# Optimisation Of The Slider Crank Mechanism Information Technology Essay

4035 words (16 pages) Essay in Information Technology

5/12/16 Information Technology Reference this

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The slider-crank is a basic rotary to linear mechanism. It is used for generating, reciprocating motion as in a motor this application is not feasible and direction reversibility is not opted. The slider-crank mechanism is used in many real systems like automobile sector and mechatronic applications. Optimizing design of any mechanism is one of important diligences of any mechanic application. The research in optimum design of mechanisms is not restricted to mechanical systems but also to electro mechanical or mechatronic system. These optimization methods are employed in the design of rigid-body systems, compliant mechanisms, and robotic functionalities, which are adopted from linear programming and interior point systems. Here we use the simplex method to perform this optimization for mechanical benefit on Slider crank mechanism. This introduces a nonlinear relation between the input and output variables in the transmission, and consequently, data-processing equipment and complex control algorithrns are required for the proper operation of the whole system.

## Introduction

The simplex-based technique does simulation testing at the vertices of the initial simplex; which is a polyhedron in a given space having N+ 1 vertex. A subsequent kind of polyhedron that is progressing towards the required optimality) is formed by three operations that are performed on the current simplex polyhedron which are the process of reflection, contraction, and expansion. At each intermediate resultant level of the search process, the point with the highest J (v) value is replaced with a new point found via reflection through the centroid of the polyhedron. This value of the polyhedron is processed up on by contraction expansion or not doing any change in the polyhedron. This simplex technique begins with a set of vertices. These points are all the same distance from the point under the current set of observation. Moreover, the distance between any two points from these set of vertices is the same. By comparison of their response values, the elimination of the factor setting least value is carried out in the centroid of the polyhedron. The resulting polyhedron either enlaces or collapses, depending on the response value at the new vertex factor. The procedure is repeated until no more elimination related value enhancement and substitution is possible and the resultant polyhedron is at it optimum smallest value. These techniques will generally perform with more efficiency for unconstrained problems; it may collapse to a point on a boundary region, thereby causing the search to come to an abrupt halt. This technique is effective for local optimal points. Hence to overcome this, use of slider-crank mechanism introduces a configuration-dependent relationship. Slider-crank mechanisms are widely used in reciprocating machinery, where transformation from rotational into translational motion (or vice versa) is required. The use of these mechanisms is quite common in robotic and mechatronic systems when complex motions are to be produced with rotational actuators. However, the velocity ratio of the slider-crank mechanism is configuration-dependent and thus, Elaborate algorithms are required to precisely control its performance. A methodology for the optimization of this mechanism is developed. First the performance of the slider-crank mechanism is analyzed and optimum geometric parameters are obtained. Then, an expression for the input-output relation of the application mechanism at hand is derived, and the corresponding displacement program of the follower is produced.

SLIDER-CRANK MECHANISM

Fig.1 shows a schematic model of the slider-crank mechanism. The slider is restricted by its direction of motion on the x-axis and the wheel-coordinate of the rotation axis is fixed at the origin. When the force is applied to the slider, the connecting-rod drives the wheel and this mechanism rotates the wheel. Thus, this mechanism can convert the linear forces into the rotational torque.

Fig. 1 Slider-crank mechanism

A mechanism is a device used to produce mechanical transformation in a machine. The apt crank-connecting rod & slider mechanism is used for converting rotational motion to linear piton like motion. The following equations describe the kinematics of this mechanism:

Where

L = connecting rod length

R = crank’ radius

Ï‰ = crank constant angular velocity

X = distance determined from crankshaft center

Xâ€² = velocity of slider

Xâ€³ = acceleration of slider

Î¸ =crank angle anti clockwise Ï† = connecting rod angular position; Ï†=0 when Î¸=0

Ï†â€² = angular velocity of connecting rod

Ï†â€³ = angular acceleration of connecting rod

All lengths are in the same units.

The dimensioning and selection of this mechanism is thus crucial in the design of the overall transmission. The slider-crank mechanism transforms rotational into translational motion (or vice versa): it is, therefore, widely used in reciprocating machine)” such as piston engines, compressors, pumps, saws, etc.

## Simplex method

The simplex method is a method for solving problems in linear programming. This method, invented by George Dantzig in 1947, tests adjacent vertices of the feasible set in sequence so that at each new vertex the objective function improves or is unchanged. The simplex method is very efficient in practice, generally taking to iterations at most (where is the number of equality constraints), and converging in expected polynomial degree and its time. For certain distributions of random inputs. However, its worst-case complexity is exponential, as can be demonstrated with carefully constructed examples.

