机械系统动力学课件
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24
6
2016-11-16
25
26
Practice
27
28
7
2016-11-16
第二章Angular velocity
• Definition
Time derivative of a unit vector in rotating systems
• It is a property for the whole body • The sum of rotations
where
A e A e A e A 1 1 2 2 3 3 t
Angular velocity and angular acceleration
Example
• The angular acceleration is the rate of change of the angular velocity
• Velocity
• Intrinsic coordinates • Acceleration
17
Байду номын сангаас
– Path Variables:
• The definition of the unit vectors and of the position relies on knowledge of the path
– A bar moving in a plane
52
13
2016-11-16
• Generalized Coordinates:
– The variables that we select to locate the current position of a system
Force and torque balance
• The velocity may be expressed in terms of either set of unit vectors
23
• Like components are grouped:
cos sin sin cos
18
Path Variables
s 0
lim
r_ P s
1 et ( s)
19
20
5
2016-11-16
21
22
Joint Kinematical Descriptions
• Relation between two sets of orthogonal unit vectors in a plane
4
2016-11-16
Spherical Coordinates
• Position • Extrinsic coordinate systems
– Rectangular Cartesian Coordinates – Cylindrical Coordinates – Spherical Coordinates
5
The equation becomes:
- mn A sin( nt ） kA sin( nt ）
2
6
Newton’s laws to obtain equations of motion
• By substituting the initial conditions:
x( t ) A sin( nt ）
12
2016-11-16
In-class Practice
第三章Degree of freedom and generalized coordinates
• DOF: the number of independent coordinates needed to completely specify the configuration of the system.
• The number of degrees of freedom = the number of generalized coordinates minus the number of constraint equations
53
Force and torque balance
• Twisting motion: k is the torsional stiffness of shaft, the mass of the shaft is ignored. represents the angular position of the shaft relative to its equilibrium position. ( 0 ) position The disk will vibrate around the equilibrium .
Virtual Displacement
• Virtual movement
– Lagrange, Hamilton, etc. – More mathematical – Lagrange formulation: treat connected bodies as a single system
• Kinetics:
– Relates forces and torques to motion
1
2
第一章
• Foundations of Dynamics
– Newton’s laws
• I. The existence of inertial reference frame • II. In an inertial frame, F=ma • III. Action and reaction forces are equal and act in opposite directions
7 8
2
2016-11-16
Lagrange’s equations
What will be covered?
• Kinematics
– Particle kinematics – Relative motion – Kinematics of rigid bodies
• Analytical Mechanics • Vibration • Matlab
• Motion of bodies>>atomic scale • Speed of motions<<the speed of light
3
• Analytical dynamics
– Lagrangian-Hamiltonian approach (scalar)
4
1
2016-11-16
Newton’s laws to obtain equations of motion
9
2016-11-16
Solution
Motion of a particle in a moving coordinate system
10
2016-11-16
Kinematics of rigid bodies
Example
11
2016-11-16
Rolling
Find o ' ao ' c ac
( t ) kx( t ) m x
Solve by assuming that
Newton’s laws to obtain equations of motion
x( t ) A sin( nt ）
Then we will have:
( t ) n A cos( nt ） x ( t ) - An 2 sin( nt ） x ( t ) kx( t ) m x
• Substitute the relation between two sets of unit vectors into the previous Eq.
e e e e
cos e sin e sin e cos e cos sin e sin cos e
Two approaches to build equations of motion
• Vectorial dynamics
– Newton’s laws – Motion is described in physical coordinates and their derivatives
– Apply to particles, systems of particles & Rigid bodies/systems of rigid bodies – Newtonian mechanics:
9 10
Chapter 1. Particle Kinematics
• Interest is on defining quantities such as position, velocity and acceleration • Need to specify a reference frame and a coordinate system in which to actually write the vector expressions. • Choose the coordinates naturally fit known aspects of the motion
k J
14
2016-11-16
Two-degree-of-freedom torsional system
Example: Side section of a vehicle
第四章Introduction to analytical mechanics
• Newton-Euler formulation for deriving equations of motion. • Analytical mechanics:
2016-11-16
Reference Books
• Advanced Engineering Dynamics (2 nd Edition, Jerry Ginsberg, Cambridge University Press 1995) • PPT (Pay attention to all the examples) • Where to find these materials?
