This document discusses modeling mechanical systems using three basic elements: springs, dampers, and masses. It describes the properties and dynamic responses of ideal spring and damper elements and provides examples of real-world springs and dampers. The document also discusses modeling nonlinear springs and damping effects in mechanical systems.
Mechanical translational rotational systems and electrical analogous circuit...SatheeshCS2
Mr. C.S.Satheesh, M.E.,
Mechanical Translational and Rotational Systems and Electrical analogous Circuits in control systems
Spring
Dash-pot
Analogous electrical elements in torque current analogy for the elements of mechanical rotational system.
Electrical systems
This Presentation explains about the introduction of Frequency Response Analysis. This video clearly shows advantages and disadvantages of Frequency Response Analysis and also explains frequency domain specifications and derivations of Resonant Peak, Resonant Frequency and Bandwidth.
Transfer Function and Mathematical Modeling
Transfer Function
Poles And Zeros of a Transfer Function
Properties of Transfer Function
Advantages and Disadvantages of T.F.
Translation motion
Rotational motion
Translation-Rotation counterparts
Analogy system
Force-Voltage analogy
Force-Current Analogy
Advantages
Example
Mechanical translational rotational systems and electrical analogous circuit...SatheeshCS2
Mr. C.S.Satheesh, M.E.,
Mechanical Translational and Rotational Systems and Electrical analogous Circuits in control systems
Spring
Dash-pot
Analogous electrical elements in torque current analogy for the elements of mechanical rotational system.
Electrical systems
This Presentation explains about the introduction of Frequency Response Analysis. This video clearly shows advantages and disadvantages of Frequency Response Analysis and also explains frequency domain specifications and derivations of Resonant Peak, Resonant Frequency and Bandwidth.
Transfer Function and Mathematical Modeling
Transfer Function
Poles And Zeros of a Transfer Function
Properties of Transfer Function
Advantages and Disadvantages of T.F.
Translation motion
Rotational motion
Translation-Rotation counterparts
Analogy system
Force-Voltage analogy
Force-Current Analogy
Advantages
Example
ppt on Time Domain and Frequency Domain Analysissagar_kamble
in this presentation, you will be able to know what is this freq. and time domain analysis.
At last one example is illustreted with video, which distinguishes these two analysis
Introduction to Mechatronics, Sensors and Transducerstaruian
Introduction: Definition, Multidisciplinary Scenario, Evolution of Mechatronics, Design of Mechatronics system, Objectives, advantages and disadvantages of Mechatronics
Transducers and sensors: Definition and classification of transducers, Difference between transducer and sensor, Definition and classification of sensors, Principle of working and applications of light sensors, proximity switches and Hall Effect sensors.
This presentation gives the information about introduction to control systems
Subject: Control Engineering as per VTU Syllabus of Aeronautical Engineering.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Disclaimer:
The contents used in this presentation are taken from the text books mentioned in the references. I do not hold any copyrights for the contents. It has been prepared to use in the class lectures, not for commercial purpose.
automatic control, Basic Definitions, Classification of Control systems, Requ...Waqas Afzal
Why automatic controls is required
2. Process Variables
controlled variable, manipulated variable
3. Functions of Automatic Control
Measurement
Comparison
Computation
Correction
4.Basic Definitions
System, Plant, Process, Controller, input, output, disturbance
5. Classification of Control systems
Natural, Manmade & Automatic control system
Open-Loop, Close-Loop control System
Linear Vs Nonlinear System
Time invariant vs Time variant
Continuous Data Vs Discrete Data System
Deterministic vs Stochastic System
6. Requirements of an ideal Control system
Accuracy, Sensitivity, noise, Bandwidth, Speed, Oscillations
State variable analysis (observability & controllability)SatheeshCS2
Mr. C.S.Satheesh, M.E.,
State Variable Analysis
Observability
Controllability
Concept of state variables
State models for linear and time invariant Systems
Solution of state and output equation in controllable canonical form
Concepts of controllability and observability
Effect of state feedback.
ppt on Time Domain and Frequency Domain Analysissagar_kamble
in this presentation, you will be able to know what is this freq. and time domain analysis.
At last one example is illustreted with video, which distinguishes these two analysis
Introduction to Mechatronics, Sensors and Transducerstaruian
Introduction: Definition, Multidisciplinary Scenario, Evolution of Mechatronics, Design of Mechatronics system, Objectives, advantages and disadvantages of Mechatronics
Transducers and sensors: Definition and classification of transducers, Difference between transducer and sensor, Definition and classification of sensors, Principle of working and applications of light sensors, proximity switches and Hall Effect sensors.
This presentation gives the information about introduction to control systems
Subject: Control Engineering as per VTU Syllabus of Aeronautical Engineering.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Disclaimer:
The contents used in this presentation are taken from the text books mentioned in the references. I do not hold any copyrights for the contents. It has been prepared to use in the class lectures, not for commercial purpose.
automatic control, Basic Definitions, Classification of Control systems, Requ...Waqas Afzal
Why automatic controls is required
2. Process Variables
controlled variable, manipulated variable
3. Functions of Automatic Control
Measurement
Comparison
Computation
Correction
4.Basic Definitions
System, Plant, Process, Controller, input, output, disturbance
5. Classification of Control systems
Natural, Manmade & Automatic control system
Open-Loop, Close-Loop control System
Linear Vs Nonlinear System
Time invariant vs Time variant
Continuous Data Vs Discrete Data System
Deterministic vs Stochastic System
6. Requirements of an ideal Control system
Accuracy, Sensitivity, noise, Bandwidth, Speed, Oscillations
State variable analysis (observability & controllability)SatheeshCS2
Mr. C.S.Satheesh, M.E.,
State Variable Analysis
Observability
Controllability
Concept of state variables
State models for linear and time invariant Systems
Solution of state and output equation in controllable canonical form
Concepts of controllability and observability
Effect of state feedback.
