EGR244 Lab 1: ELECTROMAGNETIC DAMPING Tracy Lu

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1 EGR244 ELECTROMAGNETIC DAMPING LAB EGR244 Lab 1: ELECTROMAGNETIC DAMPING Tracy Lu (Dated: 10 February 2017) In this experiment varying sizes of magnetic spheres within fixed dimension polymer shells are dropped down hollow copper tubes. Using the principles of physics, dynamics, and electromagnetic laws, equations were derived to theoretically model the motion of and forces acting on the spheres as they traveled down the tubes. I. INTRODUCTION II. THEORY This experiment of dropping a spherical magnet down a hollow copper pipe demonstrates several electromagnetic properties and laws. The goal of this report is to use these properties, laws, and dynamics to theoretically model the forces acting upon and motion of the sphere, and then compare the theoretical results to the experimentally calculated data. For example, Lenz s Law states that due to Faraday s law of induction, the direction of current induced in a conductor by a changing magnetic field will create a field that opposes the change that produced it. 1 Currents produced are called eddy currents, which exert a drag force on the traveling sphere in a magnetic field. More complex examples of the phenomena observed in this lab are found in many energy-generation systems 2. This lab serves as a great introduction to the basic principles behind them, with a compartively simple model of two forces. This drag force can be modeled and calculated given parameters that can be measured in our experimental setup. This then allows us to model a net force equation from a free body diagram shown in Figure 2, and this also allows us to find the equations modeling acceleration, velocity, position, and terminal velocity, the constant speed that the sphere eventually reaches when the resistance of the drag prevents further acceleration. As a reference, the paths of the 12.7mm diameter sphere s three data trials, theoretical terminal velocity, and trajectories are plotted across time. This experiment will compare the terminal velocities of the experimental data to the theoretically calculated expected terminal velocities for each of the different magnet sizes. The rest of the report will go into detail about deriving the theoretical models, the experimental setup, data collection, and data analysis. Overall, despite small deviations, our experimental data was very close and exhibited similar behavior to the theoretical, expected calculations. Figure 1: Diagram labeling experimental design parameters for magnet, shell, and copper tube. These values are measured and referenced throughout the experiment. Parameter Value Copper conductivity (σ) Ω 1 m 1 Gravity (g) m s 2 NdFeB density (ρ m ) 7.58 x 10 3 kg m 3 Polymer shell density ( ρ ρ ) kg m 3 Magnetization (µ 0 M r ) T Mean Diameter (D) mm Wall thickness (W) 1.03 mm Pipe Length (L) m Shell diameter (d s ) 19.1 mm Table 1: Parameter variables and corresponding values that are referenced throughout experiment. Values are either calculated experimentally through measurement, or obtained from references 1. The free body diagram of Figure 2 shows the relationship between the force of gravity and drag force created by the eddy currents, which creates a magnetic field that opposes the magnetic field that created it, which means they react back on the source of the magnetic field. The