• Constraints • Independent coordinates
f
i i
xi
m x J cx kx F m x cx kx F m x
M
oi
– Unconstrained generalized coordinates
– dynamics14@126.com – Password: dy1111
Introductory concepts
• Dynamics:
– Kinematics and kinetics of particles, rigid bodies and continua
• Kinematics:
– Studies motion without its cause
• Angular acceleration
Ex
Time derivative of a vector
If is any vector
Determine
in terms of the moving frame
8
2016-11-16
• The time rate of change of any vector described in terms of components relative to reference frame XYZ having angular velocity is
Two-degree-of-freedom system with viscous damping
1 ( c1 c2 )x 1 c2 x 2 ( k1 k2 )x1 k2 x2 0 m1 x m2 x2 c2 x1 c2 x2 k2 x1 k2 x2 0
11
Cartesian coordinates
12
3
2016-11-16
13
14
Cylindrical and polar coordinates
• Unit vectors in terms of i, j, k • Position • Velocity
For
• Acceleration
15 16
6
2016-11-16
25
26
Practice
27
28
7
2016-11-16
第二章Angular velocity
• Definition
Time derivative of a unit vector in rotating systems
• It is a property for the whole body • The sum of rotations
where
A e A e A e A 1 1 2 2 3 3 t
Angular velocity and angular acceleration
Example
• The angular acceleration is the rate of change of the angular velocity
• Velocity
• Intrinsic coordinates • Acceleration
17
Байду номын сангаас
– Path Variables:
• The definition of the unit vectors and of the position relies on knowledge of the path
– A bar moving in a plane
52
13
2016-11-16
• Generalized Coordinates:
– The variables that we select to locate the current position of a system
Force and torque balance
• The velocity may be expressed in terms of either set of unit vectors
23
• Like components are grouped:
cos sin sin cos
18
Path Variables
s 0
lim
r_ P s
1 et ( s)
19
20
5
2016-11-16
21
22
Joint Kinematical Descriptions
• Relation between two sets of orthogonal unit vectors in a plane
4
2016-11-16
Spherical Coordinates
• Position • Extrinsic coordinate systems
– Rectangular Cartesian Coordinates – Cylindrical Coordinates – Spherical Coordinates
5
The equation becomes:
- mn A sin( nt ） kA sin( nt ）
2
6
Newton’s laws to obtain equations of motion
• By substituting the initial conditions:
x( t ) A sin( nt ）
12
2016-11-16
In-class Practice
第三章Degree of freedom and generalized coordinates
• DOF: the number of independent coordinates needed to completely specify the configuration of the system.
• The number of degrees of freedom = the number of generalized coordinates minus the number of constraint equations
53
Force and torque balance
• Twisting motion: k is the torsional stiffness of shaft, the mass of the shaft is ignored. represents the angular position of the shaft relative to its equilibrium position. ( 0 ) position The disk will vibrate around the equilibrium .
Virtual Displacement
• Virtual movement
– Lagrange, Hamilton, etc. – More mathematical – Lagrange formulation: treat connected bodies as a single system
• Kinetics:
– Relates forces and torques to motion
1
2
第一章
• Foundations of Dynamics
– Newton’s laws
• I. The existence of inertial reference frame • II. In an inertial frame, F=ma • III. Action and reaction forces are equal and act in opposite directions
7 8
2
2016-11-16
Lagrange’s equations
What will be covered?
• Kinematics
– Particle kinematics – Relative motion – Kinematics of rigid bodies
• Analytical Mechanics • Vibration • Matlab
• Motion of bodies>>atomic scale • Speed of motions<<the speed of light
3
• Analytical dynamics
– Lagrangian-Hamiltonian approach (scalar)
4
1
2016-11-16
Newton’s laws to obtain equations of motion
9
2016-11-16
Solution
Motion of a particle in a moving coordinate system
10
2016-11-16
Kinematics of rigid bodies
Example
11
2016-11-16
Rolling
Find o ' ao ' c ac
( t ) kx( t ) m x
Solve by assuming that
Newton’s laws to obtain equations of motion
x( t ) A sin( nt ）
Then we will have:
( t ) n A cos( nt ） x ( t ) - An 2 sin( nt ） x ( t ) kx( t ) m x
• Substitute the relation between two sets of unit vectors into the previous Eq.
e e e e
cos e sin e sin e cos e cos sin e sin cos e
Two approaches to build equations of motion
• Vectorial dynamics
– Newton’s laws – Motion is described in physical coordinates and their derivatives
– Apply to particles, systems of particles & Rigid bodies/systems of rigid bodies – Newtonian mechanics:
9 10
Chapter 1. Particle Kinematics
• Interest is on defining quantities such as position, velocity and acceleration • Need to specify a reference frame and a coordinate system in which to actually write the vector expressions. • Choose the coordinates naturally fit known aspects of the motion
k J
14
2016-11-16
Two-degree-of-freedom torsional system
Example: Side section of a vehicle
第四章Introduction to analytical mechanics
• Newton-Euler formulation for deriving equations of motion. • Analytical mechanics:
2016-11-16
Reference Books
• Advanced Engineering Dynamics (2 nd Edition, Jerry Ginsberg, Cambridge University Press 1995) • PPT (Pay attention to all the examples) • Where to find these materials?
• Constraints • Independent coordinates
f
i i
xi
m x J cx kx F m x cx kx F m x
M
oi
– Unconstrained generalized coordinates
– dynamics14@126.com – Password: dy1111
Introductory concepts
• Dynamics:
– Kinematics and kinetics of particles, rigid bodies and continua
• Kinematics:
– Studies motion without its cause
• Angular acceleration
Ex
Time derivative of a vector
If is any vector
Determine
in terms of the moving frame
8
2016-11-16
• The time rate of change of any vector described in terms of components relative to reference frame XYZ having angular velocity is
Two-degree-of-freedom system with viscous damping
1 ( c1 c2 )x 1 c2 x 2 ( k1 k2 )x1 k2 x2 0 m1 x m2 x2 c2 x1 c2 x2 k2 x1 k2 x2 0
11
Cartesian coordinates
12
3
2016-11-16
13
14
Cylindrical and polar coordinates
• Unit vectors in terms of i, j, k • Position • Velocity
For
• Acceleration
15 16