Purpose Statement:
To provide an overview of Design for Manufacturing and Assembly (DFMA) techniques, which are used to minimize product cost through design and process improvements.
you can be friend with me on orkut
"mangalforyou@gmail.com" : i belive in sharing the knowledge so please send project reports ,seminar and ppt. to me .
We manufacture various Dynamometers to test Automobile gearboxes,Industrial gearboxes,Geared motors, Gear reducers, gear transmissions, etc
Torque: 1 Nm to 10000 Nm
Speed: 1 RPM to 1500 RPM.
Manual Dynamometer Control: To Set Braking Torque & read Torque, Speed, KW/HP on Digital Indicators.
Computerized Control: Torque Loading, Speed Control thru PC, Captured data on Torque, Speed, Power, Temperature, Time, etc is stored in MS Excel in Tabular form & custom curves between Torque, Speed, Power, Temperature, Time, Torque, Speed, Power, Temperature, or custom curves, plotted & exported to Excel.
Test Benches, for testing Performance & Endurance testing of Engines, Motors, Shafts & Axles, Gear Boxes, Automobile Gear Transmissions, Chassis, etc..
Test Beds with T slots having X, Y axis adjustment for Length, Width, & Z axis adjustments for Height. Height adjustment is either Motorized or by Mechanical Jacks. with Anti Vibration mountings are offered along with dynamometers
Er. Muhammad Zaroon Shakeel
Vibration Analysis Lectures
Book : S.S.RAO
Department of Mechanical Engineering
Faculty of Engineering (FOE)
University of Central Punjab - Lahore
Metal cutting tool position control using static output feedback and full sta...Mustefa Jibril
In this paper, a metal cutting machine position control have been designed and simulated using
Matlab/Simulink Toolbox successfully. The open loop response of the system analysis shows that the system needs
performance improvement. Static output feedback and full state feedback H 2 controllers have been used to increase
the performance of the system. Comparison of the metal cutting machine position using static output feedback and
full state feedback H 2 controllers have been done to track a set point position using step and sine wave input signals
and a promising results have been analyzed.
Comparative Analysis of PID, SMC, SMC with PID Controller for Speed Control o...IJMTST Journal
In this thesis, sliding mode control (SMC) technique is used to control the speed of DC motor. The performance of the SMC is judged via MATLAB simulations using linear model of the DC motor and known disturbance. SMC is then compared with PID controller. The simulation result shows that the sliding mode controller (SMC) is superior controller than PID for the speed control of DC motor. Since the SMC is robust in presence of disturbances, the desired speed is perfectly tracked. The sliding mode control (SMC)can adapt itself to the parameter variations and external disturbances, problem of chattering parameter, resulting from discontinuous controller, is handled by sliding with smooth control action
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
1. Mechanical System Modeling K. Craig 1
Mechanical System Modeling
Dr. Kevin Craig
Professor of Mechanical Engineering
Rensselaer Polytechnic Institute
2. Mechanical System Modeling K. Craig 2
References for Mechanical Systems
• System Dynamics, E. Doebelin, Marcel Dekker,
1998. (This is the finest reference on system
dynamics available; many figures in these notes
are taken from this reference.)
• Modeling, Analysis, and Control of Dynamic
Systems, W. Palm, 2nd Edition, Wiley, 1999.
• Vector Mechanics for Engineers: Dynamics, 7th
Edition, F. Beer, E.R. Johnston, and W. Clausen,
McGraw Hill, 2004.
3. Mechanical System Modeling K. Craig 3
Mechanical System Elements
• Three basic mechanical elements:
– Spring (elastic) element
– Damper (frictional) element
– Mass (inertia) element
• Translational and Rotational versions
• These are passive (non-energy producing) devices
• Driving Inputs
– force and motion sources which cause elements
to respond
4. Mechanical System Modeling K. Craig 4
• Each of the elements has one of two possible
energy behaviors:
– stores all the energy supplied to it
– dissipates all energy into heat by some kind of
“frictional” effect
• Spring stores energy as potential energy
• Mass stores energy as kinetic energy
• Damper dissipates energy into heat
• Dynamic Response of each element is important
– step response
– frequency response
5. Mechanical System Modeling K. Craig 5
Spring Element
• Real-world design situations
• Real-world spring is neither pure nor ideal
• Real-world spring has inertia and friction
• Pure spring has only elasticity - it is a
mathematical model, not a real device
• Some dynamic operation requires that spring
inertia and/or damping not be neglected
• Ideal spring: linear
• Nonlinear behavior may often be preferable and
give significant performance advantages
6. Mechanical System Modeling K. Craig 6
• Device can be pure without being ideal (e.g.,
nonlinear spring with no inertia or damping)
• Device can be ideal without being pure (e.g., device
which exhibits both linear springiness and linear
damping)
• Pure and ideal spring element:
• Ks = spring stiffness (N/m or N-m/rad)
• 1/Ks = Cs = compliance (softness parameter)
( )
( )
s 1 2 s
s 1 2 s
f K x x K x
T K K
= − =
= θ − θ = θ
s
s
x C f
C T
=
θ =
Ks
x f f x
Cs
7. Mechanical System Modeling K. Craig 7
• Energy stored in a spring
• Dynamic Response: Zero-Order Dynamic System
Model
– Step Response
– Frequency Response
• Real springs will not behave exactly like the
pure/ideal element. One of the best ways to
measure this deviation is through frequency
response.
