Understanding Inertia and Reflected Inertia

Understanding Inertia and Reflected InertiaThe Important Role Inertia Plays in Motion Control... Inertia Definition“1a: A property of matter by which it remains at rest or in uniform mot

Understanding Inertia and Reflected Inertia

The Important Role Inertia Plays in Motion Control

Understanding Inertia and Reflected Inertia

Inertia Definition

“1a: A property of matter by which it remains at rest or in uniform

motion in the same straight line unless acted upon by some external force.”

-Merriam-Webster Dictionary

“An object at rest will stay at rest and an object in motion will stay in motion with the same speed and direction unless acted upon by an unbalanced force.”

-Newton’s First Law of Motion

Mass is directly related to Inertia

Inertia Demonstration

Demonstration:

Force vs Inertia

Free Body Evaluation of Forces

Inertia Relative to Mass

Mass and Inertia

Inertia is the property of an object of matter to resist change in acceleration

F = ma

If it takes force to change the acceleration of an object then for linear motion inertia is directly related to mass of an object By the above equation the larger a mass is (or the more inertia it has) the more force will be required to change the acceleration of that object

Inertia Evaluation

Does this make sense?

Lead has a higher density then rubber, and for a hollow sphere of the same volume has more mass This makes sense, intuitively a tennis ball made of rubber would be lighter than a hollow lead sphere of the same geometry Weight is the result of the acceleration of gravity acting on a body of mass

Property Tennis Ball Hollow Lead Sphere

Density 0.002 slugs/in 3 0.012 slugs/in 3

Rotary Inertia Definition

Rotary Inertia

- Also known as moment of inertia

“A measure of the resistance of a body to angular acceleration about a given axis that is equal to the sum of the products of each element of mass in the body and the square of the element’s distance from the axis.”

Angular Inertia Model

Force = Mass x Acceleration

Torque = Inertia x Angular Acceleration

Angular Inertia Model

Moment of Inertia for a Rigid Body

-Assumes uniform density

𝐼𝐷 = 𝑟2 𝑑𝑚 = 𝑟2𝑑𝑚

Angular Inertia Model

Defining Moment of Inertia for a Disk

We can break the mass of the disk up into small

ring sections with reducing radii and find the

inertia of each ring.

Integrate to find the total inertia of the disk

To perform this computation the mass needs to

be related to radius of the disk

𝐼 = 𝑟2 𝑑𝑚

𝑑𝐼 = 𝑑𝑚𝑟2

𝐼 =

𝑟=0 𝑟=𝑅 𝑑𝑚𝑟2

Angular Inertia Model

Defining Moment of Inertia for a Disk continued

Solid Cylinder about central diameter

Thin rod about axis through end perpendicular

to length

Solid sphere about any axis

Thin spherical shell about any diameter

(cut away shown below)

Inertia Used with Bodies in Motion

𝑇 = 𝐼 ×∝

Applying Conservation of Energy:

If Torque is constant then angular

acceleration can be manipulated by the

moment of inertia

As inertia is increased, velocity decreases

and as inertia is decreased, velocity

increases

Inertia Ratio Definition

Inertia Ratio

In motion control the inertia ratio is defined as follows:

𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑅𝑎𝑡𝑖𝑜 = 𝐼𝑙

𝐼𝑚𝑊ℎ𝑒𝑟𝑒

Inertia Optimization Proof

Inertia Optimization Proof

Taking the derivative of αL with respect to Gr:

Inertia Optimization Proof

To find the gear ratio that results in the maximum acceleration the derivative is set equal to zero

The Impact of Inertia Ratio

Why does the inertia ratio matter in motion control?

Coupled loads are often idealized

Adding the deflection properties of the coupling introduces an element

of energy conservation in form of a spring mechanism

Servo systems can be highly dynamic and are often used in applications that require quick response with minimal overshoot and settling time.Inertia ratios help to address performance in the transient response of a system

Inertia Ratio Recommendations

Typical Inertia Ratio Industry Recommendations

Stepper Motor Driven Systems:

1:1 or as close to 1:1 as is reasonable for the system

High Inertia Ratios

High Inertia ratios can lead to the following:

