Shock and Vibration

Volume 2016 (2016), Article ID 3253178, 19 pages

http://dx.doi.org/10.1155/2016/3253178

## Analysis of Free Pendulum Vibration Absorber Using Flexible Multi-Body Dynamics

Mechanical Engineering Department, Texas Tech University, Lubbock, TX 79409, USA

Received 9 February 2016; Revised 3 July 2016; Accepted 11 July 2016

Academic Editor: Londono Monsalve

Copyright © 2016 Emrah Gumus and Atila Ertas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 wind-induced 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 beam-tip mass-free 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 mass-ball 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 wind-induced vibrations in traffic signal structures.

#### 1. Introduction

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 [1] to the rolling motion of ships [2, 3] and the rocking motion of buildings and structures under earthquakes [4].

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 [5–7]. Autoparametric vibration absorber has received considerable attention since mid-1980s and researchers published many interesting papers [8–19]. There are many practical examples of designing vibration absorber published using the concept of autoparametric resonance [20–26].

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 [27]. One of the first formalisms is given by Hooker and Margulies [28] in which they analyzed the satellites interconnected with spherical joints. Another formulation was published in 1967 by Roberson and Wittenburg [29]. Wittenburg [30] wrote the first textbook on multibody dynamics in which he explained rigid body kinematics and dynamics as well as general multibody systems. In 1988, Nikravesh [31] provided information about the computer-aided analysis of multibody systems in his textbook. Haug [32] provided basic methods of the computer-aided kinematics and dynamics for spatial and planar systems. Many more authors provided textbooks in the field of kinematic and dynamic simulations of multibody systems such as Roberson and Schwertassek [33], Huston [34], and García de Jalón and Bayo [35].

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 [36] proposed the component mode synthesis method in which each elastic component is identified by three sets of modes: rigid body, reference modes, and normal modes. Rigid body modes are used to describe the rigid body translation and large rotations of a body reference, reference modes are used to define a unique displacement field, and the normal modes are used to define the deformation relative to the body reference. An alternative formulation was proposed by Yoo and Haug [37] in which a lumped mass finite element structural analysis formulation is used to generate deformation modes. In the floating reference frame formulation, a mixed set of absolute and local deformation coordinates are used to define the configuration of the deformable body [38–40]. This method became the most widely used approach due to its straightforward nature. However, the mass matrix, centrifugal, and Coriolis forces appear to be highly nonlinear. The incremental finite element approach uses rotation angles as nodal variables, which lead to linearized kinematic equations. Therefore, models obtained by using incremental finite elements cannot describe the exact rigid body displacements [41]. In order to solve this problem in the incremental finite element approach, a different approach called the large rotation vector formulation has been proposed. In this method, finite rotations are employed instead of infinitesimal rotations, which results in an exact modeling of the rigid body displacements [42].

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 [40, 43–50]. In this formulation, instead of the angle of rotations, absolute slopes are used as nodal variables.

In this paper, the beam-tip mass-ball 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 mass-free pendulum system. This novel free pendulum device is practical and feasible passive vibration absorber in the mitigation of large amplitude wind-induced vibrations in traffic signal structures.

#### 2. Formulation of Equations of Motions for Flexible Multibody Dynamics

##### 2.1. Displacement Field

In this paper, a planar beam element is used to model flexible beam under investigation. Referring to Figure 1, the global position vector of an arbitrary point on the element is defined in terms of the nodal coordinates and the element shape function as [44]where is the global shape function and is the vector of element nodal coordinates defined asThe elements of the vector of nodal coordinates are defined as [44]where is the beam element length and is the axial coordinate that defines the position of an arbitrary point on the element in the undeformed state. , , , and are the absolute displacement coordinates and , , , and are the global slopes of the nodes.