MECH303 Lagrange’s equations
Dr. D J Walker
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Lagrange’s Equations for Non-Conservative Systems
Lagrange’s equations provide a powerful technique for obtaining the equations of motion for systems
with more than one degree of freedom. They represent arguably the most general and useful
expression of the Energy Principle. Compared with other methods of obtaining equations of motion, the
method offers the following potential advantages:
•
Accelerations do not have to be determined; only velocities. This considerably simplifies the
kinematics. Also, since velocities are squared, difficulties with algebraic sign are often avoided.
•
Required number of equations is automatically obtained.
•
Same basic method is used whatever set of co-ordinates might be chosen, which allows us to
work with the most convenient set of co-ordinates.
We shall first consider the following form of the equations, appropriate to non-conservative systems:
d  T  T
 Qi


dt   qi   qi
(1)
The terms in the equation are now described.
1.1 Generalised co-ordinates {qi}
It is necessary to define a set of i independent generalised co-ordinates {qi}. These will often be
distances and angles. In general, a system with n degrees of freedom will require n independent
generalised co-ordinates {q1, q2,..., qn}. A special class of system known as non-holonomic does exist
in which the number of generalised co-ordinates required exceeds the number of degrees of freedom;
we shall not discuss systems of this type.
Example: Double pendulum - Generalised co-ordinates are angles q1 and q2.
Lagranges Equations
D.J. Walker
1.2 Generalised forces {Qi}
When the system undergoes a virtual displacement dqi in which all other co-ordinates qj remain
unchanged, the virtual work done dW i = Qidqi. This defines the ith generalised force Qi.
1.3 Total kinetic energy (T)
The total kinetic energy of the system can be expressed as a function ‘T’ of the generalised coordinates qi and their rates of change with time
qi .
Example: Double pendulum
Position vectors of masses 1 and 2:
l cos q1 
l cos q1  l cos q2 
r1  
, r2  

l sinq1 
l sin q1  lsin q2 
Velocity vectors of masses 1 and 2:
 lq sin q1 
 lq1 sin q1  lq2 sin q2 
r1   1
 , r2  

 lq1 cos q1 
 lq1 cos q1  lq2 cos q2 
Total kinetic energy:

T  12 ml 2 q12  12 ml 2  lq1 sin q1  lq2 sin q2   lq1 cos q1  lq2 cos q2 
2
2

1.4 Application of Lagrange’s equation to single degree of freedom system
We first determine the equation of motion of a mass M under the action of a single horizontal force F.
First, define the generalised co-ordinate q1 to be the linear displacement of the mass.
F
M
q1
Then the generalised force Qi = F. (To see this, note that the virtual work done by F = Fdqi. This by
definition is equal to Q1dq1). Next, the kinetic energy is given by T 
1
2
mq12 . Hence equation (1)
above gives:
d   ( 12 mq12 )   ( 12 mq12 )
F


dt   qi 
 qi

d (mqi )
F
dt
 mqi  F .
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Lagrange’s Equations for Conservative Systems
In a conservative system, the generalised forces can be derived from a potential function V. Potential
functions can represent the effects of:
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D.J. Walker
•
Ideal springs (energy storage) - Energy storage =
•
Gravity - energy storage =
1
2
kx 2
mgh
When dealing with such systems, a so-called ‘Lagrangian’ L is defined as follows:
L  T V
(2)
The appropriate form of Lagrange’s equations for conservative systems is given by:
d L L
0


dt   qi   qi
(3)
2.1 Application to a 2-degree-of-freedom system
This is a conservative system, and so we can use form (3) of the equation above. The springs have
negligible mass, and there is no energy loss in system. We define generalised co-ordinates x
(displacement from equilibrium - this eliminates any gravity forces) and , the angular displacement of
the pulley whose moment of inertia through its axis is I.
Kinetic and potential energies given by:
T  12 I 2  12 mx 2
V  12 k1 (r ) 2  12 k2 ( x  r ) 2
(recall that energy stored in a linear spring, stiffness k, E = 1/2 kx2).
Thus the Lagrangian L 
1
2
I 2  12 mx 2  12 k1 (r ) 2  12 k2 ( x  r ) 2 . This has to be differentiated as
follows:
L
d L
 I ,  
  I

dt   
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Lagranges Equations
D.J. Walker
L
 k1r 2  k2 r ( x  r )