2 2
s s
s
C f K x
E
2 2
= =
8. Mechanical System Modeling K. Craig 8
Spring Element
( ) ( )
( )
0
s
2 2
x
s 0 s 0
s
0
Differential Work Done
f dx K x dx
Total Work Done
K x C f
K x dx
2 2
= =
= = =∫
9. Mechanical System Modeling K. Craig 9
Frequency Response
Of
Spring Elements
( )
( )
0
s 0
f f sin t
x C f sin t
= ω
= ω
11. Mechanical System Modeling K. Craig 11
More Realistic Lumped-Parameter Model for a Spring
Ks
Ks
M
B B
f, x
12. Mechanical System Modeling K. Craig 12
Linearization
for a
Nonlinear Spring
( )
( )
( )
0 0
0
22
0
0 0 2
x x x x
0 0
x x
x xdf d f
y f (x ) x x
dx dx 2!
df
y y x x
dx
= =
=
−
= + − + +
≈ + −
( )
0
0 0
x x
df
y y x x
dx
ˆ ˆy Kx
=
− ≈ + −
=
13. Mechanical System Modeling K. Craig 13
• Real Springs
– nonlinearity of the
force/deflection curve
– noncoincidence of the
loading and unloading
curves (The 2nd Law of
Thermodynamics
guarantees that the area
under the loading f vs. x
curve must be greater
than that under the
unloading f vs. x curve.
It is impossible to recover
100% of the energy put
into any system.)
14. Mechanical System Modeling K. Craig 14
• Several Types of Practical
Springs:
– coil spring
– hydraulic (oil) spring
– cantilever beam spring
– pneumatic (air) spring
– clamped-end beam spring
– ring spring
– rubber spring (shock mount)
– tension rod spring
– torsion bar spring
15. Mechanical System Modeling K. Craig 15
• Spring-like Effects in
Unfamiliar Forms
– aerodynamic spring
– gravity spring (pendulum)
– gravity spring (liquid
column)
– buoyancy spring
– magnetic spring
– electrostatic spring
– centrifugal spring
16. Mechanical System Modeling K. Craig 16
Damper Element
• A pure damper dissipates all the energy supplied
to it, i.e., converts the mechanical energy to
thermal energy.
• Various physical mechanisms, usually associated
with some form of friction, can provide this
dissipative action, e.g.,
– Coulomb (dry friction) damping
– Material (solid) damping
– Viscous damping
17. Mechanical System Modeling K. Craig 17
• Pure / ideal damper element provides viscous
friction.
• All mechanical elements are defined in terms of
their force/motion relation. (Electrical elements
are defined in terms of their voltage/current
relations.)
• Pure / Ideal Damper
– Damper force or torque is directly proportional
to the relative velocity of its two ends.
1 2dx dx dx
f B B
dt dt dt
⎛ ⎞
= − =⎜ ⎟
⎝ ⎠
1 2d d d
T B B
dt dt dt
θ θ θ⎛ ⎞
= − =⎜ ⎟
⎝ ⎠
18. Mechanical System Modeling K. Craig 18
– Forces or torques on the two ends of the
damper are exactly equal and opposite at all
times (just like a spring); pure springs and
dampers have no mass or inertia. This is NOT
true for real springs and dampers.
– Units for B to preserve physical meaning:
• N/(m/sec)
• (N-m)/(rad/sec)
– Transfer Function
( )
2
2
2
2
dx d x
Dx D x
dt dt
x x
(x)dt x dt dt
D D
⎡ ⎤
⎣ ⎦∫ ∫ ∫
Differential
Operator
Notation
19. Mechanical System Modeling K. Craig 19
• Operational Transfer Functions
• We assume the initial conditions are zero.
– Damper element dissipates into heat all
mechanical energy supplied to it.
• Force applied to damper causes a velocity in same
direction.
f BDx
T BD
=
= θ
( ) ( )
( ) ( )
f T
D BD D BD
x
x 1 1
D D
f BD T BD
θ
θ
( )( )
2
dx dx
Power force velocity f B
dt dt
⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
20. Mechanical System Modeling K. Craig 20
• Power input to the device is positive since the force
and velocity have the same sign.
• It is impossible for the applied force and resulting
velocity to have opposite signs.
• Thus, a damper can never supply power to another
device; Power is always positive.
• A spring absorbs power and stores energy as a force
is applied to it, but if the force is gradually relaxed
back to zero, the external force and the velocity now
have opposite signs, showing that the spring is
delivering power.
