Torsional vibrations in a crank shaft

Torsional vibrations in a crank shaft

Citation for published version (APA):

Molenaar, J. (1989). Torsional vibrations in a crank shaft. (IWDE Report; Vol. 89-01). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1989

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REPORT IWDE 89-01

TORSIONAL VIBRATIONS IN A CRANK SHAFT

J. Molenaar January 1989

In this report, we analyze the torsion in the crank shaft of a gas compressor with one crank. The geometry is 'as drawn in figures la and lb. Two forces are exerted on the crank. First, the crank shaft is at one end driven by a motor. We assume that this happens with uniform angular velocity, irrespective of the load of the crank. Second, the piston meets with the gas force, which is passed on to the crank via the driving shaft. The geometry of the system implies, that the moment of inertia around the crank shaft of all moving parts together depends on the rotation angle

q,

of the crank shaft. 1be equation of motion for the torsion in this shaft has consequently periodically varying coefficients. A common approach to analyze this torsion is to average these coefficients over a period [1]. The resulting modelling equation can be dealt with in terms of eigenfrequen-cies, resonances and solutions, which are periodic in time. However, the solutions of the original equation of motion might be non-periodical and even show chaotic behaviour.

The purpose of this study is to compare the behaviour of the reduced model with that of the origi-nal one in order to check the reliability of the reduction in case of realistic compressor data.

L

(a) CRANK SHAFT CRANK (b)

...__---~

MOTOR CRANK SHAFT \";

DRIVING SHAFT _{Fig 1. Sketch}

_of

_{the geometry}

_of

_crank,

driving shaft and piston in side-view (a) and top-view (b).

-2-2. Equation of Motion

The crank shaft is at one end driven with constant angular velocity w and the rotation angle 4to at this end is given by

+o(t) = {I) t. (2.1)

The rotation angle +(t) of the crank will deviate from 4to by the torsion angle .£\4>:

44>(t) = +(t)-+o(t). (2.2)

Because all movements of crank, driving shaft and piston take place in a plane perpendicular to the crank shaft, the total impulse moment J of all these moments together is directed along the crank shaft. It

can

be written as

.

.

1(4>. 4>) =

/(cp) cp

(2.3)

with the total moment of inertia I given by

(2.4) The constants lc and lp are defined in the appendix by (A.llb,c) and the dimensionless factor Z(4>) by (A.7).

The second law of Newton for rotating rigid bodies states that

(2.5)

with the summation over all momenta M; under consideration. In the present model, three

momenta are taken into account, namely the torsional, the gas and the frictional momentum. So, we may write

J (

cp , cp)

= M 10nion

+

M gas

+

M friction·

Following Hooke's law, we approximateMIOnion by Mtcnioo =-k (4>-cJto).

The torsion constant k(> 0) depends on the material properties of the shaft.,

In the appendix, expressions forM sas and M friction are derived. The fonner is given by

(2.6)

(2.7a)

(2.7b) with F₈ the force exerted on the piston by the compressed and decompressed gas. F₈ is a periodic function of

cp.

An example of the dependence of F₈ on 41 for a realistic compressor is given in figure 2a. 5 4 )(10 3 2 1

-z

-lJl a ro ~ ry I..--1

+-~

-2

t

f--3

t

"---'/

_J

• I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 0 1 2 3 4 5 6

Rotation Angle lradl

Figure 2a. Plot of F₁(4>)for the data also used in figure 7a.

The expression forM friction reads as

M friclion

=

-c(4>) 41 (2.7c)

withe(+) given in (A.15). It is a function of I Z(4>) I.

Equation (2.6) with (2.7a, band c) is a second order, inhomogeneous, non-linear, autonomous differential equation in 4>. We are interested in the behaviour of the torsion angle A+(t), defined in (2.2). This torsion angle is in practice very small ( < 0.01 radians). We may therefore linearize equation (2.6) around 41

=

+o

and use the approximations

-4-M gas(.P) =M gas(4lo)

+

M gas'(~}

Aq,

c(.p)

=

c(~) + c'(.Po) A+.

