Cheb Yshev Polynomials

8/18/2019 Cheb Yshev Polynomials

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Dymore User’s Manual

Chebyshev polynomials

Contents

1 Definition 1

1.1 Zeros and extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Orthogonality relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Derivatives of Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Integral of Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Products of Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Chebyshev approximation of functions of a single variable 4

2.1 Expansion of a function in Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Evaluation of Chebyshev expansions: Clenshaw’s recurrence . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Derivatives and integrals of Chebyshev expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Products of Chebyshev expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Clenshaw-Curtis quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Chebyshev approximation of functions of two variables 9

3.1 Expansion of a function in Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Evaluation of Chebyshev expansions: Clenshaw’s recurrence . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Derivatives of Chebyshev expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Chebychev polynomials 11

4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1 Definition

Chebyshev polynomials [1, 2] form a series of orthogonal polynomials, which play an important role in the theory of approximation. The lowest polynomials are

T 0(x) = 1, T 1(x) = x, T 2(x) = 2x2 − 1, T 3(x) = 4x3 − 3x, T 4(x) = 8x4 − 8x2 + 1, . . . (1)

and are depicted in fig. 1. The polynomials can be generated from the following recurrence relationship

T n+1 = 2xT n − T n−1, n ≥ 1. (2)

It is possible to give an explicit expression of Chebyshev polynomials as

T n(x) = cos(n arccos x). (3)

This equation can be verified by using elementary trigonometric identities. For instance, it is clear, that T 2 =cos [2 arccosx] = 2 cos2(arccos x) − 1 = 2x2 − 1, as expected from eq. (1).

1.1 Zeros and extrema

It is now easy to verify that T n(x) possesses n zeros within the interval x ∈ [−1, +1]: T n(x) = cos(n arccos x) = 0implies n arccos x = (2k − 1)π/2. Hence, the zeros of Chebyshev polynomial T n(x) are

x̄k = cos π(2k − 1)

2n , k = 1, 2, 3, . . . , n . (4)

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C H E B Y S H E V P O L

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Figure 1: The seven lowest order Chebyshev polynomials

For instance, since T 3 = x(4x2 − 3), its zeros are √ 3/2, 0, and −√ 3/2, which can be written as cos π/6 = √ 3/2,

cos3π/6 = 0, and cos 5π/6 = −√ 3/2. The value of Chebyshev polynomial T i(x) at the zeros of T n(x) is easily foundfrom eq. (3) as

T i(x̄k) = cos i(2k − 1)π

2n , i < n. (5)

It is also easy to find the extrema of a Chebyshev polynomial by imposing the vanishing of its derivative, d T n/dx =0. This leads to [n sin(n arccos x)] /

√ 1

−x2 = 0, or sin[n arccos x] = 0. The extrema of Chebyshev polynomial T n(x)

arex̂k = cos

kπ

n , k = 0, 1, 2, 3, . . . , n . (6)

For instance, dT 4/dx = x(2x2 − 1) = 0 leads to extrema cos π/4 =√

2/2, cos π/2 = 0, and cos3π/4 = −√ 2/2. Theadditional extrema, cos 0 = 1 and cos π = −1, occur at the ends of the interval. The value of Chebyshev polynomialT i(x) at the extrema of T n(x) is easily found from eq. (3) as

T i(x̂k) = cos ikπ

n , i < n. (7)

1.2 Orthogonality relationships

Chebyshev polynomials are orthogonal within the interval x ∈ [−1, +1] with a weight of (1 − x2)−1/2, i.e. +1−1

T i(x)T j(x)√ 1 − x2 dx =

0 i = jπ/2 i = j = 0π i = j = 0

. (8)

In addition to the orthogonality property defined by eq. (8), Chebyshev polynomials also enjoy the followingdiscrete orthogonality relationship

nk=1

T i(x̄k)T j(x̄k) =

0 i = jn/2 i = j = 0n i = j = 0

. (9)

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2.3 Derivatives and integrals of Chebyshev expansions