• Total Energy Dissipated
( ) ( )
2
dx dx
P dt B dt B dx f dx
dt dt
⎛ ⎞ ⎛ ⎞
= = =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
∫ ∫ ∫ ∫
21. Mechanical System Modeling K. Craig 21
Damper Element
Step Input Force
causes instantly
(a pure damper
has no inertia) a
Step of dx/dt
and a
Ramp of x
22. Mechanical System Modeling K. Craig 22
Frequency
Response
of
Damper
Elements
( )
( )
( )
0
t
0 0
0
0
f f sin t
dx
B
dt
1
x x f sin t dt
B
f
1 cos t
B
= ω
=
− = ω
⎡ ⎤= − ω⎣ ⎦ω
∫
0
x
f 0
f
A 1B
A f B
ω= =
ω
23. Mechanical System Modeling K. Craig 23
• Sinusoidal Transfer Function
– M is the amplitude ratio of output over input
– φ is the phase shift of the output sine wave with
respect to the input sine wave (positive if the
output leads the input, negative if the output lags
the input)
( )
x 1
D
f BD
= D i⇒ ω ( )
x 1
i M
f i B
ω = = ∠φ
ω
( )
x 1 1
i M 90
f i B B
°
ω = = ∠φ = ∠ −
ω ω
24. Mechanical System Modeling K. Craig 24
• Real Dampers
– A damper element is used to model a device
designed into a system (e.g., automotive shock
absorbers) or for unavoidable parasitic effects
(e.g., air drag).
– To be an energy-dissipating effect, a device
must exert a force opposite to the velocity;
power is always negative when the force and
velocity have opposite directions.
– Let’s consider examples of real intentional
dampers.
25. Mechanical System Modeling K. Craig 25
Viscous (Piston/Cylinder) Damper
A relative velocity between the
cylinder and piston forces the
viscous oil through the clearance
space h, shearing the fluid and
creating a damping force.
2 2 2
2 2 1
2 13
2
6 L h R R
B R R h
hh 2 R
2
⎡ ⎤
⎡ ⎤ ⎢ ⎥πμ −⎛ ⎞
= − − −⎢ ⎥⎜ ⎟ ⎢ ⎥
⎝ ⎠⎢ ⎥⎣ ⎦ −⎢ ⎥
⎣ ⎦
μ = fluid viscosity
26. Mechanical System Modeling K. Craig 26
Simple Shear Damper
And
Viscosity Definition
fluid viscosity
shearing stress F / A
velocity gradient V / t
μ
=
2A
F V
t
F 2A
B
V t
μ
=
μ
= =
28. Mechanical System Modeling K. Craig 28
Commercial Air Damper
laminar flow
linear damping
turbulent flow
nonlinear damping
(Data taken with valve shut)
Air Damper
• much lower viscosity
• less temperature dependent
• no leakage or sealing problem
29. Mechanical System Modeling K. Craig 29
Eddy-Current Damper
• Motion of the conducting
cup in the magnetic field
generates a voltage in the
cup.
• A current is generated in
the cup’s circular path.
• A current-carrying
conductor in a magnetic
field experiences a force
proportional to the current.
• The result is a force
proportional to and
opposing the velocity.
• The dissipated energy
shows up as I2R heating of
the cup.
32. Mechanical System Modeling K. Craig 32
• The damper element can also be used to represent
unavoidable parasitic energy dissipation effects in
mechanical systems.
– Frictional effects in moving parts of machines
– Fluid drag on vehicles (cars, ships, aircraft, etc.)
– Windage losses of rotors in machines
– Hysteresis losses associated with cyclic stresses in
materials
– Structural damping due to riveted joints, welds,
etc.
– Air damping of vibrating structural shapes
34. Mechanical System Modeling K. Craig 34
Coulomb Friction: Modeling and Simulation
• In most control systems, Coulomb friction is a
nuisance.
• Coulomb friction is difficult to model and
troublesome to deal with in control system design.
• It is a nonlinear phenomenon in which a force is
produced that tends to oppose the motion of
bodies in contact in a mechanical system.
• Undesirable effects: “hangoff” and limit cycling
35. Mechanical System Modeling K. Craig 35
• Hangoff (or dc limit cycle) prevents the steady-
state error from becoming zero with a step
command input.
• Limit Cycling is behavior in which the steady-state
error oscillates or hunts about zero.
• What Should the Control Engineer Do?
– Minimize friction as much as possible in the design
– Appraise the effect of friction in a proposed control
system design by simulation
– If simulation predicts that the effect of friction is
unacceptable, you must do something about it!
36. Mechanical System Modeling K. Craig 36
– Remedies can include simply modifying the design
parameters (gains), using integral control action, or
using more complex measures such as estimating the
friction and canceling its effect.
– Modeling and simulation of friction should contribute
significantly to improving the performance of motion
control systems.
37. Mechanical System Modeling K. Craig 37
Modeling Coulomb Friction
V
Ff
Fslip
Fstick
"Stiction" Coulomb
Friction Model
38. Mechanical System Modeling K. Craig 38
Case Study to Evaluate Friction Model
m
k
Ff
V0 V
m = 0.1 kg
k = 100 N/m
Fstick = 0.25 N
Fslip = 0.20 N (assumed independent of velocity)
V0 = step of 0.002 m/sec at t = 0 sec
42. Mechanical System Modeling K. Craig 42
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.005
0.01
0.015
0.02
0.025
time (sec)
2*position,velocity,0.1*FrictionForce
Position, Velocity, Friction Force vs. Time
43. Mechanical System Modeling K. Craig 43
Inertia Element
• A designer rarely inserts a component for the
purpose of adding inertia; the mass or inertia
element often represents an undesirable effect
which is unavoidable since all materials have
mass.
