Systems With Time-Dependent Coefficients.MATHIEU’S EQUATION

Even when the system is linear, time-dependent coefficients tend to cause great difficulties.

The situation considerably better when the terms involving the time-dependent coefficients are relatively small, because this opens the possibility for a perturbation solution.The interest lies in the case in which the support executes harmonic motion of the form:

(1)

We derive the equation of motion by first regarding θ and u as unknowns and then use eq.1 to eliminate u. Here we consider Lagrange’s equations:

(2)

where T is the kinetic energy, V is the potential energy and Θ and U are generalized Non-conservative forces.

From figure the kinetic energy and potential energy can be shown to be:

(3)

And the virtual work of the non-conservative forces is: (4) In this problem the generalized non-conservative forces are: (5) Then: (6)

Inserting eqs. (5)&(6)in (2): (7) (8)

Inserting eq.(1) into (7) , we obtain a nonlinear differential equation with one time-dependent coefficient, which can be solved for θ(t) , at least in theory. Then, introducing θ(t) thus obtained and eq.(1) in (8), we obtain the force F necessary for generating the harmonic motion u of the support .

If we consider θ(t) ～ 0 , the above equations reduce to the linearized form: (9) (10) Equation (9) may appear nonlinear due to the product , but it is not,

because u is a known quantity. Now inserting eq.(1) in (9) : (11)

This equation is known in mathematical physics as Mathieu’s equation. And then from (10) we obtain the force producing the harmonic motion of the

support : (12)

For stability of the system , it’s convenient to introduce the notation:

(13)

Moreover , it’s customary to let , so that eq.(11) reduces to the standard form of Mathieu’s equation:

(14)

If ε ＜＜ 1 equation 14 represents a quasi-harmonic system. The stability characteristics of eq.14 can be studied conveniently by means of the parameter plane ε,δ . The plane is divided into regions of stability and instability by the so-called boundary curves , or transition curves. These transition curves separating the stability regions from the instability regions, are such that a point belonging to any of these curves is characterized by a periodic solution of equation 14. To this end we assume a solution in the form:

(15) Moreover , we assume that: (16)

Inserting equations 15, 16 in equation 14 and equating coefficients of like power of ε, we obtain the sets of equations:

(17)

one set for every n . These equations must be solved recursively for the various values of n . From the first of these equations the zero-order approximation is given by:

(18)

Equations 17 yield an infinite number of solution pairs, one pair for every value of n, with the exception of the case for which there is only one solution.

Considering first the case n=0 , the second of Eqs. 17 reduces to:

(19)

For to be periodic, must be equal to zero: (20)

Then the third of Eqs. 17 becomes: (21)

For to be periodic, the constant term on the right side of Eq.21 must be equal to zero, which yields:

Hence, corresponding to n=0, there is only one transition curve, namely (22) Which, to a second-order approximation, is a parabola passing through the origin of the

parameter plane ε,δ .

Now we consider the case n=1 , in which case, there are two zero-order solutions:

(23)

Now with the first of these equations(θ0= cost ), the second of Eqs. 17 becomes:

(24)

To prevent resonance, hence the formation of secular terms we must set δ1=-1, from which it follows that the solution of (24)is: (25)

Then inserting θ0, θ1 and δ1 into the third of Eqs:17 we obtain:

(26)

For θ2 to be periodic, the coefficient of cos(t) must be zero, which yields δ2=-1/8. Hence, using Eq. (16), the transition curve corresponding to θ0= cost is:

(27) Corresponding to θ0= sint , the second of Eq.17 becomes:

(28)

The solution of this equation is periodic provided δ1 =1 , and has the form:

(29)

And then the third of equation 17 becomes:

(30) So that, for θ2 to be periodic, we must have δ2=-1/8 . Hence, the transition curve

corresponding to θ0= sint is:

(31)

Following the same pattern, it can be shown that the transition curve corresponding to

n=2 and θ0=cos2t is: (32) And the corresponding to n=2 and θ0=sin2t is:

(33)

Transition curves for n=3,4,... can be obtained in a similar fashion.

Strutt Diagram

The region terminating at δ=1, ε=0 is known as the principal instability region, and

is appreciably wider than the regions terminating at ε=0, δ=n^2 (n=2,3,…) . We observe from figure that stability is possible also for negative values of δ, which

corresponds to upright equilibrium position, θ=180° . Although this region is small, for the right choice of parameters, the pendulum can be stabilized in the upright position by moving the support harmonically.