# Stiffness

T

#### Tom Bullock

My folks used to say “stiffer than a board” when referring to things that weren’t flexible. In servos, stiffer than an I-beam would be a more appropriate comparison.

What is stiffness? How is it measured? How can it be improved? These are the questions that will be addressed here.

Stiffness is a measure of how well a servo can hold a position when some outside force is trying to move it. Consider a casting sitting on an x-y table with the x and y axes servo controlled. Another machine axis holding a large drill starts pushing against the side of the casting in the y direction to force the drill into the metal. This force is typically 10,000 pounds. How far will y move? The ratio of the force applied to the amount of movement is a measure of the stiffness.

One can think of it as a spring constant. The stiffness of a spring is determined by how much force it takes to compress it a unit of distance. For instance, if 100 pounds of force is applied to a spring and it compresses 1 inch, its spring constant is 100lb/in. Once one knows the spring constant, one can calculate how much the spring will compress for any amount of force applied.

It’s the same with a servo. If one were to put a gauge on the y axis mentioned above and measure the deflection when the 10,000 pounds of force is applied, one can compute the stiffness (spring constant). A deflection of .0001 inches would result in a stiffness of 10,000lb/.0001 in. = 100 million lb/in. And, in case this sounds high, values in the 500 million lb/in. range are common for machine tools with linear feedback devices.

The two most significant things that one can do if stiffness is important are to have a velocity loop and to take the position feedback directly from the linear motion. Since, in a servo, the direct linear feedback is compared to the command, any discrepancy between the two will result in a servo error that will force servo action. Furthermore, a velocity loop has an extra integrator, which will integrate (continue to add to itself) any error until something happens. It is not uncommon for the velocity loop to increase the stiffness by a factor of 1,000 from a bare position loop.

My associate, George Younkin, in his book entitled Industrial Servo Control Systems, has 24 pages devoted to stiffness. He has derived the equations for computing stiffness for both electrical and hydraulic systems and shows how they are affected when the feedback is switched to the motor shaft or the ball screw. He also shows the effect with and without a velocity loop.

Stiffness is proportional to the square of the gear ratio between the motor shaft and the feedback device, so the more ratio the better. It is also proportional to the position-loop gain.

In sizing motors, it is important that the gearing selected is such that the motor is near its rated speed when the load is moving at its maximum rate. This will ensure the maximum possible gear ratio. This is important not only for the stiffness, as mentioned above, but also because:
1. Inertia reflected to the motor is inversely proportional to the square of the gear ratio.
2. Resolution is improved with greater gear ratios.
3. The thrust available at the load is proportional to the ratio.

Stiffness is one of the more subtle specifications in a control system and is often overlooked, but it can have a major impact on accuracy in the face of load disturbances.

If you’re careless, your machine may not be “stiffer than a board,” but with a proper design it can, indeed, be stiffer than an I-beam.

_ _ _ _ _
Tom Bullock can be reached at Industrial Controls consulting, a division of Bull’s Eye Marketing, Inc. See them at bullseyenet.com, or contact Tom at (920) 929-6544 or e-mail: [email protected].

R

#### Ray Phillips

Tom,

What should be stiffer the load or the servo motor loop? Why? Any rules of thumb?

> My folks used to say “stiffer than a board” when referring to things that weren’t flexible. In servos, stiffer than an I-beam would be a more appropriate comparison.

What is stiffness? How is it measured? How can it be improved? These are the questions that will be addressed here.

Stiffness is a measure of how well a servo can hold a position when some outside force is trying to move it. Consider a casting sitting on an x-y table with the x and y axes servo controlled. Another machine axis holding a large drill starts pushing against the side of the casting in the y direction to force the drill into the metal. This force is typically 10,000 pounds. How far will y move? The ratio of the force applied to the amount of movement is a measure of the stiffness.

One can think of it as a spring constant. The stiffness of a spring is determined by how much force it takes to compress it a unit of distance. For instance, if 100 pounds of force is applied to a spring and it compresses 1 inch, its spring constant is 100lb/in. Once one knows the spring constant, one can calculate how much the spring will compress for any amount of force applied.

It’s the same with a servo. If one were to put a gauge on the y axis mentioned above and measure the deflection when the 10,000 pounds of force is applied, one can compute the stiffness (spring constant). A deflection of .0001 inches would result in a stiffness of 10,000lb/.0001 in. = 100 million lb/in. And, in case this sounds high, values in the 500 million lb/in. range are common for machine tools with linear feedback devices.

The two most significant things that one can do if stiffness is important are to have a velocity loop and to take the position feedback directly from the linear motion. Since, in a servo, the direct linear feedback is compared to the command, any discrepancy between the two will result in a servo error that will force servo action. Furthermore, a velocity loop has an extra integrator, which will integrate (continue to add to itself) any error until something happens. It is not uncommon for the velocity loop to increase the stiffness by a factor of 1,000 from a bare position loop.

My associate, George Younkin, in his book entitled Industrial Servo Control Systems, has 24 pages devoted to stiffness. He has derived the equations for computing stiffness for both electrical and hydraulic systems and shows how they are affected when the feedback is switched to the motor shaft or the ball screw. He also shows the effect with and without a velocity loop.

Stiffness is proportional to the square of the gear ratio between the motor shaft and the feedback device, so the more ratio the better. It is also proportional to the position-loop gain.

In sizing motors, it is important that the gearing selected is such that the motor is near its rated speed when the load is moving at its maximum rate. This will ensure the maximum possible gear ratio. This is important not only for the stiffness, as mentioned above, but also because:
1. Inertia reflected to the motor is inversely proportional to the square of the gear ratio.
2. Resolution is improved with greater gear ratios.
3. The thrust available at the load is proportional to the ratio.

Stiffness is one of the more subtle specifications in a control system and is often overlooked, but it can have a major impact on accuracy in the face of load disturbances.

If you’re careless, your machine may not be “stiffer than a board,” but with a proper design it can, indeed, be stiffer than an I-beam.