Wilberforce pendulum: modelling linearly damped coupled oscillations of a spring-mass system

Abstract

The Wilberforce pendulum is a coupled spring-mass system, where a mass with adjustable moment of inertia is suspended from a helical spring. Energy is converted between the translational and torsional modes, and this energy conversion is most clearly observed at resonance, which occurs when the damped natural frequencies of the two oscillation modes are equal. A theoretical model—with energy losses due to viscous damping accounted for—was formulated using the Lagrangian formalism to predict the pendulum mass’ trajectory. Theoretical predictions were compared with experimental data, showing good agreement. Fourier analysis of both theoretical predictions and experimental data further corroborate the validity of our quantitative model. The dependence of oscillation features like beat frequency and maximum conversion amplitude on relevant parameters such as the initial vertical displacement, initial angular displacement and moment of inertia was also investigated and experimentally verified.

Initial release condition dependent decoupling motion of a Wilberforce pendulum in the case of near resonance

“…The solutions for the special case of a = b have been reported in [11], which exhibits an initial condition dependent amplitude of normal modes, similar to equation (4). If a is not equal to b, the solution of z and θ in equations (7a) and (7b) can be written as: Because of the existence of a damping coefficient, the amplitude of ω 1 and ω 2 can't be seen directly from equation (8), but it can be extracted via making the Fourier transform.…”

“…Considering the influence of nonconservative force, we used the modified form of Euler-Lagrange equation [7,11], that is:…”

“…The detailed decouple coefficient can be obtained from numerical calculation. According to equation (25) in appendix, A, B, C, D, and f i can be expressed by the linear combination of θ 0 and z 0 , and according to equation (11), a 11 , a 12 , b 11 , and b 12 can be expressed by the linear combination of A, B, C, D, so that they can be expressed by the linear combination of θ 0 and z 0 . Hence F can be expressed only with q…”

“…The following studies focused on the motion visualization of the pendulum by optimizing experimental methods. An ultrasound distance sensor [9] was first used to record vertical motion, and an angular motion was first obtained with the aid of a digital video camera [10,11]. Further research studied the motion of a pendulum experimentally through computer-aided measurement systems, such as GY-521 module [12], and sonar and noncontact rotation sensors [13,14].…”

“…Further research studied the motion of a pendulum experimentally through computer-aided measurement systems, such as GY-521 module [12], and sonar and noncontact rotation sensors [13,14]. These experimental works further demonstrated that the system is in coupling resonance when the natural frequencies of vertical (ω z ) and angular (ω θ ) oscillations are equal [9][10][11][12]15]. In this case, if the pendulum bob is released from rest with an initial vertical displacement (z 0 ) and no rotational displacement, energy is slowly transferred between the z and θ oscillations at the beat frequency ω B [16,17].…”