DINAME 2013 - Proceedings of the XV International Symposium on Dynamic Problems of Mechanics M.A. Savi (Editor), ABCM, Buzios, RJ, Brazil, February 17-22, 2013
Modelling an Inverted Slider Crank Mechanism considering Kinematic Analysis and Multibody Aspects
Rita de C. Silva 1, Maria Alzira A. Nunes 1, João Paulo M. Bento 1 and Vinicius E. da Costa 2
1 Universidade de Brasília, Faculdade UnB Gama, Engenharia Automotiva, Área Especial de indústria Projeção A UnB, Setor Leste, 72.444-240, Gama-DF, Brasil. [email protected]; [email protected]; [email protected]. 2 Universidade de Brasília, Faculdade UnB Gama, Programa de Pós-graduação em Integridade de Materiais da Engenharia, Área Especial de indústria Projeção A UnB, Setor Leste, 72.444-240, Gama-DF, Brasil. [email protected].
Abstract: In industry, many applications of planar mechanisms such as slider-crank mechanisms (SCM) have been found in thousands of devices. A slider–crank mechanism is widely used in gasoline/diesel engines and quick-return machinery. Research works in analysis of the slider–crank mechanism have been investigated to date due to their significant advantages such as low cost, reduced number of parts, reduced weight and others. A variation of this four- bar linkage is the inverted slide-crank mechanism (ISCM) which is obtained changing the grounded link of the SCM attaching the coupler link to the ground. It kinematic analysis is little studied when compared with the original mechanism (SCM) but the ISCM may represent an important application in the automotive area, like modelling the Macpherson structure suspension. In this way this paper presents the kinematic analysis of an ISCM based in a vector and matrix formulation for future adaptation to modeling the Macpherson struct. The position, velocity and acceleration of the links are obtained as the maximum and minimum length of the coupler during it operation. In addition the same ISCM will be modeled and validated in a multibody software (ADAMS/View ®). The results agree in both models: analytical and numerical. This kinematic analysis is the initial step for the future analytical multibody modelling of the Macpherson suspension, since the ISCM may be a simplified model for this suspension with the correct adaptations and considerations. Keywords: Inverted Slider-Crank Mechanism, Kinematic, Multibody System, Macpherson Suspension.
NOMENCLATURE L = length of links, mm φ = angle associated to the maximum 3 relative to link 3 length of link 3, º 4 relative to link 4 Greek Symbols θ = angle associated to links, º Subscripts α = angle associated to the minimum 1 relative to link 1 length of link 3, º 2 relative to link 2
INTRODUCTION Mechanisms have been devised by people since the dawn of history. In last century, prior to World War II, most theoretical work in kinematics was done in Europe, especially in Germany. Since then, much new work has been done, especially in kinematic synthesis, by American and European engineers and researchers (Cervantes-Sánchez et al, 2009; Liu and McPhee, 2007). It is common to think that this subject is already totally studied and dominated, so there are many works even being developed in this area with new methods and techniques (Kim and Yoo, 2012; Talaba, 2012). The kinematic analysis is an important step to the engineering design practice in order to investigate and assay the desired motions or tasks and their consequences to the machine considered. A mechanism can be defined as a device that transforms the movement to a desirable standard and typically develops very low forces and transmits low power. A machine usually contains mechanisms that are designed to provide important strength and power transmission significantly. Mechanisms lightly loaded and run at slow speeds, they may sometimes be treated as strictly kinematic devices, i.e. they can be analyzed without regard kinematically forces. In general any machine or device that moves contains one or more kinematic elements such as linkages, cams, gears, belts, chains (Norton, 2003). There are many applications where it is desirable to design a linkage mechanism which meets a specific functional relationship. A linkage is made by attaching and fixing one link of a mechanism to the ground. The kinematic analysis consists of determining the position, velocity and acceleration of a specific link of this mechanism. There are many applications where it is desirable to design a linkage mechanism that satisfies a particular functional relationship. For example, the steering system, wheel suspensions, and piston-engine in an automobile contain linkages. Even the windshield wipers are linkage-driven (Jazar, 2009). In the automotive area the study of the kinematic mechanism is very important for engineering applications, including improvement design and optimization of subsystems. A well-known automotive mechanism which may be modeled like a closed loop linkage is the suspension subsystem. The Macpherson strut suspension is a very popular
