Linear Shaft Motor Sizing Application Note

Linear Shaft Motor Sizing Application Note By Jeramé Chamberlain One of the most straightforward tasks in the design of a linear motion system is to specify a motor and drive combination that can provide the force, speed and acceleration that is required by the mechanical design. This is all too often the most overlooked aspect of the linear motion system design. Often, the engineer will select a motor, and then design a system around it. This approach leads to improperly sized motors for the application, making this the most costly aspect of the system, not only from the perspective of the initial purchase cost, but also due to service maintenance and energy cost. The unique properties of the Linear Shaft Motor make its sizing for applications slightly different then that of other liner motors. Nevertheless, the proper sizing of a Linear Shaft Motor is rather straightforward. Nippon Pulse America provides the NPA Smart sizing software to assist in the selection of a proper motor and drive combination for your mechanical design. This application note is intended to help walk you through the steps preformed by the NPA Smart sizing software. This will assist you in understanding the process and calculation of even more complicated motion systems. The steps that will be covered include: 1 2 3 4 Define the operation conditions and motion profile Calculate forces required by your system Selecting the proper Linear Shaft Motor for your application Selecting the proper linear servo driver for your application Unless otherwise noted all values are RMS. Step #1: Define Operation Conditions and Motion Profile The first step in sizing your linear motion system is to define what your basic system operation conditions are. This would include the load to be moved (Load, table, encoder, external load forces, friction coefficient), available power (current, voltage), and environmental issues (temperature, water). The next step is to define what motion your system will be making (stroke, velocity, time). While you may not have all this information at the start, the more information you have, the better your motor selection will be. A form similar to the one below can be useful in collecting this information. 1. Operation Condition Item Load mass Load (thrust) Force Run (pre-load) Friction Moving Motor Mass Friction coefficient µ Symbol Value Unit Incline Angle α Available Voltage V Vac Available Current A Arms Max Allowable temperature 2. Motion Profile Item C Symbol Value Unit Notes M L Kg Mass of the moving part of your system less the mass of the motor. Example: Table, Encoder F L N Thrust Force is added to all segments of the motion profile. This is in addition Example of use: As the motor moves, it needs to to force needed to overcome mass, acceleration, and friction. maintain 10 lbs of force on an object. F r N Pre-load Force is considered in all moving segments of the motion profile. Example: Cable Chain, Bearing wipers, Preloaded Keep in mind all external forces that disturb the movement. Guide, springs M c Kg If you are not sure which motor you are going to need, start with a value of 1/10 of Load mass 0 is Horsonal while 90 is Vertical Notes Stroke Acceleration time Continuous time Deceleration time Settling time Waiting time V [m/s] X mm time V m/s T a T c T d T s T w T a s Required T c s force F1 F2 F1 time s F3 T d T s T w s s T a T c T d T s T w T a Note: This application note walks you through sizing with only one segment. It is recommended that for the best sizing of a Linear Shaft Motor, a complete cycle should be used for sizing. Stroke out and back. The NPA SMART sizing software allows for sizing with up to 6 segments.

To aide you in your understanding we will be working through two examples with each step. - Pick and Place application: In this application we want to move a 2 kg load point to point horizontally for a distance of 430mm, with an acceleration of 1G and a maximum velocity of 760mm/sec. There will be a 3 sec dwell after each move. - Pin insertion application: In this application we want to move a 3 kg load point to point vertically for a distance of 50 mm in 120ms. There will be a 10ms for settling after the down move and 150ms for dwell after the up move. Load conditions More than just the load (M L) must be taken into consideration when sizing your application. The drawing to the right can help to calculate a load force. Run Friction (F r ) The frictional load of the linear guide (also called preload), the resistance force of the cable carrier to motion, and any external dampening forces that are equal in both directions (an example would be a system submerged in water or a gas shock) are included in the run friction (F r) and are treated as load forces. Load Force (F L ) Forces that resist motion in one direction and assist in the other direction (an example would be a spring), or a force that the motor is to exert on an object (an example would be pushing a box) are to be included in the load force (F L). The only exception to this is gravitational forces, which are always assumed in the calculations. Forcer Mass (M c ) This will be the mass of the forcer in a moving forcer design, or shaft in a moving shaft design. If you are not sure which Linear Shaft Motor you will need, it is suggested that for your initial calculations, you use 1/10 the load mass, as the value for Forcer mass (M c). Motion Profile There are three common motion profiles: Load (M L ) Load Force (F L ) Forcer Mass (M C ) Run friction (F r ) Linear Guide Triangular Profile 1/2, 1/2 Accelerate to speed and decelerate back to original speed or zero, rest and repeat the process as needed. This is very simple and is common in applications such as pick & place. V Distance = X Trapezoidal Profile Accelerate to constant speed, travel at that constant speed, and then decelerate back to original speed or zero. This is common in applications such as scanning inspection. There are two types. (The first is below. See the second on the following page.) 1/3 rd Trapezoidal Profile 1/3, 1/3, 1/3 T/2 T/2 T Time V Distance = X Solve for Have T/3 T/3 T T/3 Time Distance Acceleration X = (1/2) * V * T X = (1/4) * A * T 2 X = V 2 / A V = 2 * (X/T) V = (A*T) / 2 V = A * X A = 4 * (X/T 2 ) A = 2 * (V/T) A = V 2 / X Solve for Distance Have V = 1.5 * (X/T) X = (2/3) * V * T X = (1/4.5) * A * T 2 V = (A*T) / 3 X = 2 * (V 2 /A) V = (A*X) / 2 1G Acceleration = 9.81 m/sec 2 Acceleration A = 4.5 * (X/T 2 ) A = 3 * (V/T) A = 2 * (V 2 /X)