• There are some applications in which mass itself
serves a useful function, e.g., accelerometers and
flywheels.
44. Mechanical System Modeling K. Craig 44
Useful Applications
of
Inertia
Flywheels are used as
energy-storage devices or as
a means of smoothing out
speed fluctuations in engines
or other machines.
Accelerometer
45. Mechanical System Modeling K. Craig 45
– Newton’s Law defines the behavior of mass
elements and refers basically to an idealized
“point mass”:
– The concept of rigid body is introduced to deal
with practical situations. For pure translatory
motion, every point in a rigid body has identical
motion.
– Real physical bodies never display ideal rigid
behavior when being accelerated.
– The pure / ideal inertia element is a model, not
a real object.
( )( )forces mass acceleration=∑
47. Mechanical System Modeling K. Craig 47
– Newton’s Law in rotational form for bodies
undergoing pure rotational motion about a single
fixed axis:
– The concept of moment of inertia J also considers
the rotating body to be perfectly rigid.
– Note that to completely describe the inertial
properties of any rigid body requires the
specification of:
• Its total mass
• Location of the center of mass
• 3 moments of inertia and 3 products of inertia
( )( )torques moment of inertia angular acceleration=∑
48. Mechanical System Modeling K. Craig 48
Rotational Inertia
J (kg-m2)
( )( )
( )( ) ( )
tangential force
mass acceleration
2 rL dr r
=
⎡ ⎤= π ρ α⎣ ⎦
( )
R 2 2
3 2
0
R MR
total torque 2 L r dr R L J
2 2
= πρ α = π ρ = α = α∫
50. Mechanical System Modeling K. Craig 50
– How do we determine J for complex shapes
with possibly different materials involved?
• In the design stage, where the actual part exists only
on paper, estimate as well as possible!
• Once a part has been constructed, use experimental
methods for measuring inertial properties. How?
51. Mechanical System Modeling K. Craig 51
Experimental Measurement
Of
Moment of Inertia
( )
2
2
2
s 2
2
s
2
0 n 0
s
n
n
n
d
torques J J
dt
d
K J
dt
Kd
0
dt J
cos t ( 0)
K
rad/sec
J
f cycles/sec
2
θ
= α =
θ
− θ =
θ
+ θ =
θ = θ ω θ =
ω
ω
π
∑
s
22
n
K
J
4 f
=
π
52. Mechanical System Modeling K. Craig 52
– Actually the oscillation will gradually die out
due to the bearing friction not being zero.
– If bearing friction were pure Coulomb friction,
it can be shown that the decay envelope of the
oscillations is a straight line and that friction
has no effect on the frequency.
– If the friction is purely viscous, then the decay
envelope is an exponential curve, and the
frequency of oscillation does depend on the
friction but the dependence is usually negligible
for the low values of friction in typical
apparatus.
53. Mechanical System Modeling K. Craig 53
Inertia Element
Real inertias may be
impure (have some
springiness and friction)
but are very close to
ideal.
( ) ( )2 2
x 1 1
D D
f MD T JD
θ
= =
Inertia Element stores
energy as kinetic energy:
2 2
Mv J
or
2 2
ω
54. Mechanical System Modeling K. Craig 54
– A step input force applied to a mass initially at
rest causes an instantaneous jump in
acceleration, a ramp change in velocity, and a
parabolic change in position.
– The frequency response of the inertia element is
obtained from the sinusoidal transfer function:
• At high frequency, the inertia element becomes very
difficult to move.
• The phase angle shows that the displacement is in a
direction opposite to the applied force.
( )
( )
2 2
x 1 1
i 180
f MM i
°
ω = = ∠ −
ωω
55. Mechanical System Modeling K. Craig 55
Useful Frequency Range
for
Rigid Model
of a
Real Flexible Body
A real flexible body
approaches the
behavior of a rigid body
if the forcing frequency
is small compared to
the body’s natural
frequency.
56. Mechanical System Modeling K. Craig 56
– Analysis:
( )
( ) ( )
i o o
2
o o i
2
o i n2 2
n
i i
2 2 2
o o
2
n
n n
2AE
x x ALx
L
L
x x x
2E
D 2E
1 x x
L
x x1 1 1
D i
Dx x i1 1 1
− = ρ
ρ
+ =
⎛ ⎞
+ = ω⎜ ⎟
ω ρ⎝ ⎠
= ω = =
⎛ ⎞ ⎛ ⎞ω ω+ + −⎜ ⎟ ⎜ ⎟ω ω ω⎝ ⎠ ⎝ ⎠
57. Mechanical System Modeling K. Craig 57
– ωmax is the highest frequency for which the real
body behaves almost like an ideal rigid body.
• Frequency response is unmatched as a technique
for defining the useful range of application for all
kinds of dynamic systems.
( )o
2
i max
n
max n
x 1
i 1.05
x
1
0.308 E
0.218
L
ω = =
⎛ ⎞ω
−⎜ ⎟ω⎝ ⎠
ω = ω =
ρ
96200 cycles/min
for a 6-inch
steel rod
58. Mechanical System Modeling K. Craig 58
Motion Transformers
• Mechanical systems often include mechanisms
such as levers, gears, linkages, cams, chains, and
belts.