Modeling an ISCM Considering Kinematics Analysis and Multibody Aspects mechanism for independent front suspension of street cars, and its equivalent model may be the four-bar linkage like the inverted slider crank mechanism (Soni, 1974). Actually many research have been developed in suspension subsystem due to its great importance in the vehicle performance [Prawoto et al , 2008; Bar David and Bobrovsky, 2011]. This subsystem is the greater responsible for isolating the chassis from irregularities frequently present in the roadway, keeping tires aligned in the desired direction and to maintain the wheels with proper camber attitudes to the road surface, to respond to the forces produced by the tires during the maneuvers and to resist roll of the chassis (Gillespie, 1992). It is easy to note that the suspension design is closely related with the dynamic performance of the vehicle and consequently with the safety requisites in the automotive area. This justify the great interest in improve this subsystem with modern techniques of modelling and simulation. Most works aims developing new numerical models for suspension subsystem, therefore the use o multibody techniques is a tendency in the researchers’ institutes (academics and industries). Liu et al (2008) uses multibody dynamics model combining with Finite Elements Analysis and these are an efficient method in optimizing the suspension design. Fallah et al (2009) proposed a new nonlinear model (2 DOF) of the Macpherson strut suspension system for ride control applications and it incorporates the suspension linkage kinematics. The results obtained were compared with a multibody model and both agree. Aydın and Ünlüsoy (2012) developed and implemented a parameter optimization methodology to improve impact harshness (IH) of road vehicles. The optimization results indicated that the selected suspension parameters were capable of improving IH performance of the full vehicle multibody model (ADAMS ®) by minimizing longitudinal and vertical acceleration responses. As mentioned above, in general, the numerical model of the suspension subsystem may considers complex models with nonlinearities, finite element model with many DOFs or complete models with joints and links (multibody model). Commonly, these models require a high computational cost, a large time of simulation in powerful computers and the use of commercial software (license or using permission is required) which requires a professional with experience to manipulate it. Although, this features needs to attend the great market competition and consumer demands which force the vehicle manufacturers to decrease the time required for development of their products with the need of maintaining quality and performance requirements. Due to the listed aspects, in this work, the authors propose to use the inverted slider crank mechanism to modeling a Macpherson suspension using therefore a simplified mechanism with appropriate considerations and adaptations. The final aim is obtaining a simplified analytical model for the suspension subsystem based in kinematics and multibody aspects. The slider crank is a well know mechanism and its description of operation can be found in the classical literature [Norton, 2003], so there are currently publications [Olyaei and Ghazavi, 2012; Wang and Chen, 2012] about this subject due to the many and important applications of it in mechanical systems. The choice of this mechanism is justified by their significant advantages such as low cost, reduced number of parts, reduced weight, less wear and clearance. As the first step of the modelling proposed, in this article is presented the kinematic analysis of the four-bar linkage ISCM using vector and matrix formulation with multibody aspects like the use of links and joints. The results show some parameters as: position, velocity and acceleration of specific links of interest, as the length variation (maximum and minimum length) of the coupler. A numerical model of the ISCM was developed in ADAMS ® software for comparison purpose.
THE INVERTED SLIDER-CRANK MECHANISM An inverted slider-crank mechanism is defined as a four-bar linkage. If the coupler link of a slider-crank mechanism is attached to the ground an inverted slider-crank mechanism is made. So, the inverted slider-crank is a simple inversion of a slider-crank mechanism. The Fig. 1 shows both mechanisms for comparison purpose.
(a) (b)
Figure 1 – (a) The slider-crank mechanism. (b) The inverted slider-crank mechanism [From Jazar, 2009].
