Geometry of five link mechanism with two degrees of freedom - PowerPoint Presentation

Slide1

Geometry of five link mechanism with two degrees of freedom

David

TavkhelidzeSlide2

Internal combustion engine

A-Crankshaft;

B-Connecting rod;

C-Slider (piston);

D-Frame;E-Valve mechanism

ESlide3

Kinematic pairs

Slide4

Degree

of

freedom

Degree of freedom for spatial mechanism W=6n-P1-2P2-3P3-4P4-5P5

Degree of freedom of planar mechanism

W=3n-2P5

3n=2P5Slide5

Kinematic chainsSlide6

Four link mechanismSlide7

Five link mechanismSlide8

Mechanisms used in technological machines

Four link slider - crank mechanism

Four link mechanism with rotating kinematic pairs

Six link gear mechanism

Five link mechanism with gear pair, reducing number degrees of freedom of the mechanical system Slide9

Mechanisms with two degrees of freedom

Kinematic scheme of five link mechanism with two degrees of freedom

Various scheme of mechanisms with two degrees of freedom Slide10

Straight geometrical task

The design diagram of five link mechanism

.

For the closed kinematic chain it is necessary that the product of matrices of transformation between coupled coordination systems connected with all incoming links has to be equal to unit matrix

(1)

For simplification of calculation it would be written

(2)Slide11

Straight geometrical task

The transformation matrix between of two sequential

i-1

and

i plain coordinate systems has the following general form:

(3)

Taking in the account the previous equations will be obtained

(4)Slide12

Straight geometrical task

The 4

th

matrix equation of five-link mechanism blockage contains full information about parameters of link motion characteristics. In order to determine relative and absolute displacement of links the respective elements of left and right parts of equation should be equated and receive system of algebraic equations the solution of which will enable to determine displacements of mechanism links.

(5)

,

.

.

Besides these equations, in order to solve the problem the subsidiary condition should be added according to which the sum of internal angles of any five link is equal to 3π.

(6)

Slide13

Straight geometrical task

After transformations we get the following quadratic equation

(7)

,

Here:

From the equation (7) we will obtain meanings of angels

ϕ

34

and

ϕ

23

that determines position of the point C of the mechanism.

(8)

(9)

.

.

And hence, in case of differentiating on time the received values of obtaining equations, we shall receive values of speeds and acceleration of the links of the mechanism.Slide14

The inverse geometrical task

In spite of the straight geometrical problem, here on the basis of the given angels of rotation of the actuators mounted on the frame of the mechanism the trajectory of the output link of the considered mechanical system is defined.

The formulation of inverse task of kinematics of five link mechanism is done in the following way: the location of C point of mechanism i.e. its coordinates in coordinate system connected with base, is given and it’s necessary to find generalized coordinates of the mechanism which provide the location of C point.

The design diagram of five link mechanism for inverse taskSlide15

The inverse geometrical task

For this we take C point radius vector from the origin of coordinates and represent it as the sum of two vectors:

(10)

Projections of these vectors in immovable coordinate system are expressed as:

(11)

In projections formula (10) will have the following form:

(12) Slide16

The inverse geometrical task

In order to find two generalized and coordinates determining the location of BC kinematic chain the expressions (12) should be squared and summed up:

(13)

The obtained expressions (13) allows to calculate values of angels

Hence, we will have:

(14)

(15)

and

Slide17

The inverse geometrical task

We behave similarly when we determine the location of

CDO

kinematic chain. We present radius vector of C point in the form of the following vectors sum:

(16)

(17)

(18)

And hence we can obtain

:

Based on derivations of the given formulas the values of velocities and accelerations of the links of the investigated mechanism have obtained . Slide18

The inverse geometrical task

Based on usage of

MATLAB

software here are given curves of alternations of phase angles of the five bar mechanism, when the two link junction point C is performing movement along the circle. Slide19

The inverse geometrical task

Curves of alternation of angles Slide20

The inverse geometrical taskSlide21

The inverse geometrical task

Values of angular velocitiesSlide22

The inverse geometrical taskSlide23

The inverse geometrical task

Values of angular accelerationsSlide24

The inverse geometrical taskSlide25

Kinetostatics of five bar planar mechanisms

On the links of mechanical system are acting two type of force factors - External forces and Internal forces.

The internal forces – forces of weight, reduction forces of inertia and moments of inertia of force couples

Forces of inertia-

Moments of inertia of force couples- Slide26

Kinetostatics of five bar planar mechanism

Reduction forces and moments of inertia acting on the links of five bar mechanism Slide27

Determination of forces and torques

Lagrange equation relatively to generalized

coordinate

Lagrange equation relatively to generalized coordinate

Slide28

Determination of forces and torques

Equitant for determination of torque acting on

A

kinematic pair.Equitant for determination of torque acting on

O kinematic pair.Slide29

Determination of force factors

Determination of torque acting on

A

kinematic pair.Slide30

Determination of force factors

Determination of torque acting on

O

kinematic pair.Slide31

Thank you