Chapter 5 Modeling and Simulation of Mechanism

Chapter 5 Modeling and Simulation of Mechanism In the present study, KED analysis of four bar planar mechanism using MATLAB program and ANSYS software has been carried out. The analysis has also been carried out by considering rigid links in the same mechanism. In the analysis of rigid links all pin joint forces, angular velocities and angular accelerations of links has been computed. The coupled solution of governing equations of motion has been obtained using MATLAB. In this MATLAB analysis each link has been considered as an element. The simulated results have been validated with the experimental results available in literature [13]. The modeling and simulations of four bar planar mechanism has been executed in ANSYS by considering rigid link and flexible links with more elements. The effect of moment of inertia of coupler and its length, flexibly of crank and rocker, and rocker length on the strain developed in coupler has been studied. After the dynamic analysis of four bar planar mechanism this methodology is extended to six bar mechanism. Multibody simulation deals with deals with study and analysis of dynamic behavior of system of flexible and/or rigid interconnected bodies. These bodies are subjected to constrain with respect to one another through a kinematic constraints modeled as joints. These systems can represent a space structure with antenna deployment capabilities, an automobile, a robot with manipulator arms, an aircraft as an assemblage of rigid and flexible parts, and so on. The components may be subjected to large displacement, large rotation, and also effects of finite strain. Multibody systems have conventionally been modeled as rigid body systems with superimposed elastic effects of one or more components. A major limitation of these methods is that non-linear large-deformation, finite strain effects or non-linear material cannot be incorporated completely into model. The FE method used in ANSYS offers an attractive approach to modeling a multibody system. The ANSYS multibody analysis method may require more 5.1

computational resources and modeling time compared to standard analysis; it has the following advantages [75]: The finite element mesh automatically represents the geometry while the large deformation/rotation effects are built into the finite element formulation. Inertial effects are greatly simplified by the consistent mass formulation or even point mass representations. Interconnection of parts via joints is greatly simplified by considering the finite motions at the two nodes forming the joint element. A general steps for FEM for non-linear analysis is as follows: (i) Build the model: A flexible mechanism usually comprises of flexible and/or rigid body parts connected via joint elements. The modeling the flexible parts with any of the 3-D solid, shell, or beam elements. The flexible and/or rigid parts are connected using joint elements. In one scenario, two parts may be simply connected to ensure that the displacements at the joints are identical. In other scenario, the two connected parts may involve joint such as the universal joint or a planar joint. While modeling these joints, a suitable kinematic constraint is implemented on the relative motion (displacement and rotation) between the two nodes that form the joint. (ii) Define element types: Simulation of a flexible multibody involving flexible and rigid components joined together subjected to some form of kinematic constraints, using appropriate joint and contact element types. (iii) Define materials: Defining the linear and non-linear material properties for each components of multibody system. (iv) Mesh the model: Mesh the all flexible components of multibody system. Two nodes define joint elements and no special meshing is required to define them. (v) Solve the model: Multibody analyses generally involve large rotations in static or transient dynamics analysis, so non-linear geometric effects must be accounted for. 5.2

(vi) Review the results of model: Results from a flexible multibody analysis consist mainly of displacements, velocities, accelerations, stresses, strains, and reaction forces in structural components. Constraint forces, current relative positions, relative velocities, and relative accelerations in joint elements are also available. 5.1 Analysis of mechanism in ANSYS The procedure for rigid and dynamic analysis of mechanism in ANSYS Workbench software is as follows [75]: 5.1.1 Selection of types of analysis In its most basic use, the ANSYS Workbench process is straight forward to select the type of analysis that is to be performed from the analysis systems group of the Toolbox and add that system to the project schematic. When the system is in place, than work through the cells in the system, generally from top-to-bottom, until completed all the required steps for analysis are completed. In most cases, data flows from top to bottom through the system as well. For example, in a mechanical system, the geometry must be defined before one defines the model; the model cell uses the geometry defined in the geometry cell as its input. 5.1.2 Renaming Systems In general, it is good practice to give each system a name which is most meaningful for analysis as shown in Fig. 5.1. When new systems are added to the project schematic, the system name initially has focus to encourage the user to enter a meaningful name. To rename a system that already exists on the project schematic, one can either double-click on the system name (shown below the system) or rightclick and select rename from the system context menu (right click on the system header, row 1 in the system, to access the system context menu). The system name will be highlighted as shown in Fig. 5.1. 5.3

