Influence of Contact Modeling on the Dynamics of ...

ECCM 2010 IV European Conference on Computational Me chanics Palais des Congrès, Paris, France, May 16-21, 2010

Influence of Contact Modeling on the Dynamics of Chain Drives 1

2

3

C. Pereira , J. Ambrósio , A. Ramalho 1

Department of Mechanical Engineering, Polytechnic Institute of Coimbra, Portugal, [email protected] Department of Mechanical Engineering, Technical University of Lisbon Portugal, [email protected] 3 Department of Mechanical Engineering, University of Coimbra, Portugal, [email protected] 2

Abstract In this work a multibody model that describes a chain as a constrained dynamic system composed of a large number of rigid bodies, links and rollers, connected to each other by revolute clearance joints, is used. The clearance revolute joint contact is further extended to handle the contact between the chain rollers and the sprocket teeth exact profiles. Penalty contact force models are considered for modeling the revolute clearance joints, and in particular those proposed by Lankarani and Nikravesh and by a new enhanced cylindrical contact model suggested by the authors. Although numerically efficient the Lankarani and Nikravesh contact model is intended for application to spherical contacting geometries under conditions of non-conformal contact. To evaluate the relationship between the indentation and the applied contact force of cylindrical contacting geometries several analytical contact force models are available in the literature. However, most of these models are iterative because the contact force is defined as an implicit function of the indentation, thus requiring the use of less efficient numerical procedures, in particular if a large number of contacting bodies are involved. To overcome this drawback, a new enhanced cylindrical contact force model was proposed as a useful alternative for implementation in a computational code for the dynamic analysis of multibody systems that experience contact/impacts between cylindrical bodies. The performance of the Lankarani and Nikravesh and the new enhanced contact models in the framework of dynamics of roller chain drives mechanisms is here presented.

1. Introduction The dynamic effects due to the presence of clearances in chain drives mechanisms have been continuously neglected in the models available in the literature that focus on the dynamics of these kind of mechanisms [1-18]. In this work a multibody formulation, involving the contact between different bodies, is used to describe the interrelation between the different elements in the chain drive system [19, 20]. The chain drive mechanism is described as planar multibody mechanical systems with multiple revolute clearance joints [20]. For the chain itself the connection between each pair of links, which is made up of the pin link/bushing link plus the bushing link/roller pairs if the chain is a roller chain instead of a bushing chain, is modeled as a revolute joint with clearance. One of the novel developments on the contact between the sprocket tooth and the roller is the systematic use of clearance revolute joint formulation to address the contact with each part of the sprocket tooth profile. In this sense, the formulation used can be understood as an extension of the formulation proposed for a generic revolute clearance joint in multibody mechanical systems for different joint components geometry [20-25]. The mathematical models that represent the contact conditions between the chain and the sprocket teeth and the contact between the pins, bushings and rollers have been recently presented [26].

1

As the contact phenomenon is described using the continuous contact force method, the dynamics of a revolute clearance joint is controlled by the contact forces developed on each contacting rigid body and included into the equations of motion during the contact-impact period, in which the contact forces on each contact pair is known at any time during the dynamic simulation [20-25]. Therefore, a suitable cylindrical continuous contact force model able to describe accurately the internal conformal contact as well as the external between cylindrical geometries is of major importance. Furthermore, the contact force model must be computationally efficient due to the great number of bodies that are involved as well as numerically stable. This important characteristic is found in the model proposed by Lanakarani and Nikravesh [27, 28], where the Hertz spherical contact law was modified by adding a term to include the energy dissipation that occurs during the impact process, reason because this model has been widely used by many researchers for modeling contact forces in multibody systems for spherical and even for cylindrical contacting geometries [16-18, 20-25, 27-32]. Unfortunately in most of the cylindrical contact models available in literature [33-37] a numerical iterative technique is required to obtain the contact force since the contact force is defined as implicit function of the indentation, or penetration, reducing substantially the efficiency of a computational code for the dynamic analysis of multibody systems [28, 38, 39]. Furthermore, these models are have all been proposed as purely elastic models and are unable to explain the energy dissipation during the impact process [40]. For these reasons and assuming a dry frictionless contact, a new enhanced alternative cylindrical contact force model was developed by the authors putting together the accuracy of the contact force evaluation with the numerical efficiency required. In addition, since in this new approach the pseudo-stiffness is defined as not dependent on the contact force, the term that accounts for energy dissipation during the impact process, in the form suggested by Lankarani and Nikravesh, is added [41]. The objective of this work is thus to establish, in the framework of dynamics of chain drives mechanisms, a comparative study between the dynamic response using both the spherical contact model proposed by Lanakarani and Nikravesh and the new enhanced cylindrical model developed by the authors and, in the process, to evaluate the numerical efficiency of using different contact modeling approaches.

