mathematics

Article Impact of a Multiple 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 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 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 , 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 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 of link i about the center of mass Ci. The generalized velocities are defined as

ui = q˙i = ωi, i = 1, ..., n. (5)

The total kinetic energy of the multiple pendulum is

T = T1 + T2 + ... + Ti + ... + Tn. (6)

3. Contact Force The impact normal force at the contact point is calculated for three distinct periods: elastic phase, elasto-plastic phase, and restitution phase [19,20]. Figure2 illustrates the normal contact force F during impact as a function of the normal elastic deformation δ. The elastic compression phase starts from the beginning of the contact where δ = 0 and ends when the maximum elastic deformation, δe, is reached. Next, the elasto-plastic compression phase begins and continues until the maximum deformation, δm, is reached. At the maximum compression the relative velocity between the bodies is zero. The restitution phase starts at the maximum deformation and ends when there is no contact and a permanent deformation, δr, is reached.

elasto-plastic compression phase F

elastic compression restitution phase phase

δ 0 δe δr δm

Figure 2. Normal contact force during impact.

For the elastic phase the Hertz method is employed. The reduced modulus of elasticity, E, is calculated with E E E = r s , (7) E (1 ν2) + E (1 ν2) s − r r − s where Er is the modulus of elasticity of the impacting rod, Es is the modulus of elasticity of the flat surface, νr is the Poisson ratio of the impacting rod, and νs are the Poisson ratio of the flat surface. The reduced radius, R, is calculated with

1 1 1 R− = + , (8) Rr Rs where Rr is the radius of curvature of tip of the impacting rod and Rs = ∞ are the radius of curvature of the surface. Mathematics 2020, 8, 1202 5 of 13

I. For the elastic phase the normal impact force is

4 ER0.5δ1.5 F = , (9) 3 where δ is the normal elastic deformation

n δ = rB  ∑ (Li sin qi0) . · − i=1

The initial impact position is given by the angles qi0. The elastic phase is ending when δ > 1.9 δc. The critical deformation, δc, is calculated with

 2 π Cj Sy δ = R , c 2 E

0.736ν where Cj = 1.295 e is a critical yield stress coefficient, and Sy is the yield strength of the weaker material . II. For the elasto-plastic phase the force is calculated with a modified Jackson and Green expression [19,20]

( 1.5 " 1.1#) 5/12   5/9   0.17(δ/δc) δ 4H ( 1/78)(δ/δc) δ F = Pc e− + 1 e − , (10) δc CjSy − δc where s  B h  i 23 Sy/E δ H a B = 0.14 e , a = R δc , = 2.84 0.92 1 cos π , 1.9δc Sy − − R  2  3 4 R πCjSy P = . c 3 E 2

III. For the restitution an elastic impact force is considered

4 ER0.5(δ δ )1.5 F = − r . (11) 3

The permanent deformation, δr, is given by [19,20]

( "   2#) δ /δ + 5.5 − δ = δ 0.8 1 m c . (12) r m − 6.5

The maximum deformation at the end of the compression phase is δm. The radius of curvature of the tip for the restitution phase is

 2 1 3Pm Rr = , (δ δ )3 4E m − r where Pm = F(δ = δm) is the maximum elasto-plastic force.

4. Impact Equations During the impact the Lagrange equations of motion are used

d  ∂T  ∂T = Qi, i = 1, ..., n (13) dt ∂q˙i − ∂qi Mathematics 2020, 8, 1202 6 of 13

where Qi are the generalized active forces. The friction force at the contact point B is

(v ı) ı F = µ F B · , (14) f − | | v ı | B · | where µ is the kinematic coefficient of friction, F = F , F is the normal impact force defined in Equations (9)–(11) for elastic compression, elasto-plastic compression and restitution phase, respectively. The gravitational forces on link i, at the mass center Ci is

G = m g , i − i where g is the gravitational acceleration. The generalized active forces associated to qi are

∂vCi ∂vB Qi = Gi + (F + F f ) , i = 1, 2, ..., n. (15) · ∂ui · ∂ui

For the dynamics of the system before and after the impact equations given by Equation (13) are used with F = 0. Numerical methods for kinematic chain dynamics have been studied extensively in the engineering and mathematics. To simulate complex mechanical systems, such as robots and walking machines, numerical methods based on ODEs and/or DAEs are used. To solve the discontinuity related to friction a regularized function can be used [21,22]. For each impact phase of the mechanical system, the contact forces were generated and feed into a system of ordinary differential equations. Numerical MATLAB methods were used to get an accurate approximate solution to the differential equations. The numerical problems have unique solutions but the continuous problem might have multiple solutions. Repeated simulations with random disturbances should be performed and sets of possible solutions can be determined [21].

