video 2.1a vijay kumar and ani hsieh - edxpennx+robo3x+2t2017+type@asset... · a system of rigidly...

Robo3x-1.3 1Property of Penn Engineering, Vijay Kumar and Ani Hsieh

Video 2.1aVijay Kumar and Ani Hsieh


Vijay Kumar and Ani HsiehUniversity of Pennsylvania

Introduction to Lagrangian Mechanics


Analytical Mechanics

• Aristotle

• Galileo

• Bernoulli

1. Principle of Virtual Work: Static equilibrium of a particle, system of N particles, rigid bodies, system of rigid bodies

2. D’Alembert’s Principle: Incorporate inertial forces for dynamic analysis

3. Lagrange’s Equations of Motion

• Euler

• Lagrange

• D’Alembert


Generalized Coordinate(s)

A minimal set of coordinates required to describe the configuration of a system

No. of generalized coordinates = no. of degrees of freedom


Virtual Displacements

Virtual displacements are small displacements consistent with the constraints

A particle of mass m constrained to move vertically

g

generalized coordinate y

y




x

l

qf

The slider crank mechanism: a single degree of freedom linkage

generalized coordinate x, q, or f




The pendulum: a single degree of freedom linkage

ql

O

P

generalized coordinate q


Virtual Work

The work done by applied (external) forces through the virtual displacement,

A particle of mass m constrained to move vertically

mg


Virtual Work


x

l

qf

The slider crank mechanism: a single degree of freedom linkage

F


Video 2.1bVijay Kumar and Ani Hsieh


Virtual Work

The pendulum: a single degree of freedom linkage

ql

O

P


mg


The Principle of Virtual Work

The virtual work done by all applied (external) forces through any virtual displacement is zero

The system is in equilibrium


Static Equilibrium

mg


x

l

qf

F

Static Equilibrium


ql

O

P

mg

Static Equilibrium


D’Alembert’s Principle

The virtual work done by all applied (external) forces through any virtual displacement is zero

The system is in static equilibrium

include inertial forces

Equations of motion for the system

principle of virtual work

inertial force = - mass x acceleration



mge1

e2

acceleration

inertial force


ql

O

P

mg


er

eq

acceleration

inertial force

equation of motion


D’Alembert’s principle with generalized coordinates

contribution from inertial

force(s)

contribution from external force(s)

generalized coordinate

Particle in the vertical plane

Simple pendulum


The Key Idea

The contribution from the inertial forces can be expressed as a function of the

kinetic energy and its derivatives

Kinetic Energy


Lagrange’s Equation of Motion


Simple pendulum


Lagrange’s Equation of Motion for a Conservative System

Conservative System

Scalar Function

There exists a scalar function such that all applied forces are given by the gradient of the potential function

Gravitational force is conservativeScalar function is the potential energy


Standard Form of Lagrange’s Equation of Motion

The Lagrangian


Lagrange’s Equation of Motion


Simple pendulum


Video 2.2Vijay Kumar and Ani Hsieh



Newton Euler Equations


Analytical Mechanics

1. Principle of Virtual Work: Static equilibrium of a particle, system of N particles, rigid bodies, system of rigid bodies

2. D’Alembert’s Principle: Incorporate inertial forces for dynamic analysis

3. Lagrange’s Equations of Motion



Recall Newton’s 2nd law of motion

net external force = inertia x acceleration

m



Newton’s equations of motion for a translating rigid body

Euler’s equations of motion for a rotating rigid body


Motion of Systems of Particles

Ox

rOPi

mi

Pi

Newton’s equations of motion

fjPi

mj

net external force = total mass xacceleration of center of mass

fj

y

z

C

Center of Mass


Newton’s Second Law for a System of Particles

The center of mass for a system of particles, S, accelerates in an inertial frame, A, as if it were a single particle with mass m (equal to the total mass of the system) acted upon by a force equal to the net external force.

Rate of change of linear momentum in an inertial frame is equal to the net external force acting on the system.

