On norm equivalence between the displacement and velocity vectors for free linear dynamical systems

Transcription

1 Kohaupt Cogent Mathematics (5 : COMPUTATIONAL SCIENCE RESEARCH ARTICLE On norm equivaence between the dispacement and veocity vectors for free inear dynamica systems Received: 8 Juy 5 Accepted: 4 September 5 Pubished: 5 October 5 *Corresponding author: Ludwig Kohaupt Department of Mathematics Beuth University of Technoogy Berin Luxemburger Str D-3353 Berin Germany E-mai: ohaupt@beuth-hochschuede Reviewing editor: Song Wang Curtin University Austraia Additiona information is avaiabe at the end of the artice Ludwig Kohaupt * Abstract: As the main new resut under certain hypotheses for free vibration probems the norm equivaence of the dispacement vector y(t and the veocity vector ẏ(t is proven The pertinent inequaities are appied to derive some two-sided bounds on y(t and ẏ(t that are nown so far ony for the state vector x(t =y T (t ẏ T (t T Sufficient agebraic conditions are given such that norm equivaence between y(t and ẏ(t hods respectivey does not hod as the case may be Numerica exampes iustrate the resuts for vibration probems of n degrees of freedom with n 3 4 5} by computing the mentioned agebraic conditions and by potting the graphs of y(t and ẏ(t Some notations and definitions of References Kohaupt (8b are necessary and are therefore recapituated The paper is of interest to Mathematicians and Engineers Subjects: Appied Mathematics; Computer Mathematics; Engineering Technoogy; Engineering Mathematics; Science; Technoogy Keywords: initia vaue probem; free vibration probem; state-space description; two-sided bounds; sufficient agebraic conditions; norm equivaence between dispacement and veocity Ludwig Kohaupt ABOUT THE AUTHOR Ludwig Kohaupt received the equivaent to the master s degree (Dipom-Mathematier in Mathematics in 97 and the equivaent to the PhD (Drphinat in 973 from the University of Franfurt/Main From 974 unti 979 Kohaupt was a teacher in Mathematics and Physics at a Secondary Schoo During that time (from 977 unti 979 he was aso an auditor at the Technica University of Darmstadt in Engineering Subjects such as Mechanics and especiay Dynamics From 979 unti 99 he joined the Mercedes-Benz car company in Stuttgart as a computationa engineer where he wored in areas such as Dynamics (vibration of car modes Cam Design Gearing and Engine Design Then in 99 Kohaupt combined his preceding experiences by taing over a professorship at the Beuth University of Technoogy Berin (formery nown as TFH Berin He retired on Apri 4 PUBLIC INTEREST STATEMENT Under certain conditions the norm equivaence of the dispacement vector y(t and the veocity vector ẏ(t for free inear dynamica systems is derived Hereby a reation of the form c y(t ẏ(t c y(t t t is understood with positive constants c and c As a consequence under norm equivaence one has that im t y(t = is equivaent to im t ẏ(t = and that boundedness of y(t is equivaent to boundedness of ẏ(t 5 The Author(s This open access artice is distributed under a Creative Commons Attribution (CC-BY 4 icense Page of 33

2 Kohaupt Cogent Mathematics (5 : Introduction In free vibration probems with one degree of freedom and mid damping the dispacement as we as the veocity have zeros in any sufficienty arge interva But with increasing dimension it is iey that not a components of the dispacement or of the veocity are zero simutaneousy in other words it is increasingy iey with increasing dimension n that y(t t t and ẏ(t t t for sufficienty arge t Then the question arises as to whether an inequaity of the form c y(t ẏ(t c y(t t t can be proven for sufficienty arge t ; this property if it is vaid wi be caed norm equivaence between y(t and ẏ(t Now the contents of this paper wi be outined In Section the state-space description ẋ = Ax x(t =x of the dynamica probem Mÿ + Bẏ + Ky = y(t =y ẏ(t =ẏ is given where M B and K are the mass damping and stiffness matrices as the case may be and where y(t is the dispacement vector y is the initia dispacement and ẏ is the initia veocity In Section 3 under certain hypotheses the above-mentioned norm equivaence between y(t and ẏ is proven for diagonaizabe system matrix A and in Section 4 the same is done for genera system matrix A The pertinent norm inequaities between y(t and ẏ are appied in Section 5 to improve Kohaupt ( Theorems and 6 In both sections aso sufficient agebraic conditions are estabished that guarantee the vaidity respectivey invaidity of norm equivaence of y(t and ẏ In Section 6 numerica exampes iustrate the resuts for vibration probems of n degrees of freedom for n 3 4 5} More precisey in Section 6 the mentioned agebraic conditions are computed and the graphs of y(t and ẏ(t are potted showing that in the exampes for n } there is no norm equivaence and that for n 3 4 5} there is norm equivaence; in Section 6 for n = 4 a case with ν x A <νa is iustrated by graphs where ν x A is the spectra abscissa of matrix A with respect to x and νa is the spectra abscissa of A; in Section 63 for n = a mode with nondiagonaizabe matrix A is constructed and anayzed in detai Section 7 contains computationa aspects and in Section 8 concusions are drawn The non-cited references 6 and 7 are given because they may be usefu to the reader The state-space description of Mÿ + Bẏ + Ky = y(t =y ẏ(t =ẏ Let M B K IR n n and y ẏ IR n Further et M be reguar The matrices M B and K are the mass damping and stiffness matrices as the case may be; y is the initia dispacement and ẏ is the initia veocity We study the initia vaue probem Mÿ + Bẏ + Ky = y(t =y ẏ(t =ẏ where y(t is the sought dispacement and z(t =ẏ(t is the associated veocity State-space description Let y y y x: = = x z ẏ : = z = y ẏ and x is caed state vector and A is caed system matrix Herewith the above second-order initia vaue probem is equivaent to the first-order initia vaue probem of doube size ẋ = Ax x(t =x ( Page of 33