(2.8)

In addition to the linearization, we bring the equation of motion into dimensionless fonn by intro-ducing the dimensionless variable

't

=

(l)t. (2.8b)

If we neglect tenns quadratic in A

q,,

we arrive at the linearized equation of motion (withy E A+

for notational ease):

y

+ a(t)

y

+ b(t) y

=

F(t). (2.9)

Equation (2.9) may be read as the equation of motion of a particle with unit mass under influence of a hannonic force -by, a frictional force -a

y

and a driving force F. Both the coefficients a and

b and the driving force F are periodic in t with period 21t. They are given by:

21' (1)2 + c (I) a(t)

=

₂ I (I) (2.10a) I" (1)2 + c' (I)+ k - M ' b(t) = gas I (1)2 (2.10b) M -I' (1)2 -

c

(I) F(t)

=

gas 2 I (I) (2.10c)

For convenience, we transfonn equation (2.9) into frictionless fonn by means of the substitution

'f

y('t) =x(t) exp(-f

J

a(s) ds)

0

If we differentiate both sides of (2.11a), we find for the torsion velocity

'f

y(t)

=

(i('t)-f x(t) a(t)) exp(-f

J

a(s) ds).

0

Transfonnation (2.11a) leads to the equation

X+

d(t) X= F(t)

with the definitions

d(t)

=

b(t)-.!. a(t)-.!. a2_{t)

2 4

1 't

F(t) =F(t) exp(+I

J

a(s) &).

0

If we introduce the vectors

(2.11a)

(2.11b)

(2.12)

{2.13a)

equation (2.12) is cast into the standard fonn

x=Ax+F

with the matrix A given by

A=(;,)~)

It is easily checked by substitution, that the solution of (2.15) is given by

't x(t) = +{t, o) x(o)

+

J

+{t, s) F(s) ds , G (2.14) (2.15) (2.16) (2.17)

with x(o) a known vector of initial values. The transition matrix • is defined as the solution of the initial value problem

+{t, a)= A(t) +{t, o)

+(cr,cr)=(~ ~]

(2.18)

In (2.18) the derivative is taken with respect to the first argument. Note, that • is completely detennined by the homogeneous version of equation (2.15), thus with F

=

0. If we take F

=

0 in (2.17), we may directly conclude that

.('t. t') +{t'. 0')

=

.('t. 0') (2.19)

for arbitrary t'. This implies, that • is non-singular for all arguments, because it is non-singular

for t

=

a. If A would be constant, • would be invariant with respect to a translation of both

argu-ments. Thus we have

(2.20) for constant A. In the following section this property is used.

-6-3. Approximation of d(t)

The coefficient d(t) in (2.12) is a strongly varying function oft mod 21t. In fig. 2b a plot of d

versus t is given for a typical data set from practice.

180

1 2

Rotation Angle (radl

/I

~

)

I i 7

Figure 2b. Plot of the coefficient d(t)for the data also used in figure 7a. The dotted line represents the corresponding staircase junction calculated from ( 3 .I).

In the literature, no method is known to determine the corresponding transition matrix analyti-cally. Instead of resorting to numerical methods, we prefer to apply a technique, which is proven to be reliable in many cases [2]. This method is based on the observation that the contribution of the Fourier components of d(t) to the calculation of • strongly decreases with increasing

fre-quency. This inspired Franks and Sandberg [3] to replace d(t) by a more regularly shaped func-tion with a Fourier spectrum, which closely coincides with that of d(t) as far as the lower

fre-quencies are concerned. We use a staircase function with N equal intervals. The height of the n-th step d, can be calculated from the formula

m (m 1t IN) imK(2n-l}IN d-. e m· sm(m1tl N) d,.=

1:

I m I <Nn.

The Fourier component

dm

of d(t) is, as usually, defined by

21t

dm

= 2 1

I

d(t) e-im-c dt. 1t 0 (3.1) (3.2)

To give an idea of this replacement procedure, we also present in fig. 2b the corresponding stair-case function for N = 16.

The transition function of the staircase function is easily obtained. At the intervals [ t,.__{1 ,} t,. ], with t,.

=

n

~

1t , n

=

1 , · · · , N, the matrix A reduces to

A.= [

~. ~]

(3.3)

with tranSition matrix

lj),.(t, a)= exp(A,lt-a)). (3.4) Evaluation of the exponent in (3.4) via its powerseries yields

[

cos ..Jd,;(t-a) _

~

sin ...Jd;;(t-a)l

-vd,.

lj),.(t' a)= . if d,. 2!: 0

-..Jd,;

sm

..Jd,; (

t- a) cos

..Jd,;

(t-o) (3.5a)

[

cosh ..Jd,;(t-a) _

~

sinh ..Jd,;(r.-a)l

-vd,.