In Fig. 1(a) the link number 1 is the ground, which is the base and reference link. The link number 2 (MA segment) is called the input link, which is controlled by the input angle θ2 and the link number 4 is the slider considered as the output link. The horizontal distance between the slider and a fixed point in the ground (usually the revolute joint M) is R.C. Silva, M.A.A. Nunes, J.P.M. Bento and V.E.Costa the output variable of this mechanism. The link number 3 (AB segment) is the coupler with angular position θ3 and it connects the input link to the output slider. The Fig. 1(b) is the inverted slider-crank mechanism illustration. As in the Fig. 1(a), the link number 1 is the ground link (reference link), the link number 2 (MA segment) is the input link which is controlled by the angle θ2 and the link number 4 is the slider (output link). The mainly difference between both mechanisms is that the slider has a revolute joint (N) with the ground and a prismatic joint with the coupler (link 3 or AB segment). Therefore, the output variable can be the angle θ4 of the slider with the horizontal or the length AB. So, there are many configurations for the ISCM in addition to that shown in Fig 1(b). In the Fig. 2 is shown others configurations for the ISCM depending of the application. The Fig. 2(c) is the same of Fig. 1(b).
(a) (b)
(c) (d)
Figure 2 - Four configurations for the ISCM [From Williams, 2012].
The Fig. 2 (a) can model a stroke internal combustion engine or an air compressor mechanism. The Fig. 2(b) represents a Whitworth quick return mechanism which converts rotary motion into reciprocating motion. This late mechanism is most commonly seen as the drive for a shaping machine. The Fig. 2(c) is the configuration of interested for this paper. It is capable to represent a Macpherson suspension [Jazar, 2009; Willians, 2012] and it can be a simplified model of this subsystem when compared with the complete models developed in multibody software. The Fig. 2 (d) is a typical representation of a pump mechanism. For a better understanding, in Fig. 3 is shown a comparative between the Macpherson suspension and the ISCM configuration used for represent this subsystem which will be the object of study of this paper.
(a) (b) (c)
Figure 3 - The Macpherson struct suspension like an inverted slider mechanism.
The Fig. 3(a) shows a Macpherson suspension in its real configuration with all mechanical components. This kind of suspension is very popular for independent front suspension of conventional cars. At the top of the structure the piston rod of the shock absorber serves as a kingpin axis. At the bottom, the shock absorber pivots on a ball joint on a single lower arm. The Fig. 3(b) is a simplified model of the structure shown in Fig. 3(a). Finally, the Fig. 3(c) shows the equivalent kinematic four-bar model that represents the suspension subsystem cited. This latest mechanism is the same of Fig. 2(c) and it is an inverted slider-crank mechanism. In the next topic the kinematic formulation of the ISCM shown in Fig. 3(c) is presented. The kinematic analysis of the position, velocity and acceleration links is developed and the length variation of the coupler is considered as a parameter of interest to the future application in the suspension structure. The numerical model developed in ADAMS ® is presented to for comparison purpose. Modeling an ISCM Considering Kinematics Analysis and Multibody Aspects
KINEMATIC MODELLING PROPOSED At this section, it is performed displacement, velocity and acceleration analysis of the coupler (link 3) and the slider (link 4) shown in Fig. 1(b). The analytical model providing displacements, velocities and accelerations can be derived in terms of the general parameters of the ISCM, like length, relative positions and angular velocity applied in the input link (link 2). Next the methodology to construct the multibody model using ADAMS/View is briefly described and, finally, the completed model is shown.
Analytical Model The general method to achieve the kinematic formulation involves basically three steps: determine the vector displacement relationship to describe the ISCM; take the time derivative of the displacement to determine velocity relationship and take the time derivative of the velocity to obtain the acceleration relationship. As said the analytical expressions are giving for the coupler and the output link (slider). The basic references used here are [Jazar (2009); Soni (1974)]. M y A 3 θθθ3 4 X 2 B E
θθθ1 θθθ2 θθθ4 O x 1 C Figure 4 – Vector configuration for an ISCM.
Figure 4 shows the vector representation of the present mechanism. Note that vectors CO , OA , BA and CB represents the links 1, 2, 3 and 4 respectively, as discussed in the previous section. Also θ1 is equal to 180º, the angle between BA and CB is 90º. θ2 is the angle between vectors OA and CO and it represents an input data. θ4 locates the slider position and θ3 locates the coupler (link 3) related to the horizontal segment BX. The segment BX is parallel to the x axis in the reference system xOy , thus the angle between the BX segment and the vector CB is (180º - θ4). This way the sum of angles at point B is: θ3 + (180º - θ4) + 90º = 360º, as a result θ3 = 90º + θ4. Table 1 summarizes the links with their corresponding length and angle position. These values will be used in the kinematic formulation.