2 EGR244 ELECTROMAGNETIC DAMPING LAB 2 two forces oppose each other, and at terminal velocity the two forces are equal. If characteristic time constant, τ = m/c where governs the rate at which the magnet reaches terminal velocity, v = c m v g = v τ g (7) Integrating with respect to time to find velocity as a function of time, and assuming the sphere begins with an initial velocity of zero, v(t) = V T (1 e t/τ ) (8) Figure 2: Free body diagram of forces acting upon magnetic sphere enclosed in plastic shell. The force of drag opposed the downward force of gravity. The net force equation modelling this based on Newton s Second Law is F net = F D + F g = ma = m v = m z. (1) Where m is the mass of the sphere, F D is the drag or damping force, F g is force of gravity, a is the acceleration vector, v is the velocity vector, and z is the position vector, in the z direction. Note that aerodynamic drag has been left out of the net force equation due to it being a negligible value in comparison to the electromagnetic braking force. The drag force created by the eddy currents is determined by the velocity and coefficient of drag, c, where 1 and c = 5π2 256 (µ 0M r ) 2 σw D 2( d ) 6 (2) D where V T is the terminal velocity at which the drag force equals the gravitational force and cv = mg, so V T = mg c = gτ (9) since the total mass of the magnetic and plastic composites is m = ρ m V m + ρ ρ (V V m ) (10) where V = πd 3 s/6 and V m = πd 3 /6 are the total volume and magnetic volume, respectively, the theoretical terminal velocity is ( ) 128g D v T = 15πσ(µ 0 M r ) 2 f(d) (11) W where based on terms that depend on the magnet and shell parameters, ( ) 3 [ ( ) 3 ] D ds f(d) = ρ ρ + ρ m ρ ρ (12) d d And integrating Eq. (8) for position gives III. z(t) = v T [t τ(1 e t/τ )] (13) EXPERIMENTAL METHODS F D = cv (3) F D is negative because it symbolizes opposition to the linear motion of velocity. f(d) = ( D d Force of gravity is just ] ) 3 (d s ) 3 [ρ ρ + ρm ρ ρ d (4) F g = mg (5) Using equations 2-5 in relation to Eq. (1) and assuming the upward z direction as positive, we have a net force equation of F net = cv mg = m v (6)

3 EGR244 ELECTROMAGNETIC DAMPING LAB 3 Figure 3: Diagram labeling experimental design setup of the copper tube that the spheres will be dropped down. a) ultrasound sensor, b) optical fork sensor The items used in the experiment were spherical magnetics with a 3D printed ABS plastic shell, a copper pipe with a small opening cut, and stand, swing mount, and retractable gate (shown in Figure 3 ). The data was collected with an ultrasonic sensor that was recorded in Lab- VIEW over a set period of time. First we dropped the sphere into the copper tube and it sat on top of a release gate until we triggered the gate to open. An optical sensor as shown in Figure 4 was oriented to find the time of the release of the gate, signaling LabVIEW when to begin data collection. Figure 5: Closer photograph of the bottom of the copper pipe where sphere is caught on metal strip. IV. RESULTS/DISCUSSION Figure 4: Closer photograph of top of copper pipe where sphere is released. The sensors, swing mount, alignment block are seen in more detail here. Then at the bottom as shown in Figure 5, we moved aside the metal strip which caught the sphere after it was dropped. Three trials were used for each sphere size, and each data point had two measurements, time and position with respect to the sensor at the top of the copper tube. The statistical methods used were mean and standard deviation when finding the experimental terminal velocity. The levels of uncertainty are represented by the standard deviations shown below in the appendix Table 2. Figure 6: Experimental position data for 12.7mm magnet plotted next to analytical trajectory and and terminal velocity as calculated by Eq. (13) and Eq. (11), respectively. The curved portion of the path represents the path before it hits terminal velocity, when it is still accelerating. This graph shows that each of the three trials were close to their expected values and it was safe to average their calculated values or derived terminal velocities.