=

_..Jd,;

.

ff~SQ

sinh ..Jd,;(t-o) cosh fi,;(t-a) (3.5b)

If we use property (2.19), we may calculate 4l for arbitrary arguments by evaluating products of lj)₁₁

matrices. This simplifies the evaluation of (2.17) considerably. It remains to evaluate the integral in (2.17) numerically, because the gas force F₈ is not given in analytical form. A further simplification in the evaluation of (2.17) is obtained by writing it in an iterative form. If we want

to evaluate the solution at grid points t; = i At, i

=

0 , 1 , 2 , · · · , with At a given increment, we may write

~+1

x(ti+t> = 4l{t;+t , 0) x(O)

+

I

4l<ti+l • s) F(s) ds

-

8-=

cp{t;+l , 0) [x(O)

+

K(O, t;+t) ] (3.6) with K defined by .( K(t, 't)

=

J

cp(O, s) F(s) ds. (3.7) 1: If we use that (3.8a) and (3.8b) we obtain the recursive fonnula

(3.9) Thus, if iP(t;, 0) and x(t;) are known at some gridpoint t;, the calculation ofx(t;+1) and cp{t;+I, 0) merely needs the analytical evaluation of 4!(ti+l , t;), the numerical evaluation of K(t;, t;+1 ), and

4. Discussion, Results and Conclusions.

To get some feeling for the possible complexity of the solutions of (2.9) with (2.10), it is instruc-tive to interprete the different tenns somewhat in detail. The frictional coefficient a contains a damping tenn proportional to c. This tenn stems from the physical friction in the familiar way. In (2.10a) also a tenn proportional to I' occurs. It changes sign and corresponds to a damping and an accelerating force alternatively. In the coefficient bin (2.10b) only the tenn proportional to k has definite sign, giving rise to a repulsive force in the familiar way. It is to be expected, that the solutions are stable if this tenn dominates the other ones. The generalized driving force F in (2.10c) contains, apart from the periodic gas force, a tenn proportional to I', which changes sign, and one proportional to c, which tends to increase the torsional angle more and more in negative direction. A constant frictional coefficient c would lead to an increase of the average level of ll

q.

via F, and to a damping of the oscillations via a. However, the damping of the piston makes c to vary periodically and a driving force F, which varies periodically via c, may even excite the sys-tem.

It is impossible to analyze the complete behaviour of the system as a function of all parameters. We shall hereafter present a few illustrative examples. The parameters of our "standard" system are I crank = 10 kg m2 md₁ =250kg md₂

=

100 kg mp =900kg R =0.15 m (4.1) Rc =0.15 m L

=

l.Om ro = 40 rad Is k =5.1<f Nm.

In the subsequent calculation we shall modify this parameter set merely slightly and study the inftuence of averaging /(t}. friction, piston movement and gas force.

4A. Results for +-independent

I

-10-2~t

_1

I

(Z)2 d'!'=l (1

+..!)

=.!...

2x ₀ 2 L 2

This leads to a constant value

f.

For the parameters in ( 4.1)

I

is equal to

I=

I crank+ Rid I R2 + t (Rid2 +mp) R2

= 10 + (250+t (100+900)) (0.15)2 =26.9/cg m2.

In this approach (2.9) and (2.10) reduce to

y+ay+by=F a= c I (I ro)

b = (k-M gas') I

(I

ro2)

F

=

(M gas -cro) I(/ ro2).

In pmctice, we may assume

Mgas' << k.

(4.2)

(4.3)

(4.4)

This can, for example, be checked from fig. 2a with definition (2.7b) forM gas· This implies. that

b is nearly constant. In that case (4.4) decribes an oscillator under a combination of a constant and a periodic force. The behaviour of the solutions of such a system are well-known [4]. The eigenfrequency

roo

of the homogeneous (F a 0) and frictionless (a =0) equation is given by

roo

=-

I-\

(I) rad Is.

'J

I ro (4.5)

The constant force -c II ro makes the system to oscillate around an avemge level fJ.

+

given by

fJ.

+

= -c ro I k rad. (4.6)

The force M gas

II

ro2 has period 2 x. If it would be sinusoidal, it could lead to amplitudes fJ.