A different type of methods for linear programming modules and problems are interior point constituent methods, whose complexity is polynomial for both average and worst case. These methods construct a sequence of strictly feasible points that converges to the solution. Research on interior point methods was spurred .In practice; one of the best interior-point methods is the predictor corrector evaluation which is competitive with the simplex method, particularly for large-scale problems.

Dantzig’s simplex method is different with the downhill simplex methodology. The latter method is an application of the unconstrained minimization case in multiple n dimensions which are by far maintained at each iteration of n+1points that ultimately define a simplex. After each iteration, this simplex is updated by the application and iteration of specific transformation methodology to it so that it keeps on iterating until it finds a minimum.

## Method

The initial value of the set of the polyhedron vertices is taken and the count of the initial variables and their count are taken. An initial vertex set of v is created and the initial values of the polyhedron are noted down. Value evaluation of the same is done to obtain the centroid of the same. The selection method of best fit vertices is applied on the same and the selected ones are inserted into a set of vertex count domain. The new vertex count i obtained after crossing over with the new current adaptation. After this analysis is performed the set is updated again. A crossing over function is applied and each set is mutated after the process of updating the optimality ratio. A checking function is deployed to check for the best case ratio and the optimality condition. Based on this the fitness value is deployed using the simplex algorithm. After this process the search space shrinks to a sub optimum value and the polyhedron is adapted to a new set of vertices. Else increase for the next value of v and continue the same algorithm adaptation. From this the best slider crank positionary individual vertex value is found out and also we have a set of constraints for the values to fit.

FLOWCHART

## Initialize the variables and the vertices count

## Create an initial vertex set.

## Evaluate their values

Apply a selection method to choose vertices

And insert them into the set of vertex count

Mutate the current vertex set after updating to reach optimality ratio.

Crossover the new count by current adaptation

Update the current set.

## Whether set is optimum and best case

## to the best individual?

No

V=v+1

Minimize the fitness function and the vertex value with the simplex algorithm.

Replace the vertex set with the

New value and shrink the search space.

Terminate?

The optimization algorithm completed.

Select the best individual of the final

Set of constraint values.

It is apparent from that force and motion are transmitted to the load by means of a slider-crank mechanism. The dimensioning and selection of this mechanism is thus crucial in the design of the overall transmission. The slider-crank mechanism transforms rotational into translational motion (or vice versa): it is, therefore, widely used in reciprocating machine)” such as piston engines. Compressors, pumps, saws, etc.

In-Line Slider-Crank Mechanism

The maximum value for s is attained when the slider lies farthest from the centre of rotation of the crank, and its minimum value when it lies closest to the centre of rotation namely, MAX = LI + L2 and MIN = LI -L2 respectively. Taking this into consideration, the inequalities yield the condition L1 >L2. In order to define the range of motion of the crank from where we choose for 4Jo the lowest value of 4> for which J-l = 1350 and for t/Jf the highest value for which J-l = 450â€¢ Therefore, the range of motion can be set to be22.5°::; fj: J::; 130.5°; we thus have all the necessary information to compute the transmission quality for different values of r.

In this picture a crank CA is constrained to rotate around the point C. Link AB is the connecting rod and B is the slider that slides along CE. We have fixed lengths AC = a and AB = b. The length CB = x varies as the crank is moved. When CA is rotating around C we can describe its position with respect to the angle ACB = that we measure from CB counterclockwise from CB to CA . Let us use some geometry and basic trigonometry here. Let us construct AD perpendicular to CB; then, from the right triangle ACD, we can determine

Note that when > 90° then D is to the right of C and CD is negative and when > 180° then A is above CE and AD is negative. Also, notice that

## .

By the Pythagorean Theorem,

## .

Thus,

## .

## 2. Offset Slider-Crank Mechanisms

A procedure similar to the one for the in Line type will be followed to analyze the offset slider-crank mechanism. We derive an expression for the output variable o in terms of the input variable d

Hence, the output variable can be obtained as a function of the input variable from the real roots of this quadratic expression, with the possible output values for given e and s defining the two conjugate configurations of the linkage.

## ,

Where d = DD’ is the distance of C from the straight line constraint for the slider. Also, notice that

Hence, the output variable can be obtained as a function of the input variable from the real roots of this quadratic expression.

By the Pythagorean Theorem, .

Thus,

Finally we can compare the two types of slider-crank mechanism, i.e. the in-line and the offset mechanisms, although it is apparent that the in-line mechanism has a lower transmission defect than the offset mechanism (and, consequently a higher transmission quality). Due to this slight difference in the performance of the in-line and the offset arrangements of the slider-crank mechanism, the use of one or the other is equivalent, with the in-line linkage being the most frequently used. As in the case of the in-line slider-crank mechanism, the performance of the offset linkage depends directly on

the ratio r; therefore, it is important to determine which value of this relation optimizes the performance of the linkage. For the offset slider-crank mechanism the value of the offset is considered to be the maximum possible, i.e., K = r – 1.

