Mechanics of Machines project

8/15/2019 Mechanics of Machines project

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MCB3043 Mechanics of MachinesSemester May 2014

SLIDER-ROCKER MECHANISM DESIGN PROJECT

GROUP 8

1. TAN HUAT CHAI 15886

2. TAN KE WEI 16330

3. LEE WEN CHING 16222

4. HEA YIH TORNG 16251

5.

KAM CHONG PING 16474

Lecturer: Azman Bin Zainuddin


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1.0 Introduction

1.1 Aim

To design a slider rocker mechanism driven by an electric motor turning at constant speed and direction

with the given parameters and requirements.To obtain 2 output motions from this mechanism which are:

A linear reciprocating motion of a slider along the horizontal axis with a stroke of S. The expected

tolerance is ± 0.5 mm.

A rocking motion with a throw angle of θ degrees. The expected tolerance is ± 1°.

To obtain the velocity of the slider with the maximum velocity between the given VMIN and VMAX mm/s.

1.2 Description

Crank Slider Mechanism

The Slider-crank mechanism is used to transform rotational motion into translational motion by means of

a rotating driving beam, a connection rod and a sliding body. The sliding mass is not allowed to rotate and

three revolute joints are used to connect the bodies. While each body has six degrees of freedom in space,

the kinematical conditions lead to one degree of freedom for the whole system.

A slider crank mechanism converts circular motion of the crank into l inear motion of the slider. In order

for the crank to rotate fully the condition L> R+E must be satisfied where R is the crank length ,L is the

length of the link connecting crank and slider and E is the offset of slider . A slider crank is a RRRP type of

mechanism i.e. It has three revolute joints and 1 prismatic joint. The total distance covered by the slider

between its two extreme positions is called the path length. Kinematic inversion of the slider crank

mechanisms produce ordinary a white work quick return mechanism.

Figure 1.1 represents il lustration of crank slider mechanism. Does not relate to text.


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Crank Rocker Mechanism

In a four bar mechanism the link on which input motion is applied is known as driver. The output motion

link is fol lower and the middle link connecting these two links is coupler link. The fourth link is grounded.

A four bar mechanism has 1 degree of freedom.

If the input link can rotate fully it is called as crank. A link that can't rotate fully is known as rocker.

Grashof's Theorem :

If S is the smallest link, L is the largest link and P,Q are the intermediate links ,then there will be at least

one link which will be able to revolve/rotate if S+ L< P+Q. The four bar mechanisms that satisfy this criteria

is known as Grashofian mechanism.

Figure 1.2 represents il lustration of crank rocker mechanism. Does not relate to text.

The Slider Rocker Mechanism

A slider rocker mechanism is a 6 linkage mechanism on which input motion is applied by the crank. This

system is a combination of both the crank slider mechanism and the crank rocker mechanism. The crank

is placed in the middle with one end the rocker and the other end is placed the s lider with both having a

coupler of their own to link them to the crank. The crank then powers the movement of both these moving

parts, the rocker and slider, to move simultaneously at the same direction located opposite to one another.

This combined mechanism satisfies both criteria which are the:

1. Grashof's Theorem :

If S is the smallest link, L is the largest link and P,Q are the intermediate links ,then there will beat least one l ink which will be able to revolve/rotate if S+ L< P+Q. The four bar mechanisms that

satisfy this criteria is known as Grashofian mechanism.

2. Crank slider mechanism lengths ratio

In order for the crank to rotate ful ly the condition L> R+E must be satisfied where R is the crank

length ,L is the length of the link connecting crank and slider and E is the offset of slider .


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1.3 Requirement

To sum up the requirement given by our lecturer, we formed a table as shown below. Kindly note that we

separate all the requirements into slider-crank mechanism and crank-rocker mechanism so that we can

obtain a clearer picture of what our lecturer wants.