+max

given by

(4.7) with A gas the maximum amplitude of M gas· From fig. 2a we see that M gas is sinusoidal only in approximation. Still, we expect expression (4. 7) to yield reasonably reliable values. From fig. 2a, (2.7b) and (A.7b) we find

A gas= 6.1o4. (4.8)

The obtained angular frequency is in exact agreement with the result of expression 4.5:

roo=

10.8 ro rad Is.

In this special case, the amplitude is completely determined by the initial values:

0)

114\nax = - = 0.092 rad.

roo

(4.9)

(4.10)

In fig. 5b, we show the effect of friction. The parameters are as in fig. Sa except for j(O) = 0 and

Cc = 104, cp = 0 (see A. IS). The average level is in exact agreement with (4.6):

111!1=-0.012 rad. (4.11)

In fig. 5c, the effect of the gas force is illustrated. The parameters are as in fig. Sa (no friction). The deviation of the sinusoidal fonn of M gas is apparently present in the solution. From (4.7)

with (4.8) we obtain for this case the value

114\nax

=

0.012 (4.12)

which is in fairly good agreement with the average amplitude in fig. 5c.

4B. Results for ~j~-dependent I

Here, we present solutions of (2.9) with (2.10) using expression (2.4) for l(~j~). To isolate the effect of the piston, we first take c e M gas e 0 (no friction, no gas force). The corresponding solution is given in fig. 6a. It clearly contains two signals with frequencies

roo

and ro respectively.

If we add friction the results are not very different. In fig. 6b crank friction is taken into account by choosing in (A.15) Cc

=

5000 Ns and Cp

=

0, while in fig. 6c piston friction is introduced by

using in (A.15) c,

=

0 and cp

=

5000 Ns.

Figures 7a,b and

c

are obtained with the same parameter data as fig. 6a,b and c respectively, except for introduction of the gas force corresponding to F gas in fig. 2a. For this parameter set the

gas force appears to have little effect on the amplitude.

4C. Comparison of the

use

of I and I for various values of k

If one uses the constant moment of inertia I, the behaviour of the amplitude can be understood

from (4.7). This expression shows resonance if

mo::::: ro.

(4.13)

If we fix the values of m and

I

in (4.5) at ro

=

40 rad Is and

I=

26.9 kg m2, we find that resonance occurs if

k

=

4.3 I

if

Nm. (4.14)

-

12-behaviour of A l!lmax

as

a function of k. we present in table 1 results for A l!lmax for various values

of k. The (a) column has been calculated using/, whereas in the (b) column the cp-dependence of I

is exactly taken into account In both cases the gas force is applied, but no friction is present (c

=

0).

A l!lmax (rad) A

IPmax

(rad) k(Nm) (a) (b) 5.0 lcf 1.210-2 4.0 10-3 (Sa) 2.51cf 2.410-2 4.0 10-2 (Sb) 1.0 1cf 6.010-2 1.010° (Sc)

t.otOS

4.710-1 _{00 (8d)} 4.3.104 00 00

Table 1. Values for the maximal amplitude A l!lmax for various values of k. The (a) column is for constant moment of inertia/, the (b) columnforcp-dependent /(cp).

The numbers (Ba-d) indicate to the figures 8a-d, in which the solutions are plotted.

From table 1 it is clear that averaging out of the cp-dependence of /leads to a considerable smaller region of resonance at the k-axis than found by exact calculations. This implies. that results obtained by the approximation are not reliable to indicate resonant behaviour of the crank-piston system. The choice of the parameters in constructing compressors should always be checked by a complete analysis as described in this report.

Figure Captions

In all figures the parameter values as given in (4.1) are used except for the values explicitly indi-cated.

