Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 2, pp. 383-394, Warsaw 2003

SIMILARITY AND MODEL DESIGNING IN A NONSCALAR

DESCRIPTION OF AN EXAMINED PROCESS

Wacław A. Kasprzak

Institute of Materials Science and Applied Mechanics, Wrocław University of Technology

e-mail: kasprzak@immt.pwr.wroc.pl

Theproblemof similarity anddesigning of a physicalmodel of an exami-
ned process is considered on the basis of generalizations of Theorem π.
It is shown that in cases of nonscalar description the similar model of a
process could be designed on the basis of a special versionofTheorem π.
Similarity scales can be obtained only for the vectormodulus, or for the
components of vectors and tensorswhen special experimental conditions
(described in this paper) are fulfilled.

Key words: similarity, dimensional analysis, physical modelling

1. Introduction

The well known notion of similarity has been used in scientific and en-
gineering activities assessing the construction or investigation of a physical
process. The basis for the establishment of the relations between observations
of a process on a model and of processes of interest to us is created by the
properties of the so called dimensional invariant and homogeneous function.
The form of this function is produced by Theorem π. Accordingly, if the pro-
cess is described by quantities Ẑ0, Ẑ1, Ẑ2, ..., Ẑs and we are interested in the
identification of the functional relationship

Ẑ0 =Φ(Ẑ1, Ẑ2, ..., Ẑs) (1.1)

in which the arguments Ẑ1, Ẑ2, ..., Ẑm created the dimensional base, then

Ẑm+j =φ
m∏

i=1

Ẑ
aji
i j=1,2, ...,r m+r= s (1.2)



384 W.A.Kasprzak

and

Ẑ0 = f(φ1,φ2, ...,φr)
m∏

i=1

Ẑ
ai
i (1.3)

In (1.2) and (1.3) the exponents aji,ai ∈ R. We can get the values of aji
and ai if we take into account that the dimensions on the left- and right-
hand side of (1.2) and (1.3) are the same (see Kasprzak et al. (1990) for
details). The course of the process or functioning of the object may be tested
on models constructed on an appropriate scale without the knowledge of the
mathematical model. It should be noted that, in the original object as well
as in the model, the same process, described by the same variables and the
same function Φ are investigated. Let us denote the quantities observed in
themodel by an asterisk. The realizations in the object are then described by
(1.1) and (1.3) and in themodel

Ẑ∗0 =Φ(Ẑ
∗

1, Ẑ
∗

2, ..., Ẑ
∗

s)= f(φ
∗

1,φ
∗

2, ...,φ
∗

r)
m∏

i=1

Ẑ
∗ai
i (1.4)

In model designing we use scales

Ẑ∗i =λiẐi λi ∈R+ i=1,2, ...,m

Ẑ∗m+j =µjẐm+j µj ∈R+ j=1,2, ...,r
(1.5)

Wewant to find the relationship between Ẑ∗0 and Ẑ0. The supposed property
of homogeneity of the function Φ guarantees that the quotient

λ=
Ẑ∗0
Ẑ0

(1.6)

exists and belongs to R+. Taking into account (1.3), (1.4) and (1.5)1, we get

Ẑ∗0

Ẑ0
=λ=

f(φ∗1,φ
∗

2, ...,φ
∗

r)

f(φ1,φ2, ...,φr)

m∏

i=1

λ
ai
i (1.7)

The real number

λφ =
f(φ∗1,φ

∗

2, ...,φ
∗

r)

f(φ1,φ2, ...,φr)
(1.8)

cannot be determined if the function f is not known. It may, however, be
easily seen that if we fulfill the so called similarity criteria

φ∗j =φj j=1,2, ...,r (1.9)



Similarity and model designing in a nonscalar description... 385

λφ will equal exactly 1, and from (1.6) we shall get

λ=
m∏

i=1

λ
ai
i and Ẑ0 =

Ẑ∗0∏m
i=1λ

ai
i

(1.10)

The dimensionless quantities φj (1.9) are well known invariants of the simila-
rity. From relationships (1.2) and (1.9) we shall get

Ẑm+j
∏m
i=1 Ẑ

aji
i

=
µẐm+j

∏m
i=1(λiẐi)

aji
(1.11)

r conditions imposed on the m+ r scales. This so called similarity criterion
can be expressed by using scales (1.5)

µj∏m
i=1λ

aji
i

=1 j=1,2, ...,r (1.12)

Obviously, we can get (1.10) when the process is described by the sca-
lar Ẑ0. In many cases, vector and tensor quantities are used to describe a
process, especially in the mechanics of continuous media. In recent years, for
example, the explanation of aerodynamic performance of flapping biofoils has
been one of the most interesting tasks in biophysics (the flying force of an in-
sect is producedby complicatedmotion of thewings, see, for example Lehman
(1999)). We shall consider a similar problem, on the base of two generalized
versions of Theorem π in which quantities modelled by tensors will be used.