Slider-Crank Mechanism Crank-Rocker MechanismStroke, S = 120mm +- 0.5mm Throw Angle, = 30⁰ +- 1⁰

1300mm/s ≤ Max Velocity of Sl ider ≤ 1500mm/s 45o ≤ Transmission Angle, ≤ 135o

3.5 < Coupler Ratio < 6

Time Ratio ≥ 1.1

2.0 Methodology

In this project, the mechanism was separated into two parts, which are slider-crank mechanism and crank-rocker mechanism. The objective of this separation was to ease the calculation and design of our

mechanism.

Crank-rocker

In designing crank-rocker mechanism L1 and L4 were randomly selected to be used in drawing. Then the

limiting positions of the mechanism were drawn using graphical method. After the drawing was done,

simultaneous equation was applied to obtained length links of L 2 and L3. In assuring the transmission angle

is between 45 and 135, they were calculated by using equation as stated bel ow.

Minimum transmission angle, min = cos-1

+ –(− )

(Eq 2.1)

Maximum transmission angle,max = cos-1 [ + −( + )

] (Eq 2.2)

With the value of link lengths obtained from the drawing and transmission angles obtained from the

calculation, designing process was proceeded to crank-slider mechanism.

Slider-crank

In designing crank-slider mechanism, there were a few constraints that need to be followed. There are as

stated below.

1. Stroke must be 120mm

2. Maximum slider speed must be between 1300mm/s and 1500mm/s

3. Coupler ratio must be between 3.5 and 6

4. Time ratio must be larger than 1.1


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First of all, a suitable coupler ratio based on the specification was selected to be used in the designing

process. By using the selected coupler ratio, a calculation by using formula as stated below was used to

obtained value of link length, L3.

Coupler Ratio =

(Eq 2.3)

Link length, L2 that was obtained previously in crank-rocker drawing was used in above calculation to get

value of L3. After that, both limiting positions were used to solve the mechanism by using trigonometric

equation and simultaneous equation. At the end of the calculation, the value for leng th L1 was obtained.

Last but not least, in order to clarify whether the design fits the specifications given or not, it was tested

by using time ratio equation as stated below.

Time Ratio = °+ °− (Eq 2.4)

The imbalanced angle of β was calculated by using trigonometric equation.Following from that, the link lengths and all the calculated data were then tabulated into excel to further

calculate the angles, displacements, velocities and accelerations. The equations used for such working are

displayed below:

θ3 =sin−[+ ] (Eq 2.5)

L4 = L cosθ L3 sinθ3 (Eq 2.6)

γ = 180° (θ θ3) (Eq 2.7) BD= √L L 2(L)(L) cos(θ) (Eq 2.8)

γ = cos− + −()() (Eq 2.9)

θ3 = 2t a n− − + +− − (Eq 2.10)

θ4 = 2t a n− + −++ − (Eq 2.11) Including Eq 2.1 and Eq 2.2 as stated previously


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3.0 Result

3.1 Link Lengths Calculations

Knowing that we are given full freedom in the length of links, we made our choice to set L 1 = 300mm and

the rocker length, L4 = 180mm. Then we draw the limiting positions of the mechanism using graphicalmethod as shown below.

Then, by measurement, we got to know that L3+L2 = 19.3cm and L3-L2 = 15.2cm. Using simultaneous

equation,

L2 = 41mm

L3 = 345mm

Next, to ensure that the transmission angle is between 45⁰ and 135⁰, we calculate them in both l imiting

position as shown below:

min = cos-1 [ + −(− )

]

= 47.23⁰

max = cos-1

[ + −(+ )

] = 73.56⁰


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After getting a set of link lengths and angles, we moved on to the crank-slider section. Based on the given

requirement:

1. Stroke must be 120mm

2. Maximum slider speed must be between 1300mm/s and 1500mm/s

3. Coupler ratio must be between 3.5 and 6

4. Time ratio must be larger than 1.1

Taking these into account, we first took the crank length, L2 from crank rocker design which is 41mm. Then

we set our coupler ratio to be 3.6. The reason is that we want our mechanism to be as mobile as we can

and besides, avoid all the joints from experiencing additional joint forces which will sho rten the lifespan

of our model.