Also true for a

rigid body

Linear momentum


Equations of Motion for a Rotating Rigid Body

• Rigid body with a point O fixed in an inertial frame

• Rigid body with the center of mass C

Angular momentum

CO

Px

y

moment of inertia


Equations of Motion for a Rotating Rigid Body

• Rigid body with a point O fixed in an inertial frame

• Rigid body with the center of mass C

The rate of change of angular momentum of a rigid body (or a system of rigidly connected particles) in an inertial frame with O or C as an origin is equal to the net external moment acting (with the same origin) on the body.


Example

O

P

ql

mge1

e2

equation of motion


Moment of Inertia of Planar Objects

m m

mm

2a

2be2

e1O

C

67 KB


Moment of Inertia of Planar Objects

C

x

y

r

dm



Dynamics and ControlAcceleration Analysis: Revisited


• Reference frame

• Origin

• Basis vectors

• Position Vectors

• Position vectors for P and Q in A

• Position vector of Q in B

Position Vectors

B

a1

a3

a2

P

Q

O

A

b1

b3

b2


General Approach to Analyzing Multi-Body System

1. Points common to

adjacent bodies

3. Pair of points on

adjacent bodies

whose relative

motions can be

easily described

2. Pair of points

fixed to the same

body

4. Write equations relating pairs of points either on same or adjacent bodies.


Position Analysis

B

a1

a3

a2

P

Q

O

A

b1

b3

b2


Velocity Analysis

B

a1

a3

a2

P

Q

O

A

b1

b3

b2

3x3 skew symmetric matrix

angular velocity of B in A


• Acceleration of P and Q in A

Acceleration Analysis

B

a1

a3

a2

P

Q

O

A


The angular acceleration of B in

A, is defined as the derivative of

the angular velocity of B in A:

Angular Acceleration

B

a1

a3

a2

P

Q

O

A


Acceleration Analysis

B

a1

a3

a2

P

Q

O

A

b1

b3

b2

centripetal (normal)acceleration

tangential acceleration


Serial Chain of Rigid Bodies

A

B

C

D

OA

OB

OC

OD



Newton Euler Equations (continued)



Newton’s equations of motion

Euler’s equations of motion

A rigid body B accelerates in an inertial frame A as if it were a single particle with the same mass m (equal to the total mass of the system) acted upon by a force equal to the net external force.

The rate of change of angular momentum of the rigid body B with the center of mass C as the origin in A is equal to the resultant moment of all external forces acting on the body with C as the origin


Moment of Inertia of 3-D Objects?

C

x

y

r

dm


Inertia Tensor of 3-D Objects

C

x

y

r

dm


Inertia Dyadic

Principal Moments of

Inertia

Products of Inertia


Example: Rectangular Plate Rotating about Axis through Center of Mass

C

B

CB

A

Is the angular momentum parallel to the angular velocity?


Principal Axes and Principal Moments

Principal axis

• u is a unit vector along a principal axis if I u is parallel to u

• There are 3 independent principal axes!

Principal moment of inertia

• The moment of inertia with respect to a principal axis, uT

I u, is called a principal moment of inertia.


Examples

b1

b3

b2

b1

b3

b2


Euler’s Equations of Motion

b1

b3

b2

C

In frame B



b1

b3

b2

C

In frame A

need to differentiate in A



b1

b3

b2

C

Transform back to frame B

angular velocity in frame B



b1

b3

b2

C

1. frame B (bi) along principal axes

angular velocity, moments of inertia, moments in frame B

2. center of mass as origin

3. all components along bi


Quadrotor

w1

w2

w3

w4

mg


Static Equilibrium (Hover)

w1

w2

w3

w4

mg

Motor Speeds

Motor Torques


w3

w1

w2

w4

F4

F3

F2

F1

M2

M3

M4

M1

Resultant Force

Resultant Moment about C

Cr4 r1

r3

r1

mg


u1

u2

Newton-Euler EquationsB

Rotation of thrust

vector from B to A

Components in the body frame along b1, b2,

and b3, the principal axes

Components in the inertial

frame along a1, a2, and a3

b1

b3

b2

video 2.1a vijay kumar and ani hsieh - edxpennx+robo3x+2t2017+type@asset... · a system of rigidly...

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