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1 University of Rhode Island Classical Dynamics Physics Course Materials Oscillations Gerhard Müller University of Rhode Island, Creative Commons License This wor is licensed under a Creative Commons Attribution-Noncommercial-Share Alie 4.0 License. Follow this and additional wors at: Abstract Part fourteen of course materials for Classical Dynamics (Physics 50), taught by Gerhard Müller at the University of Rhode Island. Entries listed in the table of contents, but not shown in the document, exist only in handwritten form. Documents will be updated periodically as more entries become presentable. Recommended Citation Müller, Gerhard, "14. Oscillations" (015). Classical Dynamics. Paper 8. This Course Material is brought to you for free and open access by the Physics Course Materials at DigitalCommons@URI. It has been accepted for inclusion in Classical Dynamics by an authorized administrator of DigitalCommons@URI. For more information, please contact digitalcommons@etal.uri.edu.

2 Contents of this Document [mtc14] 14. Oscillations Damped harmonic oscillator [mln6] Harmonic oscillator with friction [mex150] Harmonic oscillator with attenuation [mex61] Driven harmonic oscillator I [mln8] Amplitude resonance and phase angle [msl48] Driven harmonic oscillator: steady state solution [mex180] Driven harmonic oscillator: inetic and potential energy [mex181] Driven harmonic oscillator: power input [mex18] Quality factor of damped harmonic oscillator [mex183] Driven harmonic oscillator: runaway resonance [mex6] Driven harmonic oscillator II [mln9] Fourier coefficients of a sawtooth force [mex184] Fourier coefficients of periodic seuence of rectangular pulses [mex185] Driven harmonic oscillator III [mln107] Driven harmonic oscillator with Coulomb damping [mex63] Small oscillations [mln43] Transformation to principal axes [mln30] Elastic chain [mln48] Blocs and springs in series [mex13] Two coupled oscillators [mex186] Three coupled oscillators [mex187] What is the physical nature of these modes? [mex114] Small oscillations of the double pendulum [mex14]

3 Linearly Damped Harmonic Oscillator [mln6] Equation of motion: mẍ = x γẋ ẍ + βẋ + ω 0x = 0. Damping parameter: β γ/m; characteristic frequency: ω 0 = /m. Ansatz: x(t) = e rt (r + βr + ω0)e rt = 0 r ± = β ± β ω0. Overdamped motion: Ω 1 β ω0 > 0 Linearly independent solutions: e r +t, e r t. General solution: x(t) = ( A + e Ω1t + A e ) Ω 1t e βt. Initial conditions: A + = (ẋ 0 r x 0 )/Ω 1, A = (r + x 0 ẋ 0 )/Ω 1. Critically damped motion: ω 0 β = 0, Linearly independent solutions: e rt, te rt. General solution: x(t) = (A 0 + A 1 t)e βt. Initial conditions: A 0 = x 0, A 1 = ẋ 0 + βx 0. r = β Underdamped motion: ω 1 ω0 β > 0 Linearly independent solutions: e r +t, e r t. General solution: x(t) = (A cos ω 1 t + B sin ω 1 t) e βt = D cos(ω 1 t δ)e βt. D = A + B, δ = arctan(b/a). Initial conditions: A = x 0, B = (ẋ 0 + βx 0 )/ω 1. The dissipative force, γẋ, effectively represents a coupling of one lowfrequency oscillator to many high-frequency oscillators.

4 [mex150] Harmonic oscillator with friction The equation of motion of a harmonic oscillator with Coulomb damping (friction) has the form ẍ + α sgn(ẋ) + ω 0x = 0, where ω 0 = /m is the angular frequency of the undamped oscillator and sgn(ẋ) denotes the sign (±) of the instantaneous velocity. (a) Show that the solution for nπ ω 0 t (n + 1)π, n = 0, 1,,... and x(0) = A 0 + β, ẋ(0) = 0 has the form x(t) = A n cos(ω 0 t) + ( 1) n β. Find the constant β, the maximum value of n, and the amplitudes A n. (b) For the case α = 1cm/s, ω 0 = 1rad/s, and x(0) = 9cm, find the time it taes the system to come to a halt and the total distance traveled. Plot the phase portrait in the (x, ẋ/ω 0 )-plane for this particular case.