2. Generalizations of Theorem π

Weshall examine two approaches to the description of a processwith some
nonscalar quantities by the generalizations of Theorem π

• the first refers only to scalar quantities as arguments of the function Φ,
but the process will be described by complex functions for every tensor
component,

• in the second,we shall use dimensional quantitieswith internal geometry
and Theorem π satisfying postulates of invariance in relation to groups
of rotations.

Generalized Theorem π was shown for both cases after the works Kasprzak
et al. (1990), Kasprzak et al. [2], Rybaczuk (1987).



386 W.A.Kasprzak

Theorem. Theorem π for the Complex Dimensional Function. If in a dimen-
sionally homogeneous and invariant function

Ẑ =Φ(Ẑ1, Ẑ2, ..., Ẑs,X̂1,X̂2, ..., Ẑq) (2.1)

the arguments Ẑi, i=1,2, ...,m are dimensionally independent (dimen-
sional base), and the dimensionally dependent arguments can be written
in the base

X̂p = ξp

m∏

i=1

Ẑ
bpi
i p=1,2, ...,q (2.2)

Ẑm+j =Φ(Ẑ1, Ẑ2, ..., Ẑm,X̂1,X̂2, ..., X̂q)=φ(ξ1,ξ2, ...,ξq)
m∏

i=1

Ẑ
aji
i

(2.3)

j=1,2, ...,r m+r= s

then the function Φ has the form

Ẑ = f(φ1,φ2, ...,φr,ξ1,ξ2, ...,ξq)
m∏

i=1

Ẑ
ai
i (2.4)

where φ1,φ2, . . . ,φr,ξ1,ξ2, . . . ,ξq ∈ R+; ai,aji,bpi ∈ R. Conversely,
every function of form (2.4) is dimensionally homogeneous and inva-
riant.

This solution, similar to classic Theorem π, although it presents the possi-
bility of describing processes occurring in a field or in a material continuum,
cannot be considered satisfactory. Physics, for instance, requires appropriate
symmetries and invariances in relation to certain transformations. For further
deliberations let us differentiate a class of processes where the set X̂p will be
restricted to

X̂ =(X̂1,X̂2,X̂3) [X̂p] = [length] p=1,2,3
(2.5)

[X̂4] = [t̂] = [time]

Formulas (2.2) will assume the form

ξp =
X̂p

∏m
i=1 Ẑ

bi
i

p=1,2,3 τ =
t̂

∏m
i=1 Ẑ

ti
i

(2.6)



Similarity and model designing in a nonscalar description... 387

Finally, function (2.4) will become

Ẑ = f(φ1,φ2, ...,φr,ξ1,ξ2,ξ3,τ)
m∏

i=1

Ẑ
ai
i (2.7)

Consistently with this interpretation, to each point P ′ with the coordinates
X̂1,X̂2,X̂3 of a dimensional space belonging to the set D

′ (interpreted as a
configuration of the examined continuum immersed in a physical space of the
Euclidean space structure) while

D′ =X3p=1(X̂
L
p ,X̂

R
p ) (2.8)

is mapped – using formulas (2.6) – by point P ⊂ D of the Euclidean three-
dimensional space. The set D will be obtained, of course, from set (2.8) by
formulas (2.6)

D=X3p=1(ξ
L
p ,ξ
R
p ) (2.9)

It is evident that, utilizing formulas (2.6) from the set D, wemay – changing
the values Ẑi, i = 1,2, ...,m as required – generate an entire family of D

′

configurations. This is so because each point in the set D in the family of D′

configurations conforms with a hyper-surface satisfying the equations

ξp =
X̂p

∏m
i=1 Ẑ

bi
i

= const p=1,2,3 (2.10)

