The motion of two masses coupled to a finite mass spring

European Journal of Physics, 2020

Experimental analysis of the motion in a system of two coupled oscillators with arbitrary initial conditions was performed and the normal coordinates were obtained directly. The system consisted of two gliders moving on an air track, joined together by a spring and joined by two other springs to the fixed ends. From the positions of the center of mass and the relative distance, acquired by analysis of the digital video of the experiment, normal coordinates were obtained, and by a non linear fit the normal frequencies were also obtained. It is shown that although the masses of the springs are relatively small compared to that of the gliders, it is necessary to take them into consideration to improve the agreement with the experimental results. This experimental-theoretical proposal is targeted to an undergraduate laboratory.

We can describe the motion of the mass using energy, since the mechanical energy of the mass is conserved. At any position, x, the mechanical energy, E, of the mass will have a term from the potential energy, U, associated with the spring force, and kinetic energy.

ANNALS OF THE ORADEA UNIVERSITY. Fascicle of Management and Technological Engineering., 2012

This article presents the phases and the final results of the elaboration of the physical and mathematical models of 1DOF elastic mechanical systems taking into consideration the distributed mass of two kind of embedded beam spring: axial spring and torsional spring. The result and final considerations have a real utility in fast and operational calculus of the natural frequencies of this kind of mechanical models, pointing out the influence of the distributed mass of the spring beam for resonance characteristic.

American Journal of Physics, 2017

A model of a one-dimensional mass-spring chain with mass or spring defects is investigated. With a mass defect, all oscillators except the central one have the same mass, and with a spring defect, all the springs except those connected to the central oscillator have the same stiffness constant. The motion is assumed to be one-dimensional and frictionless, and both ends of the chain are assumed to be fixed. The system vibrational modes are obtained analytically, and it is shown that if the defective mass is lighter than the others, then a high frequency mode appears in which the amplitudes decrease exponentially with the distance from the defect. In this sense, the mode is localized in space. If the defect mass is greater than the others, then there will be no localized mode and all modes are extended throughout the system. Analogously, for some values of the defective spring constant, there may be one or two localized modes. If the two defected spring constants are less than that of the others, there is no localized mode. V

Proyecciones (Antofagasta), 2000

Given the sequences of real numbers (λ i ) n 1 , (µ i ) n−1 1 , satisfying the interlacing property λ i < µ i < λ i+1 , 1 ≤ i ≤ n − 1, we present a new numerical procedure to construct a spring-mass system with eigenvalues (λ i ) n 1 , where the interlaced spectrum (µ i ) n−1 1 corresponds to the modified system whose mass m r+1 , 1 ≤ r ≤ n − 2, is fixed. The method, which is a modification of the fast orthogonal reduction technique, appears to be computationally less expensive than other in the literature.

Revista Brasileira de Ensino de Física, 2013

The simple harmonic motion of a spring-mass system generally exhibits a behavior strongly influenced by the geometric parameters of the spring. In this paper, we study the oscillatory behavior of a spring-mass system, considering the influence of varying the average spring diameter Φ on the elastic constant k, the angular frequency ω, the damping factor γ, and the dynamics of the oscillations. It was found that the elastic constant k is proportional to Φ-3, while the natural frequency ω0 is proportional to Φ- 3 / 2, and γ decreases as Φ increases. We also show the differences obtained in the value of the angular frequency ω when the springs are considered as ideal (massless), taking into account the effective mass of the spring, and considering the influence of the damping of the oscillations. This experiment provides students with the possibility of understanding the differences between theoretical models that include well-known corrections to determine the frequency of oscillation...

IMA Journal of Applied Mathematics, 2011

We study some spring mass models for a structure having a unilateral spring of small rigidity ε. We obtain and justify an asymptotic expansion with the method of strained coordinates with new tools to handle such defects, including a non negligible cumulative effect over a long time: T ε ∼ ε −1 as usual; or, for a new critical case, we can only expect: T ε ∼ ε −1/2. We check numerically these results and present a purely numerical algorithm to compute "Non linear Normal Modes" (NNM); this algorithm provides results close to the asymptotic expansions but enables to compute NNM even when ǫ becomes larger.

Journal of Sound and Vibration, 1997

Arabian Journal for Science and Engineering, 2012

We consider periodic solution for coupled systems of mass-spring. Two practical cases of these systems are explained and introduced. An analytical technique, called the Hamiltonian approach, is applied to calculate approximations to the achieved nonlinear differential oscillation equations. The concept of the Hamiltonian approach is briefly introduced, and its application for nonlinear oscillators is studied. The method introduces an alternative to overcome the difficulty of computing the periodic behavior of the oscillation problems in engineering. The results obtained employing first-order and second-order Hamiltonian approach are compared with those achieved using two other analytical techniques, named the energy balance method and the amplitude frequency formulation, and also to assess the accuracy of solutions, the results were compared with the exact ones. The results indicate that the present analysis is straightforward and provide us a unified and systematic procedure which is simple and more accurate than the other similar methods. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared with the previous approaches such as the perturbation and the classical harmonic balance methods.

Nonlinear Dynamics, 2012

We investigate the dynamics of a simple pendulum coupled to a horizontal mass-spring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the massspring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dynamics of the autonomous coupled system is studied using the method of Direct Partition of Motion (DPM), in conjunction with a rescaling of fast time in a manner that is inspired by the WKB method. We particularly study the motions in which the amplitude of the motion of the harmonic oscillator is an order of magnitude smaller than that of the pendulum. In this regime, a pitchfork bifurcation of periodic orbits is found to occur for energy values larger that a critical value. The bifurcation gives rise to nonlocal periodic and quasiperiodic orbits in which the pendulum oscillates about an angle between zero and π/2 from the down right position. The bifurcating periodic orbits are nonlinear normal modes of the coupled system and correspond to fixed points of a Poincare map. An approximate expression for the value of the new fixed points of the map is obtained. These formal analytic results are confirmed by comparison with numerical integration.