Table 1 – Geometrical values used in the kinematic expressions.
Link Identification Length Angle Position
1 L1 θ1 2 L2 θ2 3 L3 θ3 4 L4 θ4 The polygon formed by the vectors in Fig. 4 can be described by:
CO + OA = CB + BA → CO + OA − CB − BA = 0 (1) where CO , OA , CB and BA are the vectors. Decomposing Eq. (1) into cosines (Eq. (2)) and sinus (Eq. (3)) components, it becomes:
L1cos θ1 + L2 cos θ2 − L3 cos θ3 − L4 cos θ4 = 0 (2)
L1sin θ1 + L2 sin θ2 − L3 sin θ3 − L4 sin θ4 = 0 (3) where L1, L2, L3 and L4 are the lengths of the corresponding links 1, 2, 3 and 4. θ1, θ2, θ3 and θ4 are the angle as previously defined. Knowing that θ1 = 180º and θ3 = 90º + θ4, substituting these values into Eq. (2) and Eq. (3), they become:
−L1 + L2 cos θ2 + L3 sin θ4 − L4 cos θ4 = 0 (4)
L2 sin θ2 − L3 cos θ4 − L4 sin θ4 = 0 (5) R.C. Silva, M.A.A. Nunes, J.P.M. Bento and V.E.Costa
According to the methodology described, the velocities and accelerations of the links 3 and 4 depend on the determination of L3 and θ4, respectively. This way, from Eq. (5) L3 is defined as the linear displacement and it can be expressed as a function of the input data θ2, L2 and L4:
L2 sin θ2 − L4 sin θ4 L3 = (6) cos θ4
Substituting Eq. (6) into Eq. (4):
L2 sin θ2 − L4 sin θ4 − L1 + L2 cos θ2 + sin θ4 − L4 cos θ4 = 0 (7) cos θ4
Simplifying Eq. (7), the resultant expression is:
(L2 cos θ2 − L1)cos θ4 + L2 sin θ2 sin θ4 − L4 = 0 (8)
The angle θ4 will be determined considering the following trigonometric relationships into Eq. (8):
θ θ 1− tan 2 4 2 tan 4 2 2 cos θ4 = ; sin θ4 = (9) θ θ 1+ tan 2 4 1+ tan 2 4 2 2
After some simplifications, Eq. (10) is obtained:
θ θ ()L − L cos θ − L tan 2 4 + ()2 L sin θ tan 4 + ()L cos θ − L − L = 0 (10) 1 2 2 4 2 2 2 2 2 2 1 4
Equation (10) is a polynomial of second degree in θ4. The coefficients are given by:
A = (L1 − L2 cos θ2 − L4) B = ()2 L2 sin θ2 (11) C = ()L2 cos θ2 − L1 − L4 and they are function of the links’ length L1, L2 and L4 and angle θ2, all of them input data.
Also Eq. (10) has two roots, r1 = + θ4 and r2 = -θ4 defined as the angular displacement of the slider block B, which are shown in Eq. (12):
2 −1 − B ± B − 4 AC θ4 = ± 2 tan (12) 2 A
Introducing the roots of Eq. (12) in Eq. (6), L3 is defined. The time derivate of the Eq. (4) and Eq. (5), respectively gives:
L2 w2 cos θ2 = L&3 cos θ4 + w4(L4 cos θ4 − L3 sin θ4) (13)
L2 w2 sin θ2 = L&3 sin θ4 + w4(L3 cos θ4 + L3 sin θ4 ) (14) where w2 and w4 are the angular velocities of the input link (link 2) and the output link (link 4), Fig. 1(b), respectively. Thus w2 is an input data and w4 will be determined. These angular velocities result from the time derivative of θ2 and θ4.
Both expressions Eq. (13) and Eq. (14) form a system of expressions that can be solve to obtain w4 and L&3 . After some mathematical operations, Eq. (15) gives L&3 that is defined as the linear velocity of the coupler relative to the changes of positions of slider block B, Fig. 1(b), or in others words, relative linear velocity of slider block B.