Consider now a function and its derivative, both expanded in Chebyshev series

f (x) =N −1i=0

ciT i(x), and f ′(x) =

N −2i=0

c′iT i(x), (25)

where the notation (·)′ indicates a derivative with respect to x. What is the relationship between the coefficients of the two expansions, ci and c

′

i? Using the formula for the derivatives of Chebyshev polynomials, eq. (13), the followingrecurrence is found

c′

N = 0, (26a)c′N −1 = 0, (26b)

c′N −2 = 2 × (N − 1) cN −1 + c′N , (26c)...

c′1 = 2 × 2 c2 + c′3, (26d)c′0 = (2 × 1 c1 + c′2)/2. (26e)

Consider finally a function and its integral, both expanded in Chebyshev series

f ′(x) =N −1i=0

c′iT i(x), and f (x) =N i=0

ciT i(x). (27)

What is the relationship between the coefficients of the two expansions, c′i and ci? In view of the relationshipestablished above, it is clear that

c1 = 2c′0 − c′2

2 , (28a)

ci = c′i−1 − c′i+1

2i , i = 2, 3, . . . , N . (28b)

Of course, c0 is the integration constant that can be selected arbitrarily.

2.4 Products of Chebyshev expansions

Let function h(x) be the product of two functions, f (x) and g (x), such that h(x) = f (x)g(x). It is assumed that the

Chebyshev expansions of functions f (x) and g(x) are known and that function h(x) is to be expanded in Chebyshevseries, i.e.,N +M −1

k=0

ckT k(x) =

N −1i=0

aiT i(x)

M −1j=0

bjT j(x)

, (29)

where ai, i = 1, 2, . . . , N − 1 and bj , j = 1, 2, . . . , M − 1 are the known coefficients of the Chebyshev expansion of functions f (x) and g (x), respectively, and ck, k = 1, 2, . . . , N + N − 2 the unknown coefficients of the expansion of h(x). With the help of identity (17), eq. (29) becomes

N +M −1k=0

2ckT k =N −1i=0

M −1j=0

aibj 2T iT j =N −1i=0

M −1j=0

aibj [T i+j + T i−j]

=

N −1i=0

M −1j=0 aibj T i+j +

N −1i=0

ij=0 aibj T i

−

j +

N −1i=0

M −1j=i+1 aibj T j

−

i.

Identification of the coefficients of the Chebyshev polynomials of same order then yields the desired coefficients

2c0 = a0b0 +

min[(N −1),(M −1)] p=0

a pb p, (30a)

2ck =

u1 p=ℓ1

a pbk− p +

u2 p=k

a pb p−k +

u3 p=0

a pb p+k, k = 1, 2, . . . , M + N − 1. (30b)

where the bound on the three summations are ℓ1 = max[0, k− (M −1)], u1 = min[(N −1), k], u2 = min[(N −1), (M −1) + k], and u3 = min[(N − 1), (M − 1) − k], respectively.

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2.5 Examples

To illustrate application of Chebyshev expansions, the following function will be approximated by Chebyshev poly-nomials

f (x) = sin x, x ∈ [0, π/2]. (31)Using the algorithm presented in section 2.1 for N = 12, the coefficients of the Chebyshev approximation werefound to be c0 = 6.0219 10

−1, c1 = 5.1363 10−1, c2 = −1.0355 10−1, c3 = −1.3732 10−2, c4 = 1.3587 10−3,

c5 = 1.0726 10−4, c6 = −7.0463 10−6, c7 = −3.9639 10−7, c8 = 1.9500 10−8, c9 = 8.5229 10−10, c10 = −3.3516 10−11,

c11 = −1.1990 10−12. Note the rapid decay in the magnitudes of the coefficients.Figures 2 shows the exact sine function, its Chebyshev approximation, and the error incurred by the approximation.

Note that the error is spread over the entire range of the approximation in a nearly uniform manner . This is due tothe fact that the extrema of Chebyshev polynomials are distributed over the entire range of the approximation andhave alternating values of plus or minus unity. These characteristics make Chebyshev polynomials an ideal basis forapproximating functions.