2. Multibody Dynamics in Chain Drive Mechanisms A chain drive, represented in Figure 1(a), is composed of a collection of components that include the sprockets, chain and eventually a tensioner and delimiters of the chain vibrations. The chain, itself, is composed of inner and outer links, articulated through a pin/bushing hinge, and, eventually, rollers hinged on the inner links bushings if the chain is a roller chain instead of a bushing chain. The chain drive is, therefore, a multibody system in which the hinges can be viewed as perfect kinematic joints or contact pairs and the sprockets, links and rollers can be represented as rigid or as flexible bodies. In this work all components of the chain are represented as rigid bodies and the hinges as contact pairs, as for instance the pin/bushing clearance revolute joint shown in Figure 1(b). Let the position and orientation of a bodies i of a multibody system be represented by qi =[rTθ]ιΤ and the position and orientation of all bodies by q=[q1 T ,q2 T … qTnb ]Τ. The equations of motion of a multibody system are [19] M  Φ q

Φ Tq  && q g   =   , 0  λ  γ

(1)

where M is the system mass matrix, Φ q the Jacobian matrix associated to the kinematic constraints, q&& the system accelerations, λ a vector of Lagrange multipliers associated to the kinematic constraints, g the vector with the velocity dependent terms of the kinematic acceleration constraints and g the vector with the forces applied to the rigid body of the system. All quantities in Equation (1) are built with the contributions of the individual rigid bodies and kinematic joints. The force vector, in 2

particular, is g=[g 1 T ,g2 T … g Tnb ]Τ being g i =[fT n]ιΤ made of the force and moment over body i resultant form the contact forces derived from the interactions with other bodies and any other external force. The contact force between two bodies i and j of the multibody system is the result of a normal force f n , perpendicular to the contacting surfaces, and of a tangential force f t , associated to friction phenomena, i.e., f i = − fn n + ft t

and

f j = −fi ,

(2)

In the description of the body forces, provided by Equation (2) the contact mechanics problem is defined as the evaluation of the normal and tangential contact forces and the identification of their point of application in each body of the contact pair. ηi ηsr

ξs r Rs

α

θs r sc r

ηj

ηi r siP

ξi

r sjP ξj

(i)

(j)

r ri

r rjP

r rc r

Pin

Pi

Pj

ξi

θi

·

θi Pitch

r e

Y

r riP

r rj

Inner Link

r ri

Y

Outer Link X X

(a)

(b)

Figure 1: Multibody representation of a chain drive: (a) Individual components; (b) Pin/Bushing revolute clearance joint.

2.1 Contact Clearance Cylindrical Joints Take the clearance cylindrical pair shown in Figure 1(b) representing the pin/bushing hinge connection. Let body i refer to the outer link that includes the busing and body j be the inner link that includes the pin. When the centers of the bushing and pin, given by points Pi and Pj respectively, separate, there is the possibility for contact to take place if their distance exceeds the existing clearance of the joint, denoted by c. With reference to Figure 1(b), let the eccentricity vector be defined as e = r Pj − r Pj . A penetration between bodies i and j exist if [20-25]

δ = eT e − c > 0 ,

(3)

When Equation (3) is fulfilled, a normal contact force between the rigid bodies included in the contact pair must be evaluated, using one of the contact models described in section 2.2 of this work, and applied and added to the bodies force vector. The contact of the bushing (or roller) on the sprocket tooth can take place in any of the regions of the tooth represented in Figure 2(a). Each one of the regions considered has a prescribed radius and center of curvature to ensure a smooth engagement of the roller on the tooth. Let body r refer to the roller (or bushing) and body i be the sprocket tooth. Depending on the position of the center of the roller with respect to each one of the centers of curvature of the tooth there is the possibility of contact between the tooth and roller. Figure 2(b) 3

represents the external contact with the topping and Figures 2(c) and 2(d) the internal contact with the topping, seating and working curves of the tooth profile, respectively. The establishment of the contact conditions between the roller and each one possible tooth profile contact regions can be found in Ref. [26]. When contact with any of the regions of the tooth profile is detected by using Equation (3) but involving the quantities shown in Figure 2, the contact models described in the next section by Equations (4) and (6) are here used to evaluate the roller/sprocket contact force. ηsr

