velocity in mechanisms

7/27/2019 Velocity in Mechanisms

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THEORY OF MACHINE BY MA HELALY 1

VELOCITIES

IN

MECHANISMS


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Introduction:Velocity

is important because:1- It affects the time required to perform a given operation.

2- Power is the product of force and velocity.

3- Friction and wear on machine parts are also dependent on

velocity.4- Further, a determination of the velocities in a mechanism is

required if an acceleration analysis is to be made.

Acceleration

is of interest becauseof its effect on inertia forces, which in turn influence stresses in the

parts of a machine, bearing loads, vibration, and noise


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VELOCITIESIN

MECHANISMS

BY METHOD OF

RELATIVE VELOCITIES


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LINEAR VELOCITY


Amplitude Direction

V

By scale from vector diagram Vc = 1.8


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Same triangle perp to original - opposite to link 2

OR Vc= VB + VC/B then find D from triangle

= constant


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Example 3:In the toggle mechanism shown in Example 2, obtain the velocity of the

ram F, The angular velocity of link EF.

Solution:

={2.=+

={4.?, ={3.?

==+={6.?, ={5.?

===+

={? , ={7.?


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7=.=12.57 .., =.=123.31


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Example 4:

For the Whitworth mechanism shown inExample 1, determine the absolute velocity of

the tool support C; also find the angular

velocities of links QB and BC.

Solution: Consider point T on link 4 under

point A which on the slider 3.

VA = 900 cm/s (OA) =+={4.?,

={?(4)

==+

={? ,={5.?

From the velocity diagram: 4=.=23.81 ..,5=.=4.65

...=.=3.44


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VELOCITIESIN

MECHANISMS

BY METHOD OF

instant Centers


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Instantaneous center of a velocity:

When two links (bodies), either both moving or one moving and one fixed,the instantaneous center is:

a- a point in both bodies,

b- a point at which the two bodies have no relative velocity (the point has

the same velocity in each body),

c- A point about which one body may be considered to rotate relative to theother at a given instant.

Notes:

- When the two links are directly connected together, the center of the

connecting joint is an instantaneous center for the two links,

- If not directly connected, an instantaneous center exists for a given phase

of the linkage.


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Instantaneous Center Notation:The system of labeling instantaneous centers is

- The instantaneous center for link (2) relative to link

(1) is labeled 21. Link(1) has the same instantaneous

center relative to link(2), when link(2) is considered

the fixed link, link(1) appears to be rotating in the

opposite sense (12 = - 21) relative to link(2).

- Since points21 and 12 are the same point, either

designation is acceptable, 12 is preferred.

-Pin connection:each pin connection is an instant center

(12, 14 remain fixed, they are called fixed centers),

(23, 34 are called moving centers, they move relative

to the frame)


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Number of instantaneous centers for a mechanisms:

Any two links in a mechanism have motion relative to

one another and have a common instantaneous

center.The number of instant centers is equal to all possible

combinations of two from the total numbers of links.

Let, n = number of links. Then the number of instant

centers is:

=(1) / 2!


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e- Primary instantaneous centers:

All instant centers which can be found by inspection are called primary instant

centers, and then we can locate the remaining by applying Kennedy's theorem.

Primary instant centers can be summarized as:

1- Instant center for pin connecting links.

2- Instant center for a sliding body.

3- Instant center for a rolling body.

4- Direct contact mechanisms:

a- For sliding contact: intersection of the common normal and the line of centers.

b- For rolling contact: at the point of contact.

f- Circle diagram method for locating instantaneous centers:

- All primary centers must be located first,

- Points are laid out along a circle, each point represent a link,

- All possible straight lines joining these points represent the instant centers,

- First, all centers which have located are drawn in as solid lines,

- The remaining are represented by dotted lines,

- In order to locate these centers, find any two triangles which a dotted linecompletes, These two triangles represent two lines their intersection is the required

instantaneous center,

- After an instant center has been located, it is drawn in as a solid line on the circle

diagram.

(The two triangles must have a common side which is dotted).


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Locating instantaneous centers in a four-bar mechanism


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Locating instant centers in the slider-crank linkage


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Velocity analysis using instantaneous centers:


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Linear velocities by instantaneous centers:

- The magnitude is equal to the product of the radius of rotation (distancebetween the point and the instant center 1i).

-It must be the radius of rotation of the point.

Angular velocities by

instantaneous centers:

From the last equation we can conclude that the angular-velocity ratio for

any two links in a mechanism is inversely as the distances from the instant

centers in the frame about which the links are rotating to the instant center

which is common to the two links.


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If the common instant center ij lies in-between 1i & 1j, then i

and jare in opposite direction, and if not, then iand jare

in the same direction.


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Example 1:For the Whitworth mechanism shown, determine the absolute velocity VC of the tool

support, when the driving link OA rotates at a speed such that VA = 900 cm/s, as

shown, also find the angular velocities of links QB and BC.

OQ = 135, OA = 270, QB = 160, and BC = 550 mm.


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velocity in mechanisms

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