theoryof machinesand mechanism lecture2wm.pollub.pl/files/65/content/files/5080_lecture_2.pdf ·...

Theory of Machines and Mechanism

Lecture 2

Łukasz Jedliński, Ph.D., Eng.

Kinematics is the study of motion without consideration of what causes the

motion. In other words, the input motion is assumed to be known and the

objective is to find the transformation of this motion.

The process of finding the mechanism parameters given the needed output

is called kinematic synthesis. If, however, the mechanism parameters are

known, then the objective is to find the motion of the output link. This

process of finding the output motion given the mechanism parameters is

called kinematic analysis.

Methods of performing kinematic analysis:

• analytical,

• graphical,

• numerical.

Basic terms and definitions

Classification of rigid body motions:

1. Plane motion

• Translation - Rectilinear / Curvilinear.

• Rotation (about a fixed axis).

• General plane motion.

2. Spherical motion

• Rotation about a fixed point.

• General motion.

A rigid body executes plane motion when all parts of the body move in

parallel planes. The plane of motion is considered, for convenience, to be the

plane which contains the mass center, and we treat the body as a thin slab

whose motion is confined to the plane of the slab. This idealization

adequately describes a very large category of rigid body motions

encountered in engineering.

Basic terms and definitions

Plane motion

Spherical motion

Rotation about a fixed point General motion

Plane motion - translation

Translation motion in which every line in the body remains parallel to its

original position at all time. That is there is no rotation of any line in the

body. The motion of the body is completely specified by the motion of any

point in the body, since all points have same motion.

Rectilinear translation - all points in the body move in parallel straight lines

of the same distance

Curvilinear translation - all points move on parallel curves of the same

distance

Plane motion - translation

vB = vC = vi, ω = 0,

aB = aC = ai, ε = 0,

v = ds/dt,

2

2

dt

sd

dt

ds

dt

d

dt

dva =

== acceleration

velocity

derivative

Plane motion - rotation

Rotation motion in which all particles move in circular paths about the axis of

rotation. All lines in the body which are perpendicular to the axis of rotation

rotate through the same angle in the same time. Circular motion of a point

helps describe the rotating motion.


ααω &==dt

d

Angular acceleration

Angular velocity

αααωε &&==

==2

2

dt

d

dt

d

dt

d

dt

d


2422

2

εω

εω

ωω

+=+=

==

==

raaa

ra

ra

OArv

tn

t

n

Tangential acceleration

Total or resultant acceleration

Normal (radial, centripetal) acceleration

Linear velocity

General plane motion

The general motion of a rigid body consists of a translation of the center of

mass with velocity and a rotation about the center of mass with all elements

of the rigid body rotating with the same angular velocity.


Method of analysis:

• Absolute motion

• Relative motion

• Instantaneous center (of zero velocity)


Absolute motion

It is an approach to the kinematics analysis. It starts with the geometric

relations that define the configuration involved. Then, the time derivatives of

the relations are done to obtain velocities and accelerations. The +/- sense

must be kept consistent throughout the analysis.

If the geometric configuration is too complex, resort to the principle of

relative motion is recommended.


Relative motion

Principle of relative motion is another way to solve the kinematics problems.

This method is usually suitable to the complex motion as it is more scalable.

Let the two points, A and B, are on the same rigid body. The consequence of

this choice is that the motion of one point as seen by an observer translating

with the other point must be circular since the radial distance to the observed

point from the reference point does not changed.


Relative motion - velocity

Movement of the rigid body is partitioned into two parts: translation and

rotation. In the figure, after the translation of the rigid body, expressed by the

motion of B, the body appears to undergo fixed-axis rotation about B with A

executing circular motion as shown in (b). Hence the relationship for circular

motion describes the relative portion of A’s motion.


Relative motion - velocity

BAAB

BAABABBA

vv

vvvvvvrr

rrrrrr

−=+=+= or

It should be noted that the direction of the relative velocity will always be perpendicular

to the line joining the points A and B.

BArv BA // ω=

The velocity of the point A relative to the point B


Relative motion - acceleration

ABBA aaarrr

+=

where aA and aB are the accelerations of A and B relative to the origin O of the

reference frame and a AB is the acceleration of point A relative to point B . Because the

point A moves in a circular path relative to the point B as the rigid body rotates, aAB

has a normal component and a tangential component

tAB

nABBA

ABtAB

AB

ABAB

nAB

tAB

nABBA

aaaa

ra

r

vra

aaa

rrrr

rrr

rrr

++=

×=

==

+=

ε

ω2

2

/

By differentiating the equation of velocities with respect to time, we obtain the

relative acceleration equation, which is



The meaning of relative acceleration equation is

indicated in the figure which shows a rigid body in plane

motion with points A and B moving along separate

curved paths with absolute accelerations aA and aB.

Contrary to the case with velocities, the accelerations aA

and aB are, in general, not tangent to the paths

described by A and B when these paths are curvilinear.

The figure shows the acceleration of A to be composed

of two parts: the acceleration of B and the acceleration

of A with respect to B. A sketch showing the reference

point as fixed is useful in disclosing the correct sense of

the two components of the relative acceleration term.