5 [mex61] Harmonic oscillator with attenuation The equation of motion of a harmonic oscillator with attenuation is written in the form ẍ + α m sgn(ẋ) ẋ m + ω 0x = 0, where ω 0 = /m is the angular frequency of the undamped oscillator and sgn(ẋ) denotes the sign (±) of the instantaneous velocity. Here we consider the three cases of Coulomb damping (m = 0), linear damping (m = 1), and quadratic damping (m = ). Use ω 0 = 1 throughout. Employ the Mathematica options of NDSolve and ParametricPlot to numerically solve the equation of motion for all three cases and to plot x versus ẋ for initial conditions x(0) = 9 and ẋ(0) = 0. Vary the attenuation parameter α m in each case and watch out for qualitative changes in the phase-plane trajectory. Present a collection of neat graphs that emphasize the differences between the three cases as well as the differences between parameter regimes for one or the other case. Describe the different types of trajectories.

6 Driven Harmonic Oscillator I [mln8] Equation of motion: mẍ = x γẋ+f 0 cos ωt ẍ+βẋ+ω 0x = A cos ωt. Parameters: β. = γ/m, ω 0. = /m, A. = F0 /m. General solution: x(t) = x c (t) + x p (t). x c (t): general solution of homogen. eq. (transients) [mln6]. x p (t): particular solution of inhomogen. eq. (steady state) [mex180]. Steady-state oscillation: x p (t) = D cos(ωt δ) amplitude: D(ω) = phase angle: δ(ω) = arctan A (ω 0 ω ) + 4ω β, ωβ ω 0 ω. Maximum amplitude realized at dd(ω) dω = 0. ωr Amplitude resonance frequency: ω R = ω0 β if β < ω0. Average energy: E(ω) = T (ω) + V (ω) = 1 4 ma ω + ω 0 (ω 0 ω ) + 4ω β. E(ω), T (ω), V (ω) are resonant at different frequencies [mex181]. Average power input: P (ω). = F 0 cos ωt ẋ(t) [mex18]. Quality factor: [mex183] driven oscillator: Q =. average energy stored π maximum energy input per period, damped oscillator: Q =. energy stored π energy loss per period. For β ω 0 the width at half maximum of the power resonance curve is ω β. Therefore, the quality factor is Q ω 0 / ω.

7 [mex180] Driven harmonic oscillator: steady state solution Consider the driven harmonic oscillator, mẍ = x γẋ + F 0 cos ωt. Show that the steady-state solution has the form x(t) = D cos(ωt δ), D = A (ω 0 ω ) + 4ω β, δ(ω) = arctan ωβ ω0, ω where we have used the parameters β. = γ/m, ω 0. = /m, A. = F0 /m..

8 [mex181] Driven harmonic oscillator: inetic and potential energy Consider the driven harmonic oscillator, mẍ = x γẋ + F 0 cos ωt, in a steady-state motion. (a) Calculate the average inetic energy T (ω), the average potential energy V (ω), and the average total energy E(ω) = T (ω) + V (ω). Use the parameters β =... γ/m, ω 0 = /m, A = F0 /m.. (b) Each quantity assumes its maximum value at a different resonant frequency: ω T, ω V, ω E. Determine each resonant frequency.

9 [mex18] Driven harmonic oscillator: power input Consider the driven harmonic oscillator, mẍ = x γẋ + F 0 cos ωt, in a steady-state motion. (a) Calculate the average power input, P (ω). = F 0 cos ωt ẋ(t). Use the parameters β. = γ/m, ω 0. = /m, A. = F0 /m. (b) Find the resonant frequency ω P and the maximum (averaged) power input P max = P (ω P ).

10 [mex183] Quality factor of damped harmonic oscillator (a) Consider the driven harmonic oscillator, mẍ = x γẋ + F 0 cos ωt, in a steady-state motion. Use the parameters β =... γ/m, ω 0 = /m, A = F0 /m. In [mex18] we have calculated the maximum (averaged) power input, P max = P (ω P ), and in [mex181] we have calculated the average energy E(ω) stored in the oscillator. Determine the quality factor of the driven oscillator defined as Q = π E(ω P ) / P (ω P ) τ with τ = π/ω P. Show that to leading or der in β/ω 0 the quality factor is equal to the amplitude ratio at resonance and at zero frequency: Q = D(ω R )/D(0). (b) Consider the harmonic oscillator, mẍ = x γẋ, with wea damping (β/ω 0 1) and no driving force. Determine the quality factor Q of the damped oscillator defined as π times the ratio of the instantaneous energy stored, E(t), and the energy loss per period, τ de/dt. Evaluate the result to leading order in β/ω 0