Fig. 5: averaged moment of inertial =26.9 kg m2;

a) md₁ = 750, md₂ = mp = 0; y(O) = 0, y(O) = 1.0; no friction; no gas force

b) as under a) withjo(O)=O and Cc = 10000.0Ns; no gas force.

c) as under a) with jo(O) = 0 and gas force; no friction

Fig. 6: +dependent moment of inertia /(lj)); no gas force. a) no friction

b) Cc = 5000.0, Cp = 0.0 Ns

c) Cc = 0.0 , Cp

=

5000.0 Ns

Fig. 7: + dependent moment of inertia I ( q> ); with gas force. a) no friction

b) Cc

=

0.0, Cp =5000.0 Ns

c) Cc = 5000.0, Cp = 0.0 Ns

Fig. 8: various k-values a) identical to fig. 7a

b) as under a) with k

=

2.5 106 Nm

c) as under a) with k = 1.0 1o6 Nm

><10- 1 1 0.9 0.6 0.4 0.2 01

-

_{g' 0} <:I c .~ -0.2 1.11 .... 0 1- -0.4 -0.6

-e.s

-1

~

I I e 1 I

I

I I I I I I I I I I 2 3 4 5 6 7

Rotation Angle !radl

Figure Sb. Averaged I, Crank Friction, No Gas Force

_,., e

xl0 ~ -0.5 iJ ttl

...

-1 01

-

0" c a: c

.£

-1.5 1.11

...

0 1--2 -2.5

~

2 4 6

e

10 12 14

x10-20.6 ~---~---~

-"0 Ill ~ 0.4 0.2 - -0.2 (11

-

g' -0.4 <I c .~ -0.6 Ill 1-0 1-

-0.s

-1 -1.2 -1.4

t

I I 0 Figure _..., ₁ ><10 .. 0.8 0.6 \ r !-0.4

-"0 Ill ~

-

0.2 (11

....

0' 0 c <I c .~ -0.2 Ill 1-0 I- -0.4 -0.6 -0.8 -1 0 I I I 20 ! I I 40 60 90 100 120 140

Rotation Angle (radl

6a. Exact I, No Friction, No Gas Fol"ce

\

~

\

\

10 20 30 40 50 60 70

FigurP 6b. Exact

I.

Crank Friction, No Gas Force

Rotation Angle (radl

Figure 6c. Exact I, Piston Friction, No Gas Force

0.2 e

-""0 Ill 1-- 1--0.2 I'll ... g'-8.4 a: c

.s

-0.6 Ill I-0 t- -0.8 -1 -1.2 -1.4 0 10 20

0 ~

.,

-e.2

1-(II -0.4

....

0' t: a: c: -0.6 0 Ill

5

-e.s

1--1 -1.2

-1.

4t

.._ .._, ...._..._...__,

-1-l ..._,

..,~...t....L•

-+I ...A.•

....!.'....!.'-ll-lji-.,..;•i-.,..;•~...o.~-•

.._, +-1 "'-' ...__, ...__, ...._, -+-1 ..I...' ..._._...._, -+1 _., ...I...L-1'-11

e

w

3 • ~ 9 ~

n

Rotation Angle Cradl

~igure

7b. Exact I, Piston Friction, With Gas rorce

(]; ~ -0.6

a

c -0.8 0 Ill

5

-1 1--1.2 -1.4 -1.6 -1.8

I

_I

70 0 40 _{50 60}

Figure 7c. Exact I, Crank Friction, With Gas Force -0.2 -0.4

:0

.,

-0.6 1-~ -0.8 0" c a: c -1 0 1./l ~ -1.2 1--1.4 -1.6

+-t

~

~

~ -1.8 0 10 20 30 40 50 60 70

0.2 -e

:~

I

II

~ -0.2

~

~

;~ I

v

~

~

1-~ -0.4 ~ 0\ c 0: c:

-e.6

-0 ,

...

IJl ~

-e.e

,_

1---1 i--1.2 i--1.4 I I I I I I 0 10 20 30 40 se 60 70

Rotation Angle lradl

Figure 8b. Exact I, No Friction, With Gas Force

- ? ₃ xH3 ~ _~ ~ 2

t

1 "'0 ro

e

1-Qj

-

01 -1 c 0: c 0 IJl -2 '-0 1---3 -4 -5

a

20 40 60

ee

100 120 140

~

-

~ c a c 0

Figure 8c. Exact I, Na Friction, With Gas Farce

-1LL~~;-~~~~~~~~~~~~~~~~~~~~

e 20 40 60 80 100 120 140

Rotation Angle !radl

Figure 8d. Exact I, Na Friction, With Gas Force

-s~~~;-~~~~~~~~~;-~~~~~~~~~

e

10 20 30 40 50 60 70

Appendix

In this appendix, we derive expressions (2.4), (2.7b) and (2.7c) used above.

