mathematics Article Impact of a Multiple Pendulum with a Non-Linear Contact Force Dan B. Marghitu *,† and Jing Zhao † Department of Mechanical Engineering, 1418 Wiggins Hall, Auburn University, Auburn, AL 36849-5341, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-3334-844-3335 † These authors contributed equally to this work. Received: 25 June 2020; Accepted: 20 July 2020; Published: 22 July 2020 Abstract: This article presents a method to solve the impact of a kinematic chain in terms of a non-linear contact force. The nonlinear contact force has different expressions for elastic compression, elasto-plastic compression, and elastic restitution. Lagrange equations of motion are used to obtain the non-linear equations of motion with friction for the collision period. The kinetic energy during the impact is compared with the pre-impact kinetic energy. During the impact of a double pendulum the kinetic energy of the non-impacting link is increasing and the total kinetic energy of the impacting link is decreasing. Keywords: impact; non-linear contact force; friction force; non-linear equations of motion; permanent deformation 1. Introduction Impact is a common and important phenomena in mechanical systems. The simplest impact analysis is based on the conservation of momentum and a kinematic coefficient of restitution defined by Newton. The kinematic coefficient of restitution is the ratio of the normal relative velocities after impact to the normal relative velocities before impact at the contact point [1]. Poisson divided impact into compression and restitution, and defined the kinetic coefficient of restitution as the ratio of the impulse during restitution to that during compression [2]. Kane and Levinson were able to extend the impact analysis and they showed for a double pendulum impacting a surface one can obtain energetically inconsistent results for friction using a kinematic coefficient of restitution [3]. Keller introduced the normal impulse as an independent variable which eliminated the normal force from the equations of motion [4]. Stronge defined a new energetic coefficient of restitution as the negative of the ratio of the work done by normal impulse during restitution to that during compression [5]. In contrast to the former two definitions, the energetic coefficient of restitution provides an energetically consistent solution for frictional collision. Stronge presented also a contact model using springs and the impulse at separation was obtained using the energetic coefficient of restitution [6]. All these previous research are using algebraic equations for the calculation of post-impact velocities. The dynamics of the systems with impacts are highly affected by the small changes of the coefficient of restitution which is estimated experimentally [7]. Impact responses of multibody systems have also been studied numerically in [8–10]. The double pendulum has been demonstrated to be chaotic for some configuration [11]. The expansion in separations of nearby trajectories was characterized by the Lyapunov exponent and an agreement between the results from experiment and numerical calculation was reached. Feedback strategies have been introduced to control chaos of the impact of an inverted pendulum in [12]. The performance of the feedback was shown to be robust with respect to the variations Mathematics 2020, 8, 1202; doi:10.3390/math8081202 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1202 2 of 13 of initial conditions and only minor information of the state of the system is required for such implementations. Litak et al. studied the effect of noise on the stability of the rotational motion of a parametric pendulum [13]. The use of the time-delayed feedback method in control of a parametric pendulum for wave energy harvesting has been explored in [14]. In practice, crane systems commonly suffer from the unexpected swings induced by the double-pendulum mechanism during transportation. In order to suppress the payload oscillations, the input shaping technique has been developed by smoothing the input commands generated by the human operator [15]. The pendulum system has been used in piezoelectric energy harvester (PEH) to amplify the energy harvested from human motion [16]. Experimental results showed that double pendulum based PEH dramatically improved the efficiency of the PEH, compared with PEH with cantilever beam and single pendulum. Experimental results were consistent with the numerical calculations which found that double pendulum system produced higher kinetic energy compared with the single pendulum. The performance of the conventional PID feedback controller and LQR optimal controller on double inverted pendulum system have been compared in [17]. Linearization and approximation were performed on the system equation using Jacobian matrix and an unstable equilibrium position. The