3 Kohaupt Cogent Mathematics (5 : In the seque we need ony the specia form of x(t 3 Norm equivaence inequaities for diagonaizabe matrix A To show the norm equivaence between y(t and ẏ(t for diagonaizabe matrix A is a very simpe tas First we formuate some hypotheses and conditions 3 Hypotheses and conditions for diagonaizabe matrix A (H m = n and A IR m m (H T AT = J = diag(λ = m where λ = λ (A = m are the eigenvaues of A (H3 λ i = λ i (A i = m (HS the eigenvectors p p n ; p p n form a basis of CI m m Remar Let (H be fufied and et Ap = λp Then we have Ap = λp where the bar denotes the compex conjugate So together with (λ p aso (λ p is a soution of the eigenvaue probem Ap = λp But if λ and p woud be rea then p and p woud not be ineary independent This situation cannot happen when hypothesis (HS is supposed Remar In the seque when the specia hypothesis (HS is chosen we do this in order to be specific in the construction of a soution basis used aready in Kohaupt (8b Other cases such as A IR 3 3 can be handed in a simiar manner however Hypothesis (H4 defined in Kohaupt (8b is not need here As a preparation to the derivation of the norm equivaence of the dispacement and veocity vectors we coect some definitions respectivey notations and representations for the soution vector x(t from Kohaupt ( 3 Representation of the basis x (r (t x(i (t = n Under the hypotheses (H (H and (HS from Kohaupt ( we obtain the foowing rea basis functions for the ODE ẋ = Ax: x (r (t =eλ(r (t t cos λ (i (t t p(r sin λ (i (t t p(i x (i (t =eλ(r (t t sin λ (i (t t p(r + cos λ (i (t t p(i = n where λ = λ (r p = p (r + iλ (i = Re λ + iim λ + ip (i = Re p + iim p = m = n are the decompositions of λ and p into their rea and imaginary parts As in Kohaupt ( the indices are chosen such that λ n+ = λ p n+ = p = n 33 The spectra abscissa of A with respect to the initia vector x IR n Let u = m = n be the eigenvectors of A corresponding to the eigenvaues λ = m = n Under (H (H and (HS the soution x(t of ( has the form ( x(t = m=n = c p e λ (t t = n c p e λ (t t + c p e λ (t t = (3 with uniquey determined coefficients c = m = n Using the reations Page 3 of 33

4 Kohaupt Cogent Mathematics (5 : c = c n+ = c = n (4 (see Kohaupt 8b Section 3 for the ast reation then according to Kohaupt (8a the spectra abscissa of A with respect to the initia vector x IR n is given by ν : = ν x A: = max = m=n λ(r (A x u } = max = m=n λ(r (A c } = max = n λ(r (A c } = max = n λ(r (A x u } (5 34 Index sets In the seque we need the foowing index sets: J ν : = IN n and λ (r (A =ν } (6 and J ν : = n} J ν = IN n and λ(r (A <ν } 35 Appropriate representation of x(t We have (7 x(t = with n = c (r x(r (t+c(i x(i (t c (r = Re c c (i = Im c = n (cf Kohaupt 8b Thus due to ( x(t = n = e λ(r (t t f (t (8 with f (t: = c (r cos λ (i (t t p(r sin λ (i (t t p(i + c (i = n sin λ(i (t t p(r + cos λ (i p(i (9 36 Appropriate representation of bdy(t and ẏ(t Let p : = q r p (r q (r : = r (r p (i q (i : = r (i ( with q r CI m m q (r r(r q(i r(i IR n = m = n Then from (8 (9 Page 4 of 33

5 Kohaupt Cogent Mathematics (5 : y(t = with n = e λ(r (t t g (t ( g (t: = c (r cos λ (i (t t q(r sin λ (i (t t q(i + c (i sin λ(i (t t q(r + cos λ (i q(i ( = n as we as z(t =ẏ(t = n = e λ(r (t t h (t (3 with h (t: = c (r cos λ (i (t t r(r sin λ (i (t t r(i + c (i sin λ(i (t t r(r + cos λ (i (t t r(i = n After these preparations for the quantities J ν g (t h (t we formuate the foowing conditions: There exists a t t such that (4 (C g (t t (C h (t t g (t t t t J ν h (t t t t J ν For these conditions there are sufficient agebraic conditions as the case may be: (A g q (r q(i J ν are ineary independent (A h r (r r(i J ν are ineary independent In the exampes of Section 6 aso the case occurs that the above conditions are not fufied For this we formuate the foowing conditions: For every t t there exists a t t such that (C g (t t (C h (t t g (t = J ν h (t = J ν For these conditions there are sufficient agebraic conditions as the case may be (see Kohaupt : (A g J ν = } and q (r q (i are ineary dependent (A h J ν = } and r (r r (i are ineary dependent Page 5 of 33