11 [mex6] Driven harmonic oscillator: runaway resonance Consider the driven harmonic oscillator with no damping, mẍ = x + F 0 cos ωt. Tae the general solution off resonance, ω ω 0 = /m, and perform the limit ω ω 0 to show that the (runaway) solution at resonance with initial condition x(0) = B cos β, ẋ(0) = ω 0 B sin β has the form x(t) = B cos(ω 0 t + β) + F 0t mω 0 sin(ω 0 t).

12 Driven Harmonic Oscillator II [mln9] Equation of motion: ẍ + βẋ + ω 0x = A(t). Parameters: β. = γ/m, ω 0. = /m, A(t). = F (t)/m. Periodic driving force: F (t + τ) = F (t). Fourier series: A(t) = 1 a 0 + [a n cos nωt + b n sin nωt], ω = π τ. Fourier coefficients: a n = τ n=1 τ 0 dt A(t) cos nωt, Linear response of system to periodic driving force: x(t) = a 0 ω 0 + ω n = πn τ, d(ω n) = b n = τ τ 0 dt A(t) sin nωt. d(ω n ) [a n cos(ω n t + δ n ) + b n sin(ω n t + δ n )], n=1 1 (ω 0 ω n) + 4ω nβ, ( ) δ ωn β n = arctan. ω0 ωn Aperiodic driving force: F (t) = ma(t) with Fourier transform: x(ω) = Inverse transform: x(t) = dt e iωt x(t), Ã(ω) = dt A(t) <. + dt e iωt A(t). dω + π e iωt x(ω), dω A(t) = π e iωt Ã(ω). Fourier transformed equation of motion is algebraic (not differential): ẍ + βẋ + ω 0x = A(t) ω x(ω) iβω x(ω) + ω 0 x(ω) = Ã(ω). Linear response of system to aperiodic driving force: x(ω) = Ã(ω) ω 0 ω iβω x(t) = + dω π e iωt x(ω).

13 [mex184] Fourier coefficients of a sawtooth driving force Find the Fourier coefficients a n, b n of a periodic sawtooth driving force F (t) = ma(t) in two renditions with different symmetries. (a) A(t) = at/τ, t < τ/; (b) A(t) = at/τ, 0 < t < τ. A(t) (a) A(t) (b) t a τ τ t

14 [mex185] Fourier coefficients of periodic sequence of rectangular pulses (a) Find the Fourier coefficients a n, b n of a periodic driving force F (t) = ma(t) in the shape of a sequence of rectangular pulses as shown. (b) Each pulse has an area c = uv. Find the Fourier coefficients in the limit u, v 0 at fixed c. A(t) v u τ t

15 Driven Harmonic Oscillator III [mln107] No damping and arbitrary driving force. Equation of motion: mẍ + ω0x = F (t), Complex variable: ξ(t) =. ẋ(t) + ıω 0 x(t) ξ(t) ıω 0 ξ(t) = F (t)/m. Ansatz: ξ(t) = B(t)e ıω 0t Ḃ(t) = 1 m F (t)e ıω 0t. x(t) = 1 [ I[ξ(t)], ξ(t) = e ıω 0t 1 ω 0 m t 0 dt F (t )e ıω 0t + ξ 0 ]. Instantaneous energy at time t or total energy absorbed at time t if oscillator is initially at equilibrium (x 0 = ẋ 0 = 0): E(t) = 1 mẋ + 1 mω 0x = 1 m ξ(t) = 1 m t 0 dt F (t )e ıω 0t. Constant force switched on: F (t) = F 0 Θ(t). x(t) = F ( ) 0 1 cos ω mω0 0 t, E(t) = F 0 ( ) 1 cos ω mω0 0 t. Force performs positive and negative wor in alternation. Fading force switched on: F (t) = F 0 e αt Θ(t). x(t) = E(t) = F 0 m(ω 0 α ) [ e αt cos ω 0 t + α ω 0 sin ω 0 t ]. F0 [ ] 1 + e αt e αt cos ω m(ω0 + α 0 t. ) Constant force switched on and then off: F (t) = F 0 Θ(t)Θ(T t). x(t) = F [ ] 0 cos ω mω0 0 (t T ) cos ω 0 t E(t) = F 0 sin ω 0T mω0 (t T ). = const (t T ).