θb d* 1

bc

*

cc *

cc

a*

c*

d a

r e

bc

c

Rb

7

6

b

b*

2

Rr

7

a

oc

θe

d

r scr

r ub

bc

5

3

r sbc 4

ξsr

(a) cc

(b) cc*

η sr

ηsr cc

oc

c*

r uo

r uc b

c

r e

r e

Rt

4

r s cc

Rr

r s oc

c

Rc Rr θe 5

r scr

r scr

θc

θe θo ξsr

ξsr

(c)

(d)

Figure 2: (a) Regions of the sprocket tooth with different radius; (b) Contact with the topping curve, region 1; (c) Contact with the seating curve, region 4; (d) Contact with the working curve, region 3.

2.2 Contact Force Models for Cylindrical Joints The simplicity of the contact model suggested by Lankarani and Nikravesh [27, 28] for implementation in a computational program for the dynamic analysis of mechanisms stems from the facts that: i) is the only that accounts for the energy dissipation during the impact process and ii) the normal contact force, fn , can be expressed as an explicit function of the indentation, δ , as 4E *  Ri Ri  fn =   3  ∆R 

1

2



δ n 1 + 

3(1− ce2 ) δ&  (−)  4 δ& 

(4)

where, E * = E 2 (1 − ν 2 ) is the composite modulus, assuming materials with similar elastic modulus and Poisson coefficients denoted by E and υ, respectively, ce is the restitution coefficient,

4

∆R, Equation (4) can be applied to internal and external contacts. When ∆R represents the sum of the bodies’ radii, (Ri +Rj ), an external contact geometry is considered. Otherwise, for internal contact, ∆R is quantified by the difference between bodies’ radii, (Ri -Rj ), corresponding to the radial clearance between the two bodies. Since the Lankarani and Nikravesh contact model results from the addition of the energy dissipation that takes place on the contact process to the contact law derived by Hertz, for spherical contacting bodies, the value for the exponent n is a constant always equal to 1.5. In addition, the application of this model is limited to spherical bodies’ geometries because the influence of the contact axial length on the contact force cannot be accounted for. ( −) δ& the relative impact velocity and δ& the actual penetration velocity. Depending on parameter

Based on the Hertz pressure distribution, several analytical contact force models to evaluate the relationship between the indentation and the applied contact force of cylindrical contacting geometries are available in the literature [33-37]. Among of these models is founded the well known Johnson model in which the total indentation of two deformable contacting cylinders of radius Ri and Rj and contact length L, made with materials with similar elastic modulus and Poisson coefficients and submitted to the action of a compressive normal load, f n , is given by [33] fn = π L E *

1   4π L E * ∆R    ln   - 1 fn    

δ

(5)

where the indentation, δ , which accounts for the contribution of both cylinders, is assumed to be measured at a point distant enough from the contact point. Parameter ∆R is quantified as described for Equation (4). Although very accurate to describe the contact between cylindrical geometries, from Equation (5) it can be concluded that three major shortcomings are associated with this model: i) is iterative because the contact force is defined as an implicit function of the indentation, thus requiring the use of less efficient numerical procedures; ii) is proposed as purely elastic models and therefore unable to explain the energy dissipation during the impact process; iii) include a logarithmic function, which impose mathematical and physical limitations on its application, particularly for conformal contact conditions with lower clearance values, which means that this has a validity domain, which depends on the clearance value and material properties [40]. These same drawbacks are associated to the most of current cylindrical contact models [34-37]. Therefore an analytical cylindrical contact force model free of mathematical and physical limitations, i.e., without domain validity problems, and defining the contact force as an explicit function of penetration avoiding the application of a numerical iterative techniques, is an useful alternative for implementation in a computational code for the dynamic analysis of multibody systems that experience contact/impacts between cylindrical bodies such as the case of chain drive mechanisms. With this purpose, a new enhanced cylindrical contact force model that has the simplicity of the Hertz spherical model and the accuracy of the Johnson cylindrical model has been recently proposed by the authors. In this model the normal contact force is evaluated as [41] fn =