6 Kohaupt Cogent Mathematics (5 : Further here and in the seque we denote by any vector norm Theorem (Norm equivaence of y(t and ẏ(t for diagonaizabe matrix A Let the hypotheses (H (H and (HS as we as the conditions (C g (t t (C h (t t or the sufficient agebraic conditions (A g (A h be fufied Then there exist constants c > and c > such that c y(t ẏ(t c y(t t t for sufficienty arge t t t Proof The proof foows immediatey from Kohaupt ( Theorems 7 and 3 or Kohaupt ( Theorems 8 and 4 4 Norm equivaence inequaities for genera matrix A In this section we prove the same statement for a genera square matrix A as in Theorem for a diagonaizabe matrix A This cannot be deduced in a simiar way as for Theorem that is it cannot be done by Kohaupt ( Theorems 9 and 5 since they contain the factor e ε(t t on the righthand side Neither can it be done by Kohaupt ( Theorems and 6 since they contain the factor e ε(t t on the right-hand side and the factor e ε(t t on the eft-hand side Nevertheess the same equivaence inequaities as in Theorem hod true in the genera case The proof however is much more invoved Again first we formuate some hypotheses and conditions 4 Hypotheses and conditions for genera square matrix A (H m = n and A IR m m (H T AT = J = diag(j i (λ i i= r where J i (λ i CI m m i i are the canonica Jordan forms (H3 λ i = λ i (A i = r (HS r = ρ and the principa vectors p ( p( m ; ; p (ρ p(ρ m ; ; p ( ρ p( m ; ; p (ρ p(ρ m form a basis of CI m m ρ We mention that for the specia hypothesis (HS simiar remars hod as for (HS in the case of diagonaizabe matrices A Let (H (H and (HS be fufied and Ap ( = λ p( the indices are chosen such that λ ρ+ = λ = ρ and p (ρ+ + p( = = p ( = The vectors p ( are the principa vectors of stage corresponding to the eigenvaue λ Hypothesis (H4 defined in Kohaupt (8b is not needed here = r where = ρ of A In the case of a genera square matrix A we aso have to coect some definitions respectivey notations and representations of x(t from Kohaupt (8b 4 Representation of the basis x (r (t x (i (t = m = ρ Under the hypotheses (H (H and (HS from Kohaupt (8b we obtain the foowing rea basis functions for the ODE ẋ = Ax: Page 6 of 33

7 Kohaupt Cogent Mathematics (5 : x (r (t =e λ(r (t t sin λ (i x (i (t =e λ(r (t t cos λ (i = = ρ where p (i sin λ (i p (r (! (! + +p (r (t t +p(r } + +p (i (t t +p(i p (r + +p (r (! +p(r } (5 p ( = p(r + ip (i is the decomposition of p ( into its rea and imaginary part 43 The spectra abscissa of A with respect to the initia vector x IR n Let u ( = m be the principa vectors of stage of A corresponding to the eigenvaue λ = r = ρ Under (H (H and (HS the soution x(t of ( has the form r=ρ x(t = c ( x( (t = = = m ρ = = c ( x( (t+c( x( (t (6 with uniquey determined coefficients c ( = m = r = ρ Using the reations c ( =(x u( = m = ρ c ( = c(ρ+ = c ( = ρ (7 (see Section 3 Kohaupt 8b for the ast reation then the spectra abscissa of A with respect to the initia vector x IR n is } ν : = ν x A: = max λ (r (A x = r=ρ M : λ (A =u( u ( m } = max λ (r (A c ( for at east one m } = r=ρ } = max λ (r (A c ( for at east one m } (8 = ρ } = max λ (r (A x = ρ M λ (A =u( u ( m 44 Index sets For the seque we need the foowing index sets: J ν : = IN ρ andλ (r (A =ν } (9 and J ν : = ρ} J ν = IN ρ and λ(r (A <ν } ( Page 7 of 33

8 Kohaupt Cogent Mathematics (5 : Appropriate representation of x(t We have m ρ x(t = x (r (t+c (i x (i (t with = = c (r c (r = Re c ( c(i = Im c ( = = ρ (cf Kohaupt 8b Thus due (8 x(t = ρ = e λ(r (t t = f ( (t ( with f ( (t: = c(r cos λ (i sin λ (i + c (i p (i sin λ (i + cos λ (i p (i = = ρ p (r (! (! p (r (! (! + +p (r (t t +p(r } + +p (i (t t +p(i + +p (r (t t +p(r } + +p (i (t t +p(i ( 46 Appropriate representation of y(t and ẏ(t Set q p ( ( : = q r ( p (r (r : = q r (r p (i (i (t: = r (i (3 with q ( r( CC n q (r Then from (4 (5 r (r q (i r (i IR n = = ρ y(t = ρ = e λ(r (t t = g ( (t (4 with g ( (t: = c(r cos λ (i sin λ (i + c (i q (i sin λ (i + cos λ (i q (i q (r (! (! q (r (! (! + +q (r (t t +q(r } + +q (i (t t +q(i + +q (r (t t +q(r } + +q (i (t t +q(i (5 Page 8 of 33

9 Kohaupt Cogent Mathematics (5 : = = ρ as we as z(t =ẏ(t = with h ( (t: = c(r ρ = e λ(r (t t = cos λ (i sin λ (i + c (i r (i sin λ (i + cos λ (i r (i h ( (t r (r (! (! r (r (! (! + +r (r (t t +r(r } + +r (i (t t +r(i + +r (r (t t +r(r } + +r (i (t t +r(i (6 (7 = = ρ After these preparations for the quantities J ν J ν g ( conditions: There exists a t t such that (C (t t g g ( (t t t t J ν = (C (t t h h ( (t t t t J ν = (t h( (t we formuate the foowing For these conditions there are sufficient agebraic conditions as the case may be (see Kohaupt However the foowing sufficient conditions are not so stringent in that ony conditions on components of eigenvectors are used and not on the set of components of a principa vectors The sufficient agebraic conditions read: (A g q(r q(i J ν are ineary independent (A h r(r r (i J ν are ineary independent The above conditions are not aways fufied For this we formuate the foowing conditions: For every t t there exists a t t such that (C (t t g J ν = (C (t t h J ν = g ( (t = h ( (t = Page 9 of 33