Fig. 5.1 Step for renaming the system 5.1.3 Define engineering data Engineering data serves as a resource related to material properties that is used in system analysis. Engineering data can be used as a repository for company or department data, such as material data libraries. While designing the engineering data workspace care is taken to allow user to create, save, and retrieve material models, and also to create libraries of data that can be saved and subsequently used in projects and by different users or in other analysis. User interface for engineering data is shown in Fig. 5.2. Engineering data can be shown as a component system or as a cell in any mechanical analysis system. As a standalone component system, the workspace accesses all material models and properties by default. Properties and material models related to system physics are shown in workspace when viewed as a cell in mechanical analysis system. The engineering data can be access by inserting an engineering data component system or a mechanical system into the project 5.4

schematic. The analyst can select edit from the engineering data cell's menu, or opt to double-click the cell. Subsequently, the engineering data workspace appears. From here, the user can navigate through the database required for analysis system, access external data sources, create new data, as well as store data for subsequent use. Fig. 5.2 Step for defining the engineering data 5.1.4 Attach geometry There are no geometry creation tools in the mechanical application so geometry must be attached to the mechanical application. The geometry can be created from either of the following sources: (i) from within Workbench using design modeler or (ii) From a CAD system supported by workbench. To import the geometry following step can be performed. From the analysis system subroutine, select the geometry cell. Browse to the CAD file from the following access points: Right-click on the geometry cell in the project schematic and choose import geometry. The model cell in the project schematic can be selected via 5.5

double click. Subsequently the mechanical application displays the geometry. The dialogue box related to geometry import as shown in Fig. 5.3. Fig. 5.3 Step for attaching the geometry 5.1.5 Define the part behavior After attaching geometry it is possible to access settings related to part behavior by right-clicking on the model cell in the analysis system schematic and choosing edit. The mechanical application opens with the environment representing the analysis system displayed under the model object in the tree, as mention in Fig. 5.4. Under this tree first branch is geometry and second is coordinate systems. In geometry list of parts or bodies with following options: (i) Stiffness behavior: In addition to making changes to the material properties of a part, it is also possible to designate a part's stiffness behavior as being flexible or rigid. The solution time is reduced significantly by setting a part behavior as rigid which in turn reduces the representation of part to single point mass. Mass of rigid part will be calculated from density of the material. In case of density being 5.6

function of temperature, it is evaluated at the reference temperature. For contact conditions, specify Young s modulus. Flexible is the default stiffness behavior. To change, simply select rigid from the stiffness behavior drop-down menu. Fig. 5.4 Step for defining the parts behaviour (ii) Coordinate systems: The coordinate systems object and its child object, global coordinate system is automatically placed in the tree with a default location of 0, 0, 0, when a model is imported. For solid parts and bodies by default, a part and any associated bodies, use the global coordinate system. If desired, it is also possible to apply a local coordinate system to the part or body. When a local coordinate system is assigned to a part, by default, the bodies also assume this coordinate system but one may modify the system on the bodies individually as desired. For surface bodies, solid shell bodies, and line bodies by default, these types of geometries generate coordinates systems on a per element type basis. It is necessary for the user to create a local coordinate system and associated it with the 5.7