• They all serve a common basic function, the
transformation of the motion of an input member
into the kinematically-related motion of an output
member.
• The actual system may be simplified in many
cases to a fictitious but dynamically equivalent
one.
59. Mechanical System Modeling K. Craig 59
• This is accomplished by “referring” all the
elements (masses, springs, dampers) and driving
inputs to a single location, which could be the
input, the output, or some selected interior point of
the system.
• A single equation can then be written for this
equivalent system, rather than having to write
several equations for the actual system.
• This process is not necessary, but often speeds the
work and reduces errors.
60. Mechanical System Modeling K. Craig 60
Motion Transformers
Gear Train Relations:
θ
θ
m
m
m
m
N
N
N
T
T
N
N N
′
= ≡
′
= ≡
2
1
1
2
1
Tm
N1
N2
θm
′Tm ′θm
61. Mechanical System Modeling K. Craig 61
Translational Equivalent
for
A Complex System
x1, x2, θ
are
kinematically related
Refer all elements and
inputs to the x1 location
and define a fictitious
equivalent system
whose motion will be
the same as x1 but will
include all the effects
in the original system.
62. Mechanical System Modeling K. Craig 62
– Define a single equivalent spring element
which will have the same effect as the three
actual springs.
– Mentally apply a static force f1 at location x1
and write a torque balance equation:
( ) 1 s2
1 1 s1 1 1 1 s2 2
1 1
1 se 1
2
2
se s1 s2 s2
1 1
x KL
f L K x L x K L
L L
f K x
L 1
K K K K
L L
⎛ ⎞
= + +⎜ ⎟
⎝ ⎠
=
⎡ ⎤⎛ ⎞
+ +⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
63. Mechanical System Modeling K. Craig 63
– The equivalent spring constant Kse refers to a
fictitious spring which, if installed at location
x1, would have exactly the same effect as all the
springs together in the actual system.
– To find the equivalent damper, mentally
remove the inertias and springs and again apply
a force f1 at x1: ( ) ( )1 1 1 1 1 2 2 2
2
2 1
1 1 1 1 2
1 1
1 e 1
2
2
e 1 2 2
1 1
f L x B L x B L B
L x
x B L x B B
L L
f B x
L 1
B B B B
L L
= + + θ
= + +
=
⎡ ⎤⎛ ⎞
+ +⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
64. Mechanical System Modeling K. Craig 64
– Finally, consider only the inertias present.
– While the definitions of equivalent spring and
damping constants are approximate due to the
assumption of small motions, the equivalent
mass has an additional assumption which may
be less accurate; we have treated the masses as
point masses, i.e., J = ML2.
( ) ( ) ( )2 21 1 1
1 1 1 1 2 2
1 1 1
1 e 1
2
2
e 1 2 2
1 1
x x x
f L M L M L J
L L L
f M x
L 1
M M M J
L L
≈ + +
≈
⎡ ⎤⎛ ⎞
+ +⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
65. Mechanical System Modeling K. Craig 65
– To refer the driving inputs to the x1 location we
note that a torque T is equivalent to a force T/L1
at the x1 location, and a force f2 is equivalent to
a force (L2/L1)f2.
– If we set up the differential equation of motion
for this system and solve for its unknown x1,
we are guaranteed that this solution will be
identical to that for x1 in the actual system.
– Once we have x1, we can get x2 and/or θ
immediately since they are related to x1 by
simple proportions.
66. Mechanical System Modeling K. Craig 66
– Rules for calculating the equivalent elements
without deriving them from scratch:
• When referring a translational element (spring,
damper, mass) from location A to location B, where
A’s motion is N times B’s, multiply the element’s
value by N2. This is also true for rotational elements
coupled by motion transformers such as gears, belts,
and chains.
• When referring a rotational element to a
translational location, multiply the rotational
element by 1/R2, where the relation between
translation x and rotation θ (in radians) is x = R θ.
For the reverse procedure (referring a translational
element to a rotational location) multiply the
translational element by R2.
67. Mechanical System Modeling K. Craig 67
• When referring a force at A to get an equivalent
force at B, multiply by N (holds for torques).
Multiply a torque at θ by 1/R to refer it to x as a
force. A force at x is multiplied by R to refer it as a
torque to θ.
– These rules apply to any mechanism, no matter
what its form, as long as the motions at the two
locations are linearly related.
68. Mechanical System Modeling K. Craig 68
Mechanical Impedance
• When trying to predict the behavior of an
assemblage of subsystems from their calculated or
measured individual behavior, impedance methods
have advantages.
• Mechanical impedance is defined as the transfer
function (either operational or sinusoidal) in which
force is the numerator and velocity the
denominator. The inverse of impedance is called
mobility.
69. Mechanical System Modeling K. Craig 69
Mechanical Impedance for the Basic Elements
( ) ( )
( ) ( )
( ) ( )
s
S
B
M
Kf
Z D D
v D
f
Z D D B
v
f
Z D D MD
v
=
=
=
70. Mechanical System Modeling K. Craig 70
• Measurement of impedances of subsystems can be
used to analytically predict the behavior of the
complete system formed when the subsystems are
connected. We can thus discover and correct
potential design problems before the subsystems
are actually connected.