10 Kohaupt Cogent Mathematics (5 : For these conditions there are sufficient agebraic conditions as the case may be (see Kohaupt However the foowing sufficient conditions are not so stringent in that we do not suppose on the agebraic mutipicity that = The sufficient agebraic conditions read: (A g J ν = } and q ( r q ( i are ineary dependent (A h J ν = } and r ( r r ( i are ineary dependent For the next emma we set: m Y(t: = J ν = m Z(t: = J ν = (t g ( (t h ( The definition of the spectra abscissa ν = ν x A of matrix A with respect to the initia vector x can aso be found in Kohaupt ( Lemma Let hypotheses (H (H and (HS be fufied Then Y(t eν (t t y(t Y(t e ν (t t t t Z(t eν (t t z(t Z(t e ν (t t t t for sufficienty arge t t Proof We prove ony the first reation The second one is proven in a simiar way One has y(t = ρ = This impies e λ(r (t t = = e ν (t t J ν = g ( (t g ( (t+ e λ (r J ν (t t = g ( (t y(t e ν (t t J ν = = e ν (t t J ν = e ν (t t J ν = g ( (t (t g ( g ( (t e λ (r J ν (t t = g ( (t J e λ(r (t t m ν = g( (t e ν (t t J ν m = g( (t for sufficienty arge t t since the fraction in the bracet tends to zero Further Page of 33

11 Kohaupt Cogent Mathematics (5 : y(t e ν (t t J ν = g ( (t + = e ν (t t m g ( (t + J ν = e ν (t t g ( (t J ν = e λ (r J ν m (t t = g ( (t J e λ(r (t t m ν = g( (t e ν (t t J ν m = g( (t for sufficienty arge t t again since the fraction tends to zero For the formuation of the next emma we introduce some abbreviations So we define } m : = max c (r + c (i > = m where the quantities c (r c (i and c( This ceary impies c ( J ν Further define m : = max J ν and J ν : = J ν = m } J ν = max c } ( J = m ν are contained in each of the quantities g( (t h( ( (t and f S (t as we as u(t: = c (r J ν cos λ (i (t t q (r sin λ (i (t t q (i After these preparations we are now in a position to state the foowing emma + c (i sin λ (i (t t q (r + cos λ (i (t t q (i v(t: = c (r J ν cos λ (i (t t r (r sin λ (i (t t r (i + c (i sin λ (i (t t r (r + cos λ (i (t t r (i + c (i sin λ (i (t t p (r S η(t: = u(t ζ(t: = v(t p(t: = p (t = (t t m m (m! + cos λ(i (t t p (i S Lemma 3 Let hypotheses (H (H and (HS be fufied Then p(t η(t Y(t p(t η(t t t p(t ζ(t Z(t p(t ζ(t t t Page of 33

12 Kohaupt Cogent Mathematics (5 : for sufficienty arge t t Proof We prove ony the first reation The second one is proven in a simiar way One has J ν = g ( This deivers J ν = (t: = g ( J ν = c (r We note that the terms containing the vectors q (r sin λ (i + c (i + cos λ (i (t = J ν = + c (i + c (r + c (i + + c (r cos λ (i q (i sin λ (i q (i q (r (! q (r (! (! + +q (i (t t +q(i (! g ( (t = c (r cos λ (i (t t J ν sin λ (i (t t cos λ (i (t t sin λ (i (t t cos λ (i (t t sin λ (i (t t + c (i q (i sin λ (i + cos λ (i q (i q (r q (r q (r + +q (r (t t +q(r and q(r with c(r = c (r and c (i m = c (i give us the m function u(t and we mention that it has the factor p(t both defined above The rest of the sum is denoted by R(t; it can be estimated from above by poynomias of degree ess than So we obtain m Y(t = g ( (t J ν = m = g ( (t J ν = = p(tu(t+r(t } + +q (r (t t +q(r + +q (i (t t +q(i + cos λ (i q (r!! q (r (t t + q (r + q (r (m! (m! (t t q (r (m! m (m! } sin λ (i (t t q (i } sin λ (i (t t + cos λ (i q (i q (i q (i }!! + +q (r (t t m +q(r } + +q (i +q(i + +q (r (t t m +q(r } + +q (i (t t m +q(i } + q (i } + q (i This entais taing into account the definition of the function η(t Y(t = p(tu(t + R(t p(tu(t p(tη(t( R(t p(tu(t p(tη(t for sufficienty arge t t since the ast fraction tends to zero as t tends to infinity Simiary Page of 33

13 Kohaupt Cogent Mathematics (5 : Y(t = p(tu(t + R(t p(tu(t p(tη(t( + R(t p(tu(t p(tη(t for sufficienty arge t t The next emma is aso important for resuts in the seque Lemma 4 Let hypotheses (H (H and (HS be fufied If additionay the sufficient agebraic condition (A respectivey g (A h is satisfied then η(t t t ζ(t t t as the case may be for sufficienty arge t t On the other hand if additionay the sufficient agebraic condition (A respectivey g (A h is satisfied then for every t t there exists a t t such that η(t = ζ(t = as the case may be meaning correspondingy Y(t = Z(t = Proof We prove ony the first reation The second one is proven in a simiar way Assume that for a t t there exists a t t such that η( t = Then u( t = so that c (r cos λ (i ( t t +c (i sin λ (i ( t t q (r J ν + Now due to (A g the vectors q(r q(i J ν are ineary independent Therefore c (r c (r c (r sin λ (i ( t t +c (i cos λ (i ( t t cos λ (i ( t t +c (i sin λ (i ( t t = sin λ (i ( t t +c (i cos λ (i ( t t = J ν or in matrix form q (i } = cos λ (i ( t t sin λ (i ( t t sin λ (i ( t t cos λ (i ( t t J ν From this we concude that c (r c (i = c (r = c (i = J m ν or c ( = J ν This deivers a contradiction since we have seen above that c ( J ν Now we state the foowing coroary Coroary 5 Let hypotheses (H (H and (HS be fufied Page 3 of 33

