lecture 2 sdf

8/3/2019 Lecture 2 Sdf

1/54

MTS-362

CONTROL ENGINEERING

Spring 2011Lecture No. 2

Department of Mechatronics Engineering

Mathematical Modeling of MechanicalSystems

Instructor: Engr. Sadaf Siddiq

Class: BEMTS 6A & 6B


2/54

2

System Modeling


3/54

3

System Modeling


4/54

4

Mathematical Models

Design of engineering systems by trying and error versus

design by using mathematical models.

Physical laws such as Newtons second law of motion is amathematical model.

Mathematical model gives the mathematical relationships

relating the output of a system to its input.


5/54

5

Mathematical Models

Control systems give desired output by controlling the

input. Therefore control systems and mathematical

modeling are inter-linked.


6/54

6


7/54

7

Mechanical Systems

If the velocity and acceleration of a body are bothzero then the body will be Static.

If the applied forces are balanced, and cancel eachother out, the body will not accelerate.

If the forces are unbalanced then the body will

accelerate and the body will be Dynamic. If all of the forces act through the center of mass

then the body will only translate- Translation

Forces that do not act through the center of mass

will also cause rotation to occur- Rotation


8/54

8

Modeling- Translational Systems

FBD: Free Body Diagrams allow us to reduce a complexmechanical system into smaller, more manageable pieces.

Common Components

gravity and other fields - apply non-contact forces

inertia - opposes acceleration and deceleration

springs - resist deflection

dampers and drag - resist motion

friction - opposes relative motion between bodies in

contact

cables and pulleys - redirect forces

contact points/joints - transmit forces through up to 3

degrees of freedom


9/54

9

Gravity and Other Fields

mgF m

mgB qvF


10/54

10

Mass and Inertia


11/54

11

Springs


12/54

12

Springs

S i


13/54

13

Springs


14/54

14

Springs in Series & Parallel

Kequiv= KS1 + KS2

D i d D


15/54

15

Damping and DragA damper is a component that resists motion. The resistive force

is relative to the rate of displacement (velocity). Springs store

energy in a system but dampers dissipate energy. Dampers andsprings are often used to compliment each other in designs.

D i S i & P ll l


16/54

16

Dampers in Series & Parallel


17/54

C P i d J i


18/54

18

Contact Points and JointsA system is built by connecting components together. These connections can be

rigid or moving. In solid connections all forces and moments are transmitted and

the two pieces act as a single rigid body. In moving connections there is at least

one degree of freedom. If we limit this to translation only, there are up to three

degrees of freedom, x, y and z. In any direction there is a degree of freedom, a

force or moment cannot be transmitted.

C t t P i t d J i t


19/54

19

Contact Points and Joints


20/54

20

The Unforced Mass-Spring SystemConsider a mass, M, suspended from a spring of natural

length l and modulus of elasticity . If the elastic limit of

the spring is not exceeded and the mass hangs inequilibrium, the spring will extend by an amount, e, such

that by Hookes Law the tension in the spring, T, will be

given by:

M

Mg

T


21/54

21

The Unforced Mass-Spring SystemIf the spring is pulled down a further distance, y,

