could now be modeled as one rotating massthat was geometrically accurate. The result was a second order system that portrayed much more stable dynamics; this is shown in Eq. (12).

/ (12)

The value for rotational inertia is calculated based on the inertia of a thin ring, I=mr2. The mass is calculated through the geometry of the array and approximated density. The value for rolling resistance is traditionally found through empirical data. In the system model rolling resistance is approximated through data previously collected on railway cars with a given weight. All values are left in variable form to ease in the application of more accurate data.

Once the system was modeled, controller selection began. Several different types of controllers were considered in the initial design. On-Off or “Bang-Bang” controllers could not be applied because they lacked a steady state value due to the nature in which they work. Continuous control was needed to eliminate the oscillations of on-off control. Proportional control was not chosen because of the resulting steady-state error. Proportional-integral controller solved the problem of unacceptable steady-state error, but was far to slow to be of great use. Therefore proportional-integral-derivative control (PID) was used to speed up the reaction time of the control system.

The control system has three major functions. The first is to maintain a constant rotational velocity. This is achieved through the implementation of a PID control system. The second function is to lead to acceleration and deceleration of the radar via activation of the gravitational drive. This was accomplished through the derivation of an electromagnetic transfer function with an input of current and an output of force. In order to measure both velocity and position, a 1/s was factored out of the system dynamics to allow for a feedback loop to be taken from both variables. The third function of the control system is to have the ability to control disturbances such as wind gusts.

The disturbances due to wind are calculated based on Eq. (13) and a maximum wind gust of 60 km/hr. Drag analysis on the radar geometry resulted in either resistive or additive forces in the direction of rotation. Further analysis revealed that the maximum disturbance occurred when the wind was at a forty five degree angle to the direction of motion.

/ (13)

The modeled system, with PID control, begins with a value of current that is the input ofthe electromagnet transfer function from Eq. (7). This value is converted to the force being produced to lift the gravitational drive. This force is then multiplied by a distance, to calculate the torque created by the gravitational drive. This torque is the input to the system model with output being angular velocity and position. Also added at the torque input is the disturbance due to wind gusts.

Figure 4 is the angular velocity versus time. The array is accelerated from a stop to a constant angular velocity of 6.28 radians per second which is equivalent to three rotations per minute around the outer track. After the constant velocity is reached a wind disturbance is introduced as a constant torque that enhances angular velocity.

Figure 4: Angular Velocity vs. Time

The system model transfer function and the electromagnetic servo transfer function with the PID control allow for an accurate model and control of the system dynamics. The input of the entire system is current and the outputs are angular velocity and position. This model is very useful when considering additional design as future designers will have the characteristic behaviors of the system without need for a large expensive prototype.

conclusions