14 Kohaupt Cogent Mathematics (5 : (i If further the conditions (C (t t g (C (t t h are satisfied then Z(t 4 Y(t z(t y(t 4 Z(t Y(t t t and Y(t > and Z(t > for sufficienty arge t t t (ii If instead the sufficient agebraic conditions (A and g (A h are fufied then ζ(t 6 η(t Z(t 4 Y(t z(t y(t 4 Z(t ζ(t 6 Y(t η(t t t and η(t > and ζ(t > for sufficienty arge t t Proof This foows from Lemmas to 4 The ast coroary is the basis for the derivation of the foowing theorem that is the main theoretica resut of this paper Theorem 6 (Norm equivaence of y(t and ẏ(t; genera square matrix A Let hypotheses (H (H and (HS be fufied (i If further the conditions (C (t t and g (C (t t h are satisfied then there exist constants c > and c > such that for sufficienty arge t t t (ii If instead (A and g (A hod then the above equivaence inequaities are vaid for t = t h Proof c y(t ẏ(t = z(t c y(t t t (i Due to Coroary 5 we have Z(t 4 Y(t z(t y(t 4 Z(t Y(t t t for sufficienty arge t t t Now Y(t and Z(t are positive for t t Thus due to the periodicity and continuity of Y(t and Z(t the extreme vaues Y min : = min Y(t t t Z min : = min Z(t t t Y max : = max Y(t t t Z max : = max Z(t t t exist and are positive Therefore Z min z(t 4 Y max y(t 4 Z max t t Y min so that Theorem 6 foows with c = 4 Z min Y max and c = 4 Z max Y min (ii Due to Coroary 5 we have ζ(t 6 η(t z(t ζ(t 6 y(t η(t t t Now η(t and ζ(t are periodic and continuous as we as positive for sufficienty arge t t Thus the extreme vaues Page 4 of 33

15 Kohaupt Cogent Mathematics (5 : η min : = min η(t t t ζ min : = min ζ(t t t η max : = max η(t t t ζ max : = max ζ(t t t exist and are positive Therefore ζ min z(t 6 η max y(t 6 ζ max t t η min so that Theorem 6 foows with c = 6 ζ min η max and c = 6 ζ max η min 5 Appications As appications we improve Kohaupt ( Theorems and 6 The corresponding resuts are nown so far ony for x(t (cf Kohaupt Theorem 7 (Improvement of Kohaupt Theorem Let hypotheses (H (H and (HS be fufied Moreover et ψ(t be defined by Kohaupt ( (4 If the conditions (C g (t t and (C h (t t are satisfied then there exist constants η > and η > such that η ψ(t y(t η ψ(t t t for sufficienty arge t t The same hods true if the sufficient agebraic conditions (A g and (A h hod Proof From Theorem 6 and the equivaence of norms in finite-dimensiona spaces it foows that c y(t ẏ(t c y(t t t Further y(t ( y(t + ẏ(t = x(t t t (8 (9 Moreover using (8 we get y(t = y(t + y(t y(t + ẏ(t c min }( y(t c + ẏ(t min } max y(t c ẏ(t } = min } x(t c (3 t t Due to the equivaence of norms in finite-dimensiona spaces from (9 and (3 we infer that there exist constants γ > and γ > such that γ x(t y(t γ x(t t t for sufficienty arge t By (3 and Kohaupt ( Theorem 6 the proof foows (3 Further we have Page 5 of 33

16 Kohaupt Cogent Mathematics (5 : Theorem 8 (Improvement of Kohaupt Theorem 6 Let hypotheses (H (H and (HS be fufied Moreover et ψ(t be defined by Kohaupt ( (4 If the conditions (C g (t t and (C h (t t are satisfied then there exist constants ζ > and ζ > such that ζ ψ(t z(t = ẏ(t ζ ψ(t t t for sufficienty arge t t The same hods true if the sufficient agebraic conditions (A g and (A h hod Proof The proof is simiar to that of Theorem 7 and is therefore omitted 6 Numerica exampes In this section we iustrate the obtained resuts by exampes We consider the muti-mass vibration mode in Figure for n 3 4 5} The associated initia vaue probem is given by Mÿ + B ẏ + Ky = y( =y ẏ( =ẏ where y =y y n T and M = m m m 3 m n B = K = b + b b b b + b 3 b 3 b 3 b 3 + b 4 b 4 b n b n + b n b n b n b n + b n n n + n n n n + n+ or in the state-space description ẋ(t =Ax(t x( =x Figure Muti-mass vibration n mode m m m n b b y y b n y n n b n Page 6 of 33

17 Kohaupt Cogent Mathematics (5 : where the state vector x is given by x =y T z T T z = ẏ and where the system matrix A has the form As in Kohaupt ( we specify the vaues as m j = j = n j = j = n + and j even b j = 4 j odd With the above numerica vaues we have M = E B = (if n is even and K = Remar We mention that in a exampes condition (H3 resp (H3 is fufied ie that a eigenvaues are different from zero Therefore the sufficient agebraic conditions (A g and (A h are equivaent (since then r ( = λ q ( J ν (see Kohaupt The same hods true for (A and g (A h for (A g and (A h and for (A and g (A h So we need ony the first sufficient agebraic condition with index g in each case The stepsize in a figures is Δt = 6 Iustration of the sufficient agebraic conditions In this subsection we iustrate the sufficient agebraic conditions that guarantee the vaidity respectivey invaidity of the equivaence inequaities as the case may be Remar In the foowing Exampes 5 we have to consider the quantities ũ j =(Ux j =(x u j j = m = n because they pay a roe in the definition of ν = ν x A (see Kohaupt Due to the numbering λ j+n (A =λ j (A =λ j (A j = n we have to study ony the quantities ũ j for j = n (and not for j = m = n see aso the definition of ν on this Exampe : n = We choose Page 7 of 33