parts and/or bodies using the coordinate system setting in the details view for the part/body if one wishes to orient those elements in a specific direction. (iii) Reference temperature: The default reference temperature is taken from the environment (by environment), which occurs when solving. This necessarily means that the reference temperature can change for different solutions. The reference temperature can also be specified for a body and will be constant for each solution (by body). Selecting by body will cause the reference temperature value field to specify the reference temperature for the body. It is important to recognize that any value set by body will only set the reference temperature of the body and not actually causes the body to exist at that temperature. (iv) Material property assignment: Once the geometry has been attached, the next step is to choose a material for the simulation. Upon selecting a part in the tree outline, the assignment entry under Material in the details view lists a default material for the part. This can be edited using material properties in the engineering data workspace. (v) Non-linear material effects: It is possible to ignore any nonlinear effects from the material properties. As default setting, all pertinent material properties are used, including non-linear properties such as stress-strain curve data. Setting non-linear effects to no will ignore any non-linear properties only for that part. This option will allow the analyst to assign same material to two different parts and also treat one of the parts as linear. (vi) Thermal strain effects: For structural analyses, it is possible to have workbench to calculate a thermal strain result by setting thermal strain effects to yes. Choosing this option enables the coefficient of thermal expansion to be sent to the solver. 5.1.6 Define connections Connections include contact regions, joints, springs, or beams. Explicit analysis connections include body interactions. Contact conditions arise where bodies meet. On importing an assembly from a CAD system, contact between various parts is automatically detected. In addition to this contact regions can also be set up manually. It is possible to transfer heat flows 5.8

across the contact boundaries and structural loads and connects the various bodies. The analysis can be linear or nonlinear, depending on the type of contact. A joint is an idealized kinematic linkage that controls the relative movement between two bodies. Joint types are characterized by their translational and rotational DOF as being fixed or free, as shown in Fig. 5.5 Fig. 5.5 Step for defining the connections 5.1.7 Apply mesh control and preview mesh Meshing is the process in which the mechanism geometry is spatially discretized into elements and nodes. This mesh along with material properties is used to mathematically represent the stiffness and mass distribution of the structure. The model is automatically meshed for further process. The element size by default is determined based on various factors including body curvature, the overall model size, the complexity of the feature and the proximity of other topologies. When required, the mesh size is adjusted up to four times (eight times for an assembly) till a 5.9

successful mesh is achieved. The dialogue box for meshing the model is shown in Fig. 5.6. If desired, it is possible to preview the mesh before solving. Mesh controls are available to assist you in fine tuning the mesh. There are some options available to modify the mesh: (i) default group, (ii) sizing group, (iv) inflation group, (v) advanced group, (vi) pinch group and (vii) statistics group. Fig. 5.6 Step of meshing the model 5.1.8 Establish the analysis setting For transient structural (ANSYS) analysis the basic controls are: Slender structures typically require large deflection. The user can use large deflection in case a slender structure has transverse displacements that are more than 10% of the thickness. Small strain and small deflection analysis assume that 5.10

displacements are small enough so that the resulting stiffness changes are insignificant. Switching ON large deflection will account for stiffness changes resulting from change in orientation and element shape due to large deflection, large strain, and large rotation. This ensures that the results will be more accurate. But this effect demands an iterative solution. In addition it may also need the load to be applied in small increments. Hence the solution may take longer time. Use of hyper elastic materials also requires large deflection to be turned on as shown in Fig. 5.7. Fig. 5.7 Step for setting the analysis steps Step controls permits to control the time step size in a transient analysis. In addition this control also makes it possible to create multiple steps. In case new loads are to be introduced or removed at different times in the load history, or if the analyst wants to change the analysis settings such as the time step size at some points in the time history, multiple steps are to be used. In case nonlinearities are present or if the applied load has high frequency content, one might be required to use a small time 5.11

step size (that is, small load increments) and compute solutions at these intermediate time steps to subsequently arrive at more accurate results. This group can be modified on a per step basis. Output controls option is useful to specify the time points at which results should be available for post processing. In a transient nonlinear analysis it may be necessary to perform many solutions at intermediate time values. However, (i) one may not be interested in all the intermediate results, and (ii) writing all the results can make the results file size unwieldy. This group can be modified on a per step basis except for calculating stress and strain. Non-linear controls feature allow the user to modify convergence criteria and other specialized solution controls. Typically one will not need to change the default values for this control. This group can be modified on a per step basis. Damping controls are used to specify damping for the structure in a transient analysis. The following forms of damping are available for a transient analysis: beta damping and numerical damping. In addition, element based damping from spring elements as well as material based damping factors are also available for the transient structural (ANSYS) analysis. Analysis data management settings make it possible to save specific solution files from the transient structural (ANSYS) analysis for other analyses. The default behavior is to only keep the files required for post processing. These controls can be used to keep all files created during solution or to create and save the mechanical APDL application database (db file). 5.1.9 Define the initial conditions For transient structural (ANSYS) analysis the initial conditions are: A transient analysis involves loads that are functions of time. The first step in applying transient loads is to establish initial conditions (that is, the condition at initial time = 0). 5.12