( a ∆ R + b) L E * ∆R



δ n 1 + 

3(1 − ce2 ) δ&  , ( −)  4 δ& 

(6)

where for external cylindrical contact a=0.39, b=0.85, n=1.094 and ∆R=Ri +Rj and for internal cylindrical contact a=0.49, b=0.10, n=Y∆R-0.005 and ∆R=Ri -Rj , being Y=1.56[ln(1000 ∆R)]-0.192 if ∆R=[0.005, 0.750[ or Y=0.0028∆R+1.083 if ∆R=[0.750, 10.0[ mm. The remaining quantities in Equation (6), have the same meaning described for Equations (4) and (5). As in this model the pseudostiffness is defined as a constant, it is possible to add the term that accounts for energy dissipation in the form suggested by Lankarani and Nikravesh. 5

A comparative assessment between the dynamic response using the spherical contact model proposed by Lanakarani and Nikravesh and the new enhanced cylindrical model models in the framework of dynamics of chain drives mechanisms is established and presented in section 4 of this work. The direction of the application of the contact force f n , given by Equations (4) and (6), is the normal to the contact surfaces in the contact point given as n = e / eT e . In the applications foreseen in this work the contact is assumed frictionless.

3. Mulibody Model of a Chain Drive The chain drive model presented in Figure 3, which corresponds to the initial positions of chain drive components, is used here to demonstrate the procedures proposed in this work. This drive uses an ASA chain nº40 that wraps around two 17 teeth sprockets. The initial distance between the sprockets is such that 38 links are required for the closure of the chain while maintaining the strands straight and the bushings centers of the chain, seated on the sprockets, coincident with the primitive circle of the sprockets. Tables 1 and 2 present the data used for the sprockets and for the bushing chain. Bushing 6

Bushing 1

Bushing 11

Bushing 30

Bushing 21

Bushing 25

Figure 3: Bicycle bushing chain drive with a nº40 ASA chain.

Units

Sprocket 1

Sprocket 2

Number of teeth

-

17

17

Pitch radius

m

0.0127

0.0127

Length

m

0.0070

0.0070

Position X

m

0.0000

0.1332

Position Y

m

0.0000

0.0000

Mass

kg

0.8170

0.8170

Moment of inertia

Kg m

0.1×10

588.29×10-6

Young modulus

Kg m-2

2.07×1011

2.07×1011

Poisson coefficient

-

0.3

0.3

Coefficient of restitution

-

0.71

0.71

2

9

Table 1: Characteristics of the sprockets.

6

Units

Pin

Bushing

Internal link

External link

Radius

m

0.00444

0.00449(*)

-

-

Length

m

0.011

0.011

-

-

Mass

kg

0.001974

0.006474

0.008265

-

Moment of inertia

Kg m2

0.4860×10

0.3220×10

8

8

0.1795×10-6

0.1174×10-6

Young modulus

Kg m-2

2.07×1011

2.07×1011

2.07×1011

2.07×1011

Poisson coefficient Coefficient of restitution Number of links

-

0.3

0.3

0.3

0.3

-

0.71

0.71

0.71

0.71

-

38

Chain pitch

m

0.0127

m/s

0.4343

N

0

Chain velocity Pré-tension (*)

0.001305 -

Internal radius of the bushing. The external radius of the bushing is 0.0079375 m. Table 2: Characteristics of the bushing chain.

The initial velocity of the chain links and sprockets is defined such a way that the driving sprocket rotates with a constant angular velocity of 120 r.p.m. A very large inertia moment is considered in the driving sprocket to ensure its rotation with a constant angular velocity throughout the analyses carried out in this work. The initial conditions of the chain drive correspond to the inexistence of pretension. When required, a tensioner, not shown in Figure 3, is used to apply a prescribed pretension to the chain. The bushing chain considered in this application has a clearance of 5.0×10-5 m on the pin/bushing joint and a restitution coefficient ce = 0.71 is assumed for the contact in any contact pair. The chain drive is subjected to gravitational forces acting downwards. The dynamic analysis is carried using the integrator DE/STEP [42] for which the numerical tolerances are set as 10-5 and the maximum time step allowed is 10-4 .

