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Transcript
Problem 10. Inverted Pendulum Problem 10. It is possible to stabilise an inverted pendulum. It is even possible to stabilise inverted multiple pendulum (one pendulum on the top of the other). Demonstrate the stabilisation an determine on which parameters this depends. Introduction • Inverted pendulum - center of mass is above its point of suspension • Achieving stabilisation – pendulum suspension point vibrating! • Principal parameters: • lenght • frequency • amplitude Introduction cont. Experimental approach • Apparatus • Construction • Measurements: • Pendulum angle in time • Stabilisation conditions: amplitude vs. pendulum length amplitude vs. frequency • Double pendulum Apparatus • Speaker (subwoofer) • Function generator • Amplifier • Stroboscope • Pendula (wooden) Apparatus cont. • Speaker – low harmonics generation • Audio range amplifier • Stroboscope – accurate frequency measurement • Point of support amplitude measured with (šubler) • Multiple measurements for error determination Construction Lengths [cm]: 4 4.5 5 5.5 6 6.5 Density [kg/m3]: 626 7 7.5 Measurements • Stability – pendulum returns to upward orientation • measurements of boundary conditions: frequency vs. amplitude length vs. amplitude angle in time (two cases); • inverted pendulum • “inverted” inverted pendulum – for drag determination Double pendulum Theoretical approach • Pendulum – tends to state of minimal energy • Upward stabilisation possible if enough energy is given at the right time • Formalism – two possibilities: • equation of motion • energy equation – Lagrangian formalism • Forces approach – more intuitive: Forces on pendulum l pendulum lenght angle between pendulum and y axis h accelerati on of suspension point Fr resistance Fy inertial force acting on the center of mass Equation of motion • In noninertial pendulum system: 1 1 I s Fy l sin Fr l 2 2 l pendulum lenght angle between pendulum and y axis I s pendulum moment of inertia Fr resistance Fy inertial force acting • Inertial acceleration: • gravity component • periodical acceleration of suspension point on the center of mass Equation of motion cont. • Resistance force – estimated to be linear to angular velocity • “inverted” inverted pendulum measurements 30 20 max ~ e angle [°] 10 0 eff 2 t eff damping coefficien t eff 3.0 s -1 -10 max angular amplitude -20 -30 -0,5 0,0 0,5 1,0 1,5 time [s] 2,0 2,5 3,0 Equation of motion cont. equation of motion: 2 A 2 eff 0 1 sin t sin 0 g 3 l 2g A suspension point amplitude suspension point angular frequency 02 parameter : 02 • Analytical solution very difficult • Numerical solution – Euler method Equation of motion cont. 0,6 0,4 l 5.0 cm 685 rad/s 2 A 4.5 mm angle [rad] 0,2 0,0 -0,2 -0,4 -0,6 0,0 0,2 0,4 0,6 0,8 time [s] 1,0 1,2 1,4 1,6 Stability conditions From equation of motion solutions stability determination: 200 180 l 5.0 cm frequency [Hz] 160 140 120 100 80 60 40 0,001 0,002 0,003 0,004 2A [m] 0,005 0,006 Stability conditions cont. 9 8 length [cm] 7 6 5 4 freq 100 Hz 3 1,8 2,0 2,2 2,4 2,6 2A [mm] 2,8 3,0 3,2 3,4 Stability conditions cont. • Agreement between model and measurements relatively good • Discrepancies due to: • errors in small amplitude measurements • speaker characteristics (higher harmonics generation) • nonlinear damping... Conclusion • we determined and experimentaly prove stability parameters • mass is not a parameter • theoretical analisis match with results • we managed to stabilise multiple inverted pendulum