The default initial condition for a transient structural (ANSYS) analysis is that the structure is at rest, that is, both initial displacement and initial velocity are zero. A transient structural (ANSYS) analysis is at rest, by default. The initial conditions object allows to specified velocity. In many analyses one or more parts will have an initial known velocity such as in a drop test, metal forming analysis or kinematic analysis. A constant velocity initial condition can be specified if required. The constant velocity could be aimed at one or more parts of the structure. The remaining parts of the structure which are not part of the horizon will be subjected to the at rest initial condition. Initial condition can also be specified using step controls, that is, by specifying multiple steps in a transient analysis and controlling the time integration effects along with activation/deactivation of loads. This is extremely useful when there are different parts of a model that have different initial velocities or more complex initial conditions. Some commonly encountered initial condition is tackled as explained below: Initial displacement = 0, Initial velocity 0 for some parts: The non zero velocity is established by applying small displacements over a small time interval on the part of the structure where velocity is to be specified. Specify second steps in the analysis. The first step is used to establish initial velocity on one or more parts. A small end time (compared to the total span of the transient analysis) is choosen for the first step. The second step will cover the total time span. Specify displacement(s) on one or more faces of the part(s) that will give the required initial velocity. This requires that one does not have any other boundary condition on the part that will interfere with rigid body motion of that part. Make sure that these displacements are ramped from a value of zero. Deactivate or release the specified displacement load in the second step so that the part is free to move with the specified initial velocity. 5.13

Initial displacement 0, Initial velocity 0: This is similar to previous case except that the imposed displacements are the actual values instead of small values. Specify second steps in the analysis. The first step is used to establish initial displacement and velocity on one or more parts. A small end time (compared to the total span of the transient analysis) is choosen for the first step. The second step will cover the total time span. The initial displacement(s) on one or more faces of the part(s) is specified, as needed. This requires that the user does not have any other boundary condition on the part that will interfere with rigid body motion of that part. Make sure that these displacements are ramped from a value of zero. Further release the specified displacement load as explained previously. Initial Displacement 0, Initial Velocity = 0: This requires the use of two steps also. The main difference between the above and this scenario is that the displacement load in the first step is not ramped from zero. Instead it is step applied as shown below with two or more sub steps to ensure that the velocity is zero at the end of step 1. Specify second steps in the analysis. The first step will be used to establish initial displacement on one or more parts. An end time for the first step is choosen that together with the initial displacement values will create the necessary initial velocity. The initial displacement(s) on one or more faces of the part(s) is specified as needed. This requires that user does not have any other boundary condition on the part that will interfere with rigid body motion of that part. Make sure that this load is step applied, that is, apply the full value of displacements at time = 0 itself and maintain it throughout the first step. Deactivate or release the specified displacement load in the second step so that the part is free to move with the initial displacement values. 5.14

5.1.10 Apply loads and supports For a transient structural (ANSYS) analysis applicable loads/supports are all inertial and structural loads, and all structural supports. Joint loads are used to kinematically drive joints. For joints in a transient structural (ANSYS) or transient structural (MBD) analysis, one has to use a joint load object to apply a kinematic driving condition to a single DOF on a joint object. Joint load objects are applicable to all joint types except fixed, general, universal, and spherical joints. For translation DOF, the joint load can apply a displacement, velocity, acceleration, or force. For rotation DOF, the joint load can apply a rotation, angular velocity, angular acceleration, or moment. The directions of the DOF are based on the reference coordinate system of the joint and not on the mobile coordinate system. A positive joint load will tend to cause the mobile body to move in the positive DOF direction with respect to the reference body, assuming the mobile body is free to move. If the mobile body is not free to move then the reference body will tend to move in the negative DOF direction for the joint load. One way to learn how the mechanism will behave is to use the configure feature. For the joint with the applied joint load, dragging the mouse will indicate the nature of the reference/mobile definition in terms of positive and negative motion. To apply a joint load: Highlight the transient environment object and insert a joint load from the right mouse button context menu or from the loads drop down menu in the environment tool bar as shown Fig. 5.8. From the joint drop down list in the details view of the joint load, select the particular joint object that has to be applied to the joint load. Apply a joint load to the mobile bodies of the joint. It is therefore important to carefully select the reference and mobile bodies while defining the joint. 5.15

