Lagrange’s Equation

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Lagrange’s Method

• Newton’s method of developing equations of

motion requires taking elements apart

• When forces at interconnections are not of

primary interest, more advantageous to derive

equations of motion by considering energies in

the system

• Lagrange’s equations:

– Indirect approach that can be applied for other types

of systems (other than mechanical)

– Based on calculus of variations – finding extremums

of quantifies expressible as integrals

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Lagrange’s Equations

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friction viscous toduen dissipatioenergy :)(

itiesular veloclinear/ang inertias, mass

masses, system of in termsenergy kinetic: )(

scoordinate of in termsenergy potential :)(

coordinateeach in loading ingcorrespond:Q

instantany at motion ssystem' describe

tonecessary scoordinatet independen:

:

2

3

2

2

1

i

i

i

i

i

i

iiii

qfR

qfT

qfU

q

where

Qq

U

q

R

q

T

q

T

dt

d

3

Deriving Equations of Motion via

Lagrange’s Method

1. Select a complete and independent set of

coordinates qi’s

2. Identify loading Qi in each coordinate

3. Derive T, U, R

4. Substitute the results from 1,2, and 3 into

the Lagrange’s equation.

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Example 11: Spring-Mass-Damper

System

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Independent coordinate: q = x

Substitute into Lagrange’s equation:

5

Example 12: Pair-Share:

Restrained Plane Pendulum

• A plane pendulum (length

l and mass m), restrained

by a linear spring of

spring constant k and a

linear dashpot of dashpot

constant c, is shown on

the right. The upper end

of the rigid massless link

is supported by a

frictionless joint. Derive

the equation of motion.

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Example 12: Pair-Share:

Restrained Plane Pendulum

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0

sin

0sin

)(2

1cos

)(2

1

2

1

)(2

1

,0,

222

222

2

22

2

kbmglcaml

smallforLinearize

kbmglcaml

QURT

)T

(dt

d

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Qq

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• One degree of freedom, q1=angle as an independent coordinate

• Velocity of bead: ; Velocity of hoop:

• Kinetic energy:

• Potential energy relative to its position at the bottom of

the hoop (when the hoop is not rotating and = 0), is

• R = 0, Q = 0

• Substitute into Lagrange’s equation:

• Solving for the angular acceleration:

Example 13: Bead on a Spinning Wire Hoop

sinR

R

.R

2 cos sin .g

R

8

2

2

1mvT 22

sin2

1 RRm

cosmgRmgRU

0sincossin222

mgRmRmR

QURT

)T

(dt

d

Example 14: Pair-Share: Copying machine

• Use Lagrange’s equation to derive

the equations of motion for the

copying machine example,

assuming potential energy due to

gravity is negligible.

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Q1 = F, Q2 = 0

9

q1=y, q2= θ

y

θ

Example 14: Pair-Share: Copying machinechp3 10

Example 15: Mass Spring Dashpot

Subsystem in Falling Container

• A mass spring dashpot subsystem

in a falling container of mass m1 is

shown. The system is subject to

constraints (not shown) that

confine its motion to the vertical

direction only. The mass m2,

linear spring of undeformed length

l0 and spring constant k, and the

linear dashpot of dashpot constant

c of the internal subsystem are

also shown.

• Derive equation(s) of motion for

the system using

– x1 and x2 as independent coordinates

– y1 and y2 as independent coordinates

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Example 15: Mass Spring Dashpot

Subsystem in Falling Container

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0)()(

0)()(

:'

)(2

1

)(2

1

2

1

2

1

0

.,

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2

2222

1212111

1

1111

2

21

2211

2

21

2

22

2

11

2,1

212211

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Qx

U

x

R

x

T)

x

T(

dt

d

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Qx

U

x

R

x

T)

x

T(

dt

d

EquationsLagrangeintoSubstitute

xxcR

gxmgxmxxkU

xmxmT

Q

groundhorizontaltorespectwithmandmofpositionsbexqxqLet

Example 15: Mass Spring Dashpot

Subsystem in Falling Container

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0

0)(

:'

2

1

)(2

1

)(2

1

2

1

0

2222212

2

2222

2122121

1

1111

2

2

21211

2

2

2

212

2

11

2,1

1222

111

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Qy

U

y

R

y

T)

y

T(

dt

d

gmgmymymm

Qy

U

y

R

y

T)

y

T(

dt

d

EquationsLagrangeintoSubstitute

ycR

yygmygmkyU

yymymT

Q

springoflengthdunstretchefromandmtorelativemofpositionbeyq

groundhorizontaltorespectwithmofpositioninertialbeyqLet

• Block mass m sliding down a wedge mass M

• Independent coordinates, q1 and q2, are shown, q1 is along the plane and

is measured relative to the (moving) wedge.