• Impedance methods also provide “shortcut”
analysis techniques.
– When two elements carry the same force they are said
to be connected in parallel and their combined
impedance is the product of the individual impedances
over their sum.
71. Mechanical System Modeling K. Craig 71
– For impedances which have the same velocity, we say
they are connected in series and their combined
impedance is the sum of the individual ones.
– Consider the following systems:
Parallel Connection
Series Connection f, v
x1
, v1
B
K
K
f, v
B
72. Mechanical System Modeling K. Craig 72
– Parallel Connection
– Series Connection
( )
K
Bf KBDD
Kv BD KB
D
= =
++
( )
f K BD K
D B
v D D
+
= + =
73. Mechanical System Modeling K. Craig 73
Force and Motion Sources
• The ultimate driving agency of any mechanical
system is always a force not a motion; force causes
acceleration, acceleration does not cause force.
• Motion does not occur without a force occurring
first.
• At the input of a system, what is known, force or
motion? If motion is known, then this motion was
caused by some (perhaps unknown) force and
postulating a problem with a motion input is
acceptable.
74. Mechanical System Modeling K. Craig 74
• There are only two classes of forces:
– Forces associated with physical contact between two
bodies
– Action-at-a-distance forces, i.e., gravitational, magnetic,
and electrostatic forces.
• There are no other kinds of forces! (Inertia force is a
fictitious force.)
• The choice of an input form to be applied to a system
requires careful consideration, just as the choice of a
suitable model to represent a component or system.
• Here are some examples of force and motion sources.
79. Mechanical System Modeling K. Craig 79
• Energy Considerations
– A system can be caused to respond only by the source
supplying some energy to it; an interchange of energy
must occur between source and system.
– If we postulate a force source, there will be an
associated motion occurring at the force input point.
– The instantaneous power being transmitted through this
energy port is the product of instantaneous force and
velocity.
– If the force applied by the source and the velocity
caused by it are in the same direction, power is supplied
by the source to the system. If force and velocity are
opposed, the system is returning power to the source.
80. Mechanical System Modeling K. Craig 80
– The concept of mechanical impedance is of some help
here.
– The transfer function relating force and velocity at the
input port of a system is called the driving-point
impedance Zdp.
– We can write an expression for power:
dp
dp
f
Z (D) (D)
v
f
Z (i ) (i )
v
=
ω = ω
2
dp dp
f f
P fv f
Z Z
= = =
81. Mechanical System Modeling K. Craig 81
– If we apply a force source to a system with a high value
of driving-point impedance, not much power will be
taken from the source, since the force produces only a
small velocity. The extreme case of this would the
application of a force to a perfectly rigid wall (driving-
point impedance is infinite, since no motion is produced
no matter how large a force is applied). In this case the
source would not supply any energy.
– The higher the driving-point impedance, the more a real
force source behaves like an ideal force source.
– The lower the driving-point impedance, the more a real
motion source behaves like an ideal motion source.
82. Mechanical System Modeling K. Craig 82
– Real sources may be described accurately as
combinations of ideal sources and an output impedance
characteristic of the physical device.
– A complete description of the situation thus requires
knowledge of two impedances:
• The output impedance of the real source
• The driving-point impedance of the driven system
83. Mechanical System Modeling K. Craig 83
Mechanical System Examples
Problem Statement
Develop the equivalent rotational
model of the rack-and-pinion gear
system shown. The applied torque T is
the input variable, and the angular
displacement θ is the output variable.
Neglect any twist in the shaft.
Bearings are frictionless. The pinion
gear mass moment of inertia about its
CG (geometric center) is Ip.
( )2 2 2
m s p rI I I m R cR kR T+ + + θ + θ + θ =
Rack-and-Pinion Gear System
84. Mechanical System Modeling K. Craig 84
Problem Statement
A load inertia I5 is driven through a
double-gear pair by a motor with inertia
I4, as shown. The shaft inertias are
negligible. The gear inertias are I1, I2,
and I3. The speed ratios are ω1/ω2 = 2
and ω2/ω3 = 5. The motor torque is T1
and the viscous damping coefficient c =
4 lb-ft-sec/rad. Neglect elasticity in the
system, and use the following inertia
values (sec2-ft-lb/rad): I1 = 0.1, I2 = 0.2,
I3 = 0.4, I4 = 0.3, I5 = 0.7. Derive the
mathematical model for the motor shaft
speed ω1 with T1 as the input.
( ) ( )
2 2 2 2
4 1 5 3 2 1 1 1
1 1 1 1
I I I I I c T
5 2 5 2
⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
+ + + + ω + ω =⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭
Multi-Gear System
85. Mechanical System Modeling K. Craig 85
Physical System
Physical Model
Problem Statement
A dynamic vibration absorber consists of
a mass and an elastic element that is
attached to another mass in order to
reduce its vibration. The figure is a
representation of a vibration absorber
attached to the cantilever support. For a
cantilever beam with a force at its end, k
= Ewh3/4L3 where L = beam length, w =
beam width, and h = beam thickness. (a)
Obtain the equation of motion for the
system. The force f is a specified force
acting on the mass m, and is due to the
rotating unbalance of the motor. The
displacements x and x2 are measured
from the static equilibrium positions
when f = 0. (b) Obtain the transfer
functions x/f and x2/f.