16 [mex63] Driven harmonic oscillator with Coulomb damping The harmonic oscillator with Coulomb damping and harmonic driving force is described by the equation of motion, ẍ + α sgn(ẋ) + ω 0x = A cos(ωt), (1) where ω0 = /m, α = µ/m, A = F 0 /m. The function sgn(ẋ) denotes the sign (±) of the instantaneous velocity. The oscillator has mass m and the spring has stiffness. The coefficient of inetic (and static) friction is µ. The natural angular frequency of oscillation is ω 0. The harmonic driving force has amplitude F 0 and angular frequency ω. In this project we consider an oscillator at resonance (ω = ω 0 = 1) launched from x(0) = 0 with initial velocity ẋ(0) = v 0 > 0. (a) Use the DSolve option of Mathematica to determine the analytic solutions of (1) with the given initial conditions, valid over a time interval with ẋ > 0. Chec whether the solution that Mathematica gives you can be further simplified by hand. Use the ParametricPlot option of Mathematica to plot this solution in the phase plane, i.e. x versus ẋ. Use A = 1, v 0 = 9 and various values of α (all in SI units). (b) Use the NDSolve and ParametricPlot options of Mathematica to generate and plot data for the solution of (1) over a larger time interval. Use again A = 1, v 0 = 9 and various values of α. Tune α to a value that yields a periodic trajectory. (c) Investigate the stability of the periodic trajectory thus found numerically. Is it a limit cycle? Vary the initial conditions and chec whether the periodic trajectory attracts or repels nearby trajectories.

17 Small Oscillations [mln43] Consider undamped small-amplitude motion about a stable equilibrium. Lagrangian: L(q 1,..., q n, q 1,..., q n ) = 1 m ij q i q j 1 ij q i q j. Mass coefficients: m ij = 3N =1 ( ) x m q i 0 ( ) V Stiffness coefficients: ij =. q i q j 0 Lagrange equations are linear: m ij q j + j=1 ij ( ) x The matrices {m ij } and { ij } are symmetric. q j. 0 ij ij q j = 0, i = 1,..., n. Ansatz for solution: q j (t) = A j cos(ωt + φ), j = 1,..., n. ( ij ω m ij )A j cos(ωt + φ) = 0, i = 1,..., n. j=1 Linear homogeneous equations: ( ij ω m ij )A j = 0, i = 1,..., n. (1) j=1 j=1 Characteristic equation (n th -order polynomial in ω ): ( 11 ω m 11 ). ( 1n ω m 1n ). ( n1 ω m n1 ) ( nn ω m nn ) = 0. The n roots ω1,..., ωn of the characteristic equation are the eigenvalues associated with n natural modes of vibration (normal modes). The normal mode with angular frequency ω is specified by a set of amplitudes A () 1,..., A () n. The amplitude ratios for this normal mode are determined from Eqs. (1): ( ij ωm ij )A () j = 0, i = 1,..., n. j=1

18 Transformation to Principal Axes [mln30] Solving the equations ( ij ωrm ij )A jr = 0, i, r = 1,..., n j=1 for the amplitudes A jr of the n normal modes amounts to finding an orthogonal matrix A, which diagonalizes the symmetric matrices m and simultaneously: A T m A = 1 = , ω1 0 0 A T A = Ω. 0 ω 0 =....., 0 0 ωn Normal coordinates: Q j = A ij q i. i=1 Lagrangian: L = 1 (m ij q i q j ij q i q j ) = 1 Lagrange equations: ij r=1 ( ) Q r ωrq r. (m ij q j + ij q j ) = 0, i = 1,..., n (coupled). j=1 Lagrange equations: Qr + ω rq r = 0, r = 1,..., n (decoupled). Applications: Blocs and springs in series [mex13] Small oscillations of the double pendulum [mex14] Two coupled oscillators [mex186]

