Model Building and Analysis of the Slider Crank Mechanism Using Hyper-Works
The four-bar mechanism has some special configurations created by making one or more links infinite in length. The slider-crank (or crank and slider) mechanism is a four-bar linkage with the slider replacing an infinitely long output link. This configuration translates a rotational motion into a translational one. Most mechanisms are driven by motors, and slider-cranks are often used to transform rotary motion into linear motion. In this project a slider crank mechanism shown below is simulated in Hyperworks by giving different inputs like angular velocity, force, BISTOP torque to the links.
•To construct this mechanism, 3 points namely P0(0,0,0), P1(0,-10,0) and P2(-20,0,0) were created using the points tools.
•After creating these points, two bodies are added; body0 would be situated between P0 and P1 and body1 would be situated between P1 and P2.
•One thing that needs to be ensured is that the gravity is set to act in the negative Y direction with the length set to metres.
•Next, 2 cylinders are added as graphics whose diameters are set to 2 metres. These cylinders are then assigned to the respective bodies as shown above. The mass and inertia properties of the two bodies are then extracted from their associated graphic cylinders.
•After creating the points, bodies and adding their associated graphics, lower pair joints were added. Joints are models of interconnections between components. A lower pair is an ideal joint that constrains contact between a point, line or plane in the moving body to a corresponding point, line or plane in a fixed body or another moving body. These have physical analogies (e.g., a revolute joint).
• Two revolute joints were added to the points P0 and P1 and were aligned in Global Z direction. This completes the model building process for the slider-crank mechanism.
After constructing this mechanism, the model was checked for the degrees of freedom it has by going to Tools>Check Model>DOF. The number of degrees of freedom can be verified by hand calculations in the following manner:
Each Rigid Body has 6 DOF thus making it a total of 6 DOF.
No of DOF that is removed by one revolute joint> 3 translational and 2 rotational> total of 5 DOF
2 revolute joints thus remove 12 DOF
No of DOF of the system not counting redundant constraints: 12 - 10= 2 DOF
With the point C not being constrained by any joint, the mechanism starts to oscillate randomly which is evident from the video below.
1) The first thing that can be done is to constrain the model so that the system becomes a kinematic one. Since the system has 2 degrees of freedom, a joint which only provides 2 degrees of restrain needs to be added. For this purpose an inline joint aligned to Global X axis is added at point C.
Each Rigid Body has 6 DOF thus making it a total of 6 DOF.
No of DOF that is removed by one revolute joint> 3 translational and 2 rotational> total of 5 DOF
2 revolute joints thus remove 12 DOF
No of DOF removed by an inline joint: 2 translational DOF
No of DOF of the system not counting redundant constraints: 12 - 10 - 2= 0 DOF
2) In this exercise, an angular velocity of 0.5 rad/s was applied at joint 0. The DOF of the system is still 0 as a system can only be constrained by adding joints, constraints or motion to the joint. This can be verified by going to the Check Model option.
An output is added between Body1 and Ground Body to measure the velocity of point P2. The mechanism was then simulated for 12 seconds. With Point C constrained by the inline joint, Body 0 rotates about Point 0 and point C behaves like a slider now. The vertical link oscillates in both the +X and -X directions (goes even beyond the mean position).
The maximum velocity obtained at point C from the velocity vs time graph is 5.02 m/s.
3) In this step, the angular velocity applied at joint 0 was removed and replaced by a force of 1000kN in the negative x-direction applied as an Action-Reaction force at Point 1. An additional output was added to measure the rection torque at Point 0 using the VTORQ function. The syntax for the VTORQ function is: VTORQ (id, jflag, comp, RM)
Arguments:
Id - The ID of the Motion_Joint or Motion_Marker element.
Jflag - jflag equal to 0 or 1 means that forces and moments are reported at the I or J marker, respectively.
comp -
1 - returns the force magnitude.
2 - returns the force x component.
3 - returns the force y component.
4 - returns the force z component.
5 - returns the torque magnitude.
6 - returns the torque x component.
7 - returns the torque y component.
8 - returns the torque z component.
RM - The reference frame in which the components are reported; RM=0 implies the global frame.
The force being applied in the negative X direction pushes the body 1 towards the left side instead of making it follow an arc towards the right side. The vertical link or body 0 now starts to oscillate and due to the reaction force the model recedes backwards before settling to its mean position.
The plot in blue shows the displacement of body 1 due to the applied force while the red plot shows the displacement of body 1 due to angular velocity. Notice how body1 is displaced even below its mean position due to the applied force whereas when there\'s only angular velocity applied the body is not displaced below its mean position.
Since there is no force, torque or motion applied at point0, there will be no reaction torque produced there. The plot below makes it evident.
4) Keeping the force still applied at point1, a BISTOP Torque is added to point0. The BISTOP function models a gap element. It can be used to model forces acting on a body while moving in the gap between two boundary surfaces. This can be used to limit the motion of a joint.
The syntax of this function is: Bistop(`x`,`x^.`,`x_1`,`x_2`,`k`,`e`,`c`,`d`)
`x`- independent variable
`x^.`- derivative of the independent variable
`x_1` - The lower bound of x -10°
`x_2`- The upper bound of x 10°
`k`– Stiffness of the boundary surface interaction
`e`– Exponent of the force deformation characteristic
`c`– Maximum damping coefficient
`d`– Penetration
The AZ and WZ functions are used as the independent variable and the derivative of the independent variable respectively within BISTOP function. The AZ and WZ measure the rotational displacement and velocity, respectively of a marker “I” with respect to the marker, “J”.
- AZ(I, J)
I - The marker whose rotational displacement is to be computed
J - The marker with respect to which the velocity is to be computed
- WZ(I, J, K)
K - The resultant velocity vector is resolved in the coordinate system of the K marker
The syntax used: BISTOP(AZ({`j_0`.joint_i.idstring},{`j_0`.joint_j.idstring}),WZ({`j_0`.joint_i.idstring},{`j_0`.joint_j.idstring}),-10*`pi`/180,10*`pi`/180,`10^4`,2,100,0.1)
The reaction torque due to the BISTOP Torque measured at Point0 was -2500Nm. A comparative study was also done to see how the last four arguments namely k,e,c and d within the BISTOP function affect the reaction torque.
In the comparison plots below, the plot in blue is for the original arguments and the plot in red is for the modified arguments.
• Changing the stiffness of the boundary surface interaction from 1e4 to 9e4: The reaction torque obtained here is in the magnitude of 22500Nm meaning that the system is pretty stiff and thus there is so much reaction torque acting on P0. The shape of the graphs are similar with the only difference being in the magnitude of reaction torque.
•Changing Exponent of the force deformation characteristic from 2 to 1.5: It was found that this argument can have a significant effect on the reaction torque being produce at P0. Increasing this value by 3 times leads to a tremendous decrease of torque from -2500Nm to below -100Nm. Thus the value of the exponent deformation was decreased instead to see the consequent increase in reaction torque.
•Changing the maximum damping co-efficient from 100 to 10000: Increasing this argument by a factor of 100 does not lead to a very significant increase in reaction torque but it changes the smooth contour of the plot to an irregular one. Close to -111Nm the reaction torque flats out and stays like that for a longer duration of time.
The animation below traces reaction torque at Point0 of the slider-crank mechanism. Notice how in all the above three cases, the mechanism slows down or moves faster at different intervals with respect to the originally defined set of arguments: