# How to model this real-world system to define transfer function for stability?

The system shown is an off-axis drive system that rotates the platform to 60° via a substantially-sized NEMA 34 stepper motor. The axial force required by the stepper/ballscrew assembly has already been determined. The maximum axial load on the ballscrew occurs at 40° and is approximately 150lb-force or ≈ 670N. The linear rail is a dual-rail quad-bearing carriage system capable of supporting over 5x the load weight. The stepper motor also will never reach a rotational velocity of more than 2.4 rev/sec and will be operated at full step to maximize torque. (200 ppr or 1.8° per step) The maximum motor torque required at the greatest load point is 1.7Nm. The stepper we have chosen provides 5.6Nm. Long story short, the system is not lacking in power or structural integrity in any way.

However, the system will be operated at specified degree points at low-stepping speeds. For example, it will be stopped at 40°, and then "stepped" in slow increments of 1.8°. I am controlling this via LabVIEW and can rotate the stepper at any number of 1.8° steps per second I choose. This positioning control is 100% open loop and determined by the operator. I am trying to determine the mechanical stability of the system as it is given this "step" input under a load. I'm not sure where to begin with deriving a transfer function or how to model it. As you can see, there are gas springs that assist the operation. Maybe it should be modeled as a spring-dampener system? I am not sure. I have Simulink and would like to be able to model they system in there...I'm just not sure where to start. Thank you.

uselessly general procedure:

write out equations for the kinematics, then the forces, then linearize the resulting differential equation, then transform the linearized version to frequency domain, which is your transfer function. This result will vary (i.e. is parametrized by) the range of positions.

A "good" mechanism would have a transfer function that is essentially constant at each input position.

If you suspect you have unwanted dynamics (i.e. ringing or oscillation from a step), coming from the drive mechanism or structure, which you should know by now since you have been using it and measuring it, you can possibly trade some of your excess drive power to reduce such unwanted dynamics, with an additional axial load on the drivescrew nut.

On the other hand, if this is an exercise to show that you can analyze these, then with a little advance planning, it would be possible to create for yourself a system that behaves in a way that is easy to analyze and "fix", by putting in something springy in a strategic location. You would of course begin by understanding the behavior of that element.

Also it looks like the angular direction in which the driving linkage, and the spring-cylinder-thing are pushing, go from being opposite sign to same sign, as you travel thru the range. That may be of interest.

A critical question is whether the drive screw can back-drive (translate an axial force at the nut into a torque on the motor). Complicates the analysis if so, as you would have to bring the stepper motor's dynamics into the picture. Besides that be wary of other sources of 'backlash', i.e. at the bearings in all the linkages. Anything like that can complicate analysis, and can usually be fixed mechanically.

• There is a 1:2.5 scale model that behaves favorably. This is indeed an exercise…to analyze the full-scale behavior. You are correct about the signs of the forces of the gas springs. Initially they work against the motor, albeit a small amount. Once the platform nears the 30° position, the gas springs begin to assist. At 40°, where the x-axis load on the linear rail is the greatest, the gas spring is perpendicular to the platform, to achieve optimal advantage. Note the gas springs are there for safety as much as functionality. Feb 20, 2021 at 21:59
• There is a power-off electromagnetic braking system on the stepper. It is meant for repeated holding applications and the power on/off is controlled via LabVIEW. Although it cannot be modeled as ideal, all components are of high grade with high tolerances of accuracy. I do have two equations: (1) the position of the linear carriage as a function of platform angle, and (2) the force required on the ballscrew as a function of the platform angle, given the load shown. Both are rather involved trigonometric functions (for me anyway) and I'm not sure how to go about linear-izing them. Feb 20, 2021 at 22:06
• @D Carlson, you linearize by using partial derivatives vs anything that can vary, i.e. the system state and inputs, to make a first order approximation local to each point - wikipedia ... within that page, see section that says multivariable function. generally covered in a 1st or 2nd year engineering math course on differential eqns and linear algebra Feb 21, 2021 at 4:32
• the transfer function, laplace transform etc, is usually done in the mech engr sequence under something like "system dynamics" or "control systems", hopefully that will help look up online resources. Feb 21, 2021 at 4:35
• start with the classic mass-on-a-spring example if you are not comfortable with this math in general. If the only problem is the complexity of the eqn with all the trig functions, you can also evaluate the partial derivatives numerically in a spreadsheet, or alternatively something like Mathematica Feb 21, 2021 at 4:38