 r 2 (k1  k2 )  rxk2
So the first equation of motion can be written as:
I  (k1  k2 )r 2  rk2 x  0
Similarly,
L
d L
 mx,  
  mx
x
dt   x 
L
  k2 (x  r )
x
which leads to the second equation of motion:
mx  k2 x  k2r  0
2.2 Application to a 2-degree-of-freedom system using generalised forces
We repeat the above example, this time using
d  T  T
 Qi . We shall need to compute the


dt   qi   qi
generalised forces Q and Qx. Note that mg appears as we have taken static equilibrium for our initial
state.
The work done during a virtual displacement d
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D.J. Walker
Q d   k2 ( x  r )  mg    k1r  mg   rd
 Q  k2 rx  (k1  k2 )r 2
Applying equation (1), we have
d   ( 12 I 2  12 mx 2 )   ( 12 I 2  12 mx 2 )
 k2 rx  (k1  k2 )r 2


dt 



 I  k2 rx  (k1  k2 )r 2 (as before).
Similarly, when the displacement is dx, the virtual work done is given by:
Qx dx  mgdx   k2 ( x  r )  mg  dx
 Qx  k2 ( x  r )
Hence
d   ( 12 I 2  12 mx 2 )   ( 12 I 2  12 mx 2 )
 k2 ( x  r )


dt 
x
x

 mx  k2 ( x  r ) .
N.B. The equations of motion form a pair of coupled linear O.D.E.’s which can be solved to yield
natural frequencies and mode shapes using eigenvalue-eigenvector methods.
2.3 Application to a 2-degree of freedom problem including a gravity term
The aim is to derive the equations of motion using Lagrange’s equations. The pendulum (of length l)
has negligible mass. The angle  is measured from the vertical, and the spring has stiffness k. The
displacement x is measured from the position of static equilibrium.
Suitable generalised co-ordinates are x and .
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D.J. Walker
We must first determine the kinetic energy of the system. That requires the (square of) the velocity of
the masses. Using axes oxy, with origin o at the equilibrium point and y measured down, the position
and velocity vectors of the pendulum bob are given by:
 x  l cos 
 x  l sin  
r1  
.
 , r1  
 l cos 
 l sin  
Therefore, total kinetic energy of the system T is given by:


T  12 (2m) x2  12 m x2  l 2 2  2lx cos .
2
There are two components to the potential energy too: energy stored in the spring ( 12 kx ) and
gravitational potential energy equivalent to ‘mgh’, where h is the height increase of the mass. Thus
V  12 kx2  mgl 1  cos  . We can now use equation (3) to write:
L
 2mx  mx  ml cos  ,
x
L
  kx ,
x
d  L
2

  3mx  ml cos   ml sin  .
dt   x 
This leads to the first equation of motion:
3mx  ml cos  ml sin  2  kx  0 .
Now differentiating with respect to the other generalised co-ordinate , one obtains:
L
 ml 2  mlx cos 

,
L
 ml x   sin    mgl sin 

,
d  L
2

  ml   ml cos x  mlx sin  .
dt   
This leads to the second equation of motion:
ml 2  ml cos x  mgl sin   0 .
N.B. In this example the equations of motion form a pair of coupled non-linear O.D.E.’s. These could
either be linearised for small , in which case eigenvector/eigenvalue methods be used to obtain the
natural frequencies and mode shapes, or else solved numerically to obtain {x, } trajectories for a given
initial condition.
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Lagranges Equations
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D.J. Walker
Summary
In all of the examples here, the equations could have been derived using Newton-Euler methods. To
do so would have required explicit calculation of both linear and angular accelerations in terms of the
generalised co-ordinates and their time-derivatives, as well as the calculation of internal forces and
moments, and then applying the laws:
F  mr
dH
Mo 
dt
to the various rigid-body components. For problems involving mechanisms consisting of more than one
coupled body, Lagrange’s equations often provide the most convenient means of arriving at the
equations of motion.
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References
Thomson WT. Theory of Vibration with Applications. Chapman & Hall, 1993. (531.32 THO)
Meriam. Dynamics, Wiley. (531.3)
Housner & Hudson. Applied Mechanics - Dynamics, Nostrand, (620.104 Hou)
Timoshenko & Young. Advanced dynamics, McGraw Hill, (531.11 Tim)
Greenwood. Prinicples of dynamics. Prentice Hall, (531.3 Gre)
Synge & Griffiths. Principles of mechanics (McGraw Hill) (531 Syn)
DJ Walker
September 2005
(Based on notes written from 1992-96)
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