0 0.5 1 1.50

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1

S I N ( X )

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0

0.01

E R R O R

0 0.5 1 1.50

0.5

1

S I N ( X )

0 0.5 1 1.5

−1

0

1

x 10−4

E R R O R

0 0.5 1 1.50

0.5

1

x

S I N ( X )

0 0.5 1 1.5

−1

0

1

x 10−12

x

E R R O R

Figure 2: Chebyshev polynomial expansion of function f (x) = sin x, x ∈ [0, π]. Function f (x): solid line; Chebyshevexpansion: circles. Figures on the left represent the function and its approximation; figures on the right show theerror associated with the approximation. Top figures: N = 3; middle figures: N = 5, bottom figures: N = 12.

Next, the sine function will be approximated using N = 3, only the terms c0 to c2 are retained in the expansion.Figure 2 also shows the results of this crude approximation. Note that the error is nearly evenly distributed over theapproximation range and that its magnitude can be estimated by looking at the magnitude of the first neglected termof the expansion: |c3| = 1.3732 10−2. The results for an approximation including 5 terms, i.e. N = 5, are presentedin fig. 2. Here again, the error is nearly evenly distributed over the approximation range and that its magnitude canbe estimated by looking at the magnitude of the first neglected term of the expansion: |c5| = 1.0726 10−4.

Finally, the algorithm presented in section 2.3 to evaluated the coefficients of the Chebyshev expansion of thederivative of the function was used to compute the coefficients of the expansion f ′(x) = cos x. The following coefficientswere found: c′0 = 6.0219 10

−1, c′1 = −5.1363 10−1, c′2 = −1.0355 10−1, c′3 = 1.3732 10−2, c′4 = 1.3587 10−3,c′5 = −1.0726 10−4, c′6 = −7.0463 10−6, c′7 = 3.9639 10−7, c′8 = 1.9500 10−8, c′9 = −8.5349 10−10, c′10 = −3.3586 10−11.Figure 3 shows the exact cosine function and its Chebyshev approximation, together with the error incurred by the

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approximation for N = 10. Note that the error is closely estimated by the magnitude of the first neglected term of the expansion: |c′10| = 3.3586 10−11.

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Figure 3: Chebyshev polynomial expansion of the derivative of function f (x) = sin x, x ∈ [0, π]. Function f ′(x): solidline; Chebyshev expansion for N = 12: circles. Figure on the left represents the function and its approximation; figureon the right shows the error associated with the approximation.

2.6 Clenshaw-Curtis quadrature

Consider the problem of evaluating the following integral ba f (x) dx. To that effect, the function is first expanded in

terms of Chebyshev polynomials,

ba f (x) dx ≈

b

a

2n

i=0 ciT i(x) dx =

2n

i=0 ci

ba T i(x) dx =

b

−a

2

2n

i=0 ci

+1−1 T i(x) dx, (32)

where the coefficients ci are found from eqs. (20a) and (20b) or eqs. (22a) and (22b). The integral of the Chebyshevpolynomials are evaluated by eq. (16) to find b

a

f (x) dx ≈ (b − a)

c0 − c23 − c4

15 − · · · − c2n

4n2 − 1

. (33)

The error, e, in the evaluation of the integral can be estimated from the last term in the series,

e ≈ (b − a) |c2n|4n2 − 1 . (34)

Assume that for a given value of n, the estimate of the integral given by eq. (33) does not satisfy a desired error

criterion, say e < ǫ, where the error estimate, e, is given by eq. (34) and ǫ a small number. A larger number of Chebyshev polynomials must then be used to approximate the function, in an attempt to meet the accuracy criterion.If the Chebyshev expansion of the function is performed based on the zeros of Chebyshev polynomials using eqs. (20a)and (20b), the abscissa at which the function will be evaluated are shown in fig. 4 for n = 2k, k = 1, 2, 3, 4, 5. Notethat for k = 2, the four abscissa all differ from those for k = 1. In fact, for two arbitrary but different values of k,all abscissa are different. On the other hand, if the Chebyshev expansion of the function is performed based on theextrema of Chebyshev polynomials using eqs. (22a) and (22b), the abscissa at which the function will be evaluatedare shown in fig. 5 for n = 2k + 1, k = 1, 2, 3, 4, 5. When using the extrema, the abscissa for n = 2k are a subsetof the abscissa for n = 2ℓ where ℓ > k. For instance, when k = 2, the n = 22 + 1 = 5 abscissa are a subset of then = 23 + 1 = 9 abscissa corresponding to k = 3. Those five abscissa are represented by black circles in figure 5.