Coupler Ratio =

3.6 =

4mm

L3 = 147.6mmNext, we solve the mechanism by listing out the equation formed by both limiting positions:


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Substituting Eqn.3 into Eqn. 1:

L12 = 21169.96 – 240X2 – (X2)

2 (Eqn. 4)

Substituting Eqn. 4 into Eqn. 2:

(106.6)2 = 21169.96 – 240X2 – (X2)2 + (X2)

2

X2 = 40.86mm (Eqn. 5)

Substituting Eqn. 5 into Eqn. 2:

(106.6mm)2 = L12 + (40.86mm)2

L1 = 98.458mm

Next, we need to check if the results we calculated are able to generate a time ratio of more than 1.1.

Thus, we need to first find the i mbalance angle, β and then use the time ratio formula as shown below:

Time Ratio = °+ °−

(120mm)2 = (106.6mm)2 + (188.6mm)2 - 2(106.6mm)(188.6mm) cos β

β = 35.99⁰

Time Ratio = 1.5 > 1.1


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Following from that, the link lengths and all the calculated data are then tabulated into excel to further

calculate the angles, displacements, velocities and accelerations.

Kindly refer to Microsoft Excel and MSC Adams for full details.

3.2 Diagrams

3.2.1 The scaled kinematic diagram of the mechanism showing the position when the slider is at the

middle of its stroke.

Figure 3.1 showing the 1cm: 30mm scaled kinematic diagram of rocker-slider mechanism drawn using

graphical method.


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3.2.2 The slider displacement curve for a single cycle.

3.2.3 The slider velocity curve for a single cycle.

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0 20 40 60 80 100 1 20 1 40 1 60 1 80 2 00 2 20 2 40 2 60 2 80 3 00 3 20 3 40 3 60

displacement (mm) versus crank angle

displacement

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0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

velocity (mm/s) versus crank angle

velocity (mm/s)


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3.2.4 The slider acceleration curve for a single cycle

3.2.5 The rocker angular displacement curve for a single cycle.

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20000.00

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0 20 40 60 80 100120140160180200220240260280300320

acceleration (mm/(s^2)) versus crank angle

acceleration (mm/(s^2))

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0 20 40 60 80 100120140160180200220240260280300320340360

angular displacement (rad) versus crank angle

angular displacement


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3.2.6 The rocker angular velocity curve for a single cycle.

3.2.7 The rocker angular acceleration curve for a single cycle .

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0.00

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200.00

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10 30 50 70 90 110 130 1 50 1 70 1 90 2 10 2 30 2 50 2 70 290 3 10 3 30 350

angular velocity (rad/s) versus crank angle

angular velocity

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10 30 50 70 90 110 1 30 1 50 1 70 1 90 2 10 2 30 2 50 2 70 2 90 3 10 3 30

angular acceleration (rad/s^2) versus crank angle

angular acceleration


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4.0 Conclusion

4.1 The lengths of all links, the slider offset

Figure 4.1 showing the 1cm: 30mm scaled kinematic diagram of rocker-slider mechanism drawn using

graphical method.

Figure 4.2 To reconfirm accuracy, we use AutoCAD to redraw our kinematic diagram.


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Figure 4.3 Drawing taken from Adams software during simulation of mechanism. The slider is in the

middle of its range of motion.

Based on calculation and measurement, we obtained the following results:

Links Length (mm)

L2 (Crank) 41

Crank-Rocker L1 300

L3, coupler 345

L4, rocker 180

Slider-Crank L1, offset 98.458

L3, coupler 147.6

4.2 The stroke of the slider and the rocker

For the sl ider, we managed to make its stroke, S= 119.9mm, which satisfied the requirement. Meanwhile,our rocker is able to generate an angular stroke (throw angle) of 30.31⁰.