4 EGR244 ELECTROMAGNETIC DAMPING LAB 4 down hollow copper tubes, data collected was compared to their theoretically derived counterpart models of the motion and forces acting on the spheres by using the principles of physics, dynamics, and electromagnetic laws. Figure 7: Theoretical terminal velocity is plotted as a function of drag next to the experimental terminal velocity points. Each trial is offset for visual clarity. Error bars represent the standard deviation across the three trials for each sphere size. The experimental terminal velocity points were calculated with the diff command in Matlab (code can be found in appendix). Within an appropriate time interval of when the sphere reaches terminal velocity (before sphere hits the bottom but after the curve in Figure 7 ), the difference between each position data point is calculated, forming an array, which is then divided by a scalar time step of.02 seconds. Now the array represents the velocity for each time step. The average of this was taken to achieve an average terminal velocity for a single trial. The to obtain the mean terminal velocity of one sphere size, the three data trials average terminal velocity was averaged. The standard deviation across three trials for a single sphere size was calculated and drawn as an error bar. Standard deviations can be found in the tables section in the appendix. The linear nature of the graph makes sense, since by Eq. (11), terminal velocity is related to f(d) by a constant. V. CONCLUSION In this experiment of dropping varying sizes of magnetic spheres within fixed dimension polymer shells Overall our data matched our theoretical models pretty closely and exhibited similar general trends. There were slight differences, which are most likely due to some assumptions and experimental inaccuracies. For example, we assumed that there was no aerodynamic drag. Also, when we triggered the release gate, it shook the copper pipe a bit, which would affect the path of the magnet. To increase the accuracy of the experiment and report we could conduct more than three trials for each of the sphere sizes, modify the release gate to move smoother and with less impact, tape down the stand holding the copper pipe to prevent shaking, and consider doing calculations for and factoring in aerodynamic drag. The theories and results from this experiment provide insight on electromagnetic damping which has many applications to electric motors, magnetic levitation trains, energy generation, etc. 1 Our analysis shows us that larger magnets have smaller terminal velocities which is most likely due to the fact that they create greater magnetic fields experience greater damping, which would be an essential fact to apply when designing better electromagnetic dampers. VI. ACKNOWLEDGEMENTS Lab Partners for data collection: William Willis. TA s: Roozbeh and Sarah Land. Hailey Prevett, 1 Land, Sara, Patrick Mcguire, Nikhil Bumb, Brian P. Mann, and Benjamin B. Yellen. Electromagnetic braking revisited with a magnetic point dipole model. American Journal of Physics 84, no. 4 (2016): doi: / Electrical4u. Lenz Law of Electromagnetic Induction. Lenz Law of Electromagnetic Induction Electrical4u. Accessed February 10,

5 Appendix February 10, 2017 A Tables of Data Magnet Size (in.) Outer Diameter (in.) Mass (g) Density ( kgm 3 ) (ρ) 3/ based on given value 7/ based on given value 1/ based on given value 5/ based on given value 3/ based on given value Solid Table 2: Measurements for Each Sphere Size Magnet Size (in.) Average Terminal Velocity (m/s) Standard Deviation 3/ / / / / Table 3: Terminal Velocity and Standard Deviation for Varying Magnetic Sphere sizes Matlab code %P o s i t i o n vs time %% 38 time38=a38 ( 1 : 1 5 0, 1 ) ; posa38=a38 ( 1 : 1 5 0, 2 ) ; posb38=b38 ( :, 2 ) ; posc38=c38 ( :, 2 ) ; pos38=(posa38+posb38+posc38 ). / 3 ; p l o t ( time38, pos38 ) %% 716 time716=a716 ( 1 : 1 5 0, 1 ) ; posa716=a716 ( 1 : 1 5 0, 2 ) ; posb716=b716 ( :, 2 ) ; posc716=c716 ( :, 2 ) pos716=(posa716+posb716+posc716 ). / 3 ; 1