The cognizance of f in the set D is decisive in the knowledge of the dimen-
sional description of the function Φ in all generated D′ configurations – for
fixed φ1,φ2, ...,φr,τ parameters, of course.Considering thevariables ξ1,ξ2,ξ3,
function (2.7) describes the scalar field. If we regard that φ1,φ2, ...,φr are also
functions of ξ1,ξ2,ξ3,τ in this field, then (2.7) can be written differently

Ẑ = f∗(ξ1,ξ2,ξ3,τ)
m∏

i=1

Ẑ
ai
i (2.11)

Physics and technology operatewithvectors and tensors; their components are
elements of a dimensional space of the same dimension. Thus, for example, for
a vector or tensor with the components Ẑ1, Ẑ2, Ẑ3: [Ẑ1] = [Ẑ2] = [Ẑ3]. Each
of these components Ẑν is expressed by formulas (2.7) or (2.11), i.e.

Ẑν = fν(φ1,φ2, ...,φr,ξ1,ξ2,ξ3,τ)
m∏

i=1

Ẑ
ai
i

(2.12)

Ẑν = f∗ν(ξ1,ξ2,ξ3,τ)
m∏

i=1

Ẑ
ai
i



388 W.A.Kasprzak

Corollary 1. When the functions φj, j = 1,2, ...,r of the variables
ξ1,ξ2,ξ3,τ are known and have been fixed, then the function f

∗ and
functions f∗ν describe, on the basis of the similarity relations, the pro-
cess in every D′ (2.8) configuration generated through (2.6) from subset
(2.9).

This essential conclusion allows us to transfer obtained experimental re-
sults to geometrically similar spatial configurations (fields of a similar geo-
metry and similar geometrical bodies). Let us return to the formulation of
Theorem π for the dimensional functionwith nonscalar values and arguments.

Theorem (Theorem π). Let arguments and values of a dimensional function
belong to generalized dimensional spaces presented by Rybaczuk (1987)
and the function be invariant with respect to the action of the rotation
group, then the formof this function, according toKasprzak et al. (1990),
Rybaczuk (1987) (see also Rychlewski, 1978, 1991), can be presented as

Ẑ0 = Φ(Ẑ1, Ẑ2, ., Ẑs)=
(2.13)

=
l∑

i=1

fi(φp(i)+1,φp(i)+2, ...,φr;γ
r
k=1)× ĝi(Ẑ1, Ẑ2, ..., Ẑs)

p(i)∏

t=1

|Iit|bti

where

• ĝi – are well known generators supplied by a suitable representation
theorem for O(3) – invariant function (Wang, 1971a,b,c),

• Iq are formed from Ẑ1, Ẑ2, ..., Ẑs scalar invariants,
• φw, w = 1,2, ...,r are dimensionless numbers (invariants of the
gauge group) formed from the scalars Iq,

• γrk=1 denotes the sequence of signs of the scalars Iq,
• p(i) is the numerator of a dimensional base (chosen from Iq) for fi.

3. Similarity and model designing according to generalized

Theorem π

Letus examine the similarity andmodel designingaccording toTheorem π
formulated for a nonscalar description of a process. The equivalent expression



Similarity and model designing in a nonscalar description... 389

for the scale λ, if Ẑν is a vector or tensor component, could be written in the
form

λν =
m∏

i=1

λ
ai
i (3.1)

when conditions (1.5)1 and (1.9) are fulfilled.The scale for the vectormodulus
(if the configurations of all acting vectors is similar – meaning that all angles
between the vectors are respectively the same) according to (2.4) formulation
of Theorem π could be expressed as

λ=

√√√√√√√√

3∑
ν=1
f2ν
∏m
i=1λ

2ai
i

3∑
ν=1
f2ν

=
m∏

i=1

λ
ai
i (3.2)

When Ẑν is a tensor component and formula (2.13) is used as the presentation
of Theorem π, we shall obtain a quite different result. For the description of
the objectwe shall get the expression as in (2.13). The value of the component
Ẑν0 in the model can be expressed as

Ẑν∗0 =
l∑

i=1

fνi
(
φ
(i)
p(i)+1

,φ
(i)
p(i)+2

, ...,φ(i)r ;γ
r
k=1

)
×g(i)ν λg(i)ν

p(i)∏

t=1

|Itλt|bti (3.3)

The scale λ of the investigated process could be assumed only for the

tensor component or, if Ẑ0 is a vector, for its modulus (specifically for
√
Z20).

