Structures which are commonly used in our infrastructures are becoming lighter with progress in material science. These structures due to their light weight and low stiffness have shown potential problem of windinduced vibrations, a direct outcome of which is fatigue failure. In particular, if the structure is long and flexible, failure by fatigue will be inevitable if not designed properly. The main objective of this paper is to perform theoretical analysis for a novel free pendulum device as a passive vibration absorber. In this paper, the beamtip massfree pendulum structure is treated as a flexible multibody dynamic system and the ANCF formulation is used to demonstrate the coupled nonlinear dynamics of a large deflection of a beam with an appendage consisting of a massball system. It is also aimed at showing the complete energy transfer between two modes occurring when the beam frequency is twice the ball frequency, which is known as autoparametric vibration absorption. Results are discussed and compared with findings of MSC ADAMS. This novel free pendulum device is practical and feasible passive vibration absorber in the mitigation of large amplitude windinduced vibrations in traffic signal structures.
Many mechanical systems can be modeled as a beam with a lumped mass, such as a wing of an airplane with a mounted engine, a robot arm carrying a welding tool, or a traffic light. Understanding the dynamics of those systems having flexible and slender beams is of great importance in vibration analyses to prevent catastrophic failures of the structures. Therefore, there is an extensive amount of experimental and numerical work on the responses of beams in the nonlinear dynamics and vibration field.
There is widespread interest in pendulum modeling and the use of the pendulum as a vibration absorber. This interest ranges from the dynamics of Josephson’s Junction in solid state physics [
Autoparametric vibration absorber is a device designed to absorb the energy from the primary mass (main mass) at conditions of combined internal and external resonance. Autoparametric resonance is a special case of parametric vibration and is said to exist if the conditions at the internal resonance and external resonance are met simultaneously due to external force [
The first studies in multibody systems were on the dynamics of the rigid bodies which were related to gyrodynamics, the mechanism theory, and biomechanics. A good review of this topic is given by Schiehlen [
Until now, we discussed papers and textbooks that were related to the multibody systems consisting of rigid bodies. However, in many applications, bodies undergo large deformations, which necessitate the modeling of the flexible bodies. Flexible multibody systems have attracted many researchers and several flexible multibody formulations have been established such as the floating frame of reference method, incremental finite element corotational method, and the large rotation vector method. Agrawal and Shabana [
Most of the methods explained above suffer from highly nonlinear terms inside the mass matrix, centrifugal, and Coriolis forces. Therefore, a new approach called the absolute nodal coordinate formulation (ANCF) was proposed for the solution of large deformation problems [
In this paper, the beamtip massball structure is treated as a flexible multibody dynamic system and the ANCF formulation is used to demonstrate the coupled nonlinear dynamics of a large deflection of a beam with an appendage consisting of a massfree pendulum system. This novel free pendulum device is practical and feasible passive vibration absorber in the mitigation of large amplitude windinduced vibrations in traffic signal structures.
In this paper, a planar beam element is used to model flexible beam under investigation. Referring to Figure
Planar beam element.
The element shape function can be defined as [
Kinetic energy of the finite element can be written as
In order to develop the equations of motion of the beam element, generalized elastic forces,
The longitudinal strain,
Let
Using (
Let
Using the principles of virtual work in dynamics and the expression of the kinetic and strain energies given in (
Referring to Figure
Beam coordinate system.
The beamtip massfree pendulum system consists of three bodies, among which the beam is assumed to be flexible, and the tip mass and ball are assumed to be rigid. The ANCF beam is modeled using three finite elements. Referring to Figure
Global nodes.
The connection between the free end of the beam and the tip mass is modeled using a fixed joint. Referring to Figure
Tip mass displacement vectors.
Constraint equations between the ball and the tip mass can be defined such that the velocity of the contact point C on the ball has to be equal to the velocity of the contact point C on the tip mass. Therefore, referring to Figure
Ball displacement vectors.
The system has five dependent coordinates and seventeen independent coordinates. For the numerical analysis, the vectors of independent and dependent coordinates are selected as
Referring to Figure
Impact condition.
Numerical integration parameters, rigid body parameters, and flexible body parameters are given in Table
System parameters.
Numerical integration parameters  Rigid body data  Flexible beam data  

Newmark parameters  NewtonRaphson parameters  Tip mass  Ball  Flexible beam 





The equations of motion of a multibody system consisting of interconnected rigid and deformable bodies are a combined set of ordinary differential and algebraic equations. These kinds of equation sets are called differential algebraic equations (DAEs) in literature. The solutions to DAEs are not as straightforward as ordinary differential equations. Specialized numerical techniques have been developed for the solution of DAEs. In this paper, the direct integration approach based on the Wehage coordinate partitioning technique [
Computational algorithm for dynamic analysis.