6 hold on p l o t ( time716, pos716 ) %% s o l i d time38=a38 ( 1 : 1 5 0, 1 ) ; posa38=a38 ( 1 : 1 5 0, 2 ) ; posb38=b38 ( :, 2 ) ; posc38=c38 ( :, 2 ) ; pos38=(posa38+posb38+posc38 ). / 3 ; p l o t ( time38, pos38 ) %% 12 time12=a12 ( 1 : 2 5 0, 1 ) ; posa12=a12 ( 1 : 2 5 0, 2 ) ; posb12=b12 ( :, 2 ) ; posc12=c12 ( :, 2 ) ; %avg pos12= ( posa12+posb12+posc12 ). / 3 ; p l o t ( time12, pos12 ) %% 58 time58=a58 ( 1 : 5 0 0, 1 ) ; posa58=a58 ( 1 : 5 0 0, 2 ) ; posb58=b58 ( 1 : 5 0 0, 2 ) ; posc58=c58 ( 1 : 5 0 0, 2 ).. pos58=(posa58+posb58+posc58 ). / 3 ; p l o t ( time58, pos58 ) %% 34 time34=a34 ( 1 : 8 0 0, 1 ) ; posa34=a34 ( 1 : 8 0 0, 2 ) ; posb34=b34 ( 1 : 8 0 0, 2 ) ; posc34=c34 ( 1 : 8 0 0, 2 ).. pos34=(posa34+posb34+posc34 ). / 3 ; p l o t ( time34, pos34 ) legend ( 3/8 in, 7/16 in, 1/2 in, 5/8 in, 3/4 in ) legend ( Location, south ) x l a b e l ( time ( seconds ) ) y l a b e l ( Z(m) ) hold o f f t i t l e ( P o s i t i o n vs Time f o r each sphere s i z e ) %%12 exp Vt timecut =1.08; step =.02; n= 5 0 : 9 0 ; VtExpa12i = d i f f ( posa12 ( n ) ). / ( step ) ; VtExpa12 = mean( VtExpa12i ) ; VtExpb12i = d i f f ( posb12 ( n ) ). / ( step ) ; VtExpb12 = mean( VtExpb12i ) ; VtExpc12i = d i f f ( posc12 ( n ) ). / ( step ) ; VtExpc12 = mean( VtExpc12i ) ; VtExp12= mean ( [ VtExpa12, VtExpb12, VtExpc12 ] ) VtStd12= s t d ( [ VtExpa12, VtExpb12, VtExpc12 ] ) %%38 exp Vt timecut =1.08; step =.02; 2

7 n= 2 0 : 3 5 ; VtExpa38i = d i f f ( posa38 ( n ) ). / ( step ) ; VtExpa38 = mean( VtExpa38i ) ; VtExpb38i = d i f f ( posb38 ( n ) ). / ( step ) ; VtExpb38 = mean( VtExpb38i ) ; VtExpc38i = d i f f ( posc38 ( n ) ). / ( step ) ; VtExpc38 = mean( VtExpc38i ) ; VtExp38= mean ( [ VtExpa38, VtExpb38, VtExpc38 ] ) VtStd38= s t d ( [ VtExpa38, VtExpb38, VtExpc38 ] ) %%58 exp Vt step =.02; n= : ; VtExpa58i = d i f f ( posa58 ( n ) ). / ( step ) ; VtExpa58 = mean( VtExpa58i ) ; VtExpb58i = d i f f ( posb58 ( n ) ). / ( step ) ; VtExpb58 = mean( VtExpb58i ) ; VtExpc58i = d i f f ( posc58 ( n ) ). / ( step ) ; VtExpc58 = mean( VtExpc58i ) ; VtExp58= mean ( [ VtExpa58, VtExpb58, VtExpc58 ] ) VtStd58= s t d ( [ VtExpa58, VtExpb58, VtExpc58 ] ) %%34 exp Vt step =.02; n= : ; VtExpa34i = d i f f ( posa34 ( n ) ). / ( step ) ; VtExpa34 = mean( VtExpa34i ) ; VtExpb34i = d i f f ( posb34 ( n ) ). / ( step ) ; VtExpb34 = mean( VtExpb34i ) ; VtExpc34i = d i f f ( posc34 ( n ) ). / ( step ) ; VtExpc34 = mean( VtExpc34i ) ; VtExp34= mean ( [ VtExpa34, VtExpb34, VtExpc34 ] ) VtStd34= s t d ( [ VtExpa34, VtExpb34, VtExpc34 ] ) %%716 exp Vt step =.02; n= 5 0 : 6 5 ; VtExpa716i = d i f f ( posa716 ( n ) ). / ( step ) ; VtExpa716 = mean( VtExpa716i ) ; VtExpb716i = d i f f ( posb716 ( n ) ). / ( step ) ; VtExpb716 = mean( VtExpb716i ) ; VtExpc716i = d i f f ( posc716 ( n ) ). / ( step ) ; VtExpc716 = mean( VtExpc716i ) ; VtExp716= mean ( [ VtExpa716, VtExpb716, VtExpc716 ] ) VtStd716= s t d ( [ VtExpa716, VtExpb716, VtExpc716 ] ) %% graph1 o f r. 5 in g= ; mag = ; 3