Let us investigate the value of the scale

λ(i) =λ
g
(i)
ν

p(i)∏

t=1

λ
bti
t (3.4)

It is easy to notice that the dimensions in (3.4) and (2.13)

[
g(i)ν

pi∏

t=1

|It|bti
]
=
[
giνλg(i)ν

p(i)∏

t=1

λ
bti
t |It|bti

]
=
[
Ẑν0

]
(3.5)

irrespective of the value of the index i. From (3.5) we can see, after having
noticed relations (1.12) anddonesomealgebraic transformation, that λ(i) (3.4)

λ(i) =
m∏

i=1

λ
ai
i (3.6)



390 W.A.Kasprzak

It is easy to showthat in this case thedimensionof [Zν] canbeexpressed in the
dimensional base exactly as in (1.3), and the scale λν is equal to the scale λ in
(1.10). We can get formally the similarity conditions for the nonscalar model
(2.13) if we divide, as was previously stated, instead of vectors or tensors,
vectormoduli or values of vector and tensor components.We shall present the
results of three different approaches to the model construction on the base of
Theorem π in versions (1.3), (2.4) and (2.13) (numerically the same), using a
process well known in the field of applied mechanics.

Fig. 1. Relative angular motion ω2 of point A on the disk rotating disk with
angular velocity ω1

Let us investigate the relative motion accelerations presented in Figure 1.
Accordingly, point A rotates with the constant angular velocity ω2 along the
edge of the disk. The disk is in angular motion ω1 around its diameter. We
assume that the acceleration Ẑ depends on

Ẑ =Φ(ω1,ω2, t,r,ρ) (3.7)

The dimensions of the function Φ value and the arguments in the SI system
of units are: [Ẑ] = [ms−2], [ω1] = [ω2] = [s

−1], [t] = [s], [r] = [ρ] = [m],
ρ = r sinω2t. If we wish to design the model on the base of Theorem π
expressed by formula (1.3), accepting the dimensional base r,ω1 (modulus of
vectors r,ω1) and the scales for the variables in the model

ω∗1 =λ1ω1 r
∗ =λ2r ω

∗

2 =µ1ω2

t∗ =µ2t ρ
∗ =µ3ρ

(3.8)

we shall get for the process

Ẑ = f
(ω2
ω1
,
ρ

r
,tω1
)
rω21 (3.9)



Similarity and model designing in a nonscalar description... 391

and for the model

Ẑ∗ = f
(µ1ω2
λ1ω1
,
µ3ρ

λ2r
,µ2tλ1ω1

)
λ2rλ

2
1ω
2
1 (3.10)

The scale

λ=
Ẑ∗

Ẑ
=λ2λ

2
1 (3.11)

if

µ1 =λ1 µ2 =
1

λ1
µ3 =λ2 (3.12)

We shall get the same result using formula (2.4) (of course, for the vector mo-
dulus, or the vector or tensor component). Let us now investigate the process
description according to formulation (2.13) of Theorem π using variables as
in (3.7). We shall get the scalar invariants

ω21, ω
2
2, ω1ω2, t, r

2, ρ2, rρ, ω1r, ω2r, ω1ρ, ω2ρ (3.13)

and generators

g1 =ω1× (ω×ρ) g2 =ω2× (ω2×r)
(3.14)

g3 =ω1× (ω2×r)

Let us accept the dimensional base as previously
(√
ω21,
√
r2
)
and scales (3.8),

we shall get for (3.4)

Z
∗

= f∗1λ
2
1µ3g

∗

1+f
∗

2µ
2
1λ2g

∗

2+f
∗

3λ1µ1λ2g
∗

3

f∗i =f
∗

i

(µ21ω
2
2

λ21ω
2
1

,
λ1µ1ω1ω2
λ21ω

2
1

,µ2µ1tω2,
µ23ρ
2

λ22r
2
,
λ2µ3rρ

λ22r
2
,
µ1λ2ω2r

λ1λ2ω1r
,
λ1ω1µ3ρ

λ1ω1λ2r
,
µ1ω2µ3ρ

λ1ω1λ2r

)

(3.15)

g∗i = giλ
i
g i=1,2,3

(please notice that at this stage we know the values of f∗i , f1 = f
∗

1 = f2 =
f∗2 =1, f3 = f

∗

3 =2), and for object (2.13)