The Clenshaw-Curtis quadrature scheme based on the extrema of Chebyshev polynomials is well suited for adaptiveintegration. At stage k of the procedure, the function to be integrated is evaluated at n = 2k + 1 abscissa, and its

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1.5

2

2.5

3

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4

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5

x

P O W E R O F N

Figure 4: Abscissa of the Clenshaw-Curtis integrationscheme using the zeros of Chebyshev polynomials, forn = 1, 2, 3, 4, 5.

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1.5

2

2.5

3

3.5

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x

P O W E R O F N

Figure 5: Abscissa of the Clenshaw-Curtis integrationscheme using the extrema of Chebyshev polynomials, forn = 1, 2, 3, 4, 5.

Chebyshev expansion is computed using eqs. (22a) and (22b). Equations (33) and (34) then yield an estimate of the

integral and of the error, respectively. If the error satisfies the accuracy criterion, the process stops. If not, stagek + 1 starts and the function is evaluated at n = 2k+1 + 1 abscissa to evaluate its new Chebyshev expansion. Of these n = 2k+1 + 1 evaluations, however, n = 2k + 1 where already performed at stage k , leaving n = 2k new functionevaluations to be performed for stage k + 1. At every stage of the computation, all function evaluations are used toobtain the new Chebyshev expansion of the function. The process stops when the accuracy criterion is met.

3 Chebyshev approximation of functions of two variables

3.1 Expansion of a function in Chebyshev polynomials

Section 2 describes the expansion of arbitrary functions of a single variable in series of Chebyshev polynomials. Clearly,functions of two variables can be similarly expanded in double series of Chebyshev polynomials

f (x, y) =M −1i=0

N −1j=0

cijT i(x)T j(y). (35)

To find these coefficients given function f (x, y), the above relationship is expressed at x = x̄k, y = ȳℓ, where x̄k and ȳℓthe zeros of T M (x) and T N (y), respectively, as given by eq. (4). This yields f (x̄k, ȳℓ) ≈

M −1i=0

N −1j=0 cijT i(x̄k)T j(ȳℓ).

Multiplying both sides of this equation by T p(x̄k)T q(ȳℓ) and summing the resulting equations expressed at all zerosof T M (x) and T N (y) leads to

M k=1

N ℓ=1

f (x̄k, ȳℓ)T p(x̄k)T q(ȳℓ) =

M −1i=0

N −1j=0

cij

M k=1

T i(x̄k)T p(x̄k)

N ℓ=1

T j(ȳℓ)T q(ȳℓ)

. (36)

In view of the discrete orthogonality relationship of Chebyshev polynomials, eq. (9), it then follows that

c00 = 1

M N

M k=1

N ℓ=1

f (x̄k, ȳℓ), (37a)

ci0 = 2

M N

M k=1

N ℓ=1

f (x̄k, ȳℓ)T i(x̄k), (37b)

c0j = 2

M N

M k=1

N ℓ=1

f (x̄k, ȳℓ)T j(ȳℓ), (37c)

cij = 4

M N

M k=1

N ℓ=1

f (x̄k, ȳℓ)T i(x̄k)T j(ȳℓ). (37d)

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In some cases, function f (x, y) is partially expanded in Chebyshev series. For instance, the function dependencyon the y variable is in the form of a Chebyshev expansion, whereas its dependency on the x variable is not, i.e.

f (x, y) =N −1j=0

gj(x)T j(y). (38)

The coefficients of the complete Chebyshev expansion are found by introducing the above expression into eqs. (37a)and (37d) to find

c0j =

1

M

M k=1 g

j(xk), (39a)

cij = 2

M

M k=1

gj(xk)T i(xk). (39b)

