2nd Order Differential Equations

8/20/2019 2nd Order Differential Equations

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Ship Structural Dynamics (NM 404) 1/7

2nd Order Differential Equations (DE)

1. Introduction

Modelling of physical phenomena is often performed with 2nd order DE. The general

form of this type of equation is:

)x(F)x(cf dx

)x(df b

dx

)x(f da

2

2

=++ (1)

where the coefficients a, b and c are constant . The variable of interest is x and the

function f is some real function of x. On the RHS of equation (1) we have some other

function F. Equation (1) is the general form of the type of equation we are interested in

the course of Marine Dynamics. In the following, we will examine its properties and its

solution process.

2. An interesting property

Assuming two solutions of equation (1), say f(x) = u and f(x) = v, then it is obvious that u

and v should satisfy (1). In order to simplify things, we further assume F(x) = 0. So we

get:

0cvdx

dvb

dx

vda

0cudx

dub

dx

uda

2

2

2

2

=++

=++

If we take these two equations a step further and add them together, we derive a very

interesting result which is used extensively in the study of dynamical systems. That is,

( ) 0vucdx

dv

dx

dub

dx

vd

dx

uda

2

2

2

2

=++

++

+

but( )

dx

vud

dx

dv

dx

du +=+ and

( )2

2

2

2

2

2

dx

vud

dx

vd

dx

ud +=+




So, we can rewrite equation (1) in the following form:

0)vu(cdx

)vu(dbdx

)vu(da2

2

=+++++ (2)

This means that the summation of two solutions of the equation (1) is also a solution of

it!

3. Solution process, F(x) = 0

The solution process of a 2nd order DE starts with the selection of a suitable function.

That is, a function with the characteristic of repeating itself (or some part of it) after we

take its 1st and the 2

nd derivative successively.

Obviously, function with this property are the trigonometric functions sin(x) and cos(x)

and the exponential function Aemx. The exponential function appears more attractive in

any case, so it is the one that we will use.

Here is how it works:

mx2

2

2

mx

mx

eAmdx

)x(f d

Amedx

)x(df Ae)x(f

=

=

=

where A and m are real numbers.

Substituting in (1) and crossing out the common terms, we get the following:

0cbmam

or0cAebAmeeaAm

2

mxmxmx2

=++

=++ (2)

We ended up by a quadratic equation for solving equation (1). This result proves the

reasoning behind the selection of a suitable function.

Say that the roots satisfying the quadratic equation are m1 and m2. Then, the solution (in

general form) of equation (1) has the form of:






Also, the following abbreviations will be used:

• CF: complementary function

• PI: particular integral

In order to obtain function X, we follow a similar process as before, i.e. we select a

suitable function depending on the form of F, as is presented in the following. The

important thing here is not why we specifically select these functions but how they reflect

and satisfy our needs in the study of marine dynamics in general. This will be explained

in the following paragraph.

So, here is a list of options:

… if … we select

(i) F(x) = k (i.e. constant) g(x) = C

(ii) F(x) = kx g(x) = Cx + D

(iii) F(x) = kx2 g(x) = Cx2 + Dx + E

(iv) F(x) = k sin(x) or k cos(x) g(x) = C cos(x) + D sin(x)

Once the form of F is established, then we select the appropriate function from the right

column and successively calculate its 1st and 2nd derivatives. Substituting these results in

equation (1) will allow us to calculate the involved constants and therefore the exact form

of g(x).

Here is an example of the process: say that the equation has the form of

)x4sin(2)x(f 6dx

)x(df 5dx

)x(f d2

2

=+−

CF is calculated by assuming two things: (i) the RHS is equal to zero and (ii) f(x) = Aemx

.

These two will lead to m2 – 5m + 6 = 0 or (m – 2)(m – 3) = 0 and therefore: f(x) = Ae

2x +

Be3x.

For PI we get the following: assume g(x) = C cos(4x) + D sin(4x). The 1st and 2nd

derivatives of g(x) will be:




)x4sin(D16)x4cos(C16dx

)x(gd

)x4cos(D4)x4sin(C4dx

)x(dg

2

2

−−=

+−=

Now, by substitution to the LHS of equation (1), we get the following:

( ) ( )

( ))x4sin()x4cos(225

1)x(gtherefore

251D25

2C

or0D20C10

2D10C20

)x4sin(2)x4cos(D20C10)x4sin(D10C20or

)x4sin(2)x4sin(D6)x4cos(C6

)x4cos(D20)x4sin(C20)x4sin(D16)x4cos(C16

−=

−=

=

=−−

=−

→

=+−−

=++

−+−−

Finally, the general solution will be:

( ))x4sin()x4cos(225

1BeAe)x(f x3x2

−++=

Calculation of constants A and B is based on the available information of the problem we

solve or the system we are modelling.

5. Application in marine dynamics

Solution of equation (1) is a rather trivial topic for books of mathematics. For example,

reference [1] can provide a lot more information about it along with plethora of examples

and exercises. In the context of marine dynamics though the aim is not the equation itself

but rather what it represents.

Bearing in mind the importance of the time element in the study of dynamic systems, we

quite naturally set the variable x to be equal to t. In turn, the function f(x) transforms into

the function of displacement with respect to time, say q(t). That is, the 1st derivative of q

will be speed and 2nd

derivative will be acceleration.




The fact that the study of the dynamic behaviour of a body essentially addresses its

motion in space, which in turn is attributed to a set of forces acting on it directly

highlights the importance of Newton’s 2nd law of motion: the sum of forces acting on the

system equals the product of its mass times its acceleration .

Realisation of the last statement makes us understand that equation (1) in the course of

studying the dynamic behaviour of floating bodies is essentially an equation of

equilibrium of forces.

So, here is how we utilise equation (1) in the study of dynamics:

• Factor a is equal to the mass of the system

• Factor b is related to the damping of the system

• Factors c is related to the stiffness of the system

• qdx

)x(dqandq

dx

)x(qd2

2

&&& ≡≡ are the acceleration and velocity of the system

respectively provided of course that q is the displacement of the system.

• F(t) is the external force acting on the system (usually called excitation

force, which in the general case is time-dependent)

Considering for example that F(t) = constant, this means that there is a constant force

acting on the system irrespective of time. On the other hand, by considering an excitation

force of the type 3sin(0.5t – φ) then we introduce a periodic force of magnitude 3 (units

of force), of circular frequency 0.5 and phase angle φ. Obviously, selecting of a different

function for the excitation of our system will create an equally different response of the

system. This way of thinking allows us to perceive mathematical modelling of a system

as a tool rather than some complicated mathematical process. All we need to know is how

to use this tool in order to extract the results we need.

Concluding this small introduction, it is necessary to mention that far more details

regarding the models described above can be found in numerous books like [2] and [3]

which are included here for reference and are strongly recommended for further study.




6. References

[1]. Engineering Mathematics, K. A. Stroud (with additions by D. J. Booth), 5th

edition,

2001

[2]. Vibrations and Waves, A. P. French, 1971

[3]. Theory of vibration with applications, W. T. Thomson, 4 th edition, 1993

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2nd Order Differential Equations