System parameters given in Table
Before having detailed discussions on the unlocked ball cases, one can refer to Figure
Ball locked/detailed dynamics of the beam for the forcing frequency of 4.13 Hz.
Ball locked/phase plane curves for the forcing frequency of 4.13 Hz at steady state.
Figure
Frequency response curves for an excitation amplitude of 1 mm peaktopeak.
As shown in Figure
Figures
Detailed dynamics of the system for the forcing frequency of 3.70 Hz.
Phase plane curves for the forcing frequency of 3.70 Hz.
Among the numerical analysis performed for the system, the most important results are given in Figures
Impact details for the forcing frequency of 4.13 Hz.
System trajectories for the forcing frequency of 4.13 Hz.
Phase plane curves for the forcing frequency of 4.13 Hz at steady state.
FFT of the system for the forcing frequency of 4.13 Hz.
Energy curves of the ball and the tip mass for the forcing frequency of 4.13 Hz.
Beam energy curves for the forcing frequency of 4.13 Hz.
Comparison of the beam kinetic energy and the ball kinetic energy for the forcing frequency of 4.13 Hz.
Referring to Figure
Comparison of the beam tip displacement,
Comparison of ball rotation angle in active (ball unlocked) and passive (ball locked) cases.
Referring to Figure
FFT of the system for the forcing frequency of 4.25 Hz.
Figure
Beating cycle (ball response curve).
Moreover, equations for the beating period and the oscillation period in terms of the beating frequencies
Impact details for the forcing frequency of 4.25 Hz.
Since the system has complete energy transfer at the forcing frequency of 4.13 Hz, ADAMS simulation is performed at this frequency, and the results are given in Figures
Beam tip displacement,
Beam kinetic energy curve for the forcing frequency of 4.13 Hz.
Beam potential energy delta curve for the forcing frequency of 4.13 Hz.
Beam strain energy curve for the forcing frequency of 4.13 Hz.
Ball potential energy delta curve for the forcing frequency of 4.13 Hz.
Ball kinetic energy curve for the forcing frequency of 4.13 Hz.
Comparing the transverse displacement curves of the beam shown in Figures
Figures
In conclusion, the results obtained from ADAMS and ANCF are observed to be in good quantitative and qualitative agreement even though two methods used different solution approaches.
This paper is concerned with the dynamics of a flexible beam with a tip massball arrangement. The system is treated as a flexible multibody system interconnected with joints. The tip mass and the ball are assumed to be rigid, and the beam is treated as a flexible body. Connection between the tip mass and the free end of the beam is modeled using a fixed joint, and the contact between the ball and the tip mass is modeled using the geometry of the bodies.
The absolute nodal coordinate formulation (ANCF) is used to determine the mass matrix, stiffness matrix, and generalized forces of the system. Generalized elastic forces for the flexible beam are found using the continuum mechanics approach. Nonlinear equations of motion of the system are found using the Lagrangian Formulation, in which constraints are treated explicitly as extra equations by using Lagrange Multipliers. The resulting differential algebraic equations are solved using a twoloop sparse matrix numerical integration method, in which the kinematic constraint equations are satisfied at the position, velocity, and acceleration levels.
The detailed system dynamics including frequency response curves, time history curves, FFT curves, phase plane curves, and energy curves are plotted for various base excitation frequencies. Numerical results are compared with the results of previously studied similar systems and a good qualitative agreement is observed. Moreover, the same system with the same parameters is modeled using the mechanical analysis software, ADAMS, and the results are observed to be in good quantitative agreement, although the two methods use different formulations. Therefore, in view of the numerical results, it is found that the free pendulum can be considered a suitable autoparametric vibration absorber under periodic excitation.
Futuristic structures will be made of materials like fiber reinforced polymers which are much lighter than steel and hence the vibration problem will be more acute. For example, traffic signal light structures, highway signs, and luminaires are observed to vibrate regularly at steady winds of 10 to 30 mph. The amplitude of vibration depends upon the characteristics of the wind like mean speed, mean direction, and gustiness; dynamic characteristics of the structures; and shape and size of the structure. Vortex shedding and buffeting are the two predominant windstructure interaction phenomena which could cause vibrations in this class of structures, consisting mainly of a vertical pole and horizontal arm and lights or signs attached to the arm.
This study will provide useful information for designing passive vibration control devices and systems in an exposed environment. Also it will provide important information for the designiteration process leading to an optimum passive vibration absorber for use in the real world. Results obtained from this study will generate knowledge to develop (a) better understanding of the working principles of control systems, (b) design guidelines and standards, and (c) practical approaches for design, fabrication, and field installations.
The authors declare that they have no competing interests.