And the Variable Trapezoidal Profile. V Total Distance = X Xa Xc Xd Settling and Waiting time. In a motion profile the waiting time (T W) plus the settling time (T S) is the total amount of pause time. Waiting time is the amount of time the load is not moving, while settling is the amount of time it takes the load to stop moving. Solve for Distance Xa (m) Have Acceleration Aa (m/sec 2 ) Xa (m) Ta (sec) Ta Tc Td T Ta (sec) Aa (m/sec 2 ) Ta (sec) Aa (m/sec 2 ) Xa (m) X = (V * T) / 2 X = (A * T 2 ) / 2 X = V 2 / (2*A) V = (2*X ) / T V = A * T V = (2*A) / X A = (2*X) / T 2 A = V / T A = V 2 / (2*X) These formulas can be used for Acceleration ( a) or Deceleration ( d) only. To get total distance traveled use the Following formula: X = Total Distance traveled X = ½*(V*T a) + (V*T c) + ½*(V*T d) Time In servo-based systems the actual motor (load) movement will lag behind the commanded operation of the controller. The distance of this lag is called the Thermal Following Error (TFE). The amount of time it takes the motor (load) to move from the TFE position to the desired position to within a specified error band is called the settling time (T S). First we must find out how large the TFE is. This is largely dependent upon the servo bandwidth and deceleration rate. f 0 = Servo Bandwidth in Hz (50 Hz is Typical) ω 0 = Servo constant ω 0 = 2πf 0 A d = Movement Deceleration A d = V/T d The Trapezoidal Profile 1/3, 1/3, 1/3 is the most power efficient motion profile for Linear Servo motor applications. TFE = Thermal Following Error TFE = 4*A d /ω 0 2 Variable Trapezoidal Profile V=760mm/sec A=9.81 m/sec 2 X a=x d=0.760 2 / (2 * 9.81) =29.4mm X c = 430 (29.4*2) = 371.2 mm T a= X a / (0.5 * V) = 29.4 / (0.5 * 760) T a=t b=77ms T c = X c/v = 371.2 / 760 = 487ms Next we must define the error band that we will settle within. For purpose of this document we will call this the point at which the proportional servo loop cannot move the motor due to friction. We will call this Minimum Following Error (minfe). It can be calculated as follows: SS = Servo stiffness SS = ( (M c + M L ) * ω 0 2 )/4 minfe =Minimum Following Error minfe = F f / SS We can now calculate the settling time. It is calculated as follows: τ = Settling time constant τ = 1/ω 0 T=120ms X=50mm 1/3 rd Trapezoidal Profile T s = Settling Time T s = Ln( TFE / minfe ) * τ T a= T/3 T a=t b= T c =40ms V = 1.5 * 50 / 0.120 V = 625 mm/sec If the waiting time is greater then 100 ms then you can get by without calculating settling time, since it will most likely be less than the waiting time.