Fig. 5.8 Step for applying the load The unconstrained DOF has to be selected for applying the joint load, based on the type of joint. This selection can be made from the DOF drop down list. For joint types that allow multiple unconstrained DOF, a separate joint load is necessary to drive each one. Joint load objects that include velocity, acceleration, rotational velocity or rotational acceleration are not applicable to static structural analyses. Type of joint load has to be selected from the type drop down list. The list is filtered with choices of displacement, velocity, acceleration, and force if case of selection of a translational DOF in step 3. The choices are rotation, rotational velocity, rotational acceleration, and moment if you selected a rotational DOF. The magnitude of the joint load type is specified in step 4 as a constant, in tabular format, or as a function of time using the same procedure as is done for most loads in the mechanical application. 5.16

5.1.11 Description of solve tool When performing a nonlinear analysis, one may encounter convergence difficulties due to a number of reasons. Some examples may be initially open contact surfaces causing rigid body motion, large load increments causing non-convergence, material instabilities, or large deformations causing mesh distortion that result in element shape errors. Dialogue box for setting solve method and result tracker is shown in Fig. 5.9. Solution output continuously updates any listing output from the solver and also specifies useful information related to behavior of the structure during the analysis. Fig. 5.9 Step for setting solve method and result tracker It is possible to view contour plots of Newton-Raphson residuals in a nonlinear static analysis. Such a capability can be useful when user experience convergence issues in the middle of a step, where the model has non-linearties and a 5.17

large number of contact surfaces. When the solution diverges, identifying regions of high Newton-Raphson residual forces can provide insight into possible problems. Result tracker is also a useful tool that permits monitoring of displacement and energy results as the solution moves ahead. This is typically useful when structures that go through convergence difficulties owing to buckling instability. 5.1.12 Post processing of analysis results The analysis type determines the results available for user to examine after solution. For example, in a structural analysis, one may be interested in maximum shear results or equivalent stress results, while in a thermal analysis, the user may be interested in total heat flux or temperature. The result in the mechanical application section lists various results available for post processing. In order to add result objects in the mechanical application: Highlight a solution object in the tree structure. Select the pertinent result from the solution context toolbar or opt for the right-mouse click option. To review results in the mechanical application: Click on a result object in the tree structure. After the solution has been obtained, it is possible to review and interpret the output as explained below: Contour plots - Displays a contour plot of a result such as stress over geometry. Vector Plots - Displays some results in the form of vectors (arrows). Probes - Displays a result at a single time point, or as a variation over time, using a table as well as a graph. It is also possible to set up various probes to review results as shown in Fig. 5.10. Charts Shows various results over period of time, or displays one result versus another, for example, force versus displacement. 5.18

Animation - Animates the change of results over geometry along with deformation of structure. Stress Tool - to access a design using different failure theories. Fatigue Tool - to carry out advanced life prediction calculations. Contact Tool - to review contact zone behavior in complex assemblies. Beam Tool - to study stresses in line body representations. Fig. 5.10 Step to set the different probe to review results 5.2 Modeling of four bar planar mechanism The kinematic and dynamic analysis using MATLAB and ANSYS software have been carried with different considerations and also tabulated in Table 5.1. 5.19