4.3 The crankshaft speed in rpm

After trying different crankshaft speed in the multiplication of 10rpm, we finally decided to use 180rpm

as it is able to push the sliding mechanism to achieve a Vmax of 1497.95mm/s. This achieved the objective

of the design.


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4.4 The time ratios of the slider and the rockers

For the slider mechanism,

To find the time ratio, we need to f irst find the imbalance angle, . Figure below shows 2 l ines produced

when the coupler and crank forms a straight line at both limiting positions. The third line (120mm) is the

path of sliding which is also known as the stroke of the sl ider.

Using Cosine Rule:

(120mm)2 = (106.6mm)2 + (188.6mm)2 – 2(106.6mm)(188.6mm) cos

= 35.99⁰

≈ 36⁰

Time Ratio =+ −

= 1.5

For the crank-rocker mechanism,

The imbalance angle, can be obtained as shown below:


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Time Ratio =+

−

= 1.091

Thus, time ratio of sl ider-crank mechanism is 1.5 while time ratio of crank-rocker mechanism is 1.091.

4.5 The coupler ratio of the slider

Coupler ratio =3 (upler legh)

(rk legh) = 47.6mm

4mm

= 3.6

Thus, the coupler ratio is 3.6 and it meets the requirement. This will ensure that no excessive force will be

experienced in the joints, therefore enhancing the lifespan of our mechanism.


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4.6 The range of transmission angles of the rocker

Based on the transmission angle calculated in Excel, the related table is shown below:

Thus, it can be clearly seen that our transmission angle is:

45o ≤ Transmission Angle, ≤ 135o

4.7 The working area required by the mechanism (in terms of L x W)

To calculate the working area needed by our slider-rocker mechanism, we have to first calculate the

maximum possible expansion of our mechanism vertically and horizontally.

In terms of vertical length, which is the height,

We can calculate i t simply by adding the L4, rocker and L1, offs et. This is because when the rocker is in the vertical

position, it expand the most vertically. Besides that, for the vertical length below the rocket pivot, wechose L1, offset because it’s the fi xed length from the pivot of rocker to the ground where the sl ider moves.

Thus:

Vertical Length = L4, rocker + L1, offset= 180mm + 98.458mm

= 278.458mm

0.00

10.00

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30.00

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0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

Transmission Angle vs Crank Angle


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Then speaking about the horizontal expansion, we can actually divide the mechanism into three portion

to analyze. This division can be shown below:

The left part is the horizontal length between the rocker pivot to the point where rocker can stretch the

most horizontally.

Hence, L4,rocker cos (4 min) = 180mm cos (75.76⁰)

= 44.28mm

The middle part is the horizontal length between the rocker pivot to crank pivot. It is L1, which is 300mm.

The right part is the horizontal length where the slider can stretch out the most to the right from the crank

pivot. Thus, we take the maximum value of L4, slider which is 160.84mm

Thus:

Horizontal Length = 44.28mm + 300mm + 160.84mm

= 505.12mm

Therefore, the size of the mechanism in terms of L x W is 505.12mm x 278.458mm. However, i f this value

is to be used for casing design, we wil l strongly recommend it to be 10% larger which is approximately

550mm x 310mm because the slight increment in area will prevent our mechanism from knocking our

casing due to tolerance. Besides that, slight increment of area will also not occupy much space and

increase much initial cost.

4.8 The maximum speed of the slider and the rocker

By installing a motor which has an operating speed of 180rpm, we are able to make the rocker to rock at

a maximum angular speed of 307.79rad/s. On the other hand, the maximum speed of the slider is

1497.95mm/s. Hence, the requirement of maximum slider velocity between 1300mm/s to 1500mm/s is

fulfilled.

Therefore, all the requirements are fulfilled.

Mechanics of Machines project

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