Step #2: Calculating force required by your system Calculating required force for Acceleration (F i ) Acceleration force (also referred to as force to overcome inertia) is simply Newton s Second Law. It states that force is equal the amount of mass times the amount of acceleration. F = M * A The definition of the standard metric unit of force is stated by the above equation. One Newton is defined as the amount of force required to give a 1-kg mass an acceleration of 1 m/s 2. Since acceleration can also be stated as divided by Time. A = V / T We can also use the following formula: F = (M c + M L ) * V / T a V=760mm/sec T a=77ms M L=2 kg M c=0.15kg "g" is the acceleration of gravity. "α" is the angle of incline. S160D forcer F i = (0.15 + 2) * (0.76 / 0.077) F i = 2.15 * 9.87 F i = 21.2 N F f = (0.15 + 2) * 9.81*[sin(0)+0.1 * cos(0)] + 0 F f = 2.15 * 9.81* 0.1 + 0 F f = 2.1 N Calculating required force due to Friction (F f ) Here, we calculate force required to move the load. We must know: The angle of the move (α ) Vertical = 90 and Horizontal = 0 The Friction Coefficient of the sliding surfaces (µ) Mass being moved. (M c + M L) Run friction (F r) g = 9.81 m/s 2 (acceleration of gravity) For horizontal or incline against gravity moves: F f = (M c + M L ) * g *[ µ * cos(α) + sin(α)] + F r And for incline moves with gravity: F fd = (M c + M L ) * -g *[ µ * cos(α) - sin(α)] + F r V=625mm/sec T a=40ms M L=3 kg M c=1.1kg F i = (1.1 + 3) * (0.625 / 0.04) F i = 4.1 * 15.625 F i = 64.1 N S250T Shaft 100mm stroke F f = (1.1 + 3) * 9.81*[0.1 * cos(90) + sin(90)] + 0 F f = 4.1 * 9.81* 1 + 0 F f = 40.2 N F fd = (1.1 + 3) * -9.81*[0.1 * cos(90) - sin(90)] + 0 F fd = 4.1 * -9.81 * -1 + 0 F fd = 40.2 N

Calculating force for each segment The force required for each segment is different. The force for each segment will be a made up of one or more of the following four forces we have calculated. Load Force (F L ) As can be seen by the drawing below, the load force may stay constant over all parts of the motion. It will be added when it works against a motion and is subtracted when it works with the motion. As shown in the below example, the force can increase in some cases (example would be pushing against a spring). Acceleration force (F i ) This force is only seen during Acceleration and Deceleration. Friction force (F f ) (F fd ) This force is seen only when the load is in motion. It will assist the load in during deceleration. Dwell force (F d ) This is made up of only gravitational forces; these are not seen in horizontal moves. F i = 21.2 N F f = 2.1 N F L = 0 N F1 = 21.2 + 2.1 + 0 F1 = 23.3 N F2 = 2.1 + 0 F2 = 2.1 N F3 = 21.2 (2.1 + 0) F3 = 19.1 N F4 = (0.15 + 2) * 9.81*[sin(0)] + 0 F4 = 2.15 * 9.81* 0 + 0 F4 = 0 N It is not possible for the motor to produce a negative force. Force calculations that produce negative results mean the motor is producing that force opposite the direction of travel. For simplicity reasons, any forces that are negative should be multiplied by -1 to make them positive. Load Force Required force Acceleration Force F1 = F i + F f + F L Continuous Force F2 = F f + F L Deceleration Force F3 = F i - (F f + F L ) Dwell Force F4 = (M c + M L ) * g *[sin(α)] + F L In the above formulas: For vertical or incline moves use F f for againstgravity moves and F fd for with-gravity moves. When the load force is with motion change the sign of F L from + to -. F i = 64.1 N F f = 40.2 N F fd = 40.2 N F L = 0 N F L Down Up F1 = 64.1 + 40.2 + 0 F1 = 64.1 (40.2 + 0) F1 = 104 N F1 = 23.8 N F2 = 40.2 + 0 F2 = 40.2 + 0 F2 = 40.2 N F2 = 40.2 N F3 = 64.1 (40.2 + 0) F3 = 64.1 + 40.2 + 0 F3 = 23.8 N F3 = 104 N F4 = (1.1 + 3) * 9.81*[sin(90)] + 0 F4 = 4.1 * 9.81* 1 + 0 F4 = 40.2 N V [m/s] T a T c T d T s T w time time F1 F2 F1 time T a T c F3 T s T w T a T d