8 mass =10.67E 3; %mass =6.92E 3; s i g = 4.795E7 ; W = 1.03E 3; D = 21.16E 3; d = ; %d =.0095; c = ( ( 5 ( pi ). ˆ 2 ) / ). ( ( mag ). ˆ 2 ). ( s i g ). W. (D. ˆ 2 ). ( d/d). ˆ 6 ; T= mass/ c ; Vt=g. T; %%Predicted T r a j e c t o ry %%t r a t ) Vt. ( t (T. (1 exp( t /T ) ) ) ) ; t r a j=@( t ) ( g. T. t ) (g. (T.ˆ2). (1 exp( t. /T ) ) ) ; time12=a12 ( 1 : 2 5 0, 1 ) ; %p o s i t i o n s, s t a g g e r e d posa12=a12 ( 1 : 2 5 0, 2 ) ; posb12 =((b12 ( 1 : 2 5 0, 2 ) ) ) ; posc12 =(( c12 ( 1 p l o t ( time12, posa12,. ) hold on p l o t ( time12, posb12,. ) p l o t ( time12, posc12,. ) y=t r a j ( time12 ) ; p l o t ( time12, y, k ) p l o t ( time12, Vt. time12, c ) p l o t ( time12, y +.05, k ) p l o t ( time12, y +.1, k ) p l o t ( time12, Vt. time , c ) p l o t ( time12, Vt. time12 +.1, c ) legend ( t r i a l 1, t r i a l 2, t r i a l 3, a n a l y t i c a l, terminal, Location, a x i s ( [ 0,. 4, 0,. 2 5 ] ) hold o f f y l a b e l ( z (m) ) x l a b e l ( Time ( s ) ) t i t l e ( Path o f 1/2 in or 12.7mm magnet ) %%graph 2 t e r m i n a l V e l o c i t y g= ; mag = ; s i g = 4.795E7 ; W = 1.03E 3; D = 21.16E 3; dm = : : ; 4

9 pm=7.58e3 ; pp= 1.02E3 ; ds =19.1E 3; f da ) ( (D. / da ). ˆ 3 ). ( pp. ( ( ds. / da ).ˆ3)+pm pp ) sub= f (dm) ; Vtg2= ( ( g ). / ( 1 5. pi. s i g. ( mag ). ˆ 2 ) ). (D. /W). sub ; f i g u r e ( 3 ) c l f ; p l o t ( sub, Vtg2 ) hold on dt = [ ]. 10ˆ( 3) Vts= [ VtExp38, VtExp716, VtExp12, VtExp58, VtExp34 ] ; Stds= [ VtStd38, VtStd716, VtStd12, VtStd58, VtStd34 ] ; f ( dt ) s c a t t e r ( f ( dt ), Vts ) e r r o r b a r ( f ( dt ), Vts, Stds ) t e x t ( f ( dt (1))+7000, Vts ( 1 ), 9. 5 mm, HorizontalAlignment, l e f t ) t e x t ( f ( dt (2))+7000, Vts (2)+.1, mm, HorizontalAlignment, r i g h t ) t e x t ( f ( dt (3))+7000, Vts ( 3 ), mm, HorizontalAlignment, l e f t ) t e x t ( f ( dt (4))+7000, Vts ( 4 ), mm, HorizontalAlignment, l e f t ) t e x t ( f ( dt (5))+7000, Vts ( 5 ), mm, HorizontalAlignment, l e f t ) hold o f f y l a b e l ( V T, (m/ s ) ) x l a b e l ( f ( d ) ) legend ( T h e o r e t i c a l, Experimental, Location, southeast ) t i t l e ( T h e o r e t i c a l and Experimental Terminal Velocity ) 5