• Velocity of the wedge is , but

velocity of the block has components from both

q1 and q2:

• Total kinetic energy is

• The potential energy of the wedge can be taken as zero, and the block is

• Substitute into Lagrange’s equations and differentiate wrt to q1 and q2

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Example 16: A Block Sliding on a Wedge

q2

q1

a

2q

2 1 cos .m

q qM m

a

12

sin.

1 cos

gq

m

M m

a

a

14

2

1

2

12

222sincos aa qqqvvv yxblock

2

1

2

12

2

2 sincos2

1

2

1aa qqqmqMT

α

asin1mgqU

Example 17: Pair-Share:

Mass Pendulum Dynamic System

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• A simple plane pendulum of mass m0 and

length l is suspended from a cart of mass m as

sketched in the figure. The motion of the cart

is restrained by a spring of spring constant k

and a dashpot constant c; and the angle of the

pendulum is restrained by a torsional spring of

spring constant k, and a torsional dashpot of

dashpot constant ct. Note that the constitutive

relations for the torsional spring and torsional

dashpot are linear expressions τ=ktθ and τ=ctθ,

respectively, where τ is the torque and θ is the

change in the angle across the element from its

undeformed configuration.

• The torsional spring is undeformed when the pendulum is in its downward-

hanging equilibrium position. Also, m0 is acted upon by a known dynamic

force F0sinωt, which remains horizontal, regardless of the angle θ and the

motion of the cart. The system is constrained to remain in the plane of the

sketch and the cart remains on the bed throughout its motion. Derive the

equation(s) of motion for the system.

.

Example 16: Pair-Share:

Mass Pendulum Dynamic System

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cos)sin(cos sin

sin)sincos()(m

:Equations sLagrange' into Substitute

cos2)(2

1)sin()cos(

2

1

2

1

)cos1(2

1

2

1

2

1

2

1

cos)sin(sin

, :scoordinatetindependenwo

0000

0

2

00

22

0

22

0

2

0

22

22

0201

21

ltFlxmglmkclm

tFlmkxxcxm

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pendulum

cart

pendulumcart

t

tx

16

Comparison of Newton’s and

Lagrange’s MethodsNewton’s (Direct Approach) Lagrange’s (Indirect Approach)

Accelerations required Velocities required

Generally vectors required Generally scalars required

Free-body Diagrams useful Free-body diagrams not useful

All forces considered Workless forces (constraints) forces

not considered

All forces handled via same

expression

Conservative and non-conservative

forces handled separately

Intermediate forces more readily

available

Intermediate forces less readily

available

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Case Study: Feasibility Study of a

Mobile Robot Design

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Preliminary Design Model

• Read Handout 1

• Develop a simplified model of the base

and the arm by neglecting the reaction

forces that occur at the pivot

• When do the reaction forces become

significant?

=> at high accelerations and speeds of the

base and the arm

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Preliminary Design Modelchp3 20

Preliminary Design Modelchp3 21

Motion Profiles

• Read Handout 2

• Plot the motor force versus time, and

motor torque versus time, and determine

whether the motor is powerful enough

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Motion Profileschp3 23

A More Detailed Model

• The simplified model appears feasible

• Develop a detailed model, including the

reaction forces to see whether they

significantly change the results

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A More Detailed Modelchp3 25

A More Detailed Modelchp3 26

A More Detailed Modelchp3 27

Matlab Simulation Example:

Solving Systems of Equations

• From Matlab website: http://www.mathworks.com/support/tech-notes/1500/1510.html#time

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Matlab Simulation Example:

Solving Systems of Equations

• From Matlab website: http://www.mathworks.com/support/tech-notes/1500/1510.html#time

• More on ode45 commands: http://www.mathworks.com/access/helpdesk/help/techdoc/ref/ode45.html

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Homework 3: chapter 3

• 3.14, 3.21,3.24,3.25,3.31,3.33,3.36

• Case study simulation:

– Simulate the detailed model for 10 seconds with initial

conditions and for three

cases:

1. T = 0, f =0

2. T=370 Nm, f = 0

3. T=0, f = 263 N

– For each case, plot state vector versus time

– Describe the behavior of the system for each case

and discuss the stability of the system

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0,10,0,0 0

0

000 xx

Case Study Simulation Model

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References

• Woods, R. L., and Lawrence, K., Modeling and

Simulation of Dynamic Systems, Prentice Hall, 1997.

• Williams, J. H., Fundamentals of Applied Mechanics,

Wiley, 1996

• Close, C. M., Frederick, D. H., Newell, J. C., Modeling

and Analysis of Dynamic Systems, Third Edition, Wiley,

2002

• Lecture notes: web.njit.edu/~gary/430/

• Palm, W. J., Modeling, Analysis, and Control of Dynamic

Systems

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