Understanding Inertia and Reflected Inertia

Understanding Inertia and Reflected Inertia The Important Role Inertia Plays in Motion Control Understanding Inertia and Reflected Inertia Overview  Inertia Definition  Inertia Ratio  Reflected Inertia  Reflected Inertia of Mechanical Motion Components 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 Force vs Inertia Fgravity Fdrag x Fgravity Fdrag x Fthrow x Fthrow y The ball continues to move on the x-axis even though there is no longer a propulsion force present 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? Property Tennis Ball Hollow Lead Sphere Diameter 2.7 in 2.7 in Wall Thickness 0.1in 0.1in Volume 1.1 in3 1.1 in3 Density 0.002 slugs/in3 0.012 slugs/in3 Mass 0.0022 slugs 0.0132 slugs Weight (Force) 1.0 oz 6.8 oz 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 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.” -Merriam-Webster Dictionary “An object at rest will stay at rest and an object in angular motion will stay in motion with the same speed and direction unless acted upon by an unbalanced torque.” -Newton’s First Law of Motion (applied to rotary motion) Mass and distance to the axis of rotation is directly related to the moment of inertia Angular Inertia Model Force = Mass x Acceleration Torque = Inertia x Angular Acceleration 𝑇 = 𝐼 ×∝ For a single particle of mass m 𝐼𝐷 = 𝑟 × 𝑚 Units 𝐼 = 𝑚2 × 𝑘𝑔 ∝= 𝑟𝑎𝑑𝑠 𝑠2 𝑇= 𝑚2 ×𝑘𝑔 𝑠2 ∗∗ 𝑁 = = Nm 𝑘𝑔×𝑚 𝑠2 Angular Inertia Model Moment of Inertia for a Rigid Body -Assumes uniform density 𝐼𝐷 = 𝑟 𝑑𝑚 = 𝑟 𝑑𝑚 Stored Energy of a Coupling Deflection of a Rigid Coupling Modeled as a Hollow Shaft 𝐿×𝑇 𝜃= 𝐺 × 𝐼𝑜 𝑊ℎ𝑒𝑟𝑒: 𝜃 = 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑠ℎ𝑎𝑓𝑡 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑇 = 𝑡𝑜𝑟𝑞𝑢𝑒 𝐿 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝐺 = 𝑠ℎ𝑒𝑎𝑟 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑟𝑖𝑔𝑖𝑑𝑖𝑡𝑦 𝐼𝑜 = 𝑠𝑒𝑐𝑜𝑛𝑑 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝜃= 32 × 𝐿 × 𝑇 𝐺 × 𝜋 × 𝐷4 − 𝑑4 𝑊ℎ𝑒𝑟𝑒: 𝐷 = 𝑠ℎ𝑎𝑓𝑡 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑑 = 𝑠ℎ𝑎𝑓𝑡 𝑖𝑛𝑠𝑖𝑑𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝜃= 32 × 1𝑖𝑛 × 5𝑖𝑛𝑙𝑏 10.9 × 106𝑝𝑠𝑖 × 𝜋 × 754𝑖𝑛 −.3754𝑖𝑛 −4 𝜃 = 1.58 × 10−5 𝑟𝑎𝑑𝑠 = × 10𝑑𝑒𝑔𝑠 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 × 𝜃 𝑊ℎ𝑒𝑟𝑒 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 JM to JL Full article and evaluation can be viewed at the following link posted 02/10/2015 http://www.motioncontrolonline.org/content-detail.cfm/MotionControl-Technical-Features/Understanding-the-Mysteries-of-InertiaMismatch This has been provided by Motion Control Online, by contributing Editor Kristin Lewotsky Coupling Evaluation JM to JL Expressions for angular acceleration: 𝑇 − 𝐵𝑀 𝜃𝑀 − 𝐵𝑀𝐿 𝜃𝑀 − 𝜃𝐿 − 𝐾𝑆 𝜃𝑀 − 𝜃𝐿 = 𝐽𝑀 𝜃𝑀 −𝐵𝐿 𝜃𝐿 + 𝐵𝑀𝐿 𝜃𝑀 − 𝜃𝐿 + 𝐾𝑆 𝜃𝑀 − 𝜃𝐿 = 𝐽𝐿 𝜃𝐿 Where: 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 Coupling Evaluation JM to JL The following equations can be derived from transfer functions defined: 𝜔𝐴𝑅 = 𝐾𝑠 𝐽𝐿 𝜔𝑅 = 𝐾𝑠 𝐽𝑀 + 𝐽𝐿 𝐽𝑀 𝐽𝐿 **Increasing the stiffness