Mechatronics Modeling and Analysis of Dynamic Systems Case-Study Exercise

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1 Mechatronics Modeling and Analysis of Dynamic Systems Case-Study Exercise Goal: This exercise is designed to take a real-world problem and apply the modeling and analysis concepts discussed in class. As the design engineer, your goal at this stage is to understand the physical system, develop a physical model of it, derive the mathematical model, and predict the dynamic behavior. Physical System: Rotor Stator Gear Train Solar Array Inboard Body Outboard Body +x Solar Alpha Rotary Joints Solar Array Figure 1. Space Station Freedom The Solar Alpha Rotary Joint (SARJ) on Space Station Freedom is a single-axis, angular-positioning mechanism used to point the solar arrays toward the sun. Ultimately, it is desired to design a control system which ensures the outboard body tracks a commanded-position input. However, the first step in the control system design is to understand and model the physical system. The SARJ is the structure which connects the Space Station's outboard and inboard truss segments as shown in the schematic of Figure 1. It allows for controlled rotation about the x-axis and constrains the segments about the other five degrees of freedom. The outboard body is moved with respect to the inboard body using a brushless DC torque motor which drives a gear train. The stator of the motor is connected rigidly to the inboard body and can be assumed to be mechanically grounded (i.e., inboard body and stator do not move) because the mass moment of inertia of the inboard body is significantly greater than the inertia of outboard body. The intermediate gear train inertia is very small compared to Case-Study Exercise 1

2 the rotor inertia. However, the torsional flexibility of the gear train is important because it has a relatively small stiffness. The solar arrays have an even lower torsional stiffness and are also important to the dynamic response of the system. Both of these stiffnesses are very lightly damped (0.5% damping, ζ =0.005); this low damping results because the structures are relatively light weight in order to minimize the costs of placing them in orbit. All the system parameters are approximately constant over time. Friction from the bearings and gear-train backlash are important nonlinear phenomena for this type of system. However, during nominal operation of the SARJ, it rotates at a constant velocity in a single direction. In fact it rotates 360 every 90 minutes (same period as the Orbiter) to cancel out the orbital rotation rate of the Space Station and keep the solar arrays pointed at the sun. Because it moves in a constant direction, the gear train will always remain engaged and thus backlash will not be a problem. The friction losses will be approximately constant and can be initially neglected from the dynamic model providing the control system is eventually designed to allow a constant motor torque output to cancel friction (i.e., the controller will need an integrator or feed forward in the position loop). All environment noise can be neglected except for the plume loads on the solar arrays. When the Orbiter (Space Shuttle) docks with the Space Station, the reaction control jets firing from nose and tail of the Orbiter will induce a torque on the solar arrays. During one of the Shuttle missions, measurements of these plume loads were taken in order to quantify the magnitude and direction of these disturbances. Such tests are performed because designers, like yourself, require this information in order to design control systems which are robust to disturbances. Step 1 - Physical Model: R L V in + - i m + e b J m Stator Rotor q m - Stator mechanically grounded N:1 Gear Ratio K 1 K 2 B 1 J ob J sa B 2 q m q ob q sa T d +x Figure 2. Physical Model of SARJ Dynamics Case-Study Exercise 2

3 Figure 2 shows the physical model of the SARJ after making all the appropriate simplifications. The notation is defined as follows: Symbol Meaning Units V in input voltage to armature volts i m armature current amps R motor resistance ohms L motor inductance henrys e b back emf voltage volts K t motor torque constant ft-lb/amp K e back emf constant volts-sec/rad N gear ratio unitless J m motor inertia slug-ft 2 θ m motor position radians m motor position reflected through gear train radians J ob outboard truss inertia slug-ft 2 θ ob outboard body position radians J sa solar array inertia slug-ft 2 θ sa solar array position radians K 1 gear train and truss torsional stiffness ft-lb/radian B 1 gear train and truss torsional damping ft-lb-s/radian K 2 solar array torsional stiffness ft-lb/radian B 2 solar array torsional damping ft-lb-s/radian T d plume disturbance torque ft-lb T m motor torque ft-lb T m motor torque reflected through gear train ft-lb Q1: List all approximations of the physical system required to generate the physical model shown in Figure 2. Describe not only the approximation and why it is valid, but also describe the corresponding mathematical simplification. Step 2 -Mathematical Model: Q2: Looking at Figure 2, what are the constitutive physical relations (physical laws) which will be used to write the equations of motion for the system? What are the equilibrium and compatibility relations you will need? Case-Study Exercise 3

4 Q3: Draw a free-body diagram and write the differential equations of motion for the following groups in Figure 2: a) motor electronics b) rotor inertia c) outboard body inertia d) solar array inertia Are they all Ordinary Differential Equations? Are they linear? Q4: Draw a block diagram of the physical model. What are the inputs to the physical model? Let's make one more approximation in order to further simplify the model. In practice, a servo amplifier (i.e., power amp) is added to the DC motor. This effectively compensates for the back emf effect and allows the motor to be accurately modeled using just the torque constant (i.e., Tm = imkt, where im is now an input rather than a state variable and Vin, L, and R are no longer included in the model). Thus, for the remaining questions assume T m = i m K t where i m is an input. Remember, our objective is to develop a physical model which will predict the behavior of the actual system. Thus, we will need to solve these linear ODE's. The easiest way is to write the equations in state space form and solve them numerically using MATLAB. Q5: Rewrite the linear ODEs in state space form: L M m ob ẋ = Ax + Bu y = Cx + Du θsa im x = u y θ m Td θ ob M P θ N θ sa O P Q where: L = N M O L Q P = NM θ θ m ob sa O QP Case-Study Exercise 4

5 Step 3 - Predict Dynamic Behavior: Use the following values for your simulation: Symbol Value Units i m input amps K t 8.26 ft-lb/amp N 283 unitless J m 9.4E-3 slug-ft 2 m output radians J ob 400 slug-ft 2 θ ob output radians J sa 7.0E5 slug-ft 2 θ sa output radians K E6 ft-lb/radian B ft-lb-s/radian K E5 ft-lb/radian B E3 ft-lb-s/radian T d input ft-lb Q6: Code the state space equations into MATLAB. a) What are the poles of the system matrix? What are the corresponding frequencies and damping ratios (ζ) of these poles? b) Generate all six transfer functions which map the two inputs to the three outputs. Also, generate all six corresponding Bode plots. Plot only the frequency range from 0.1 rad/sec to 1000 rad/sec. Label the frequency of the lightly-damped resonances and anti-resonances. Do the frequencies of the resonances correspond with the lightlydamped poles you got in (a)? c) Develop simplified physical models for motion at the following frequencies: 0.62 rad/s, rad/s, and rad/s. Draw free-body diagrams and determine the natural frequencies of these systems. d) Perform a time simulation with a cosine current input of frequency 0.62 rad/s and plot the position of the motor, outboard body and solar array. Do these results correspond with your predictions in part (c)? Case-Study Exercise 5

6 e) Perform a time simulation with a cosine current input of frequency rad/s and plot the position of the motor, outboard body and solar array. Do these results correspond with your predictions in part (c)? f) Perform a time simulation with a cosine plume torque disturbance of frequency rad/s and plot the position of the motor, outboard body and solar array. Do these results correspond with your predictions in part (c)? Case-Study Exercise 6

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