BAAB

BAAB

aa

aaarr

rrr

−=+=

Alternatively, we may express the acceleration of B in

terms of the acceleration of A, which puts the

nonrotating reference axes on A rather than B. This

order gives


Relative motion

Solution of the relative motion equation may be carried out by :

1. Vector algebra approach

2. Graphical analysis approach

3. Vector/Graphic approach

Because the normal acceleration components depend on velocities, it is

generally necessary to solve for the velocities before the acceleration

calculations can be made.


Relative motion

1. Vector algebra approach

Write each term in terms of i- and j-components; two scalar equations; at

most two unknowns.

2. Graphical analysis approach

Known vectors are constructed using a convenient scale. Unknown vectors

which complete the polygon are then measured directly from the drawing.

This is suitable when the vector terms result in an awkward math expression.

3. Vector/Graphic approach

Scalar component equations may be written by projecting the vectors along

convenient directions. Simultaneous equations may be avoided by a careful

choice of the projections.


Instantaneous Center

Instantaneous center: a point of rigid body whose velocity is zero at a give instant.

Principle of relative motion

Find the velocity of a point on a rigid body by adding the relative velocity, due to

rotation about a reference point, to the velocity of the reference point. If the

reference point has zero velocity momentarily, the body may be considered to be in

pure rotation about an axis, normal to the plane of motion, passing through this point.

This point is called ICZV, which aids in visualizing and analyzing velocity in plane

motion.


Instantaneous Center


Motion of a point (on a rigid body) that moves relative to another

point (on another rigid body)

There are two particles B and C, which are on different rigid bodies. Imagine that there

is a rotating guide 1 on which particle B is located. On this guide there is a slider 2 with

a point C, which is currently coincident with point B. Relationship between vC and vB is

indicated by

CBBC vvvrrr

+=

vCB is the velocity of point C relative to point B, and

is tangent to the path (guide).

Since point C is on the different body, vC ≠ vB


The acceleration of the point C relative to the point B is:

cCB

tCB

nCBCB aaaa

rrrr++=

CBBC aaarrr

+=

aCB is the acceleration of point C relative to point B




ρ is the curvature (radius) of guide.

If the guide is rectilinear (ρ = ∞) acceleration is

equal 0

n

CBa normal acceleration

ρ

2CBn

CB

va =

02

=∞

= CBnCB

var

Direction of this acceleration is toward the center of

curvature of the guide (relative path)




t

CBa tangential acceleration

dt

dva CBt

CB = Direction of this acceleration is tangent to guide

(relative path)




c

CBa Coriolis acceleration

CBcCB va

rrr×= ω2

Direction of this acceleration can be found by rotating

velocity vector vCB in the direction accordance with

angular velocity of the guide ω of right angle (90°)

Coriolis acceleration is equal 0 when angular velocity

of guide is zero ω = 0 – guide makes translation

motion

or slider doesn't move relative to the guide vCB = 0



Analytical method of kinematic analysis – Vector Loop Technique

Analytical methods are suitable for mechanism that must be analyzed for multiple

positions.

The loop closure equations are fundamental to modeling mechanisms. The vectors

that describe the components must add to zero when links form a loop:

How many scalar equations for planar mechanism can be written from the vector loop-

closure equation?:

Two scalar equations could be written from a vector equation

How many unknowns can be solved for:

Two unknowns: case 1) two unknown angles; case 2) one unknown angle, one

unknown length; case 3) two unknown lengths

0321

1

∑ =++++=

=

n

n

i

i lllllrrrrr

...

where ln are vectors in the chain


Example – four bar mechanism

We want to determine two angles in four bar mechanism φ2 and φ3 and also

velocities and accelerations. Lengths of links and angles φ1, φ4 are known.

Let’s define the x-axis to be along length l4. Angles are measured from x-axis in

counter-clockwise direction (this is a assumption, we can of course measure angle in

clockwise direction, but we can’t change thin assumption during calculations)

Step 1: Draw and label vector loop for mechanism


Step 2: Write vector equation:

04321 =+++ llllrrrr

Solve this equation. As a result we should obtain two equation:

0∑1

=

=

n

i

ilr

axis-X cos 0∑1

=

=

i

n

i

il ϕ axis-Y s 0∑1

=

=

i

n

i

i inl ϕ

0ss s

0coscos cos

332211

4332211

=++=−++

ϕϕϕϕϕϕ

inlinlinl

llll

−−++−=

γθγθθcoscos

sinsin2

42231

4223

LLLL

LLarctg

−−+−=

γθγθθ

coscos

sinsin2

31422

3224

LLLL

LLarctg

and scalar equations:


To obtain velocities we need to differentiate one of the scalar equation and to

obtain acceleartions we need to differentiate velocity equation

( )

−−=γ

θθωωsin

sin

3

24223

L

L

( )

−−=γ

θθωωsin

sin

4

23224

L

L

( ) ( ) ( )( )343

343

2

34

2

4422

2

242223

sin

coscossin

θθθθωωθθωθθεε

−−+−−+−=

L

LLLL

( ) ( ) ( )( )344

3

2

3344

2

3322

2

232224

sin

coscossin

θθωθθωθθωθθεε

−+−−−+−=

L

LLLL

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