5. Application and Results The method in this study can be applied to a system with n links. For the numerical application we selected the impact of a double pendulum with a horizontal surface. Figure3 shows a double pendulum with homogeneous links made of steel with the length L1 = L2 = L, diameter d = 2 R and mass m1 = m2 = m. The impact flat surface is also made of steel. The density of the material is ρ, the Young’s modulus is E, the Poisson ratio is ν, and the yield strength is SY. The kinematic coefficient of friction is µ. The initial impact angles of the links are q10, q20. The impact point is defined by the position vector rB0 = (L1 cos q10 + L2 cos q20) ı + (L1 sin q10 + L2 sin q20) . The initial impact angular velocities of the links are ω10, ω20. The material properties, geometries and initial conditions used for the simulation are shown in Tables1 and2.

Table 1. Material properties, geometries for the double pendulum and the impact flat surface.

Double Pendulum Flat ρ 7800 (kg/m3) ρ 7800 (kg/m3) E 210 (GPa) E 210 (GPa) ν 0.29 ν 0.29 SY 1.12 (GPa) SY 1.12 (GPa) µ 0.2 µ 0.2 L 1 (m) R 0.005 (m) m 1 (kg) Mathematics 2020, 8, 1202 7 of 13

The analytical expressions for the Lagrange equations of motion are obtained using symbolic MATLAB. The numerical solution of the non-linear ordinary differential equations is calculated with the MATLAB ode45 function. The end of elastic compression, maximum compression, and the end of restitution are detected with MATLAB event functions.

y general configuration B 2

A q  1 2 q ı 1 x O 20

C1 q 1 impact configuration sin 2 L

A + 10 q sin

C2 1 2 L

radius of curvature R B (impact point)

impact surface Figure 3. Impact of a double pendulum.

Table 2. Initial conditions of the double pendulum.

Link 1 Link 2 q 70 q 60 10 − ◦ 20 − ◦ ω 0.1 (rad/s) ω 0.2 (rad/s) 10 − 20 −

Figure4 shows the generalized coordinates q1 and q2 during impact. The magnitude of the angle q1, of the link 1, is increasing, as seen in Figure4a. The magnitude of the angle q2, of the impacting link 2, is increasing initially and then is decreasing, as depicted in Figure4b. The maximum angle of the impacting link is obtained during the elasto-plastic phase.

(a) (b) -70.0000 -59.9975 elastic elastic elasto-plastic elasto-plastic -70.0005 restitution -59.9980 restitution

-70.0010 -59.9985

-70.0015 -59.9990

-70.0020 -59.9995 (deg) (deg) 1 2

q -70.0025 q

-60.0000 -70.0030

-60.0005 -70.0035

-70.0040 -60.0010

-70.0045 -60.0015 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 t (s) 10 -4 t (s) 10 -4

Figure 4. Generalized coordinates during impact: (a) q1(t) and (b) q2(t). Mathematics 2020, 8, 1202 8 of 13

Figure5 shows the angular velocities ω1 and ω2 during impact. The non-impacting link 1 has a pure rotational motion and is connected with the impacting link 2 at a frictionless revolute joint, A. The impact force does not act directly on link 1. The motion of link 1 is influenced by the joint reaction force at A (the reaction force of the impacting link 2 on link 1) and the weight G1. The magnitude of the angle q1 is increasing during the impact and the magnitude of the angular velocity of link 1 will increase during the impact [3,23]. Figure5a shows the increase of the magnitude of the angular velocity of link 1, ω1. The impacting link 2 has a general motion (translation and rotation) and the normal impact force at B, the tangential friction force at B, the weight G2, and the joint reaction force of link 1 on link 2 at A are considered for this element. The angular velocity ω2 has a complex motion: due to the impact and friction force the angular velocity is decreasing until zero, then is changing the sign, and next is increasing [23]. For the given geometrical configuration, Figure5b shows the angular velocity of the impacting link 2, ω2. The angular velocity of the impacting link is changing the sign during the elasto-plastic phase and its magnitude is increasing to a final value larger than the initial value. Simulations for another configuration of the double pendulum with q = 90 , q = 78 10 − ◦ 20 − ◦ shows that final angular velocity is ω2 f = 0.178 rad/s and it is less than the initial magnitude value (ω = 0.2 rad/s). For this case the angular velocity ω changes sign at maximum compression. 20 − 2 The evolution of the angular velocity of the impacting end is dependent on the initial conditions.

