Math is Not a Four Letter Word FTC Kick-Off. Andy Driesman FTC4318 Green Machine Reloaded

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1 1 Math is Not a Four Letter Word 2014 FTC Kick-Off Andy Driesman FTC4318 Green Machine Reloaded andrew.driesman@gmail.com

2 2 Agenda Why calculate rather than build? Electric Motors and Servo Specifications Gears and Gear Ratios Example calculating raising the flag Chains Drives Torque to Linear force Example lifting a 20 pound robot The use of Margin Battery Health Other types of calculations.

3 3 Why Do Math? Many FTC design iterations are done by trial and error. Trial and error takes time and money. Doing a calculation allows the following without building hardware: To see if a design will work (or not) Select components Calculate performance Bound a design Test various design ideas quickly

4 4 Andy s Golden Rule You don t ever get something for nothing!!!

5 5 Electrical Motors 101 Motors are electro-mechanical devices that convert electrical energy to mechanical energy. They have the following characteristics: Torque Rotational force. SI Unit is: Newton*meter (N*m) though sometimes kilogram*centimeter (kgf*cm) English Unit are: ounce*inches (oz*in) or Pound*feet (lbf*ft) Motor Speed (velocity) Rotational rate. SI unit is: radians per sec English units: Degrees per second or Rotations per minute (RPM) or Rotations per Second (RPS) One rotation = 360 Speed is inversely proportional to Torque

6 Tetrix Motor Specifications 6

7 7 Specifications Tetrix Motor No Load Speed = 139 ± RPM Output Speed at max power = 76 RPM Output Torque at max power (t) = 11.8 kgf*cm=1.2 N*m=10 lbf*in Stall Torque = 23.4 kgf*cm (Speed = Zero)=2.3 N*m HS485HB Servo (180 ) No Load Speed = 45 RPM (0.66 sec/180 rotation) Stall Torque = 4.8 kg*cm Continuous Rotation Servo No Load Speed = 43 RPM Stall Torque = 2.8 kg*cm Available Tetrix Gears 40 tooth (diameter ~34 mm) 80 tooth (diameter ~66 mm) 120 tooth (diameter ~97 mm) Tetrix Gear Specs: Diametrical Pitch: 32 teeth/in Pressure Angle: 20 degrees Face Width: ¼ inch Tetrix Ratios: 1.0:2.0:3.0 Gears from other vendors are allowed. Available Tetrix Sprockets 16 tooth (diameter ~36 mm) 24 tooth (diameter ~52 mm) 32 tooth (diameter ~68 mm) Sprockets from other vendors are allowed. Ratios 1.0:1.5:2.0

8 8 Gears, Gear Ratios and Units Gears are used: To transmit power from one place to another. Change direction that power is applied. To transform the motor output: Increase/decrease rotation (angular) rate Increase/decrease torque ( a measure of turning force) Rotational Rate or Gear Speed Measured in Rotations per minute (RPM) 1 RPM = 360 per minute = 6 per second 1 RPM equivalent to 1 minute per rotation Torque Torque is a unit of rotational force at a distance. Units are: N*m, kgf*cm or lbf*inch. 1 kgf*cm is the equivalent of placing a 1 kg weight at a 1 cm distance from the axis of rotation.

9 9 Using Gears to Change Speed and Torque Output Speed = Input Speed x Gear Ratio Output Torque = Input Torque / Gear Ratio OUTPUT W=76 RPM T=11.8 kg*cm INPUT (drive) S=76 RPM T=11.8 kg*cm 40 tooth gears OUTPUT S=38 RPM T=23.6 kg*cm 80 tooth gear Gear Ratio 40 tooth input :80 tooth output Ratio is 40:80 or 1:2 or ½. Output Speed (RPM) Calculation Input Speed is multiplied by the gear ratio of ½ to get output speed 76 RPM* ½ =38 RPM Output Torque (T) Calculation Input torque is divided by the gear ratio to get output torque 11.8 kg*cm/(½) =23.6 kg*cm Gears introduce inefficiency (due to friction), so there is always a loss.