4. Dynamics of a Chain Drive To start a dynamic simulation and to ensure the accuracy prediction outcomes of the dynamical behavior of any multibody systems, a proper set of initial conditions on the positions and velocities is required. The methodologies adopted to achieve the suitable initial conditions of chain drive components ensuring not only that the kinematic constraints of the model are fulfilled, in terms of positions and velocities, but also that the contact in any of the contact pairs starts properly, i.e. without effective contact between roller and sprocket or between pin and bushing or even between bushing and roller can be found in Ref. [43]. In this reference a comprehensive procedure to build multibody models of chain drives with minimal information given by the user as well as the strategy used to specify the contact pairs and their update during the dynamic analysis, which optimizes the computational efficiency on the contact search, are presented. The dynamic analysis of the bushing chain is perform considering that the chain drive is only subjected to gravitational forces with no pretension applied to the chain. When the contact on the bushing/sprocket contact pairs is handled with the Lankarani and Nikravesh contact model, given by 7

Equation (4), the contact forces that develop between any particular tooth of any of the sprockets and the bushings that seat on it during the analysis have the same time behavior regardless of the contacting tooth. The contact forces on bushings due to the sprockets are particularly illustrated, for two selected bushings, by Figure 5. It is observed that a first impact, with higher force, takes place before the bushing finally starts seating on the sprocket tooth as seen in Figure 5. This force peak is higher or lower depending on the velocity of the bushing when actual contact takes place. As the chain strands change length, when a bushing enters or leaves it, and the polygonal effect excites the transversal vibration of the chain, a particular bushing may seat on a sprocket with a high normal impact velocity, thus leading to higher peaks for initial contact. 50

50

Sprocket 1/Tooth 9 40

Sprocket 1/Tooth 4

Sprocket 2/Tooth 7 Contac t F orc e [N]

Co ntact F orce [N]

40

30

20

10

30

S procket 2/Tooth 2 20

10

0

0

0

0 .5

1

1 .5

2

2.5

0

0.5

1

1.5

Time [ s]

2

2.5

Tim e [s ]

a)

b)

Figure 5: Time history of the contact forces on the sprockets due to the interaction of a) Bushing #6 and b) Bushing #11.

50

50

40

40

C ontact F orce [N]

C ontac t F orc e [N]

The contact force that develops in typical contact pairs bushing/sprocket are presented in Figure 6 to illustrate not only their indentation length but also their energy dissipating characteristics. In Figure 6 the markers on the force-penetration curves identify the value of the pair reported with a time step of 10-4 s. Each contact spans several time steps indicating in this form the continuous nature of the contact model. It is also confirmed that although the initial contact leads to contact forces of about 40 N, the peaks of the forces during the contact are in the range of 10-20 N for the scenario considered in this application.

30

20

10

0 0. 00

30

20

10

0 .50

1. 00

1.50

2 .00

0 0.00

2.50

-6

0. 50

1 .00

1. 50

2. 00

2.5 0

-6

Penet ration [10 m]

P enetr ation [10 m ]

a)

b)

Figure 6: Contact forces vs. penetration on the contact with the sprockets during the dynamic analysis for: a) Bushing #6 and b) Bushing #11.

8

120

12 0

100

10 0

80

80 Co ntac F or c e [N ]

Cont ac t F or c e [N ]

The time history of the internal contact forces on the clearance joints between pin and bushing are depicted in Figure 7, while the relation between these forces and the indentation pin/bushing is shown in Figure 8. A first observation that Figure 8 suggests is that although each period of contact is rather short, the separation between the bodies takes some time to happen. In any case, a complete contact is made by a sequence of short contacts and separations, as seen in Figure 7, rather than by a longer smoother contact.

60

40

60

40

20

20

0

0 0

0.5

1

1.5

2

0

0. 5

1

Tim e [s ]

1 .5

2

Tim e [s ]

a)

b)

80

80

60

60

Co ntac t F orc e [N]

Co ntac t F orc e [N]

Figure 7: Contact forces vs. time on clearance joints for the contact pairs: a) Bushing #6 and Pin#6 and b) Bushing #11 and Pin#11.

40

20

40

20

0

0 0

0.1

0.2

0.3

0. 4

0.5

0. 6

0. 7

0

-6

0.1

0. 2

0. 3

0 .4

0. 5

0.6

0.7

-6

Pene trat io n [10 m ]

Pe netr ation [ 10 m]

a)

b)

Figure 8: Contact forces vs. penetration on clearance joints between: a) Bushing #6 and Pin#6 and b) Bushing #11 and Pin#11, during the time period between 1.5s and 1.6s. The trajectories of the pin on the inner surface of the bushing, depicted by Figure 9, are mostly in the neighborhood of the circle of contact, which suggests that if either friction forces are considered or if some pretension is used on the chain, longer periods of contact would be observed. Moreover, the typical force-penetration curves associated to the contact law exhibit, in some of the contacts, a rather irregular loop. This is due to the fact that the bodies in contact are not only acted by the contact forces that develop in the particular pin/bushing pair, but also acted by the forces on the joint with the other link.

9

12

8

8

4

4 EY [10 -6 m]

EY [ 10 -6 m]

12

0 -12

-8

-4

0

4

8

12

0 -12

-8

-4

0

-4

-4

-8

-8

-12

-12

EX [10 -6 m]

EX [10 -6m]

a)

b)

4

8

12

Figure 9: Trajectory of center of the: a) Pin #6 and b) Pin#11with respect to the Bushing #6 and Bushing #11, respectively. The dynamic performance of the bushing chain is now studied with the new enhanced cylindrical contact force model for the bushing/sprocket and the pin/bushing contact. As before, no pretension is applied to the chain being only subject to gravitational forces. Figure 10 shows that, for the same penetration value, the contact forces between bushings and sprockets predicted by the new enhanced contact model are larger than those obtained using the Lankarani and Nikravesh spherical contact model. This means that the cylindrical contact is stiffer than the spherical. This behavior is due to the fact the new enhanced contact model presents a stiffer force-indentation relation, mostly due to the length of the bushing that is considered for this model and that is not included in the Lankarani and Nikravesh contact model. Moreover, and based on Equations (5) and (6) the contact force increases with the axial length of the contacting cylinders, which cannot be predicted by the spherical model, since it does not account for the influence of the length dimension on the contact force. The Lankarani and Nikravesh contact model represents therefore a rough approximation to evaluating the contact between cylindrical bodies, since the contact force values are underestimated. The contact bushing/sprocket is characterized by a sequence of impacts with a short duration as depicted by Figure 10. As demonstrated by the small dispersion of the force-penetration slopes the contacts between bushings and sprockets occur in the same areas of the tooth profile.

50

50 Lan k arani & Nik r aves h m odel

Lank ara ni & N ik raves h mod el

Ne w E nhanc ed mode l

New Enhanc e d m odel 40

Conta c t F or c e [N ]

Conta c t F or c e [N ]

40

30

20

10

30

20

10

0 0. 00

0. 50

1.00

1. 50

2.00

0 0. 00

2.50

-6

0. 50

1.00

1. 50

2.00

2.50

-6

P enetr ation [10 m]

P enetr ation [10 m]

a)

b)

Figure 10: Contact forces vs. penetration with the Lankarani and Nikravesh and new enhanced contact models for the bushing/sprocket contact: a) Pin/bushing #6; b) Pin/bushing #11.

10

The contacts between the pins and the bushings are also characterized by a higher stiffness of the new enhanced contact model than the Lankarani and Nikravesh contact model, as shown in Figure 11. The trajectories of the pins inside the bushings, depicted in Figure 12, show that the contact of these clearance joints is characterized by many free-flight modes, confirming the higher stiffness of the contact of the clearance joints.

120 120

L ank ara ni & N ik raves h model

La nk aran i & N ik r aves h model

N ew Enhanc ed mo del

100

N ew E nhanc ed mod el

100

80 C ontac t F orc e [N ]

C ontac t F orc e [N ]

80

60

40

60

40

20 20 0 0

0 .05

0 .1

0.15

0.2

0 .25

0 .3

0.3 5

0 0

-6

P enetr ation [ 10 m]

0.05

0. 1

0. 15

0. 2

0. 25

0.3

0.35

-6

Penet ration [10 m]

a)

b)

Figure 11: Contact forces vs. penetration with the Lankarani and Nikravesh and new enhanced contact models for clearance joints: a) Pin/bushing #6; b) Pin/bushing #11, during the time period between 0.4s and 0.5s.