18 Kohaupt Cogent Mathematics (5 : y = ẏ = Here ũ j =(Ux j j = and λ (A = i λ (A = i = λ (A =λ (A (3 Thus Sufficient agebraic condition (A g : ν = ν x A =νa =Re λ (A = 375 We have q (r = q (i = Since q (r q (i are ineary dependent the equivaence inequaities between y(t and ẏ(t do not hod This is consistent with the fact that y(t and ẏ(t have zeros (see Figures and 3 Exampe : n = We choose y = T ẏ = T Here ũ j =(Ux j j = and λ (A = i λ (A = i λ 3 (A = i = λ (A =λ (A λ 4 (A = i = λ (A =λ (A (33 Thus Figure y = y(t for n = 9 n= 8 7 y= y(t = y(t 6 y t Page 8 of 33

19 Kohaupt Cogent Mathematics (5 : Figure 3 y = ẏ(t for n = 9 8 y= z(t = z(t n= 7 6 y t ν = ν x A =νa =Re λ (A =max Re λ j= j (A = 5 Sufficient agebraic condition (A g : We have q (r = Since q (r q (i are ineary dependent the equivaence inequaities between y(t and ẏ(t do not hod This is consistent with the fact that y(t and ẏ(t have zeros (see Figures 4 and 5 Exampe 3: n = 3 We choose q (i = Figure 4 y = y(t for n = 5 n= y= y(t y t Page 9 of 33

20 Kohaupt Cogent Mathematics (5 : Figure 5 y = ẏ(t for n = 6 4 y= z(t n= y t y = T ẏ = T Here ũ j =(Ux j j = 3 and λ (A = i λ (A = i λ 3 (A = i λ 4 (A = i = λ (A =λ (A λ 5 (A = i = λ (A =λ (A λ 6 (A = i = λ 3 (A =λ 3 (A Thus ν = ν x A =νa =Re λ 3 (A =max Re λ j=3 j (A = Sufficient agebraic condition (A g : We have q (3r = Since q (3r q (3i are ineary independent the equivaence inequaities between y(t and ẏ(t hod This is consistent with the fact that y(t and ẏ(t do not have zeros for t > (see Figures 6 and 7 Exampe 4: n = 4 We choose q (3i y = T ẏ = T = Page of 33

21 Kohaupt Cogent Mathematics (5 : Figure 6 y = y(t for n = y= y(t n= t Here ũ j =(Ux j j = 4 is not true since ũ 4 =(Ux 4 =(x u 4 = We have λ (A = i λ (A = i λ 3 (A = i λ 4 (A = i λ 5 (A = i = λ (A =λ (A λ 6 (A = i = λ (A =λ (A λ 7 (A = i = λ 3 (A =λ 3 (A λ 8 (A = i = λ 4 (A =λ 4 (A Figure 7 y = ẏ(t for n = 3 5 y= z(t n=3 5 y y t Page of 33

22 Kohaupt Cogent Mathematics (5 : Thus ν = ν x A νa =Re λ 4 (A = max Re λ j= 4 j (A = Since ũ 3 =(Ux 3 =(x u 3 it foows ν = ν x A =Re λ 3 (A = max Re λ j= 4 j (A x u j } = Sufficient agebraic condition (A g : We have q (3r = q (3i = Since q (3r q (3i are ineary independent the equivaence inequaities between y(t and ẏ(t hod This is consistent with the fact that y(t and ẏ(t do not have zeros for t > (see Figures 8 and 9 Remar Here we have a nontrivia exampe of a case with ν x A <νa i i i i Exampe 5: n = 5 This mode was often used before by the author We choose y = T ẏ = T Here ũ j =(Ux j j = 5 and Figure 8 y = y(t for n = y= y(t n=4 4 y t Page of 33

23 Kohaupt Cogent Mathematics (5 : Figure 9 y = ẏ(t for n = 4 5 y= z(t n=4 5 y t λ (A = i λ (A = i λ 3 (A = i λ 4 (A = i λ 5 (A = i λ 6 (A = i = λ (A =λ (A λ 7 (A = i = λ (A =λ (A λ 8 (A = i = λ 3 (A =λ 3 (A λ 9 (A = i = λ 4 (A =λ 4 (A λ (A = i = λ 5 (A =λ 5 (A Thus ν = ν x A =νa =Re λ 5 (A = max Re λ j= 5 j (A = Sufficient agebraic condition (A g : We have q (5r = q (5i = Since q (5r q (5i are ineary independent the equivaence inequaities between y(t and ẏ(t hod This is consistent with the fact that y(t and ẏ(t do not have zeros for t > (see Figures and Page 3 of 33

24 Kohaupt Cogent Mathematics (5 : Figure y = y(t for n = 5 5 y= y(t n=5 5 y t 6 Iustration of a case with ν x A <νa In most of the Exampes 5 of Section 6 one has ν x A =νa However in Exampe 4 of Section 6 we have seen that ν x A <νa To iustrate this resut we empoy Kohaupt ( Theorems 7 and 3 where we restrict ourseves to the upper bounds y(t Y e ν (t t and z(t Z e ν (t t with the abbreviation ν = ν x A For comparison reasons however first we pot the upper bounds y(t Y e νa(t t and z(t Z e νa(t t We have νa =Re λ 4 (A 68 ν x A =Re λ 3 (A 8596 Figure y = ẏ(t for n = y= z(t n=5 y t Page 4 of 33

25 Kohaupt Cogent Mathematics (5 : In what foows we give the point of contact t su between curve and upper bound as we as the optima constants Y and Z computed by the differentia cacuus of norms For the upper bound y = Y e νa(t t we obtain t su 7579 Y 64 The curve y = y(t and its upper bound can be seen in Figure For the upper bound y = Z e νa(t t we obtain t su Z 4699 The curve y = z(t and its upper bound can be seen in Figure 3 For the upper bound y = Y e ν (t t we obtain t su 548 Y 68 The curve y = y(t and its upper bound can be seen in Figure 4 For the upper bound y = Z e ν (t t we obtain t su Z 657 The curve y = z(t and its upper bound can be seen in Figure 5 Figure y = y(t and upper bound y = Y e νa(t t 5 n=4 y=y e νa(t t 5 y x= y(t t Page 5 of 33