3.2 Evaluation of Chebyshev expansions: Clenshaw’s recurrence

If the coefficients of the two dimensional Chebyshev expansion are known, the function can be evaluated using eq. (35).However, here again, rather than computing the polynomials then summing all contributions, it is preferable to useClenshaw’s recurrence defined by eq. (24). To that effect, eq. (35) is rewritten as

f (x, y) =M −1

i=0

N −1

j=0

cijT j(y)T i(x) =M −1

i=0

di(y)T i(x). (40)

Clenshaw’s recurrence, eq. (24), is first used M times to compute the coefficients di, i = 0, 1, . . . , M − 1. Finally, onemore application of Clenshaw’s recurrence yields the desired value of the function. Of course, it is also possible torecast eq. (35) as

f (x, y) =

N −1j=0

M −1i=0

cijT i(x)

T j(y) =

N −1j=0

gj(x)T j(y). (41)

At first, N applications of Clenshaw’s recurrence yield the coefficients gj , j = 0, 1, . . . , N −1, and one additional stepyields the desired function value.

Using this second option, Clenshaw’s recurrence, characterized by eqs. (23a) to (23e), is rewritten as

yM +1,j = 0, (42a)

yM,j = 0, (42b)

yM −1,j = cM −1,j − yM +1,j + 2x yM,j , (42c)...

y1,j = c1,j − y3,j + 2x y2,j, (42d)y0,j = c0,j − y2,j + 2x y1,j. (42e)

The coefficients, gj(x), now simply become

gj(x) = (c0,j − y2,j) + y1,j x = y0,j − x y1,j. (43)

Clenshaw’s recurrence applied to the coefficients gj(x) then yields the desired function value.

3.3 Derivatives of Chebyshev expansions

Consider now a function of two variables and its derivative with respect to x, both expanded in Chebyshev series

f (x, y) =M −1i=0

N −1j=0

cijT i(x)T j(y), and f ′(x, y) =

M −2i=0

N −1j=0

c′ijT i(x)T j(y), (44)

where the notation (·)′ indicates a derivative with respect to x. What is the relationship between the coefficientsof the two expansions, cij and c

′

ij? Using the formula for the derivatives of Chebyshev polynomials, eq. (13), the

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following recurrence is found

c′M,j = 0, (45a)

c′M −1,j = 0, (45b)

c′M −2,j = 2 × (M − 1) cM −1,j + c′M,j , (45c)...

c′1,j = 2 × 2 c2,j + c′3,j, (45d)c′0,j = (2 × 1 c1,j + c′2,j)/2. (45e)

Consider next a function of two variables and its derivative with respect to y , both expanded in Chebyshev series

f (x, y) =M −1i=0

N −1j=0

cijT i(x)T j(y), and f +(x, y) =

M −1i=0

N −2j=0

c+ijT i(x)T j(y). (46)

where the notation (·)+ indicates a derivative with respect to y. What is the relationship between the coefficientsof the two expansions, cij and c

+ij? Using the formula for the derivatives of Chebyshev polynomials, eq. (13), the

following recurrence is found

c+i,N = 0, (47a)

c+i,N −1 = 0, (47b)

c+

i,N −2 = 2 × (N − 1) ci,N −1 + c+

i,N , (47c)...

c+i,1 = 2 × 2 ci,2 + c+i,3, (47d)c+i,0 = (2 × 1 ci,1 + c+i,2)/2. (47e)

4 Chebychev polynomials

Chebychev polynomials [1, 2] are defined in the following manner

T 0(x) = 1T 1(x) = x

T 2(x) = 2x2 − 1T 3(x) = 4x3 − 3x

T 4(x) = 8x4 − 8x2 + 1. . .

T n+1(x) = 2xT n(x) − T n−1(x).

(48)

The variable x ∈ [−1, 1]. An explicit formula in terms of transcendental functions is also available: T n(x) =cos(n arccos x).