Table: 5.1 Detail of cases simulated in present study Sr. Cases considered for No. simulation ANSYS MATLAB Importance of study 1 Position analysis No Yes Necessary for dynamic 2 Velocity analysis Yes Yes analysis 3 Acceleration analysis Yes Yes 4. To study effect of No Yes Necessary for joint transmission angle force and torque 5. Determination of joint force Yes Yes calculation which is useful for selection of drives 6. Flexible dynamic analysis of Yes Yes Important for light coupler weight links, which affect the mechanism performance due to 7. To study effect of link orientation and cross section 8. To study the effect of rocker arm length 9. To study the effect of coupler length 10. To study the effect of links flexibility on coupler strain 11. Flexible dynamic analysis of six bar Watt s mechanism deflection of links Yes No Useful for proper selection of link cross section Yes No Useful for proper Yes No selection of link length and transmission angle Yes No Important for light weight links, which affect the mechanism performance due to deflection of links Yes No Important for light weight links, which affect the mechanism performance due to deflection of links Two type of analysis have been carried out in ANSYS software, (i) Rigid analysis and (ii) Flexible analysis. The links are connected to each other through revolute joints. The effect of gravity is taken into consideration. The crank, follower and coupler are modeled using beam elements. A constant time step was chosen for the simulation. The strain of coupler at various time intervals is calculated. Flexible analysis is carried out by considering the coupler as flexible and remaining links are to be rigid. Geometry has been prepared in Pro-E as per the specification mention in Table 5.2 and exported to ANSYS of analysis. Meshed 5.20

model of four bar planar mechanism has been presented in Fig. 5. 11. Detail related to modeling and parameters to be selected for analysis have been mention in Table 5.3. Table 5.2 Specification of four bar planar mechanism [12, 13] Parameters Fixed link (1) Crank (2) Coupler (3) Follower (4) Length (mm) 250 110 280 260 C/S Area (mm 2 ) - 108 40 40 Area moment of Inertia (mm 4 ) - 160 9 9 Modulus of Elasticity, E = 7.10 10 4 MPa Density = 2770 kg/m 3 Crank speed = 32.3 rad/sec Fig. 5.11 Meshed model of four bar planar mechanism 5.21

Table 5.3 Detail of finite element model of four bar planar mechanism Object Name LINK1 (Fix) LINK2 (Crank) LINK3 (Coupler) LINK4 (Rocker) Definition Stiffness Behavior Reference Temperature Assignment Rigid Flexible Rigid By Environment Material Aluminum Alloy Nonlinear Effects Thermal Strain Effects Properties Yes Yes Volume(mm 3 ) 51571 22451 21903 21103 Mass (kg) 0.14285 0.06219 0.06067 0.058455 Moment of Inertia Ip1 kg mm² Moment of Inertia Ip2kg mm² Moment of Inertia Ip3kg mm² 5.9932 4.4632 3.9738 3.8673 799.36 124.38 784.46 665.22 802.97 122.08 782.45 663.31 Statistics Nodes 1 924 1 Elements 1 102 1 Modeling of four bar mechanism has been done for varying different parameter i.e. cross section and orientation of cross section of coupler, length of the rocker and coupler, and flexibility of other links to study the effect on the strain produce in flexible coupler. Therefore, different models have been prepared in Pro-E 5.22

and exported ANSYS for analysis as presented in Figs 5.12-5.18 and detail of model i.e. number of nodes and element are presented in Tables 5.4-5.8. Fig. 5.12 Coupler link having rectangular cross section with orientation 1 Fig. 5.13 Coupler link having rectangular cross section with orientation 2 5.23

Figures 5.15 to 5.18 show the rectangular, circular and elliptical cross section with different orientation of coupler link with cross section area of 40 mm 2. Due to the change in orientation of coupler has changed its moment of inertia from I xx to I yy (rectangular cross section). It is worth to mention here, that in case of I xx, the width of coupler is parallel axis of rotation while, it is perpendicular to axis of rotation in case of I yy. Fig. 5.14Coupler link having circular cross section Fig. 5.15Coupler link having elliptical cross section 5.24