19 Elastic Chain [mln48] Consider the elastic chain consisting of n blocs and n + 1 springs as shown. m m m q q 1 n Equations of motion: m q j + (q j q j 1 q j+1 ) = 0, j = 1,..., n. Boundary conditions: q 0 (t) = q n+1 (t) = 0. Ansatz for solution: q j (t) = C sin(αj) cos(ωt + φ), j = 1,..., n. Chec boundary condition: sin[(n + 1)α] = 0 α = πr, r = 1,..., n. n + 1 [ ( )] Chec eqs. of motion: mω πr + 1 cos C sin(αj) = 0. n + 1 Normal mode frequencies: ω r = m sin πr, r = 1,..., n. (n + 1) ( ) πr Normal mode amplitudes: A jr = sin n + 1 j, j, r = 1,..., n. General solution: superposition of normal modes q j (t) = = r=1 r=1 ( πr C r sin n + 1 j q ) cos(ω r t + φ r ) ( ) πr sin n + 1 j [d r cos ω r t + e r sin ω r t], with d r = e r = n + 1 r=1 1 n + 1 ω r ( πr q j (0) sin r=1 ), n + 1 j ( πr q j (0) sin n + 1 j ).

20 [mex13] Blocs and springs in series Consider a system of two blocs of mass m attached by springs of stiffness to each other and to a rigid wall. The blocs can slide without friction along the x-axis. When the springs are relaxed, the blocs are at the positions x 1 = a and x = a. (a) Find the Lagrangian L(q 1, q, q 1, q ) for the system, where q 1, q are the displacements of the two blocs from their equilibrium positions. (b) Find the angular frequencies ω 1, ω of the two normal modes by solving the characteristic equation. (c) Find the amplitude ratios A (), = 1, for the two normal modes. 1 /A() m a m a x

21 [mex186] Two coupled oscillators Consider a system of two blocs of mass m attached by springs of stiffness to rigid walls on two sides and by a spring of stiffness c to each other. The blocs can slide bac and forth without friction. (a) Find the Lagrangian L(q 1, q, q 1, q ). (b) Write the equations of motion for q 1, q. (c) Find the normal mode frequencies ω 1, ω by solving the characteristic equation. (d) Rewrite the Lagrangian in the form L = 1 ij [m ij q i q j + ij q i q j ]. Then find the orthogonal matrix A ij which diagonalizes the symmetric matrices m ij and ij simultaneously: lm AT il m lma mj = δ ij, lm AT il lma mj = ω i δ ij. (e) Find the normal mode coordinates Q j = i A ijq i. m c m q q 1

22 [mex187] Three coupled oscillators Consider the elastic chain consisting of three blocs and four springs as shown. (a) Show that the equations of motion for the generalized coordinates q j can be brought into the form m q j + (q j q j 1 q j+1 ) = 0, j = 1,, 3 with boundary conditions q 0 (t) = q 4 (t) = 0. (b) Use the ansatz q j (t) = A j cos(ωt) and find the three normal-mode frequencies ω r, r = 1,, 3. (c) Find the normal coordinates Q j = i A ijq i, j = 1,, 3. (d) Illustrate each normal mode Q j modes by plotting q i (0) versus i. m m m q 1 q q 3

23 [mex114] What is the physical nature of these modes? Three beads of mass m each are constrained to slide without friction along parallel wires. The beads are connected to each other by rubber bands of negligible mass which are stretched considerably (L L 0 ). (a) Describe the physical nature of the modes specified by the generalized coordinates q 1, q, q 3, where x 1 = q 1 + q + 1 q 3, x = q 1 q 3, x 3 = q 1 q + 1 q 3. Give a quantitative description of the motion that ensues if the system is initially at rest with only one the generalized coordinates diplaced infinitesimally: (b) 0 < q (0) 1 L, q (0) = q (0) 3 = 0, (c) 0 < q (0) L, q (0) 1 = q (0) 3 = 0, (d) 0 < q (0) 3 L, q (0) 1 = q (0) = 0. x 1 x x 3 L L

24 [mex14] Small oscillations of the double pendulum Consider a plane double pendulum consisting of two equal point masses m and two rods of negligible mass and equal lengths l. The Lagrangian L(φ 1, φ, φ 1, φ ) of this system is nown from [mex0]. (a) Expand L(φ 1, φ, φ 1, φ ) to quadratic order in the dynamical variables and derive the Lagrange equations from it. They describe the small oscillations about the stable equilibrium position. (b) Find the angular frequencies ω 1, ω of the two normal modes by solving the characteristic equation. (c) Find the amplitude ratios A (), = 1, for the two normal modes. 1 /A() φ 1 l m φ l m