## Results

Two numerical methods are proposed to obtain the integral of the ODE at hand, namely, i) the direct numerical integration of the ODE by means of a Runge-Kutta method, and ii) a continuation method based on the numerical solution of the quadratic polynomial. To this end, the following recommendation should be considered:

â€¢ The length of the links of the slider-crank mechanism: L1 and L2 although the selection of the dimensions of the linkage are case-dependent~ it is convenient as seen to keep the ratio T = ll/l2 in the neighborhood of 3.

â€¢ The pitch of the screw: p. It is common knowledge that the torque T required to produce a force F parallel to the displacement of the nut can be computed.

It is apparent that the smaller the pitch the less torque is required~ which is desirable. However, since the angular displacement of the follower driving the screw is limited, the pitch must be selected so that a reasonable stroke of the nut is permitted. Hence, this parameter is also case-dependent, and a balance between the requirements of torque and displacement of the nut must be sought.

The results obtained when applying the two methods were essentially the same for bath are identical up to the 13th decimal place. Hence, the accuracy of the solution can be presumed.

## Discussion

Slider-crank mechanisms are widely used in reciprocating machinery, where transformation from rotational into translational motion (or vice versa) is required, the use of these mechanisms is quite common in robotic and mechatronic systems when complex motions are to be produced With rotational actuators. However, the velocity ratio of the slider-crank mechanism is configuration-dependent and thus, elaborate algorithms are required to precisely control its performance.

There are a plenty of numerical and mechatronic simulations using the slider crank optimization method of evaluation. Here we consider a system of vehicle and pedestrian system in real life CAE which are used to develop a detailed understanding of how pedestrian injuries relate to documented vehicle damage in a given scenario of mechanical sabotage or damage. In relation to the complexity of the mechanism, modeling the simulations of this slider crank typically involves objective evaluations of the pre-impact conditions using a limited number of constraints and optimized value of the simulations. The goal of this study is to develop a robust methodology for analyzing the pre-impact pedestrian positioning in relation to the angle of inclination and vehicle speed utilizing positional values and optimization of these simulations techniques. It can be applied as a deflector device for motor vehicles, where the front part of the machine which comprises a left-hand and right-hand beam, is supported and set up before a front wheel and configured by a deflector those coming out from the longitudinal beam in a considerably horizontal manner and backward at a specific curve to protect the discussed part and the aid wheel or the vehicle under consideration in the case of a collision. The simulated deflection unit prevents the vehicles from being stuck and being pushed off the road in the event of an expected pedestrian and vehicle overlapping frontal collision. The simulated deflection unit comprises a deflection element which can be displaced in a longitudinal direction from a position under rest to an operational position, and the deflection unit consisting of a slider guided on or in the crank and a forward direction stilt. The part projecting forward from the deflection element is a direction stilt, the location at which it is connected to the deflection unit being counterbalance from the external end of the latter toward the centric position of the vehicle. It is for an instance that counterbalance so far inward that the connecting location is somewhat in front of the front vehicle. Hence it is so that, even the direction stilt acts on the front vehicle of the other element involved in the collision. This is so in particular if the base of the direction stilt is aligned such that, in the rest position and at the beginning of a possible accident or collision, it is in a transverse direction to the traveling parameter. The direction stilt is preferably squeezable, in order that the contortion of the slider is delayed in the case of a vehicle pedestrian collision. A continuous sequence of the pedestrian features based on the existing data and simulations may be developed for use as a design parameter during the simulation process after optimization. Then, the robustness and efficiency of two optimization algorithms were evaluated in mock crash situations. The pre-impact values of the pedestrian and the vehicle models were taken up as unknown design variables for the purpose of validating the optimization technique. While all algorithms found solutions in close vicinity of the exact solution. The resultant data noted showed a more sensitive to the non vehicular posture and its relative position with respect to the mechanical making than to the speed for the chosen design analytics. In the post validation process, the simulation with the mock reconstruction, a true situational dimensional scenario was re enabled using the data obtained from the mechanism vector analysis and the optimization mapping scheme. A set of pedestrian and vehicle initial conditions capable of matching all observed contact points was determined. Based on the hypothetical and real-world simulations, this study indicates that mechanical slider crank simulations in complementation with optimization algorithms can be used to analyze and provide a mapping pedestrian and vehicle damage and pre impact situational conditions. Determining the minimum and maximum allowable speed reduction should also be given due attention.

The parameters under consideration are:

Data: This menu allows for the selection of the user data. When selected, list box containing three options will be displayed: the aforementioned options are MATLAB files that the user must program to generate the information required for the follower curves.