L&3 = L2 w2 cos (θ4 −θ2)− L4 w4 (15) Modeling an ISCM Considering Kinematics Analysis and Multibody Aspects
Next the angular velocity w4 is defined as shown in Eq. (16):
L2 w2 w4 = sin ()θ2 −θ4 (16) L3
Thus Eq. (15) becomes:
L2 w2 L&3 = ()L3 cos ()()θ4 −θ2 − L4 sin θ2 −θ4 (17) L3
The acceleration expressions are the time derivative of Eq. (13) and Eq. (14). The derivate of Eq. 13 results in:
2 2 2 L&&3 cos θ4 + ()L4 cos θ4 − L3 sin θ4 α4 = −L2 w2 sin θ2 + L2 α2 cos θ2 + L3 w4 cos θ4 + L4w4 sin θ4 + 2w4 L&3 sin θ4 (18) and of the Eq. (14):
2 2 2 L&&3 sin θ4 + ()L4 sin θ4 + L3 cos θ4 α4 = L2 w2 cos θ2 + L2 α2 sin θ2 + L3 w4 sin θ4 − L4w4 cos θ4 − 2w4 L&3 cos θ4 (19)
The solution of the system of equations between both Eq. (18) and Eq. (19) gives L&&3 and α4, which are defined as the linear acceleration of the coupler relative to the changes of positions of slider block B, Fig. 1(b), and the angular acceleration of link 4 (output link). These angular velocities result from the time derivative of w2 and w4. This way the linear acceleration is defined as shown in Eq. (20):
2 2 2 L2 w2 cos θ2 + L2 α2 sin θ2 + L3 w4 sin θ4 − L4 w4 cos θ4 − L&3 w4 cos θ4 − ()L3 cos θ4 + L4 sin θ4 α4 L&&3 = (20) sin θ4 and Eq. (21) defines the angular acceleration:
2 2 L2 w2 cos ()()θ4 −θ2 − L2 α2 sin θ4 −θ2 − 2 L&3 w4 − L4 w4 α4 = (21) L3
In the moment that the link 4 (output slider) stops while link 2 (input link) continue to turn, it is verified that the slider is at a limit position; these positions for the ISCM are shown in Fig. 5 (a) and Fig. 5 (b). Figure 5 (a) represents the minimum position for the coupler; it corresponds to the segment BA , where A is the location of a revolute joint to be placed between the input link and the coupler. At this situation, the angle between both links is 180º. In the other hand, Fig. 5 (b) denotes a maximum position for the coupler; it is also represented by the same segment BA , but the angle between the slider and the coupler is now 0º. Notice that the point A (revolute joint) turns 180º to occupy a maximum position. The α angle corresponds to the minimum position of the coupler as shown in Fig. 5 (a). From Fig. 4, it is established that the angle between BA and CB is 90º. Thus, in the Fig. 5 (a), α is determined as shown in Eq. (22):
CB L4 −1 L4 sin α = → sin α = → αmin = sin (22) OC L1 L1 where CB and OC according to Fig. 5 are the lengths of the link L4 and L1, respectively. Thus, the angle dictated by the maximum position of the coupler, Fig. 5 (b), corresponds to the trajectory of the point A as shown in Eq. (23):
φ = α min +180 º (23)
Based on Fig. 5 (a), the following Eq. (23) can be defined:
2 2 2 OB + CB = OC ; OB = L2 + L3 (24)
Substituting the lengths of the links in Eq. (24): R.C. Silva, M.A.A. Nunes, J.P.M. Bento and V.E.Costa
2 2 2 2 2 ()L2 + L3 + L4 = L1 → L2 + L3 = ± L1 − L4 (25)
Then,
2 2 2 2 L3min + L2 = L1 − L4 → L3min = L1 − L4 − L2 (26) 2 2 2 2 2 2 L3max + L2 = − L1 − L4 → L3max = − L1 − L4 − L2 → L3max = − L1 − L4 − L2
As said, L3 min corresponds to the segment BA in Fig. 5 (a) and L3 max corresponds to the segment BA in Fig. 5 (b).
E y B OC – Link 1 OA – Link 2 A AE – Link 3 CB – Link 4 ααα O x C (a)
E y B OC – Link 1 OA – Link 2 AE – Link 3 CB – Link 4
O x C φ A (b)
Figure 5 – Limit positions for the ISCM. (a) distance AB in minimum position (b) distance AB in maximum position.
