VIBRATING BASE PENDULUM. Math 485 Project team Thomas Bello Emily Huang Fabian Lopez Kellin Rumsey Tao Tao

Transcription

1 VIBRATING BASE PENDULUM Math 485 Project team Thomas Bello Emily Huang Fabian Lopez Kellin Rumsey Tao Tao

2 Agenda Midterm Recap Equation of Motion & Energy Modeling Effective Potential Stability Analysis Experiment Error Analysis

3 Background Inverted Vibrating Pendulum Application

4 Recap--Vertical Angle Equation of Motion: K = 1 m( θ l + d 0 ω sin (ωt) θl (sin θ)d 0 ω sin(ωt) ) U = mg(l cos θ + d 0 sin(ωt)) Lagrangian: L = K U L = 1 l θ + d 0 ω θsinθ sin ωt gcosθ + 1 l ω d L L dt θ θ = 0 (Euler-Lagrange Equation) Ӫ + d 0ω cos ωt l g l sinθ = 0 Separate into fast and slow motion θ t = X t + ξ(t)

5 Recap Vertical Angle Averaging ξ = d 0ω l sin X cos ωt dt = d 0 l x = d dθ g l cos θ Effective Potential U eff = g l cos θ Stability Analysis d 0 ω l d 0 ω l sin θ sin θ sin X cos ωt

6 Objectives Lagrangian Effective Potential Stability Analysis Experiment Horizontal Angle Arbitrary Angle

7 Variables d 0 = amplitude of base oscillations ω = angular frequency of base oscillations l = length of pendulum θ = counterclockwise angular displacement of pendulum φ = counterclockwise angle of base g = gravitational constant 9.81 m/s K = kinetic energy U = potential energy l d 0 cos(ωt) θ φ

8 Arbitrary Angle of Base d 0 cos(ωt) X & Y Coordinates: x = lsin θ + d 0 cos ωt cos φ y = l lcos θ + d 0 cos ωt sinφ φ θ l Velocities: v x = l θ cos θ d 0 ω sin ωt cos φ v y = l θsin(θ) d 0 ω sin ωt sinφ

9 Lagrangian for Arbitrary Angle Lagrangian for any physical system is defined as Kinetic Energy minus Potential Energy L = K U Kinetic Energy: K = 1 mv = 1 m(v x + v y ) K = 1 ml θ md 0 ωl θcos θ φ sin(ωt) Potential Energy: U = mgh Lagrangian: U = mgl mgl cos θ + gd 0 cos ωt sin φ L = 1 ml θ md 0 ωl θcos θ φ sin ωt + mgl cos θ gd 0 cos ωt sin φ L = 1 l θ + d 0 ω sin θ φ cos ωt + g cos θ

10 Effective Potential Derivation Use Euler-Lagrange Equation to write Equation of Motion L θ d dt L θ = 0 Separate variables into rapid oscillations due to vibrating base and slow motion of pendulum θ = X + ξ Final differential equation can be written as a total derivative in position, which corresponds to the effective potential energy of the system X = X g cos θ d o ω l 4l sin (θ φ) General equation of motion relates position and potential energy x = U(x) x

11 Effective Potential Stable equilibria Comparing equation of motion to general form suggests concept of effective potential U eff = g l cos θ d o ω 4l sin (θ φ) Separation of variables treats motion of the pendulum as one smooth motion with periodic perturbations Averaging technique smooths out rapid oscillations by averaging over the period of the rapid motion, like a strobe light, creating an idealized model Effective potential is the hypothetical potential energy of the idealized model Effective potential of a pendulum with base angle of 45,

12 Stability Analysis Stability occurs at local minima of potential energy, including effective potential Stability positions appear for frequencies above a minimum frequency gl ω > d 0 For Horizontal this occurs at an angle: θ s = cos 1 gl d 0 ω For an arbitrary angle of the base of 45, theoretical stable angle is19, or 39 above the horizontal ω = 75.6 rad l = m d sec 0 = 0.0 m

13 Stability Analysis Stable equilibria ω = 75.6 rad sec l = m d 0 = 0.0 m φ = 45

14 Physical Pendulum Theoretical model used simple pendulum where all mass is concentrated at single point at end of rod Physical pendulum is realistic model Must incorporate center of mass and moment of inertia into calculations Simple pendulum model U eff = 6 g 36 d o ω cos θ 7 l 49 4l sin (θ φ) Shifts stability points to more realistic locations Physical pendulum model

15 Experimental Overview Use Cannon High Speed Camera to observe the pendulum s motion On screen measurement Raw Data Table Measurement Length of Pendulum (m).187 Diameter of Pendulum (m) Amplitude (m) 0.00 Minimum Maximum Frequency (rad/s) 75.6 Angle of Base 51 Moment of Inertia

16 Experimental Results ω = 75.6 rad sec l = m d 0 = 0.00 m φ = 51

17 Error Analysis Error in Measurements When taking measurements, there will be some error that depends on the accuracy of the instrument. This absolute error is called the Least Count. Relevant Measurements: Measured Value Absolute Error Percent Error Length of Pendulum (l) l = meters δl = meters.54% Amplitude of Base (d ) d = 0.00 meters δd = meters 5.0% Period for 60 Oscillations (T*) T* = 1.35 seconds δt* = 0.05 seconds 3.7% Note that Angular Frequency (ω) cannot be directly measured. Instead, the variable T* is introduced and defined as the period of time required for the pendulum s base to make 60 oscillations. ω = π 60 T = 10π T

18 Error Analysis Error Propagation For a function f(x,y) with absolute errors δx and δy, there is sure to be some propagation of absolute error δf. This error is given by the Variance Formula: δf x, y, z = f x δx + f y δy Relevant Equations For Pendulum + f z δz Critical Angles for Vertical Pendulum θ c (l, d 0, ω) = ± cos 1 glt 700π d 0 = ±97 Stability Angle for Horizontal Pendulum θ c (l, d 0, ω) = cos 1 glt 700π d 0 = 83

19 Error Analysis Error for Theoretical Critical Angles The Variance Formula for θ(l, d 0, ω): δθ l, d 0, ω = f l δl θ c (l, d 0, ω) = ± cos 1 + f d 0 δd 0 + f ω δω α lt d 0 = ±97 By letting α = g = 1.38 x 700π 10 4 the following partial differential equations can be obtained: θ l = d 0 αt 1 αlt d 0 θ T = d 0 αlt 1 αlt d 0 θ d 0 = d 0 3 αlt 1 αlt d 0 **Notice that since each of these quantities will be squared, the sign doesn t matter. This is why the error analysis will be the same for the stability angle for the Horizontal Pendulum.

20 Error Analysis δθ l, d 0, ω = f l δl + f d 0 δd 0 + f ω δω If we plug in our measured values, we obtain: δθ l, d 0, ω = δθ =.0148 radians =.86 Percent Error = δθ θ x 100 This means: Critical Angles for the Vertical Pendulum: θ c = ± 97 ± 0.86 = ± 97 ± 0.89% Stability Angle for the Horizontal Pendulum: θ s = 83 ± 0.86 = 83 ± 1.04%

21 Thank You Questions?