These polynomials are orthogonal polynomials that form an ideal basis for the approximation of functions. Anarbitrary function F (s) can be approximated as

F (s) =N

i=1

ci T i−1(x), (49)

where T i are the Chebychev polynomials, ci the coefficients of the expansion, N the number of terms in the expansion,and x a non dimensional variable defined as

x = 2s− (shi + slo)

shi − slo . (50)

slo and shi are the lower and upper bounds defining the range over which the approximation is valid.The physical characteristics of dampers and springs are accurately and efficiently approximated by Chebychev

polynomials. For linear springs or dampers, the elastic or viscous force, respectively, is approximated in terms of thestretch or stretch rate, respectively. For torsional springs and dampers, the elastic or viscous moment, respectively,is approximated in terms of the rotational stretch or rotational stretch rate, respectively.

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4.1 Examples

1. Example 1. Consider the simple example of a linear spring of stiffness constant k , i.e. defined as F = ks. Atfirst, the approximation range is assumed to be between slo = −1 and shi = 1. It then follows from eq. (50)that x = s, and

F = ks = kx = 0 × 1 + k × x = 0 × T 0(x) + k × T 1(x). (51)Hence, the representation of this spring is slo = −1, shi = 1, c1 = 0, and c2 = k. The input sequence givenbelow defines a torsional damper for k = 1.2e + 04. The moment-rotational stretch rate is depicted in fig. 6.

@DAMPER DEFINITION {

@DAMPER NAME { damperRvjTeeter } {@DAMPER TYPE { TORSIONAL }

@DAMPER DEFINITION TYPE { CHEBYCHEV }

@APPROXIMATION RANGE { -1.0, 1.0 }

@CHEBYCHEV COEFFICIENTS { 0.0, 1.2e+04 }

}

}

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5

0

0.5

1

1.5x 10

4 damperRvjTeeter

ANGULAR VELOCITY [rad/sec]

M O M E N T [ N . m ]

Figure 6: Torsional damper with linear moment versus angular velocity characteristics.

2. Example 2. Suppose now that the approximation range of the same spring is defined as slo = −10 and shi = 10.It then follows from eq. (50) that x = s/10 and

F = ks = 10kx = 0 × 1 + 10k × x = 0 × T 0(x) + 10k × T 1(x). (52)

Hence, the representation of this spring is slo = −10, shi = 10, c1 = 0, and c2 = 10k. The spring constant is kin both cases, but the Chebychev coefficients are different due to the different approximation range. The inputsequence given below defines a torsional damper for k = 1.2e + 04. The resulting viscous moment versus angular

velocity is depicted in fig. 7.@DAMPER DEFINITION {

@DAMPER NAME { damperRvjTeeter } {

@DAMPER TYPE { TORSIONAL }

@DAMPER DEFINITION TYPE { CHEBYCHEV }


@CHEBYCHEV COEFFICIENTS { 0.0, 1.2e+05 }

}

}

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−10 −8 −6 −4 −2 0 2 4 6 8 10−1.5

−1

−0.5

0

0.5

1

1.5x 10

5 damperRvjTeeter

ANGULAR VELOCITY [rad/sec]

M

O M E N T [ N . m ]

Figure 7: Torsional damper with linear moment versus angular velocity characteristics.

3. Example 3. As a more elaborated example, consider a nonlinear spring defined as F (s) = k1s + k3s3 with an

approximation range slo =

−5 and shi = 5. It then follows from eq. (50) that x = s/5, and

F = k1s + k3s3 = 5k1x + 125k3x

3 = 5k1T 1(x) + 125k31

4 [T 3(x) + 3T 1(x)] . (53)

Collecting the coefficients of the various polynomials then yields

F =

5k1 +

3 × 1254

k3

T 1(x) +

125

4 k3

T 3(x). (54)

Hence, the representation of this spring is slo = −5, shi = 5, c1 = 0, c2 = 5 k1 + 375/4 k3, c3 = 0, andc4 = 125/4 k3. The resulting force versus stretch is depicted in fig. 8 for k1 = 1000 and k3 = 80.