(a) (b) -0.10 0.40 elastic elastic elasto-plastic elasto-plastic restitution restitution 0.30

0.20 -0.15

0.10 (rad/s) (rad/s)

1 2 0.00

ω

ω

-0.20 -0.10

-0.20

-0.25 -0.30 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 t (s) 10 -4 t (s) 10 -4

Figure 5. Angular velocities during impact: (a) ω1(t) and (b) ω2(t).

The velocity of the contact point B, during impact, is represented in Figure6. The normal velocity vBy is decreasing and for the maximum compression vBy = 0, as seen in Figure6a. The tangential velocity of the impact point, vBx, is shown in Figure6b. The tangential velocity is decreasing and changes the sign during restitution phase. Figure7a depicts the deformation δ during impact. The maximum deformation is at the end of compression. Figure7b shows the normal force F during impact. The normal force is maximum at maximum compression. Mathematics 2020, 8, 1202 9 of 13

(a) (b) 0.15 0.15 elastic elastic elasto-plastic elasto-plastic restitution 0.10 restitution 0.10 0.05

0.05 0.00

-0.05

(m/s) 0.00 (m/s) By Bx

v v -0.10

-0.05 -0.15

-0.20 -0.10 -0.25

-0.15 -0.30 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 t (s) 10 -4 t (s) 10 -4

Figure 6. Velocity of the contact point, B, during impact: (a) normal velocity vBy(t) and (b) tangential velocity vBx(t).

-5 (b) 10 (a) 2.0 900 elastic elastic elasto-plastic elasto-plastic 1.8 restitution 800 restitution

1.6 700

1.4 600

1.2 500 1.0 (m) F (N) F δ 400 0.8

300 0.6

200 0.4

0.2 100

0.0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 t (s) 10 -4 t (s) 10 -4

Figure 7. (a) Deformation, δ, and (b) normal impact force, F.

Figure8a shows the increase of the kinetic energy of the non-impacting link 1 during collision. 2 The kinetic energy of link 1 is dependent on the square of the angular velocity ω1 and that is why the kinetic energy of link 1 will increase with the increase of the magnitude of ω1, as seen in Figures5a and8a. For the impacting link the kinetic energy is decreasing as shown in Figure8b. The total kinetic energy, shown in Figure8c, is decreasing during impact. Figure9 shows the total kinetic energy for q = 70 , q = 60 , and ω = 0.1 rad/s. 10 − ◦ 20 − ◦ 10 − The initial angular velocities of the impacting link are ω = 0.2, 0.3, 0.4, 0.5 rad/s. The energy 20 − − − − loss is defined as the difference between the initial kinetic energy and the final kinetic energy, ∆T = T T . The energy loss, ∆T, is increasing with the magnitude of the initial angular velocity, 0 − f ω , of the impacting link: ∆T = 0.005 J for ω = 0.2 rad/s, ∆T = 0.010 J for ω = 0.3 rad/s, 20 20 − 20 − ∆T = 0.016 J for ω = 0.4 rad/s, and ∆T = 0.025 J for ω = 0.5 rad/s. For this case impact 20 − 20 − duration is decreasing with respect to the magnitude of the initial angular velocity, ω20. The energy loss is proportional with the permanent deformation δr as seen in Figure 10a: ∆T = 0.005 J for 6 6 5 δr = 5.49 (10− ) m, ∆T = 0.010 J for δr = 8.64 (10− ) m, ∆T = 0.016 J for δr = 1.25 (10− ) m, and 5 ∆T = 0.025 J for δr = 1.55 (10− ) m. Figure 10b shows the increase of the permanent deformation with the increase of the magnitude of the initial angular velocity, ω20. This is one of the reason for the increase of the energy loss with the magnitude of the initial angular velocity of the impacting link. Mathematics 2020, 8, 1202 10 of 13

(a) (b) 0.01 0.022 elastic 0.020 elasto-plastic 0.008 restitution elastic 0.018 elasto-plastic restitution 0.006 0.016

(J) (J) 0.014 1 2 T T 0.004 0.012

0.010 0.002 0.008

0 0.006 0.0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 5.0 t (s) 10 -4 t (s) 10 -4

(c) 0.024 elastic 0.022 elasto-plastic restitution

0.020

0.018 T (J) T

0.016

0.014

0.012 0.0 1.0 2.0 3.0 4.0 5.0 t (s) 10 -4

Figure 8. Kinetic energy during impact: (a) non-impacting link 1, (b) impacting link 2, and (c) total kinetic energy.