Multibody Model The multibody model was constructed using the ADAMS/View. For the ground (link 1), the length between OC is equal to 3 cm, Fig. 4. The dimensions, for the input link (link 2), are length 1 cm, width and depth 0.2 cm; for the output link (link 4), length is 0.5 cm and width and depth are equal to 0.2 cm. The angle θ2, Fig. 4, is supposed of 45º. Using expressions, Eq. (12) and Eq. (6) respectively, it is possible to define the values of the outputs angles θ4 = 84.88º; θ4 = - 119.17º and, the length of the coupler link (link 3) around ± 2.34 cm. Thus, admitting θ4 = 84.88º ≅ 85º, the dimensions of the output link (link 4) are: length + 2.34 cm and width/depth supposed equal to 0.2 cm. Note that the slider block (link 4), Fig. 1(b) and Fig. 4, is composed by a link attached on the ground by a revolute joint in point C and, by a block in the other extremity (point B). This block is modeled using the tool: Rigid body: Box. The dimensions are 0.8 cm of length, 0.4 cm of height and 0.2 cm of depth. After their creation separately, they are transformed in a unique body using the tool Boolean: Unite two solids. To provide movement to the mechanism four joints are inserted: three revolute joints and one translational. The revolute joints are applied in points O, A and C, according to Fig. 4, allowing rotational path. The translational joint is inserted in point B to guarantee the movement of link 3 into the slider block. Figure 6 illustrates the multibody model in front and isometric views based on the geometrical dimensions previously presented.
To carry out the kinematics analysis of the proposed model, an angular velocity w2 in a counterclockwise direction equal to 15 rad/s is applied at the point O, in the input link (link 2). Thus, the linear and angular velocities and accelerations of the output and coupler links are compared with the analytical expression determined in the previous section, Eq. (16), Eq. (17), Eq. (20) and Eq. (21). Also, it is verified the limit positions for the ISCM according to the Eq. (26).
Modeling an ISCM Considering Kinematics Analysis and Multibody Aspects
(a) (b)
Figure 6 – Multibody model of the ISCM. (a) front view (b) isometric view.
RESULTS This section presents some graphics that denotes the accordance between the analytical and multibody models. This way, Fig. 7 (a) and (b) show the variation of the slider angle θ4 versus the input angle θ2. The θ2 angle can be assumed to be θ2 = w2 t , where w2 is the angular velocity of the input link (link 2) and its value is 15 rad/s and t is given in seconds and, it varies from 0 to 0.5s. From the figure, it is seen that θ4 varies from 80º approximately, to 120º. As shown in Eq. (12), θ4 can have values positive and negative both with the same amplitude. In this paper only the values positive are considered.
(a) (b)
Figure 7 – Slider angle variation in function of the input angle θθθ2 (a) positive values (b) negative values
Figure 8 illustrates the coupler position in function of θ2, θ4 as shown in Eq. (6). It can be seen that the maximum and minimum values determined from Eq. (26), are in accordance with those obtained from Eq. (6), shown in Fig. 8. At this case, values are: L3 min ͌ 19.58 mm and L3 max ͌ 39.58 mm. The θ4 angle in both situations is about of 80º in accordance to Fig. 7 (a).
Figure 8 – Variation of the coupler position in function of time. R.C. Silva, M.A.A. Nunes, J.P.M. Bento and V.E.Costa
Figure 9 (a) and (b) shows the comparison of analytical results of angular and linear velocities of the ISCM (Eq. (16) and Eq. (17) with those obtained with the multibody model. The angular velocity refers to the slider link and the linear velocity to the coupler. The offset observed in curves is t = 0.05s, at this case curves are coincident. In Fig. 9 (a) and (b), the extreme values (amplitude) at t = 0.225s and t = 0.425s are very close.
(a) (b)
Figure 9 – Angular and linear velocities (a) slider block link (b) coupler link.
Figure 10 (a) and (b) shows the comparison of analytical results of angular and linear accelerations of the ISCM (Eq. (20) and Eq. (21) with those obtained with the multibody model. The angular acceleration refers to the slider link and the linear acceleration to the coupler. The offset observed in curves is t = 0.05s, at this case slider angular acceleration, Fig. 10 (a), is coincident. In the other hand, linear acceleration of the coupler, shown in Fig. 10 (b), is not so close but they are similar. Also, the extreme values of linear acceleration (amplitude) at t = 0.225s and t = 0.425s keep comparable values.
