An Introduction To Kinematics of Mechanisms

An Introduction To Kinematics of Mechanisms

Introduction:

Theory of Machines: may be defined as that branch of engineering science, which deals with the study of relative motion between the various parts of machine, and forces which act on them. The knowledge of this subject is very essential for an engineer in designing the various parts of a machine.

Topics to be considered:
1. Kinematics and Kinetics of Mechanisms.
2. Balancing of Machines.
        a. For Rotating masses.
        b. For Reciprocating masses.
3. Cams and Followers.
4. Flywheel.
5. Gears and Gears trains.
6. Friction Clutches.
7. Belt drives and Band Brakes.
8. Power Screw.
9. Universal Joint.
10. Gyroscope.
11. Speed Governors.
12. Steering mechanism.

Sub- divisions of theory of Machines:

They Theory of Machines may be sub- divided into the following four branches:

1- Kinematics: is that branch of theory of machines which is responsible to study the motion of bodies without reference to the forces which are cause this motion, i.e it’s relate the motion variables (displacement, velocity, acceleration) with the time.

2- Kinetics: is that branch of theory of machines which is responsible to relate the action of forces on bodies to their resulting motion.

3- Dynamics: is that branch of theory of machines which deals with the forces and their effects, while acting upon the machine parts in motion.

4- Statics: is that branch of theory of machines which deals with the forces and their effects, while the machine parts are rest.

There are some definitions which are concerned with this subject, must be known:

Mechanism: is a combination of rigid bodies which are formed and connected together by some means, so that they are moved to perform some functions, such as the crank- connecting rod mechanism of the I.C. engines, steering mechanisms of automobiles……. etc.

The analysis of mechanisms is a part of machine design which is concerned with the kinematics and kinetics of mechanisms (or the dynamics of mechanisms).

Rigid Body: is that body whose changes in shape are negligible compared with its overall dimensions or with the changes in position of the body as a whole, such as rigid link, rigid disc…..etc.

Links: are rigid bodies each having hinged holes or slot to be connected together by some means to constitute a mechanism which able to transmit motion or forces to some another locations.

Absolute motion: the motion of body in relative to another body which is at rest or to a fixed point located on this body.

Relative motion: the motion of body in relative to another moved body.

Scalar quantities: are those quantities which have magnitude only e.g. mass, time, volume, density etc.

Vector quantities: are those quantities which have magnitude as well as direction e.g. velocity, acceleration, force etc.

Kinematics of Mechanisms:

1- The connection of mechanism parts:

The mechanism is a combination of rigid bodies which are connected together using different methods:

1-1: Hinged part:

The hinge connection may be used to connect the links together or connect a link to a fixed point, piston, disc ….. etc, the connection is achieved using pin, which is pass through the hinge holes.

1-2: Sliding Parts:

The sliding connection may be used to connect two links rotate about fixed points by means of slot, pin and hinge.

1-3: Rolling without slipping parts:

2- Translated bodies:

There are some bodies in the mechanism which are constrained to move in translation manner, such as the piston of crank- connecting rod mechanism, the body is used to be in translation motion, if any line remain in some configuration during motion; then all the points have the same absolute velocity and acceleration.

Velocity diagram:

- the motion is absolute, then select any fixed point such as o be as a reference point (i.e point of zero velocity).

- Draw the path of translation.

- If vB is known, select a scale factor to draw the velocity diagram (denoted by SFv)

SFv= draw value in mm/actual value of velocity in (m/s) = ob/𝒗𝖡

The draw a line ob=(vB)(SFv) in direction of vB parallel to the path of translation.

Then all points on the piston have the same velocity, such as point D, i.e on the velocity diagram, the point d coincide on the point b.

Acceleration diagram:

- the motion is absolute, then select any fixed point such as o be as a reference point (i.e point of zero acceleration).

- Draw the path of translation.

- If aB is known, select a scale factor to draw the acceleration diagram (denoted by SFa)

SFa = draw value in mm / actual value of acceleration in (m/s²)

In which ob=(aB)(SFa).

Then all points on the piston have the same acceleration value.

Note: the base (ref.) point o of vo =0, ao=0.

Dynamic review:

Translation motion can by treatment by the dynamics of particles i.e body B can be treatment as a particle moved on straight or curved path.

Then: 𝒗= ds/dt = , 𝒶 = d𝒗/dt =d²s/dt²  ↣  𝒗d𝒗=a ds

Where:

s: displacement ,v: velocity , a: acceleration

𝒗₂-𝒗₁  = 𝒶 (t₂-t₁)

𝒗₂²-𝒗₁² =  2 𝒶 ( 𝗌₂- 𝗌₁)

𝗌₂- 𝗌₁ = 𝒗₂ (t₂-t₁) + (t₂²-t₁²)

3-Bodies rotate about fixed point:

Consider the link shown which is rotate about the fixed point o, the motion of this link can be analyzed using the principle of absolute motion as follow:

If θ: angular displacement about fixed rotation centre.

ω: angular velocity about fixed rotation centre.

α: angular acceleration about fixed rotation centre

ω = 𝑑θ/𝑑𝒕 , 𝛼 = 𝑑ω / 𝑑𝒕 , ↣ ω 𝑑ω = 𝛼 𝑑θ

𝛚₂  - 𝛚₁ = 𝛼 (t₂-t₁)

𝛚₂²  - 𝛚₁² = 2 𝛼 ( θ₂-θ₁)

θ₂-θ₁  =  2  𝒗₁ (t₂-t₁) + 1/2  𝛼 (t₂²-t₁²)

4-Bodies under general plane motion:-
If a body under general plane motion, then it’s motion can be analyzed using the principle of relative motion.
The motion of any point can be discretized into translation and rotation, if consider the link shown under general plane motion, the ends , B of absolute velocities vA, vB, and absolute accelerations aA, aB t