@SPRING DEFINITION {

@SPRING NAME { SpringTest } {

@SPRING TYPE { LINEAR }

@SPRING DEFINITION TYPE { CHEBYCHEV }


@CHEBYCHEV COEFFICIENTS { 0.0, 4.25e+04, 0.0, 1.25e+04 }

}

}

4. Example 4. Consider now a spring whose force-stretch relationship is defined by data points, typically exper-imental measurements. The data points as shown by symbols in fig. 9. An expansion in terms of 4 Chebychevpolynomials [1] is then performed and the resulting approximation is shown in fig. 9. It is important to ap-propriately chose the number of Chebychev polynomials to be used in the expansion: fig. 10 shows the result

of the Chebychev approximation using 48 terms in the expansion. Since the spring characteristics are linearlyinterpolated between the data points, the expansion produces a set of nearly linear segments between the datapoints.


@SPRING NAME { SpringNew } {

@SPRING TYPE { TORSIONAL }

@SPRING DEFINITION TYPE { DATA POINTS }

@TABLE ENTRIES {

@VARIABLE { -4.0e-03 } @FUNCTION VALUE { -5.0 }

@VARIABLE { -3.0e-03 } @FUNCTION VALUE { -3.5 }

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-15000

-10000

-5000

0

5000

10000

15000

-5 -4 -3 -2 -1 0 1 2 3 4 5

F O R C E S [ N ]

STRETCH [m]

SpringTest

Force

Figure 8: Linear spring with nonlinear force versus stretch characteristics.

@VARIABLE { -2.0e-03 } @FUNCTION VALUE { 0.0 }

@VARIABLE { -1.0e-03 } @FUNCTION VALUE { 2.0 }

@VARIABLE { 0.0e-03 } @FUNCTION VALUE { 2.5 }





}

@NUMBER OF CHEBYCHEV COEFFICIENTS { 4 }}

-6

-5

-4

-3

-2

-1

0

1

2

3

-0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

M O M E N T S [ N . m ]

ROTATION [rad]

SpringNew

MomentData

Figure 9: Torsional spring with nonlinear moment versusrotation characteristics. The symbols indicate the datapoints, the solid line the Chebychev approximation using4 Chebychev polynomials.

-5

-4

-3

-2

-1

0

1

2

3

-0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

M O M E N T S [ N . m ]

ROTATION [rad]

SpringNew

MomentData

Figure 10: Torsional spring with nonlinear moment versusrotation characteristics. The symbols indicate the datapoints, the solid line the Chebychev approximation using48 Chebychev polynomials.

5. Example 5. Since Chebychev polynomials are continuous, they are not well suited for the approximationof functions presenting discontinuities, or very sharp gradients. Consider the data points as shown in fig. 11.An expansion in terms of 4 Chebychev polynomials shows poor correlation with the data points, whereas anexpansion with 16 terms, see fig. 12, exhibits the Gibbs phenomenon, violent oscillations of the approximationin the vicinity of the region of the discontinuity.


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@SPRING NAME { SpringDiscontinuity } {

@SPRING TYPE { LINEAR }

@SPRING DEFINITION TYPE { DATA POINTS }

@TABLE ENTRIES {

@VARIABLE { -5.0 } @FUNCTION VALUE { -5.0 }

@VARIABLE { -0.25 } @FUNCTION VALUE { -4.0 }

@VARIABLE { 0.25 } @FUNCTION VALUE { 4.0 }

@VARIABLE { 5.0 } @FUNCTION VALUE { 5.0 }

}

@NUMBER OF CHEBYCHEV COEFFICIENTS { 4 }}

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

F O R C E S [ N ]

STRETCH [m]

SpringDiscontinuity

ForceData

Figure 11: Linear spring with nonlinear force versusstretch characteristics. The symbols indicate the datapoints, the solid line the Chebychev approximation using4 Chebychev polynomials.

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

F O R C E S [ N ]

STRETCH [m]

SpringDiscontinuity

ForceData

Figure 12: Linear spring with nonlinear force versusstretch characteristics. The symbols indicate the datapoints, the solid line the Chebychev approximation using16 Chebychev polynomials.

References

[1] W.H. Press, B.P. Flannery, S.A. Teutolsky, and W.T. Vetterling. Numerical Recipes. The Art of Scientific Com-puting . Cambridge University Press, Cambridge, 1990.

[2] M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions . Dover Publications, Inc., New York, 1964.

Cheb Yshev Polynomials

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