26 Kohaupt Cogent Mathematics (5 : Figure 3 y = ẏ(t and upper bound y = Z e νa(t t 5 y=z e νa(t t n=4 5 y= z(t t Comparing the corresponding figures it is evident that the spectra abscissa with respect to the initia vector x ie ν = ν x A has not ony theoretica meaning but sometimes aso practica significance 63 Iustration of a case with non-diagonaizabe matrix A In this subsection we first construct an exampe with n = degrees of freedom so that A IR 4 4 is not diagonaizabe The aim is then to appy Theorems 7 and 8 where we restrict ourseves to the upper bounds y(t η ψ(t and z(t ζ ψ(t (i Construction of a non-diagonaizabe matrix Figure 4 y = y(t and upper bound y = Y e ν (t t 5 n=4 y=y e ν A(t t 5 y y y= y(t t Page 6 of 33

27 Kohaupt Cogent Mathematics (5 : Figure 5 y = ẏ(t and upper bound y = Z e ν (t t 3 5 y=z e ν A(t t n=4 y 5 y= z(t t A In the case n = we have so that the pertinent characteristic equation reads For the construction of a case with non-diagonaizabe matrix A we choose b = m = m = b 3 = b 3 = Then λ m + λb +( + =s with s + } Hence with m = λ = b ± ( b + s Now in order to get one rea soution we set : = ( b Page 7 of 33

28 Kohaupt Cogent Mathematics (5 : This impies λ = b s =+ b ± i s = As numerica vaues for the quantities are not yet specified we choose b = 4 = On the whoe this deivers the foowing data: m = m = ; b = 4 b = b 3 = 4; = 64 = 4 = 3 = 64 = 4 which eads to Now The jordan routine of MATLAB gives V J =jordan(a with and Further the Matab command V s J s =jordan(a deivers J s = J After rearranging the eigenvaues of A such that λ (A =λ (A = 3 and caing the rearranged J s now J A and the rearranged V s now U we obtain and Page 8 of 33

29 Kohaupt Cogent Mathematics (5 : We have The next step is to repace the principa vector of stage p (3 by a principa vector of stage w(3 with (w (3 u(3 = Foowing the method of 7 for this we see w (3 in the form w (3 = p (3 + α(3 p(3 From (w (3 u(3 = we obtain α (3 = (p (3 u(3 (p (3 to p = p (p u p = p (p u p(3 = p (3 (p(3 u(3 p p p (3 p(3 to p p p(3 p(3 as the case may be Then u(3 Moreover we normaize according p (3 = w (3 (w(3 u(3 and rename The numerica vaues of the new p p p (3 p(3 are: (ii Compex basis functions Simiar as in Kohaupt (8b we obtain as compex basis functions x (t =p e λ (t t x (t =p e λ (t t = x (t x (3 (t =p(3 eλ (t t 3 x (3 (t =p(3 (t t +p(3 eλ (t t 3 The genera soution of ẋ = Ax is given by x(t =c x (t+c x (t+c (3 x(3 (t+c(3 x(3 (t where we prefer the usage of the doube index for the first coefficient (c with = in the notation of Kohaupt (8b The boundary condition x(t =x is met for t = t deivering x = c p + c p + c (3 p(3 + c(3 p(3 Page 9 of 33

30 Kohaupt Cogent Mathematics (5 : Scaar mutipication by the coumns of U eads to c =(x u c(3 =(x u (3 c (3 =(x u (3 (iii Rea basis functions As in Kohaupt (8b for the spitting in the rea and imaginary parts we set = ; = r where r = 3 and m = m = and m 3 = and where we have set = p = and so on Then the soution with rea basis is given by p ( with p ( = p(r + ip (i e λ (t t = e (λ(r +iλ(i (t t = e λ(r (t t cos λ (i (t t +isin λ(i (t t } x(t =c (r x(r (t+c(i x(i (t+c(3r x (3r (t+c (3r x (3r (t c (r = Rec } c (i = Imc } c (3r = c (3 c (3r = c (3 and with the rea basis functions x (r (t =eλ(r (t t cos λ (i (t t p(r sin λ(i (t t p(i } x (i (t =eλ(r (t t sin λ (i (t t p(r + cos λ(i (t t p(i } x (3r (t =e λ(r 3 (t t p (3r x (3r (t =e λ(r 3 (t t p (3r +p (3r } Figure 6 y = y(t and upper bound y = η ψ(t 5 y=η ψ(t n= y 5 y= y(t t Page 3 of 33

31 Kohaupt Cogent Mathematics (5 : Figure 7 y = ẏ(t and upper bound y = ζ 5 y=ζ ψ(t n= y 5 y= z(t t (iv Vector ψ(t From Kohaupt ( we have ψ (t =(x p e λ(r (t t ψ (t =(x p e λ(r (t t ψ (3 (t =(x p(3 eλ(r (t t 3 ψ (3 (t =(x p(3 (t t +p(3 eλ(r (t t 3 and thus ψ(t =ψ (t ψ (t ψ (3 (3 (t ψ (tt Remar Since ψ (t =ψ (t we coud aso use the vector ψ(t without component ψ (t Then merey the constants in the upper bounds woud change (v Optima upper bounds on y(tand z(t =ẏ(t The dispacement y(t and the veocity ẏ(t can be computed from x(t =y T (t ẏ T (t T For comparison reasons we have determined x(t aso by x(t =e A (t t x and obtained numericay identica resuts The advantage of the representation of x(t by the rea basis functions is that we get more insight into its vibration behavior than without it We restrict ourseves to the upper bounds y(t η ψ(t and z(t ζ ψ(t As initia condition we choose y = T ; ẏ = z = T In the seque we denote by t su the point of contact between the considered curve and optima upper bound Page 3 of 33