(a) (b)
Figure 10 – Angular and linear velocities (a) slider block link (b) coupler link.
CONCLUSION The present paper aims to show the similarity in results of analytical and multibody models, related to angular and linear velocities and accelerations of the slider block and coupler link of an inverted slider crank mechanism. Also, it presents the limit positions (minimum and maximum) of the coupler link and the associated angle of each situation. In this context, the results seem to be very suitable validating the multibody model proposed. This study represents a first approach to develop an analytical and multibody models more adapted to a Macpherson structure.
ACKNOWLEDGMENTS The authors would like to thank CAPES and UnB-Gama Faculty for their financial support.
REFERENCES Aydın , M. and Ünlüsoy, Y. S. , 2012, "Optimization of suspension parameters to improve impact harshness of road vehicles", The International Journal of Advanced Manufacturing Technology, Vol. 60, N. 5-8, pp. 743-754. Bar David, S., Bobrovsky, B., 2011, Actively controlled vehicle suspension with energy regeneration capabilities , Vehicle System Dynamics, Vol.49(6), pp.833-854. Modeling an ISCM Considering Kinematics Analysis and Multibody Aspects
Cervantes-Sánchez, J.J., Gracia, L., Rico-Martínez, J.M., Medellín-Castillo, H.I., González-Galván, E.J., 2009, "A novel and efficient kinematic synthesis approach of the spherical 4R function generator for five and six precision points", Mechanism and Machine Theory J., Vol. 44, Issue 11, pp 2020–2037. Fallah, M. S., Bhat, R., Xie, W. F., 2009, "New model and simulation of Macpherson suspension system for ride control applications", Vehicle System Dynamics, Vol. 47 Issue 2, pp. 195-220. Gillespie, T.D., 1992, “Fundamentals of Vehicle Dynamics”, Ed. SAE International, Warrendale, USA, 519 p. Jazar, R. N., 2009, "Vehicle Dynamics: Theory and Application", Springer, New York. Kim, B. S., Yoo, H. H., 2012, "Unified synthesis of a planar four-bar mechanism for function generation using a spring- connected arbitrarily sized block model", Mechanism and Machine Theory J., Vol. 49, pp. 141–156. Liu, J., Zhuang, D.J., Yu, F. and Lou, L.M., 2008, "Optimized design for a MacPherson strut suspension with side load springs", International Journal of Automotive Technology, Vol. 9, N. 1. Liu, Y., McPhee, J., 2007, "Automated Kinematic Synthesis of Planar Mechanisms with Revolute Joints", Mechanics Based Design of Structures & Machines, Vol. 35 Issue 4, pp. 405-445. Norton, R. L., 2003, "Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines”, Mcgraw-Hill College, 3rd edition. Olyaei, A. A., Ghazavi, M. R., 2012, “Stabilizing slider-crank mechanism with clearance joints”, Mechanism and Machine Theory, Vol. 53, pp. 17–29. Prawoto, Y., Ikeda, M., Manville, S.K. , Nishikawa, A., 2008, Design and failure modes of automotive suspension springs, Engineering Failure Analysis,Vol.15, Issue 8, December 2008, pp. 1155–1174. Soni, A. H., 1974, "Mechanism Synthesis and Analysis", McGraw-Hill Book Co. Talaba, D., 2012, "The angular capacity of spherical joints used in mechanisms with closed loops and multiple degrees of freedom", Robotics and Computer Integrated Manufacturing, Vol.28(5), pp.637-647. Wang, Y., Chen, C., 2012, “The dynamics of a slider-crank mechanism with a Fourier-series based axially periodic array non-homogeneous coupler”, Journal of Sound and Vibration,Vol. 331, Issue 22, pp. 4831–4847. Williams, B., 2012, "Atlas of Structures, Mechanisms, and Robots. Department of Mechanical Engineering", Ohio University. Available in: http://www.ohio.edu/people/williar4/html/PDF/MechanismAtlas.pdf (Accessed in August/2012).
RESPONSIBILITY NOTICE The authors are the only responsible for the printed material included in this paper.