A brief discussion on the rigidity and inertia of servo motors

To talk about rigidity, we must first talk about stiffness. Stiffness refers to the ability of a material or structure to resist elastic deformation when subjected to force, and is a representation of the ease of elastic deformation of a material or structure. The stiffness of a material is usually measured by the elastic modulus E. Within the macroscopic elastic range, stiffness is the proportional coefficient of the load and displacement of a part, that is, the force required to cause a unit displacement. Its reciprocal is called flexibility, that is, the displacement caused by a unit force. Stiffness can be divided into static stiffness and dynamic stiffness.

The stiffness (k) of a structure refers to the ability of an elastic body to resist deformation and stretching. k=P/δ, where P is the constant force acting on the structure and δ is the deformation caused by the force. The rotational stiffness (k) of a rotating structure is: k=M/θ, where M is the applied torque and θ is the angle of rotation. For example, we know that steel pipes are relatively hard and generally deform less when subjected to external forces, while rubber bands are relatively soft and deform more when subjected to the same force. In this case, we say that steel pipes have strong rigidity and rubber bands have weak rigidity, or that they are more flexible.

In the application of servo motors, using a coupling to connect the motor and the load is a typical rigid connection; using a synchronous belt or belt to connect the motor and the load is a typical flexible connection. Motor rigidity refers to the ability of the motor shaft to resist external torque interference, and we can adjust the rigidity of the motor in the servo controller. The mechanical stiffness of a servo motor is related to its response speed. Generally, the higher the rigidity, the higher the response speed, but if it is adjusted too high, it is easy for the motor to produce mechanical resonance. Therefore, there is an option to manually adjust the response frequency in the general servo amplifier parameters. It needs time and experience to adjust it according to the resonance point of the machine (in fact, it is to adjust the gain parameter).

In the servo system position mode, force is applied to deflect the motor. If the force is large and the deflection angle is small, the servo system is considered to have strong rigidity. Otherwise, the servo is considered to have weak rigidity. Note that the rigidity I am talking about here is actually closer to the concept of response speed. From the perspective of the controller, rigidity is actually a parameter composed of the speed loop, position loop and time integral constant, and its size determines the response speed of the machine.

Servo motors like Panasonic and Mitsubishi have automatic gain functions and usually do not need to be adjusted. Some domestic servos can only be adjusted manually.

In fact, if you do not require fast positioning but only accurate positioning, when the resistance is not large and the rigidity is low, accurate positioning can also be achieved, but the positioning time is long. Because low rigidity leads to slow positioning, when fast response and short positioning time are required, there will be an illusion of inaccurate positioning.

While inertia describes the inertia of an object's motion, rotational inertia is a measure of the inertia of an object's rotation around an axis. Rotational inertia is only related to the rotation radius and the mass of the object. Generally, if the load inertia exceeds 10 times the motor rotor inertia, it can be considered to have a large inertia.

The rotational inertia of the guide rail and the lead screw has a great influence on the rigidity of the servo motor transmission system. Under a fixed gain, the larger the rotational inertia, the greater the rigidity, and the easier it is to cause the motor to shake; the smaller the rotational inertia, the smaller the rigidity, and the less likely the motor will shake. The motor can be made non-shaky by replacing the guide rail and lead screw with a smaller diameter to reduce the rotational inertia and thus reduce the load inertia. We know that when selecting a servo system, in addition to considering parameters such as the motor's torque and rated speed, we also need to first calculate the inertia of the mechanical system converted to the motor shaft, and then select a motor with a suitable inertia size based on the actual mechanical action requirements and the quality requirements of the workpiece.

During debugging (in manual mode), correctly setting the inertia ratio parameters is a prerequisite for fully utilizing the best performance of the mechanical and servo systems.

So what is "inertia matching"? In fact, it is not difficult to understand. According to Newton's second law: the torque required by the feed system = system moment of inertia J × angular acceleration θ. Angular acceleration θ affects the dynamic characteristics of the system. The smaller θ is, the longer it takes for the controller to issue a command to complete the execution of the system, and the slower the system response. If θ changes, the system response will be fast and slow, affecting the processing accuracy. After the servo motor is selected, the maximum output value remains unchanged. If you want the change of θ to be small, J should be as small as possible. In the above, the system moment of inertia J = servo motor rotational inertia momentum JM + motor shaft converted load inertia momentum JL. The load inertia JL is composed of the inertia of the workbench and the fixtures and workpieces, screws, couplings and other linear and rotating moving parts installed on it converted to the inertia of the motor shaft. JM is the servo motor rotor inertia. After the servo motor is selected, this value is a fixed value, while JL changes with the change of the workpiece and other loads. If you want a smaller change rate of J, it is best to make JL a smaller proportion. This is what is commonly known as "inertia matching".

Generally speaking, motors with small inertia have good braking performance, quick response to starting, accelerating and stopping, and good high-speed reciprocating performance, and are suitable for some light load and high-speed positioning occasions. Motors with medium and large inertia are suitable for large loads and occasions with relatively high stability requirements, such as some circular motion mechanisms and some machine tool industries. Therefore, if the servo motor is too rigid or insufficient, it is generally necessary to adjust the controller gain to change the system response. Excessive inertia and insufficient inertia refer to a relative comparison between the inertia change of the load and the inertia of the servo motor.