( )[ ]
( )[ ]
2
2 2
4 2
2 2 2 2 2
2 2
4 2
2 2 2 2 2
m D kx
F mm D m k k mk D kk
x k
F mm D m k k mk D kk
+
=
+ + + +
=
+ + + +
Dynamic Vibration Absorber
86. Mechanical System Modeling K. Craig 86
Rigid Body Dynamics: Kinematics
Reference Frames
R - Ground xyz
R1 - Body x1y1z1
( )1 1 1
1 1 1
R R RR P R A R R AP R AP
R R RP R P
a a r r
a 2 v
⎡ ⎤ ⎡ ⎤= + ω × ω × + α ×⎣ ⎦⎣ ⎦
⎡ ⎤+ + ω ×⎣ ⎦
y
z O
P
xR
x1
y1
z1
R1
A
( )1 1R RR P R A R AP P
v v r v= + ω × +
Note: For any vector q
1
1
RR
RRdq dq
q
dt dt
= + ω ×
87. Mechanical System Modeling K. Craig 87
R
R1 R2
O θ = 30º
r = 0.06 m
Rigid-Body Kinematics Example
Given:
Find:
Reference Frames:
R → ground: xyz
R1 → shaft: x1y1z1
R2 → disk: x2y2z2
φ x1
y1
x2
y2
O
z1
y
z
y1
O
α
1
1 2
RR
R R
1
ˆ5i constant
ˆ4k constant
ω = =
ω = =
R P
a
1
1
1
ˆ ˆi i1 0 0
ˆ ˆj 0 cos sin j
ˆ ˆ0 sin cosk k
⎡ ⎤ ⎡ ⎤⎡ ⎤
⎢ ⎥ ⎢ ⎥⎢ ⎥= α α⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥− α α⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦
88. Mechanical System Modeling K. Craig 88
( )2 2 2
2 2 2
R R RR P R O R R OP R OP
R R RP R P
a a r r
a 2 v
⎡ ⎤ ⎡ ⎤= + ω × ω × + α ×⎣ ⎦⎣ ⎦
⎡ ⎤+ + ω ×⎣ ⎦
2
2
R O
R P
R P
a 0
a 0
v 0
=
=
=
Point O at end of rotating shaft fixed in R
Point P fixed in R2 (disk)
( )
( )
( )
2 1 1 2
2
2
1
R R R RR R
1
RR R R
RR
1
R
RR1
1
1 1 1
ˆ ˆ5i 4k
d d ˆ ˆ5i 4k
dt dt
dk ˆ0 4 4 k
dt
ˆ ˆ ˆ4 5i k 20j
ω = ω + ω = +
ω
⎡ ⎤α = = +⎣ ⎦
= + = ω ×
= × = −
( )ˆ ˆ20 jcos ksin= − α + α
( ) ( )OP
1 1
ˆ ˆr rcos i rsin j= θ + θ
89. Mechanical System Modeling K. Craig 89
After Substitution and Simplification:
( ) ( ) ( )R P
1 1 1
ˆ ˆ ˆa 16rcos i 41rsin j 40rcos k= − θ + − θ + θ
Alternate Solution:
( )1 1 1
1 1 1
R R RR P R O R R OP R OP
R R RP R P
a a r r
a 2 v
⎡ ⎤ ⎡ ⎤= + ω × ω × + α ×⎣ ⎦⎣ ⎦
⎡ ⎤+ + ω ×⎣ ⎦
1
1
1
R O
RR
RR R
RR
a 0
ˆ5i constant
d
0
dt
=
ω = =
ω
α = =
( ) ( )OP
1 1
ˆ ˆr rcos i rsin j= θ + θ
90. Mechanical System Modeling K. Craig 90
( )1 1 1 2 1 2 1 2R R R R R R R RP O OP OP
a a r r⎡ ⎤ ⎡ ⎤= + ω × ω × + α ×⎣ ⎦⎣ ⎦
(P is fixed in R2)
( )
1
1 2
1 1 2 1
1 2
1 1 1 2
1
R O
R R
1
R R R R
R R
1
R R R RP O OP
R O
a 0
ˆ4k
d d ˆ4k 0
dt dt
v v r
v 0
=
ω =
ω
⎡ ⎤α = = =⎣ ⎦
= + ω ×
=
( ) ( )OP
1 1
ˆ ˆr rcos i rsin j= θ + θ
After Substitution and Simplification:
( ) ( ) ( )R P
1 1 1
ˆ ˆ ˆa 16rcos i 41rsin j 40rcos k= − θ + − θ + θ
(same result)
91. Mechanical System Modeling K. Craig 91
Rigid Body Dynamics: Kinetics
Linear Momentum
Angular Momentum about point C
Equations of Motion
Point C: mass center of a rigid body of mass m.
Reference Frames
R - Ground xyz
R1 - Body x1y1z1
R C
L m v= y1
y
z O
’
xR
x1
z1
R1
A
C
y1
1
1 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1 1
RR
x x x x y x z x
RR
y y x y y y z y
RR
z z x z y z z z
H I I I
H I I I
H I I I
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ω
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
= ω⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ω⎣ ⎦ ⎣ ⎦ ⎣ ⎦
1 1 1x 1 y 1 z 1
ˆ ˆ ˆH H i H j H k= + +
R R C
R
d v
F m
dt
dH
M
dt
∑ =
∑ =