simulation results showed that a desired response was obtained by implementing a LQR controller. Udwadia and Koganti proposed an analytical approach for controlling a multi-body inverted pendulum system without any linearazations or approximations [18]. An explicit expression of the control force was obtained by imposing a user-prescribed Lyapunov constraint on the system which minimized the control cost simultaneously. To compensate for the uncertain knowledge of the properties of the actual system, an additional control force was designed based on the generalization of the concept of a sliding surface. Numerical simulations verified that the system was able to move from an initial position to various inverted configurations using their methodology. There has been a large number of publications that focused on the dynamics of double pendulum [11,12]. For the mechanical impact of a double pendulum previous researchers conducted studies to investigate the post-impact velocities. The developments have led to algebraic equations and three definitions of the coefficient of restitution (kinematic, kinetic, and energetic) [3–5]. Using a newtonian approach for some cases the total kinetic energy at the end of the impact is greater than the total kinetic energy at the beginning of the impact for a double pendulum [3]. Until now there is not a standard procedure for the calculations of the post-impact velocities of a double pendulum. We propose a differential model based on three phases of the impact: elastic compression, elasto-plastic compression, and restitution. For each impact intervals non-linear contact forces are developed. This model is different from previous impact algebraic and differential studies. The previous models did not consider the three periods and we propose a new elasto-plastic force and new permanent deformations for the impact of the double pendulum. With this method we solve the post-impact velocities without introducing a coefficient of restitution. The energy loss of a mechanical system is a measure of utmost importance for impacting links and represents a new problem too. The kinetic energy during the impact is compared with the pre-impact kinetic energy. In this study we focus also on the kinetic energy of the impacting and non-impacting links. There are cases for which the kinetic energy of the non-impacting is increasing with respect to the initial impact kinetic energy. The total kinetic energy is decreasing during impact. This research provides a new strategy for analysis and design of impacting systems. The reminder of the paper is organized as follows. In Section2, the mathematical model of the general multiple pendulum in impact with a solid surface is formulated. The contact force expressions for the three impact phases are proposed in Section3. The impact equations of motion are derived in Section4. The numerical simulations and results for the impact of a double pendulum with a horizontal surface are discussed in Section5. Mathematics 2020, 8, 1202 3 of 13 2. Mathematical Model Figure1 represents a general multiple compound pendulum during the impact with a solid surface. The plane of motion will be designated the (x y) plane. The y-axis is vertical, with the positive sense directed vertically upward. The x-axis is horizontal and is contained in the plane of motion. The z-axis is also horizontal and is perpendicular to the plane of motion. These axes define an inertial reference frame. The unit vectors for the inertial reference frame are ı, , and k. The links i are homogenous bars and have the lengths Li and the masses mi. At O and Ai there are pin joints. The mass center of links i is Ci. At a certain instant the free end B of the chain impacts the horizontal surface. Ai Ci i q n i Cn y A 2 impact point 2 B C2 A q impact surface 1 1 2 C1 0 q ı 1 x O Figure 1. Impact of a multiple pendulum. To characterize the configuration of the multiple pendulum, the generalized coordinates qi(t), i = 1, ..., n are selected. The coordinate qi denotes the radian measure of the the angle between link i and the horizontal x axis. The position vector and the velocity vector of the mass center C of − j link j is " # " # j 1 j 1 ı rCj = ∑ (Li 1 cos qi 1) + Lj cos qj + ∑ (Li 1 sin qi 1) + Lj sin qj , (1) i=1 − − 2 i=1 − − 2 where L0 = 0. The velocity vector of the mass center Cj of link j is " # " # j 1 j 1 ı vCj = ∑ (Li 1 q˙i 1 sin qi 1) + Lj q˙j sin qj + ∑ (Li 1 q˙i 1 cos qi 1) + Lj q˙j cos qj . (2) − i=1 − − − 2 i=1 − − − 2 The position vector and the velocity vector of the impact point B are n n rB = ∑ Li cos qiı + ∑ Li sin qi , (3) i=1 i=1 n n vB = ∑ Li q˙i sin qiı + ∑ Li q˙i cos qi . − i=1 i=1 The kinetic energies of the link i is 1 1 T = I q˙2 + m v v , (4) i 2 Ci i 2 i Ci · Ci Mathematics 2020, 8, 1202 4 of 13 where ICi is the mass moment of inertia of link i about the center of mass Ci.