0.08 ω = 0.5 rad/s 20 − 0.07

0.06

ω = 0.4 rad/s 20 − 0.05 T (J) T 0.04 ω = 0.3 rad/s 20 −

0.03

ω = 0.2 rad/s 20 − 0.02

0.01 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 t (s) 10 -4

Figure 9. Total kinetic energy during collision for different initial angular velocities, ω20, of the impacting link.

Figure 11 depicts the total kinetic energy of the double pendulum, during collision, for different initial impact angles of the impacting link: q = 50 , q = 55 , q = 60 , and q = 65 . 20 − ◦ 20 − ◦ 20 − ◦ 20 − ◦ Figure 12 shows the permanent deformation as a function of initial angle of the impacting link: δ = 5.30 (10 6) m and ∆T = 0.005 J for q = 50 m, δ = 5.33 (10 6) m and ∆T = 0.005 J for r − 20 − ◦ r − q = 55 m, δ = 5.49 (10 6) m and ∆T = 0.005 J for q = 60 m, and δ = 5.82 (10 6) m and 20 − ◦ r − 20 − ◦ r − ∆T = 0.005 J for q = 65 m. The permanent deformation is not influenced by the initial angle of the 20 − ◦ Mathematics 2020, 8, 1202 11 of 13 impacting link, and the energy loss is not influenced by the initial impact angles of the impacting link, ∆T 0.005 J. The duration of the impact is increasing with the magnitude of the initial impact angle. ≈ Another reason of the increase of the kinetic energy loss with the magnitude of the impact angular velocity as seen in Figure9 is that the kinetic energy is dependent on the square of the angular velocity, 2 ω2. The kinetic energy is also dependent on the square of the velocity of the center of mass of link 2, v2 . The impact angle, q , has a small influence on the square of the velocity of the center of mass of C2 2 link 2 and that is why the kinetic energy and the kinetic energy loss are less dependent on the impact angle as depicted in Figure 11.

(a) (b) 10 -5 0.030 1.6

0.025 1.4

0.020 1.2

(m) 1.0 T (J) T

0.015 r

δ

0.8 0.010

0.6 0.005

0.4 0.549 0.864 1.200 1.550 0.1 0.2 0.3 0.4 0.5 0.6 δr (m) 10 -5 | ω20 | (rad/s)

Figure 10. (a) Total kinetic energy loss, ∆T, as a function of permanent deformation, δr;(b) permanent deformation, δ for different magnitudes of initial angular velocities, ω |, of the impacting link. r | 20

0.024

0.022

q20 = 65 ◦ 0.02 − q = 60 20 − ◦ q = 55 20 − ◦ q20 = 50 ◦ 0.018 − T (J) T

0.016

0.014

0.012 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t (s) 10 -4

Figure 11. Total kinetic energy during collision for different initial angles, q20, of the impacting link. Mathematics 2020, 8, 1202 12 of 13

10-5 1.6

1.4

1.2

(m) 1.0 r

0.8

0.6

0.4 -65 -60 -55 -50 q (deg) 20

Figure 12. Permanent deformation, δr for different initial angles, q20, of the impacting link.

6. Conclusions This article presents a method to solve the impact of a kinematic chain in terms of a nonlinear contact force. The non-linear differential equations for the impact are obtained using Lagrange equations. The contact force is calculated for elastic compression, elasto-plastic compression, and elastic restitution. The final impact time is obtained from the permanent deformation of the material. The angular velocities of the double pendulum are increasing during the impact. The tangential linear velocity of the contact point is changing the sign during restitution phase. The kinetic energy of the non-impacting link is increasing during the impact. The total kinetic energy of the pendulum is decreasing during the impact period. To validate the proposed model, experimental investigation of the impact of the double pendulum will be developed. A high-speed camera can be used to capture the motion of the double pendulum before, during and after impact by placing markers on the links. The permanent deformation after impact can be scanned and measured with an optical profilometer. The presented approach can be applied to complex dynamical, systems such as robots, walking machines, and animal locomotion.