Table 5.4 Detail of FE model for different cross section and orientation of coupler Object Name For Fig. 5.12 For Fig. 5.13 For Fig. 5.14 For Fig. 5.15 Definition Stiffness Behavior Flexible Reference Temperature Assignment Nonlinear Effects By Environment Material Aluminum Alloy Yes Thermal Strain Effects Properties Yes Volume(mm³) 21903 22386 22014 14102 Mass (kg) 0.060672 0.06201 0.060979 0.039063 Moment of Inertia Ip1 (kg mm²) Moment of Inertia Ip2 (kg mm²) Moment of Inertia Ip3 (kg mm²) 3.9738 4.1438 2.7188 2.5527 784.46 800.83 785.77 660.59 782.45 799.84 786.37 661.19 Statistics Nodes 924 1595 1858 1151 Elements 102 723 958 504 Table 5.5 Detail of FE model for different length of coupler Coupler length (mm) 260 270 280 290 300 Statistics Nodes 883 883 924 921 921 Elements 96 96 102 100 100 5.25

Fig. 5.16 Meshed model of mechanism with flexible coupler and rocker Table 5.6 Detail of finite element model with flexible coupler and rocker Object Name LINK1 LINK2 LINK3 LINK4 Stiffness Behavior Rigid Flexible Reference Temperature By Environment Material Assignment Aluminum Alloy Nonlinear Effects Thermal Strain Effects Yes Yes Statistics Nodes 1 886 845 Elements 1 98 92 5.26

Fig. 5.17 Meshed model of mechanism with flexible crank and coupler Table 5.7 Detail of finite element model with flexible crank and coupler Object Name LINK1 LINK2 LINK3 LINK4 Stiffness Behavior Rigid Flexible Rigid Reference Temperature By Environment Material Assignment Nonlinear Effects Thermal Strain Effects Aluminum Alloy Yes Yes Statistics Nodes 1 904 924 1 Elements 1 114 102 1 5.27

Fig. 5.18 Meshed model of mechanism with flexible crank, coupler and rocker Table 5.8 Detail of finite element model with flexible crank, coupler and rocker Object Name LINK1 LINK2 LINK3 LINK4 Stiffness Behavior Rigid Flexible Reference Temperature By Environment Material Assignment Nonlinear Effects Thermal Strain Effects Aluminum Alloy Yes Yes Statistics Nodes 1 904 886 845 Elements 1 114 98 92 5.28

5.3 Modeling of Watt s mechanism The strain developed in links of Watt s mechanism (six bar planar mechanism) has been investigated using the ANYSYS. In this analysis two links are to be considered as flexible with numbers of beam elements. Specifications of six bar mechanism for analysis are mentioned in Table 5.9 and mesh model of six bar mechanism is shown in Fig. 5.19. Table 5.9 Specification of six bar mechanism Parameters Link1 Link 2 Link 3 Link 4 Link 5 Link 6 Length (mm) 250 110 280 260 270 200 C/S Area (mm 2 ) - 108 40 40 45 40 Area moment of Inertia (mm 4 ) - 160 9 9 10 9 Modulus of Elasticity, E = 7.1 10 4 MPa Density = 2770 kg/m 3 Crank speed = 32.3 rad/sec Fig. 5.19 Meshed model of six bar mechanism 5.29

Table 5.10 Detail of finite model of six bar mechanism Object Name LINK 1 LINK2 LINK3 LINK4 LINK5 LINK6 Stiffness Behavior Rigid Flexible Rigid Flexible Rigid Reference Temperature By Environment Material Assignment Aluminum Alloy Nonlinear Effects Yes Yes Thermal Strain Effects Yes Yes Properties Volume (mm³) 1.0015e+005 22451 37872 3.9238e+005 26751 28067 Mass (kg) 0.27742 0.06219 0.1049 1.0869 0.074101 0.077746 Moment of Inertia Ip1 (kg mm²) Moment of Inertia Ip2 (kg mm²) Moment of Inertia Ip3 (kg mm²) 227.69 4.4632 6.2678 5411.9 4.6573 5.2736 5129.4 124.38 1036.6 2914.7 799.88 839.61 5352.4 122.08 1032.5 8308.5 797.21 836.42 5.30

Object Name LINK 1 LINK2 LINK3 LINK4 LINK5 LINK6 Statistics Nodes 1 152 1 240 1 Elements 1 42 1 88 1 Mesh Metric None 5.31