Z = f1g1+f2g2+f3g3
(3.16)

fi = fi
(ω22
ω21
,
ω1ω2
ω21
, tω1, tω2,

ρ2

r2
,
ρr

r2
,
ω2r

ω1r
,
ω1ρ

ω1r
,
ω2ρ

ω1r

)
i=1,2,3



392 W.A.Kasprzak

Dividing the arguments of f∗i by fi for the same i, we shall get

µ21
λ21
=1 µ2λ1 =1

µ23
λ22
=1 (3.17)

and µ1 = λ1, µ2 = 1/λ1, µ3 = λ2 as in (3.12). Now, we can calculate the
scales for the generators:

• for g1 we get λ
(1)
g =λ

2
1µ3 =λ

2
1λ2

• for g2 we get λ
(2)
g =λ

2
1µ3 =λ

2
1λ2

• for g3 we get λ(3) =λ1µ1λ2 =λ21λ2

At the end we come to the following conclusion:

• the construction of a model of the investigated process produces in all
considered formulations of Theorem π the same results for the values of
the scales,

• the cognitive possibilities of experimental investigations on the base of
Theorem π expressed by (1.3) or (2.4) are quite different in comparison
to Theorem π formulated by formula (2.13).

Fig. 2. Relative motion of point Awith velocity v on the rotating plane

In the last case, we shall get all generators (in our example all accelera-
tions) without any empirical investigations. If, for example, we do not know
that there is – in the relative motion a centripetal acceleration and Coriolis
acceleration, it is difficult to obtain pertinent knowledge on an experimental
basis only. In some cases we can measure only the sum of all accelerations as
was shown in Figure 2, where point Amoves with a constant velocity v on a
plane which rotates around point O with a constant angular motion ω – in
this case ω×v is parallel to ω×(ω×r). Themain difficulties in investigations



Similarity and model designing in a nonscalar description... 393

on the model of a process, when the investigated quantity is modelled by a
vector or tensor, are connected withmeasurements, exactly speaking with the
knowledge about the proper direction of the acting generator. Such knowled-
ge can only be obtained from formulation (2.13) of Theorem π (in the cases,
whenweare unable to get such information fromthe theory of the investigated
process).

References

1. Kasprzak W., Lysik B., Rybaczuk M., 1990,Dimensional Analysis in the
Identification of Mathematical Models, World Scientific Singapore

2. Kasprzak W., Lysik B., Rybaczuk M., Measurement, Dimensions, Inva-
riant Models and Similarity, available at the http://www.immt.pwr.pl/kniga

3. Lehman F.O., 1999, The aerodynamic basic of insect flight, Proc. of 2-nd
International Workshop on Similarity Methods, Stuttgart

4. Rybaczuk M., 1987, π Theorem for SO(3)×Hom(W,R+) from dimensional
symmetry,Bulletin of the Polish Academy of Sciences, Mathematics

5. Rychlewski J., 1978, Mathematical foundation of the theory of dimensions,
analogies and similarity, Trends in Applications of Pure Mathematics to Me-
chanics, G. Fichera (ed.), Pitman, London, 333-364

6. Rychlewski J., 1991,Wymiary i podobieństwo, PWN,Warszawa

7. WangC.C., 1971a,On the representationof isotropic functions. Part 1. Scalar
functions,Arch. Rat. Mech. Anal., 36, 166

8. Wang C.C., 1971b,On the representation of isotropic functions. Part 2. Non-
scalar functions,Arch. Rat. Mech. Anal., 36, 198

9. Wang C.C., 1971c, Corrigendum to my paper on representation of isotropic
functions,Arch. Rat. Mech. Anal., 43, 392

Podobieństwo i projektowanie modeli dla procesów zależnych od

nieskalarnych zmiennych

Streszczenie

Znana z literatury teoria podobieństwa modelowego i algorytmy projektowania
modeli opracowano przy założeniu, że proces zależy od zmiennych, które są skalara-
mi. W pracy podaje się rozwiązanie tych problemów dla nieskalarnych zmiennych,



394 W.A.Kasprzak

od których zależy przebieg procesu. Podaje się odpowiednią wersję twierdzenia π dla
funkcji wymiarowych zależnych od zmiennych, które mogą być tensorami i sposób
wyznaczania skal. Pokazuje się, że otrzymuje się takie same rezultaty, jak dla mode-
li skalarnych, ale otrzymuje się pełną informację o składowych tensora opisującego
badany proces, a więc pełne informacje pozwalające na opracowanie projektu badań
empirycznych (nie otrzymywano ich w tradycyjnymmodelu).

Manuscript received May 22, 2002; accepted for print November 26, 2002