 Sub satisfactory performance

 Vibration/Noise

 Unstable operating condition

These all reflect poor control of the systems transient response

The mechanical components degree of compliance will be a factor as well

 Stiff mechanics improve response

 Soft mechanics reduce response

Stored Energy of a Coupling

Deflection of a Rigid Coupling Modeled as a Hollow Shaft

Connecting a Load

A rigid coupling has little deflection and can optimize system response, but generally is not as forgiving on shaft alignment and manufacturing tolerances

 Alternative coupling technologies add compliance

 Compliance effects the dynamic system response

 Steady-state operation is less critical of inertia ratio

For a given system performance target the stiffness of the coupling will allow for varied degrees of inertia ratio Coupling, in this statement refers to any mechanical component between the load and the motor

Inertia Transmission

Coupling Modeled as a Spring

 Servo controlled assembly

 High acceleration and deceleration

 Coupling deflection stores energy

The deflection recovery can be modeled as a

spring

𝑇 = −𝑘 × 𝜃𝑊ℎ𝑒𝑟𝑒

𝑇 = 𝑡𝑜𝑟𝑞𝑢𝑒

𝑘 = 𝑠𝑝𝑟𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝜃 = 𝑡𝑜𝑟𝑠𝑖𝑜𝑛𝑎𝑙 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛

Inertia's Effect on System Control

Let’s consider inertia’s effect on torque and acceleration

𝑇 = 𝐼 ×∝

If the system performance goal for acceleration is fixed then:

 Higher inertia leads to higher torque

 Higher torque leads to higher deflection

 Higher deflection leads to a longer settling time, or unstable

conditions

This may explain why inertia miss-match for direct drive, rigidly coupled loads has not been of much concern in servo systems

Coupling Evaluation J M to J L

𝑇 − 𝐵𝑀𝜃𝑀 − 𝐵𝑀𝐿 𝜃𝑀 − 𝜃𝐿 − 𝐾𝑆 𝜃𝑀 − 𝜃𝐿 = 𝐽𝑀𝜃𝑀

−𝐵𝐿𝜃𝐿 + 𝐵𝑀𝐿 𝜃𝑀 − 𝜃𝐿 + 𝐾𝑆 𝜃𝑀 − 𝜃𝐿 = 𝐽𝐿𝜃𝐿Where:

JM = rotor inertia of the motor

JL = the load inertia

KS = coupling elasticity

T = applied torque

BML = viscous damping of the coupling

BM = viscous damping between ground and rotor

BL = viscous damping between ground and load Expressions for angular acceleration:

Reflected Inertia Definition

Tangential drive

Screw drive

Direct Driven Reflect Inertia

Reflected Inertia of a Gear Drive

Gear Drive

 Speed reducing device

 Gears make up mechanical linkage

Reflected Inertia of a Belt or Rack Drive

Tangential Drive

 Belt & pulley linkage etc

 Load transmitted to motor off of pulley tangent

Reflected Inertia of a Screw Drive

Screw Drive

 Screw and nut linkage etc

 Load transmitted to motor from screw

Reflected Inertia Example

Reflected Inertia example 1:

A belt and pulley driven linear axis has a 15lb load and a pulley

diameter of 2in It is a two pulley configuration with both the drive

pulley and idler pulley having an inertia of 3.1x10-5 slug-ft2 The motor

directly coupled to the drive pulley has a rotor inertia of 1.5x10-5

slug-ft2 What is the inertia ratio of the system?

𝐼𝑙 = 15𝑙𝑏𝑠 ×.083𝑓𝑡

2 32.2 𝑓𝑡

𝑠2

+ 3.1 × 10−5𝑠𝑙𝑢𝑔𝑓𝑡2 + 3.1 × 10−5𝑠𝑙𝑢𝑔𝑓𝑡2 = 3.27 × 10−3𝑠𝑙𝑢𝑔𝑓𝑡2

𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑟𝑎𝑡𝑖𝑜 = 3.27 × 10−3𝑠𝑙𝑢𝑔𝑓𝑡2

1.5 × 10 −5 𝑠𝑙𝑢𝑔𝑓𝑡 2 = 218: 1

This will not be a well controlled system, what

can be done to improve the inertia ratio?

Reflected Inertia Example

Reflected Inertia example 1 continued:

Adding a 10:1 gearbox between the motor and drive pulley of the belt driven system

The new inertia ratio is:

What adverse affect might this have on the systems performance?

-Possibly speeding limiting either by the motor or gearbox

Speaker Contact Details

Keith Knight