32 Kohaupt Cogent Mathematics (5 : For the coefficients of the soution we obtain c =c c c (3 c(3 T = i 5 65i T and thus c (r = Re c = c(i = Im c = 5 c(3r = c (3 = c (3r = c (3 = The curve y = y(t and its optima upper bound y = η ψ(t can be seen in Figure 6 One has t su 4354 η 5389 The curve y = z(t = ẏ(t and its optima upper bound y = ζ ψ(t are drawn in Figure 7 One gets t su η 578 Since c (3 =(x u (3 = and c (3 =(x u (3 = it foows that the soution part corresponding to λ 3 (A is suppressed Thus the soution behaves ie a one-mass mode with eigenvaues λ (A and λ (A It is evident that the representation of the soution by the rea basis functions offers much more insight into the vibration behavior than the representation x(t =e A (t t x that does not aow such an interpretation We mention that here the upper bounds y = η ψ(t and y = ζ ψ(t are numericay identica with those of y = Y e νa(t t and y = Z e νa(t t as the case may be The vaues t su and best constants η are obtained by the differentia cacuus of norms 7 Computationa aspects In this subsection we say something about the used computer equipment and the computation time (i As to the computer equipment the foowing hardware was avaiabe: a Pentium D 94 (3 GHz and 64 GB mass storage faciity and a 48 MB DDR-SDRAM 533 MHz (x4 MB highspeed memory As software pacage for the computations we used 368-Matab Version 4c; for the generation of the figures Version 65 in order to be abe to caption them; and for the jordan routine iewise Version 65 (ii The computation time t of an operation was determined by the command sequence t=coc; operation; t=etime(coct; it is put out in seconds rounded to two decima paces by MATLAB For exampe to compute the points of contact and to generate the tabe of vaues t y(t y u (t y (t t = (5 for Figure 6 we obtained t 6 = 69 s 8 Concusion In a one-mass vibration mode with no or mid damping the dispacement y(t and the veocity ẏ(t cannot satisfy the equivaence reation c y(t ẏ(t c y(t t t for sufficienty arge t since y(t = respectivey ẏ(t = for some t occurs in any sufficienty arge interva which is not a disadvantage because then the ower bound for both functions is simpy the time axis On the other hand for muti-mass vibration modes one can imagine that the case y(t ẏ(t t t occurs; the probabiity for this to happen increases intuitivey with increasing dimension since then it wi be uniey that a components of y(t or ẏ(t wi be zero simutaneousy at any time In the case y(t ẏ(t t t naturay the question of norm equivaence between the Page 3 of 33

33 Kohaupt Cogent Mathematics (5 : quantities y(t and ẏ(t arises In this paper sufficient conditions are given under which the norm equivaence of y(t and ẏ(t can be proven Whereas the case of diagonaizabe matrices A is simpe to treat the case of genera square matrices needs much more effort As appication improvements of some theorems of Kohaupt ( are presented Moreover the agebraic conditions for norm equivaence are iustrated for severa exampes of diagonaizabe and non-diagonaizabe matrices A and their vaidity is underpinned by the graphs of y = y(t and y = ẏ(t Acnowedgements The author woud ie to than the referees for evauation of this paper and for suggesting some improvements that ed to a better presentation Funding The author received no direct funding for this research Author detais Ludwig Kohaupt E-mai: ohaupt@beuth-hochschuede Department of Mathematics Beuth University of Technoogy Berin Luxemburger Str D-3353 Berin Germany Citation information Cite this artice as: On norm equivaence between the dispacement and veocity vectors for free inear dynamica systems Ludwig Kohaupt Cogent Mathematics (5 : References Coppe W A (965 Stabiity and asymptotic behavior of differentia equations Boston MA: DC Heath Kohaupt L (7 Construction of a biorthogona system of principa vectors for the matrices A and A* with appications ẍ = Ax x to = x Journa of Computationa Mathematics and Optimization Kohaupt L (8a Soution of the matrix eigenvaue probem VA + A*V = μv with appications to the study of free inear systems Journa of Computationa and Appied Mathematics Kohaupt L (8b Soution of the vibration probem MŸ + Bẏ +Ky = y(to =y ẏ (to = ẏ without the hypothesis BM K = KM B or B =αm = βk Appied Mathematica Sciences Kohaupt L ( Two-sided bounds on the dispacement y(t and the veocity ẏ(t of the vibration system MŸ + Bẏ + Ky = y(to = y ẏ (to= ẏ with appication of the differentia cacuus of norms The Open Appied Mathematics Journa 5 8 Müer P C & Schiehen W O (985 Linear vibrations Dordrecht: Martinus Nijhoff Thomson W T (97 Theory of vibration with appications Prentice-Ha NJ: Engewood Ciffs 5 The Author(s This open access artice is distributed under a Creative Commons Attribution (CC-BY 4 icense You are free to: Share copy and redistribute the materia in any medium or format Adapt remix transform and buid upon the materia for any purpose even commerciay The icensor cannot revoe these freedoms as ong as you foow the icense terms Under the foowing terms: Attribution You must give appropriate credit provide a in to the icense and indicate if changes were made You may do so in any reasonabe manner but not in any way that suggests the icensor endorses you or your use No additiona restrictions You may not appy ega terms or technoogica measures that egay restrict others from doing anything the icense permits Cogent Mathematics (ISSN: is pubished by Cogent OA part of Tayor & Francis Group Pubishing with Cogent OA ensures: Immediate universa access to your artice on pubication High visibiity and discoverabiity via the Cogent OA website as we as Tayor & Francis Onine Downoad and citation statistics for your artice Rapid onine pubication Input from and diaog with expert editors and editoria boards Retention of fu copyright of your artice Guaranteed egacy preservation of your artice Discounts and waivers for authors in deveoping regions Submit your manuscript to a Cogent OA journa at wwwcogentoacom Page 33 of 33