12

8

8

4

4

-6

0 -1 2

-8

-4

0

4

8

E Y [10

E Y [1 0

-6

m]

m]

12

12

0 -1 2

-8

-4

0

-4

-4

-8

-8

-1 2

-1 2

E X [10 - 6 m]

E X [1 0- 6 m ]

a)

4

8

12

b)

Figure 12: Trajectory of the pin inside the bushing for the clearance joints with new enhanced contact models: a) Pin/bushing #6; b) Pin/bushing #11. Although not shown here, the higher contact stiffness exhibited by the new enhanced contact model is also indirectly observed by the behavior of the numerical integrator used. In fact, the computer time required to simulate 2 seconds of the chain drive dynamics with the new enhance contact model is larger than what is required when the Lankarani and Nikravesh model is used. This is due to the smaller time steps required to integrate the equations of motion of the chain drive model that uses the new enhanced contact model, because it induces higher frequencies on the dynamic response of the chain drive. The effect of the chain pretension on the dynamic response of the drive using the new enhanced 11

contact model has been therefore investigated. Instead of using a pretension for the chain, at the start of the analysis, a mechanism applying it gradually is devised [44]. The consequence of the pretension increasing on the chain is that the trajectories of the pins inside the bushings are mostly in the contact mode, as shown in Figure 13 for the clearance joint made up by Pin/bushing #6 for the different levels of pretension tested. 12

8

EY [ 1 0

-6

m]

4

0 -1 2

-8

-4

0

4

8

12

-4

-8

-1 2 -6

E X [ 10 m ]

(a)

(b)

(c)

Figure 13: Trajectory of the pin #6 inside the bushing #6 for a pretension of (a) 0 N; (b) 25 N; (c) 100 N. In Figure 13(a) the free-flight of the pin is occurs quite frequently, opposed to what happens in Figures 13(b) and (c) in which only few free-flight trajectories exist. If no pretension is applied the contact is mostly discontinuous, characterized by a large number of impacts and rebounds. With pretension the contact is mostly continuous being the tension force oscillating about the pretension force for the links on the strands and almost null when the bushings seat on the sprockets. Just as for the pin/bushing contact for the contact between bushings and sprockets the increase of the pretension level leads to the decrease of contact losses maintaining the continuous contact on the clearance joints during longer periods. The performance of a new enhanced cylindrical contact force model proposed by the authors is demonstrated here through its application to the running dynamics of a chain drive. It is shown that the used o spherical contact force model proposed by Lankarani and Nikravesh to describe the contact between cylindrical bodies, in which the influence of the length dimension on the contact force cannot be accounted for, leads to a rough approximation since contact forces are underestimated. In addition, it is verified that the novel cylindrical continuous contact force model is precise and efficient in representing the meshing contact phenomena between the chain bushings and the sprocket teeth when pretension is used. If the pretension of the chain is null, i.e., in the case of non-tightened chains, the meshing phenomena is characterized by a succession of impacts and rebounds, instead of continuous contact. However, and for the multibody model of chain drive developed to become more realistic, both the tangential friction forces and the hydrodynamic forces needed to be added to the model. It is expected that the development of these forces on the contact will contribute for the increase of the continuous contact modes decreasing the free-flight trajectories. Furthermore, the implementation of a lubrication model is an attractive issue, useful for the design of roller chain drives taking into account fatigue and wear and to create standards and maintenance procedures.

Conclusions The study of the dynamics of a bushing chain drive, in which the interaction between all components of the drive is modeled using clearance revolute joints, instead of the traditional kinematic constraints, has been outlined in this work. The procedure presented includes the use of a new enhanced cylindrical contact model, which does not present the limitations of the standard continuous