Author Contributions: All authors contributed equally in this research paper. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Whittaker, E.T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies; Dover Publications: Mineola, NY, USA, 1944. 2. Routh, E.J. Dynamics of System of Rigid Bodies; Macmillan: London, UK, 1905. 3. Kane, T.R.; Levinson, D.A. Dynamics: Theory and Applications; McGraw-Hil: New York, NY, USA, 1985. 4. Keller, J.B. Impact with friction. ASME J. Appl. Mech. 1986, 53, 1–4. [CrossRef] 5. Stronge, W.J. Rigid body collisions with friction. Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 1990, 431, 169–181. 6. Stronge, W.J. Planar impact of rough compliant bodies. Int. J. Impact Eng. 1994, 15, 435–450. [CrossRef] Mathematics 2020, 8, 1202 13 of 13

7. Blazejczyk-Okolewska, B.; Kapitaniak, T. Dynamics of impact oscillator with dry friction. Chaos Solitons 1996, 7, 1455–1459. [CrossRef] 8. Ma, D.; Lankarani, H.M. A nonlinear finite-element approach for kineto-static analysis of multibody systems. Nonlinear Dyn. 1995, 8, 237–250. [CrossRef] 9. Liu, C.; Zhao, Z.; Brogliato, B. Frictionless multiple impacts in multibody systems. ii. numerical algorithm and simulation results. Proc. R. Soc. A Math. Phys. Eng. Sci. 2009, 465, 1–23. [CrossRef] 10. Khulief, Y.; Shabana, A. Impact responses of multi-body systems with consistent and lumped masses. J. Sound Vib. 1986, 104, 187–207. [CrossRef] 11. Shinbrot, T.; Grebogi, C.; Wisdom, J.; Yorke, J.A. Chaos in a double pendulum. Am. J. Phys. 1992, 60, 491–499. [CrossRef] 12. Lenci, S.; Rega, G. Numerical control of impact dynamics of inverted pendulum through optimal feedback strategies. J. Sound Vib. 2000, 236, 505–527. [CrossRef] 13. Litak, G.; Borowiec, M.; Wiercigroch, M. Phase locking and rotational motion of a parametric pendulum in noisy and chaotic conditions. Dyn. Syst. 2008, 23, 259–265. [CrossRef] 14. Vaziri, V.; Najdecka, A.; Wiercigroch, M. Experimental control for initiating and maintaining rotation of parametric pendulum. Eur. Phys. J. Spec. Top. 2014, 223, 795–812. [CrossRef] 15. Huang, J.; Liang, Z.; Zang, Q. Dynamics and swing control of double-pendulum bridge cranes with distributed-mass beams. Mech. Syst. Signal Process. 2015, 54, 357–366. [CrossRef] 16. Izadgoshasb, I.; Lim, Y.Y.; Tang, L.; Padilla, R.V.; Tang, Z.S.; Sedighi, M. Improving efficiency of piezoelectric based energy harvesting from human motions using double pendulum system. Energy Convers. Manag. 2019, 184, 559–570. [CrossRef] 17. Banerjee, R.; Dey, N.; Mondal, U.; Hazra, B. Stabilization of Double Link Inverted Pendulum Using LQR. In Proceedings of the 2018 International Conference on Current Trends towards Converging Technologies (ICCTCT), Coimbatore, India, 1–3 March 2018; pp. 1–6. 18. Udwadia, F.E.; Koganti, P.B. Dynamics and control of a multi-body planar pendulum. Nonlinear Dyn. 2015, 81, 845–866. [CrossRef] 19. Jackson, R.L.; Green, I. A finite element study of elasto-plastic hemispherical contact against a rigid flat. J. Trib. 2005, 127, 343–354. [CrossRef] 20. Gheadnia, H.; Cermik, O.; Marghitu, D.B. Experimental and theoretical analysis of the elasto-plastic oblique impact of a rod with a flat. Int. J. Impact Eng. 2015, 86, 307–317. [CrossRef] 21. Stewart, D.E. Rigid-body dynamics with friction and impact. SIAM Rev. 2000, 42, 3–39. [CrossRef] 22. Brogliato, B. Nonsmooth Mechanics: Models, Dynamic and Control; Springer Science & Business Media: Cham, Switzerland, 2016. 23. Stronge, W. Generalized impulse and momentum applied to multibody impact with friction. Mech. Struct. Mach. 2001, 29, 239–260. [CrossRef]

c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).