Moment of Inertia

To simplify the derivations, we model the physical driving shaft with length L by a dumb bell of the same length, with all mass concentrated at the ends. See fig. 3.

md

2

l2

L

CENTRE OF

~t

MASS CRANK SHAFT

_md1

Figuur 3. The driving shaft is modelled as a dumb bell with the same rotational properties. To assure that the dumb bell has the same rotational properties as the driving shaft, the following conditions have to be satisfied:

-15-mdl +md2 = md

mdt Lt

=

md2 L2 (A.l)

with md the total mass of the driving shaft. To obtain the moment of inertia corresponding to the parts, which move horizontally, we have to express the position and velocity of the piston in tenns of the rotation angle ell and the angular velocity ell of the crank. For the notation, see fig. 4.

-X

Figuur 4. Indication of the meaning of the symbols R , L , x , ell, a,

P

used in the text.

The position x of the piston is given by

By differentiating both sides of (A.2), we find an expression for the velocity

x

of the piston

x

= -(R • sin ell+ L

ix

sin a).

.

.

(A.2)

(A.3)

It remains to express a, cos a, and sin a in tenns of ell and •· From the geometry in fig. 4 we immediately derive that

. R . "' A. . "'

sma=Lsm'l'

=

sm'l'. (A.4a)

Thus

cosa="l-A.2 sin2cll. (A.4b)

If we differentiate both sides of (A.4a), we find

• A.coscll •

a= _{cos a} •· (A.5)

with

which, for A.

< < 1, reduces

to

Z(¢1)

=

sin¢1+t A. sin2¢1.

The kinetic energy of the horizontally moving masses m~wr is 1 . 2

Em _{= 2} mhor x

If we equate this to the general expression for kinetic energy

1 •2

Em=-Ioorll> , 2

we obtain for the moment of inertia of the horizontally moving parts

lhor =mhor R2 Z2(cjl).

So. the total moment of inertia I ( ¢1) of all moving parts can be written as /(¢1) = lc + lp Z2(¢1)

with lc and IP constants, given by

lc =I crank+ mdl Rz (A.7a) (A.7b) (A.8) (A.9) (A.lO) (A.lla) (A.llb) (A.llc) I crank is the moment of inertia of the crank itself and mp the mass of the piston together with all

parts rigidly connected to it.

Gas Momentum

Because the driving shaft is, of course, not rigidly connected to the piston or the crank, it can only pass on a force directed along itself. The component of the force F₁(+), exerted on the piston by the gas, along the driving shaft is (see fig. 4)

F₁ cos

a.

(A.12a)

The impact of this force on the impulse momentum of the crank is given by

Mgas =F₁ R cos a sin~. (A.12b)

sin~= sin(1t-(4l+a.)) = sin(4l +a.)

= sin

41

cos a. + cos

41

sin a..

-17-Expressions for sin a. and cos a. are given in (A.4). We arrive at the expression M sas = F₈ R(l-l2 sin2 cp) Z(cp)

which for l

< < 1 reduces

to M gas

=

F₈ R Z(cp). Frictional Momentum

(A.13)

(A.14a)

(A.14b)

We assume the friction to be composed of two pans, one of which being proportional to the velo-city of the crank shaft and the other proportional to the velocity of the piston. We introduce there-fore two frictional constants cc and cp~ 0), which have to be determined experimentally, and write

=-(Cc Rc

+

Cp R I Z(cjl) I) 4>

= -

c(cf>) cjl.

(A.lS)

Note, that R is the radius of the crank and Rc the radius of the crank shaft. We used expression

(A.6) fori, but introduced the absolute value of Z(4>) rather than Z(cjl) itself, to make sure that the

friction is oppositely directed to the crank movement. This implies, that c(cp) is not differentiable if

41

=

0 or n:. At these points, Z vanishes as long as l < 1. Because this derivative is used in the linearized equation of motion, the function

Z(41)

is numerically smoothed around these points.

References

1. Hafner K..E., Maas H., Torsionsschwingungen in der Verbrennungskraftmaschine, 1985, Springer Verlag Wien, ISBN 3-211-81793-X.

2. Richards J.A., Analysis of Periodically Time-Varying Systems, 1983, Springer Verlag Ber-lin, ISBN, 3-540-11689-3.

3. Franks, L.E., Sandberg I.W., Bell Syst. Techn. J. 89, 1960, pp. 1321-1350.