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cylindrical contact force models based on the Hertz elastic contact theory, and the ability to address very large and complex chain drive systems. The results shown, and the computational times not presented, illustrate the dynamic responses that are possible to obtain with the methodology presented. It is observed that the contact model represents correctly the continuous contact phenomena, but it has limitations for cases with many impacts, as for instance when the chain has a null pretension, because when the new enhanced contact model is used, the contact is stiffer than what the Lankarani and Nikravesh model exhibits. In fact, the contact forces developed in the pin/bushing and bushing/sprocket contact pairs are characterized by a sequence of contacts and rebounds with a short duration that do not help the smoothness of the dynamic response of the system. Although it has been demonstrated that the use of some pretension level changes the characteristics of the contact by enforcing a continuous contact with only a small number of contact losses, the contact between revolute clearance joints components has been treated here as a dry contact. The implementation of friction and lubrication models is therefore required not only to get a more realistic model description of the contact phenomena but to deliver a smoother response of the dynamic system. In this form the contact modeling presented in this research will be used with advantage in the framework of multibody models. References [1] R.A Morrison, Poligonal Action in Chain Drives, Machine Design, 24(9), 155-159, 1952. [2] R.C. Binder, Mechanics of the Roller Chain Drive, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1956. [3] G. Bouillon and G.V.Tord ion, On Polygonal Action in Roller Chain Drives, Journal of Engineering for Industry, ASME Transactions, 87, 243-250, 1965. [4] G.K. Ryabov, Inertia Effects of Impact Loading in Chain Drives, Russian Engineering Journal, 48(8), 1719, 1968. [5] S.R. Tu rnbull and J.N. Fawcett, Dynamic Behavior of Ro ller Chain Drives, Mechanisms, 29-35, 1972. [6] M.R. Naji and K.M. Marshek, Analysis of Sprocket Load Distribution, Mechanism and Machine Theory, 18(5), 349-356,1983. [7] M. Chew, Inert ia Effects of a Roller Chain on Impact Intensity, Journal of Mechanisms, Transmissions, and Automation in Design, 107, 123-130, 1985. [8] C.K. Chen and F. Freudenstein, Towards a More Exact Kinematics of Roller Chain Drives, Journal of Mechanisms, Transmissions, and Automation in Design, 110(3), 123-130,1988. [9] N.M. Veikos and F. Freudenstein, On the Dynamic Analysis of Roller Chain Drivers: Part I – Theory, Mechanical Design and Synthesis, 46, 431-439, 1992. [10] M.S. Kim, Dynamic Behavior of Roller Chain Drives at Moderate and High Speeds, Ph.D. Dissertation, University of M ichigan, 1990. [11] K.W. Wang, S.P. Liu, S.I. Hayek, F.H.K. Chen, On the Impact Intensity of Vibrating A xially Moving Roller Chains, Machinery Dynamics and Element Vibrations, 36, 97-104, 1991. [12] K.W. Wang and S.P. Liu, On the Noise and Vib ration of Chain Drive Systems, The Shock and Vibration Digest, 23(4), 8-13, 1991. [13] M.S. Kim and G.E. Johnson, Mechanics of Ro ller Chain-Sprocket Contact: A General Modeling Strategy, Proceedings of the 1992 International Power Transmission and Gearing Conference, 43(2), 689-695, 1992. [14] M.S. Kim and G.E. Johnson, A General Mult ibody Dynamic Model to Predict the Behavior of Roller Chain Drives at Moderate and High Speeds, Advances in Design Automation, 1, 257-268, 1993. [15] W. Choi and G.E. Johnson, Transverse Vibrations of a Ro ller Chain Drive with a Tensioner, Vibrations of Mechanical Systems and the History of Mechanical Design, 63, 19-28, 1993. [16] S.L. Pedersen, Chain Vibrations, M.S. Dissertation, Department of Mechanical Engineering, Solid Mechanics, Technical University of Den mark, Lyngby, Den mark, 2001. [17] S.L Pedersen, Simulation and Analysis of Roller Chain Drive Systems, Ph.D. Dissertation, Department of Mechanical Engineering, Solid Mechanics, Technical University of Den mark, Lyngby, Denmark, 2004. [18] S.L. Pedersen, J.M. Hansen, J.A.C. A mb rósio, A Roller Chain Drive Model Including Contact with Gu ideBars, Multibody System Dynamics, 12, 285-301, 2004. [19] P. Nikravesh, Computer-Aided Analysis of Mechanical Systems, Ed. Prentice-Hall, Englewood-Cliffs, New Jersey, 1988. [20] P. Flores, J. A mbrósio, J. Pimenta Claro, H. Lankaran i, Kinematics and Dynamics of Multibody Systems with Imperfect Joints: Models and Case Studies, Springer, Dordrecht, The Netherlands, 2008. [21] P. Flores and J. Ambrósio, Revolute Joints with Clearance in Mult ibody Systems, Computers & Structures, 82, 1359-1369, 2004.

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