10 10 More Complicated Illustration Output Gear Small gears are 40 tooth gears Large gears are 120 tooth gears Input Gear Gears can be stacked into a Gear Chain Overall ratio is calculated by multiplying the individual ratios 120 tooth:40 tooth= 120:40=3:1=3/1 (3/1)x(3/1)x(3/1)=27:1=27 Speed (RPM) Calculation Input Speed is multiplied by the gear ratio of 27/1 to get output speed 76 RPM* 27 =2052 RPM Torque (T) Calculation Input torque is divided by the gear ratio to get output torque 11.8 kg*cm/(27) =0.44 kg*cm

11 11 Illustration from Last Year s Game How long will it take to raise the flag in the Block Party Challenge using a Tetrix Motor with a 120 tooth gear driving a 40 tooth gear? Parameters Motor Speed = 100 RPM (load assumed) ½ PVC Outer Diameter = 0.84 Height to top Flag = ~58 Calculation: Circumference of PVC pipe = π*d π*0.84 =2.6 Number of rotations needed to raise flag 58 = 58/2.6 = 22.3 rotations Gear Ratio = 120:40=3:1 Output Speed = Input Speed x Gear Ratio. Output Speed = 3*100 RPM=300 RPM MPR= 1/(RPM) = 1/300 =0.003 MPR Time to execute 21.5 rotations = 22.3*0.003 = min = 4.4 sec

12 Chains and Sprockets Output Speed = Input Speed x Sprocket Ratio Output Torque = Input Torque / Sprocket Ratio Chains and sprockets calculations are very similar to gears. 24 tooth Drive Sprocket 32 tooth Load Sprocket Sprocket Ratio 24 tooth drive :32 tooth load Ratio is 24:32 or 3:4 or ¾. Speed (RPM) Calculation Input Speed is multiplied by the gear ratio of ¾ to get output speed. 76 RPM* ¾ =50 RPM Torque (T) Calculation Input torque is divided by the gear ratio to get output torque 11.8 kg*cm/(¾) =15.7 kg*cm 12

13 Converting Rotational Motion to Linear Motion Linear Force = Torque of final gear / Radius of final gear Torque = Force x Radius Radius = Torque/Force TORQUE Radius (r) Force = Torque / Radius Units of Force are Newtons. Kgf and lbf Torque = torque of the motor after gearing. Radius is the radius of the final gear. The bigger the radius, the less force. TORQUE Radius (r) 13 FORCE FORCE

14 14 Illustration from Last Year s Game What radius is required to lift this robot 3 inches? Parameters: Input Torque: 11.8 kgf*cm = 164 oz*inches 1:12 Gear/Sprocket Ratio Robot Weight: 25 lbf

15 15 Calculations Output Torque = Input Torque / Gear Ratio Output Torque = 164 oz*in/(1/12) = 1,968 oz*in Radius = Torque/Force Force = 25 lbf = 400 oz Radius = 1,968 oz*in/400 oz = 4.92 inches Radius <= 4.92 inches

16 16 Margin Margin is a factor added to a calculation to account for real world conditions Motors wear out Batteries are not fully charged Gear train and chain drives are not 100% efficient. Imprecise construction techniques. Where practical, I use a factor of 2. For example: if a calculation says, we need a given torque, I multiply it by 2.

17 17 Battery Health Nameplate capacity of battery is 3,000 mamp*hour (good surrogate for battery energy). Under perfect conditions battery will provide 3 amps for one hour before it is completely discharged. Batteries do wear out. They slowly loose their capacity. Measuring the health of a battery requires specialized equipment. A rule of thumb (for batteries used in this environment) is that a battery s usable capacity is about 30% - 50% of the name-plate or 1,000 ma*hr to 1,500 ma*hr. How much energy does a 2.5 minute match use? Dependent on #of motors, servos, activity of play, friction, etc. Hard to calculate actuals, but easy to calculate maximum. Each battery has a 20A fuse. Meaning max current draw is 20A. A*Hr used = 20A*2.5min/60 min/hour = 0.83 A*H = 830 ma*hr. Good policy to switch out batteries every match or every other match.

18 18 Other things of interest that can be calculated Force applied by surgical tubing Used during Ring It Up to calculate the amount of assist provided to a scissor lift Dependent on the material modulus, inner and outer diameters of the tubing. Axle or beam strength Used in Block Party to understand why axles kept bending and identify a replacement material. Dependent on Young s modulus, area and length of axle/beam and type of support. Scissor lift forces Used to calculate the amount of force required to lift a another robot during Block Party Dependent on geometry of scissor lift. Etc

19 19 Summary of Formulas Gears: Output Speed = Input Speed x Gear Ratio Output Torque = Input Torque / Gear Ratio Chains and Sprockets: Output Speed = Input Speed x Sprocket Ratio Output Torque = Input Torque / Sprocket Ratio Converting Rotational Motion to Linear Motion Linear Force = Torque applied to final gear / Radius of final gear Converting Linear Motion to Rotational Motion Torque = Linear Force x Radius

20 20 References Understanding DC Motors Gears, Chains and Sprockets tml

21 21 Tetrix Motor Spec Sheet