(with y positive downwards) the restoring force

will now be the new tension in the spring, T

F


22/54

22

The Unforced Mass-Spring-Damper System

tCoefficiendampingBdt

dyBFForceDamping

D

,

dt

dyBky

dt

ydM

dt

dyB

l

y

dt

ydM

2

2

2

2

FD

This time, the net downward force will beMg - T - FD

using Newtons 2nd Law, this results in

dt

dyB

l

y

dt

dyB

l

yeMgFForceNet

)(


23/54

23

The Forced Mass-Spring-Damper System

)(2

2

tfdt

dyBky

dt

ydM

dt

dyB

l

yetfMgFForceNet

)()(

The net downward force =Mg + f(t)- T- FD

using Newtons 2nd Law, this results in

FD

dt

dyBkytf

dt

dyB

l

ytfFForceNet )()(

S t E l


24/54

24

System Examples

S t E l


25/54

25

System Examples

S t E l


26/54

26

System Examples

S t E l


27/54

27

System Examples

S t E l


28/54

28

System Examples

S stem E amples


29/54

29

System Examples

System Examples


30/54

30

System Examples

System Examples


31/54

31

System Examples

System Examples


32/54

32

System Examples

System Examples


33/54

33

System Examples

System Examples


34/54

34

System Examples


35/54

35

Modeling- Rotational Systems

Basic properties of rotation


36/54

36

Modeling- Rotational Systems

FBD: Free Body Diagrams (FBDs) are required

when analyzing rotational systems, as they were

for translating systems.

Common Components

inertia - opposes acceleration and deceleration springs - resist deflection

dampers and drag - resist motion

friction - opposes relative motion between bodies in

contact

levers - rotate small angles

gears and belts - change rotational speeds and torques

Inertia


37/54

37

InertiaWhen unbalanced torques are applied to a mass it will begin to

accelerate, in rotation, the sum of applied torques is equal to the

inertia forces

The mass moment of inertia will be used when dealing with

acceleration of a mass. The areamoment of inertia is used for torsional

springs

Inertia


38/54

38

InertiaThe center of rotation for free body rotation will be the centroid. Moment of

inertia values are typically calculated about the centroid. If the object is

constrained to rotate about some point, other than the centroid, the moment ofinertia value must be recalculated. The parallel axis theorem provides the

method to shift a moment of inertia from a centroid to an arbitrary center of

rotation

Springs


39/54

39

SpringsTwisting a rotational spring will produce an opposing torque. This

torque increases as the deformation increases. The angle of rotation is

determined by the applied torque, T, the shear modulus, G, the areamoment of inertia, JA, and the length, L, of the rod. The constant

parameters can be lumped into a single spring coefficient similar to

that used for translational springs.

This calculation uses the area moment of inertia (JA)

Springs


40/54

40

SpringsIf the object is constrained to rotate about some point, other

than the centroid, the moment of area value must be

recalculated. The parallel axis theorem provides the method toshift a moment of area from a centroid to an arbitrary center of

rotation

Damping


41/54

41

Damping

Rotational damping is normally caused by viscous fluids, such

as oils, used for lubrication.

The equation used for a system with one rotating and one stationary

part is given by:

The equation used for damping between two rotating parts is given

by:

Friction


42/54

42

Friction

Friction between rotating components is a major source of

inefficiency in machines. It is the result of contact surface

materials and geometries. Calculating friction values in rotating systems is more

difficult than translating systems.

Normally rotational friction will be given as static and

kinetic friction torques.

Levers


43/54

43

LeversThe levers can be used to amplify forces or motion. Although

theoretically a lever arm could rotate fully, it typically has a limited

range of motion.The amplification is determined by the ratio of arm lengths to the left

and right of the center.

Gears and Belts


44/54

44

Gears and BeltsWhile levers amplify forces and motions over limited ranges of motion, gears can

rotate indefinitely. Some of the basic gear forms are:

Spur - Round gears with teeth parallel to the rotational axis.Rack - A straight gear (used with a small round gear called a pinion).

Helical - The teeth follow a helix around the rotational axis.

Bevel - The gear has a conical shape, allowing forces to be transmitted at angles.

Basic Gear Relationships

Rack & Pinion


45/54

45

Rack & Pinion

Rack and pinion gear sets are used for converting rotation

to translation. A rack is a long straight gear that is driven

by a small mating gear called a pinion.

Relationships for a rack and pinion gear set


46/54

46

System Examples


47/54

47

System Examples

System Examples


48/54

48

System Examples

System Examples


49/54

49

System Examples

System Examples


50/54

50

System Examples

System Examples


51/54

51

System Examples

System Examples


52/54

52

System Examples

System Examples


53/54

53

System Examples


54/54

54

lecture 2 sdf

Documents