of the system (KS) will raise the resonance frequency and allow for a low pass filter (or system operation below resonance and anti-resonance) Reflected Inertia Definition Reflected Inertia: The inertia from the load that is translated through the drive components back to the drive input of the axis of motion Direct drive Gear drive Tangential drive Screw drive Direct Driven Reflect Inertia Direct Drive  Simplest  No mechanical linkages  Load directly transmitted to motor 𝐼𝑡 = 𝐼𝑚 + 𝐼𝑙 𝑇𝑚 = 𝑇𝑙 𝜔𝑚 = 𝜔𝑙 𝑊ℎ𝑒𝑟𝑒: 𝐼𝑡 = 𝑡𝑜𝑡𝑎𝑙 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝐼𝑚 = 𝑚𝑜𝑡𝑜𝑟 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝐼𝑙 = 𝑙𝑜𝑎𝑑 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑇𝑚 = 𝑚𝑜𝑡𝑜𝑟 𝑡𝑜𝑟𝑞𝑢𝑒 𝑇𝑙 = 𝑙𝑜𝑎𝑑 𝑡𝑜𝑟𝑞𝑢𝑒 𝜔𝑚 = 𝑚𝑜𝑡𝑜𝑟 𝑠𝑝𝑒𝑒𝑑 𝜔𝑙 = 𝑙𝑜𝑎𝑑 𝑠𝑝𝑒𝑒𝑑 Reflected Inertia of a Gear Drive Gear Drive  Speed reducing device  Gears make up mechanical linkage 𝐼𝑙 𝐼𝑡 = + 𝐼𝑚 𝑁 𝑇𝑙 𝑇𝑚 = 𝑁 𝜔𝑚 = 𝜔𝑙 × 𝑁 𝑊ℎ𝑒𝑟𝑒: 𝑁 = gear ratio Reflected Inertia of a Belt or Rack Drive Tangential Drive  Belt & pulley linkage etc  Load transmitted to motor off of pulley tangent 𝑊𝑙 × 𝑅2 𝐼𝑡 = + 𝐼𝑝1 + 𝐼𝑝2 + 𝐼𝑚 𝑔 𝑇𝑚 = 𝐹𝑙 × 𝑅 𝑉𝑙 𝜔𝑚 = 2×𝜋×𝑅 𝑊ℎ𝑒𝑟𝑒: 𝑊𝑙 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑙𝑜𝑎𝑑 𝑅 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑚𝑜𝑡𝑜𝑟 𝑝𝑢𝑙𝑙𝑒𝑦 𝑔 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝐼𝑝1 = 𝑝𝑢𝑙𝑙𝑒𝑦 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝐼𝑝2 = 𝑝𝑢𝑙𝑙𝑒𝑦 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑉𝑙 = 𝑙𝑖𝑛𝑒𝑎𝑟 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑙𝑜𝑎𝑑 𝐹𝑙 = 𝐹𝑜𝑟𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑜𝑎𝑑 Reflected Inertia of a Screw Drive Screw Drive  Screw and nut linkage etc  Load transmitted to motor from screw 𝑊𝑙 𝐿 𝐼𝑡 = × + 𝐼𝑙𝑠 + 𝐼𝑚 𝑔 2×𝜋 𝜋 × 𝐿 × 𝜌 × 𝑅4 𝐼𝑙𝑠 = 2×𝑔 𝑃×𝐿 𝑇𝑚 = 2×𝜋×𝑒 𝑉𝑙 𝜔𝑚 = 𝐿 𝑊ℎ𝑒𝑟𝑒: 𝑊𝑙 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑙𝑜𝑎𝑑 𝐿 = 𝐿𝑒𝑎𝑑 𝑜𝑟 𝑙𝑖𝑛𝑒𝑎𝑟 𝑡𝑟𝑎𝑣𝑒𝑙 𝑝𝑒𝑟 𝑟𝑒𝑣 𝑔 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 ρ = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑃 = 𝐿𝑖𝑛𝑒𝑎𝑟 𝑡ℎ𝑟𝑢𝑠𝑡 𝑉𝑙 = 𝑙𝑖𝑛𝑒𝑎𝑟 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑙𝑜𝑎𝑑 𝑒 = 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 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 slugft2 What is the inertia ratio of the system? 15𝑙𝑏𝑠 ×.083𝑓𝑡 𝐼𝑙 = + 3.1 × 10−5 𝑠𝑙𝑢𝑔𝑓𝑡 + 3.1 × 10−5 𝑠𝑙𝑢𝑔𝑓𝑡 = 3.27 × 10−3 𝑠𝑙𝑢𝑔𝑓𝑡 32.2𝑓𝑡 𝑠2 3.27 × 10−3 𝑠𝑙𝑢𝑔𝑓𝑡 𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑟𝑎𝑡𝑖𝑜 = = 218: 1.5 × 10−5 𝑠𝑙𝑢𝑔𝑓𝑡 This will not be a well controlled system, what can be done to improve the inertia ratio? Reflected Inertia Example Reflected Inertia example continued: Adding a 10:1 gearbox between the motor and drive pulley of the belt driven system 𝐼𝑙 3.27 × 10−3 𝑠𝑙𝑢𝑔𝑓𝑡 𝐼𝑅 = = = 3.27 × 10−5 𝑠𝑙𝑢𝑔𝑓𝑡 2 𝑁 10 The new inertia ratio is: 3.27 × 10−5 𝑠𝑙𝑢𝑔𝑓𝑡 𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑟𝑎𝑡𝑖𝑜 = = 2.18: 1.5 × 10−5 𝑠𝑙𝑢𝑔𝑓𝑡 What adverse affect might this have on the systems performance? -Possibly speeding limiting either by the motor or gearbox Speaker Contact Details Keith Knight