Follower curves: Once an option has been selected from the Data menu, the user will be able to see, in the plotting area, the corresponding follower curves by entering this menu, Le., a list-box will appear from where the options Displacement, Velocity. Acceleration, and Jerk, can be selected. If Displacement is selected, the menus Rise phase and Return phase will be enabled; otherwise they are inactive.

Rise phase: When selected, the displacement program on the plotting area will be divided by vertical dashed lines, indicating the phases of rise. Then the user will specify through a list-box, which phase to analyze.

Return phase: This menu is similar to the previous one. The return phases will be indicated on the displacement program; the user must select one.

OK: If the user data rise and return phases, and pressure-angle bounds are properly selected, then the user must press this button to continue the design procedure: otherwise, the aforementioned information can be changed at anytime.

Optimize: If solutions exist for the given problem, the user may continue by selecting values. After selecting Optimize, a point must be selected with the Points of the solutions.

## Data

## Follow curve

## Displacement

No

Rise phase Yes

## Return phase

## Angle bound

## Ok

Change No

Solution?

Yes

No

## Yes

## Optimize

## Minimum value

Mechanism Flowchart

From the above flowchart it can be inferred that input data vertex position of the pedestrian is taken in as the initial input parameter and the curve is followed. In case of a displacement, the rise phase of the crank and its return phase angular motion is calculated from the linear displacement that is recorded. The angular bound of the same is also found out .In case of no displacement the loop re trace back to detect any displacement. Post calculation of this phase and the angular bounds any anomaly or specific data change I inferred. If any kind of change I found, the solution feature of it is established. If it appears to be the apt solution it is applied to an optimization algorithm and the minimum value is recorded, else for all the change recorded, processing takes place to determine the final expected out come of this slider crank data set. A methodology for the optimization of this mechanism is developed. First the performance of the slider-crank mechanism is analyzed and optimum geometric parameters are obtained.

This method anticipates the existence of solutions it cannot predict if the extreme values of the simulation will be global or local! As it can be learned, this however does not represent a real disadvantage because all possible solutions can be verified by the optimizer within a few seconds. Determining the minimum and maximum allowable speed reduction should also be given due attention. Further research is needed regarding the separation phenomenon: i.e.: the separation of members of the optimization mechanism: if the vehicle is to be driven at high speeds. Also, the issue of contact stress between the vehicle and the pedestrian needs to be analyzed. The range of motion of the crank link of the slider-crank mechanism cans be enlarged by means of an amplifier mechanism. However, the practicality and complexity of the resulting system is worth further investigation.

Slider crank application

Mechatronic applications study

Contrary to industrial robots, that is installed on a fluxed base, and hence, uses conventional industrial facilities for their power supply and their controls, mechatronic systems, such as walking machines, rovers, and the like, call for autonomous operations. Mechtronic systems, thus, carry their own power-supply)’ and control subsystems. In these cases, then, weight and speed of response become crucial design criteria. To ease the control of these systems, and consequently, to lower the demands on the control hardware, we propose to rectify the configuration-dependent velocity ratios in their transmissions, by means of simple and reliable mechanisms. It can be argued that the addition of a transmission stage will offset the benefit of a constant velocity ratio. Nevertheless current research work at the Robotic Mechanical Systems Laboratory of the aims at developing mechanical transmissions that integrate, in one single unit, two functions, speed reduction and speed rectification. Therefore, the transmissions that we are proposing here will be an integral part of the actuator. In other words, we aim at systems that will do away with conventional gear trains for speed reduction, while replacing these with more efficient, stiffer, and more reliable and lighter multifunction transmissions.

## Conclusion

A methodology for the design of Optimisation of the Slider Crank Mechanism Using Simplex method was formulated to simplify the control of slider-crank mechanisms used in robotic and mechatronic systems. An important index in linkage optimization, that measures the performance of the linkage globally as opposed to locally, is the transmission quality. The transmission quality “, as defined by Angeles and Bernier (1987-a) as a positive-definite quantity Mechanism to rectify this nonlinear behavior seerns to be a novel Idea, and is claimed to be a major contribution an in-line or an offset slider-crank linkage should be used to ensure optimum force-and-torque transmission characteristics. By comparing the transmission quality of the two types of arrangements it was apparent that the use of one or the other is equivalent; because of its compactness the in-line configuration was selected. Hence, two numerical approaches were proposed. This was successfully used to complete the analysis of the optimum system to reduce and rectify the given slider-crank mechanism by simplex method. The combination of cam mechanisms with linkages has been used in the past to improve their independent performance or to produce motions with suitable dynamic behavior. In future study the detailed analysis of the dynamics of the whole transmission device is required to determine the effects of masses and forces in the reducer-rectifier cam mechanism can be initiated. Further research is needed regarding the separation phenomenon: i.e.: the separation of members of the cam mechanism.

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