Calculating effective (RMS) and peak force Peak Force - The largest segment force is your peak force. Effective (RMS) Force The root mean square average of all the force segments is the Effective force required from the motor. It is found by using the following formula: F eff = Down F1 = 104.3 N F2 = 40.2 N F3 = 23.9 N F4 = 40.2 N T a=t b= T c =40ms T s + T w = 10 ms (F1 2 * Ta) + (F2 2 * Tc) + (F3 2 * Td) + (F4 2 * {Ts + Tw}) (T a + T c + T d + T s +T w ) Up F1 = 23.9 N F2 = 40.2 N F3 = 104.3 N T s + T w = 150 ms F1 = 23.3 N F2 = 2.1 N F3 = 19.1 N F4 = 0 N T a=t b=77ms T c = 487ms T s + T w = 3000 ms.. F eff = (23.3 2 * 77)+(2.1 2 * 487)+(19.1 2 * 77)+(0 2 * 3000).. (77 + 487 + 77 + 3000) F eff = 72197.96 3619 F eff = 19.949 F eff = 4.47 N F peak = 23.3 N.. F eff = (104.3 2 * 40)+(40.2 2 * 40)+(23.9 2 *40)+(40.2 2 * 10)+(23.9 2 *40)+(40.2 2 * 40)+(104.3 2 * 40)+(40.2 2 * 150).. (40+40+40+10+40+40+40+150) F eff = 1303825.6 400. F eff = 3259.564 F eff = 57.01 N F peak = 104.3 N Step #3: Selecting the proper Linear Shaft Motor for your system In selecting a Linear Shaft Motor for your application, you must confirm that effective force (F eff) is less than the continuous rated force (F rated) of the motor. If the mass of the required moving part is larger then the value used in Mc in Step #1 then redo the calculations with the larger value. S160D Moving Forcer F eff = 4.44 N < F rated - 10N Good The follow information in needed from the selected motor. Resistance R (Ω) Inductance L (mh) Force Constant K f (N/A rms) Back EMF K e (V/m/sec) Magnetic Pitch (N-N) E c (mm)5 The follow information in needed about the application. Peak V peak (m/sec) Peak Force F peak (N) F peak = 23.3 N V peak= 0.76 m/sec R = 21 Ω L = 8.2 mh K f = 16.1 N/A rms K e = 5.37 V/m/sec E c = 60 mm S250T Moving Shaft 100mm stroke F eff = 57.01 N < F rated - 60N Good F peak = 104.3 N V peak= 0.625 m/sec R = 12 Ω L = 15 mh K f = 46.88N/A rms K e = 15.63 V/m/sec E c = 90 mm

Next, we will calculate the required Voltage and Current needed for your application, and confirm that your power source can handle it. We will now calculate the following: Resistance Hot R hot = R * 1.423 Peak Current I p = F peak / K f Voltage due to Back Emf V bemf = K e * V Voltage due to I * R V ir = 1.225 * R hot * I p Voltage due to Inductance R hot = 21 * 1.423 R hot = 29.9 Ω I p = 23.33 / 16.1 I p = 1.45 A V bemf = 5.37 * 0.76 V bemf = 4.08 V V ir = 1.225 * 29.9 * 1.46 V ir = 53.11 V V L = 7.695 * 0.76 * 8.2 * 1.45 / 60 V L = 1.16 V V bus = 1.15 ( 4.08 + 53.11 ) 2 + 1.16 2 V bus = 65.78 V V L = 7.695 * V * L * I p / E c Minimum Bus Voltage needed for application V bus = 1.15 ( V bemf + V ir ) 2 + V L 2 Check the following 5 points: R hot = 12 * 1.423 R hot = 17.1 Ω I p = 104.3 / 46.88 I p = 2.22 A 1 2 3 4 5 Is Moving Mass used in Step #1 correct for this selected LSM? Is F rated larger then F eff? Is there safety margin? NPA suggest that F eff be no more then 80% of F rated for a 20% safety margin. 30% to 50% safety margin is optimal. Is there sufficient Voltage supplied? Is there sufficient Current? V bemf = 15.63 * 0.625 V bemf = 9.77 V V ir = 1.225 * 17.1 * 2.22 V ir = 46.5 V V L = 7.695 * 0.625 * 15 * 2.22 / 90 V L = 1.78 V V bus = 1.15 ( 9.77 + 46.5 ) 2 + 1.78 2 V bus = 64.74 V If you can answer YES to these five questions then congratulations: the LSM you have selected will work for your application. Note: This application note walks you through sizing with only one segment. It is recommended that for the best sizing of a Linear Shaft Motor, a complete cycle should be used for sizing. Stroke out and back. The NPA SMART sizing software allows for sizing with up to 6 segments. Now let s find a Servo Driver.

Step #4: Selecting the proper linear servo driver for your application At this point you have most of the information needed to select the servo driver. You have the following application requirements (RMS Values): Minimum AC Bus Voltage and Peak Current. Now let s calculate the Continuous Current (RMS) and peak of sin or DC values. I c = 4.47 / 16.1 I c = 0.278 A Icrms = Continuous Current I crms = F eff / K f RMC to Peak (of sin) values Peak = RMS * 2 For AC amplifiers we use the RMS Values V DC = 65.78 * 2 V DC = 93.03 V dc I pp = 1.45 * 2 I pp = 2.05 A I cp = 0.278 * 2 I cp = 0.393 A For DC amplifiers we use the Peak of Sin Values I c = 57.01 / 46.88.1 I c = 1.22 A V DC = 64.74* 2 V DC = 91.56 V dc I pp = 2.22 * 2 I pp = 3.14 A I cp = 1.216 * 2 I cp = 1.72 A