Equivalent Linearization of Bouc–Wen Hysteretic Model with Harmonic Input

Abstract

Equivalent linear stiffness and damping coefficient are calculated for the Bouc–Wen hysteretic model with the input as a harmonic function. Analytical estimates of these quantities are provided after constructing the hysteretic loops. For two cases, the hysteretic loops are found in closed form, and for the other cases limiting forms of the loops under high amplitude and low amplitude excitations are studied. Correct scaling laws for the equivalent parameters are derived analytically which are subsequently verified by numerical simulation.

Appendices

Appendix 1

Bouc–Wen Hysteretic Loop for n = 2

The loop consists of the following branches:

• 1. Branch-I$\left( { - \varDelta \le x \le a;\quad x^{\prime} > 0} \right)$

  $$r = \sqrt {\frac{1}{{\left( {\beta + \gamma } \right)}}} \tanh \left\{ {\left( {x + \varDelta } \right)\sqrt {\beta + \gamma } } \right\},$$

  (52a)

• 2. Branch-II$\left( {\varDelta \le x \le a;\quad x^{\prime} < 0} \right)$

  $$\begin{aligned} & r = \sqrt {\frac{1}{{\left( {\beta - \gamma } \right)}}} \tan \left\{ {\left( {x - \varDelta } \right)\sqrt {\beta - \gamma } } \right\}\quad ({\text{if}}\;\beta \ne \gamma ) \\ & {\text{or}}\;r = x - \varDelta \quad ({\text{if}}\;\beta = \gamma ) \\ \end{aligned}$$

  (52b)

• 3. Branch-III$\left( { - a \le x \le \varDelta ;\quad x^{\prime} < 0} \right)$

  $$r = \sqrt {\frac{1}{{\left( {\beta + \gamma } \right)}}} \tanh \left\{ {\left( {x - \varDelta } \right)\sqrt {\beta + \gamma } } \right\},$$

  (52c)

• 4. Branch-IV$\left( { - a \le x \le - \varDelta ;\quad x^{\prime} > 0} \right)$

  $$\begin{aligned} & r = \sqrt {\frac{1}{{\left( {\beta - \gamma } \right)}}} \tan \left\{ {\left( {x + \varDelta } \right)\sqrt {\beta - \gamma } } \right\}\quad ({\text{if}}\;\beta \ne \gamma ) \\ & {\text{or}}\;r = x + \varDelta \quad ({\text{if}}\;\beta = \gamma ) \\ \end{aligned}$$

  (52d)

  where $\varDelta$ is obtained by solving the following equation

  $$\tanh \left\{ {\left( {a + \varDelta } \right)\sqrt {\beta + \gamma } } \right\} = \sqrt {\frac{\beta + \gamma }{\beta - \gamma }} \tan \left\{ {\left( {a - \varDelta } \right)\sqrt {\beta - \gamma } } \right\}\quad {\text{if}}\;\beta \ne \gamma$$

  (53)

  or

  $$\tanh \left\{ {\left( {a + \varDelta } \right)\sqrt {2\beta } } \right\} = \sqrt {2\beta } \left( {a - \varDelta } \right)\quad {\text{if}}\;\beta = \gamma$$

  (53')

It can be seen that branch-I and branch-III are asymptotic to $r = \sqrt {\frac{1}{\beta + \gamma }}$ and $r = - \sqrt {\frac{1}{\beta + \gamma }}$, respectively. The variation in the shape and size of the loop follows the same trend as in the previous case. For small value of the amplitude a, branch-I and branch-II can be approximately written as

$${\text{Branch - I:}}\quad r(x) = (\varDelta + x) - \frac{1}{3}\left( {\varDelta + x} \right)^{3} \left( {\beta + \gamma } \right)$$

(54)

$${\text{Branch - II:}}\quad r(x) = (x - \varDelta ) + \frac{1}{3}\left( {x - \varDelta } \right)^{3} \left( {\beta - \gamma } \right)$$

(55)

Here $\varDelta$ satisfies the following equation

$$\varDelta = \frac{\beta }{3}\left( {a^{3} + 3a\varDelta^{2} } \right) + \frac{\gamma }{3}\left( {3a^{2} \varDelta + \varDelta^{3} } \right).$$

(56)

When a is small, the approximate solution is $\varDelta \approx \frac{\beta }{3}a^{3}$.

For large value of a, Eqs. (53) and (53′) can be written approximately as

$$\varDelta = a - \frac{1}{{\sqrt {\beta - \gamma } }}\tan^{ - 1} \left\{ {\sqrt {\frac{\beta - \gamma }{\beta + \gamma }} } \right\}\quad {\text{if }}\; \, \beta \ne \gamma$$

(57)

and

$$\varDelta = a - \frac{1}{{\sqrt {2\beta } }}\quad {\text{if}}\; \, \beta = \gamma ,$$

(57')

respectively. The shape of the loop becomes nearly rectangular.

Appendix 2

The functions appearing in Eq. 24(a)–(d), obtained after carrying out integration [20], are as follows:

$$F_{1} \left( x \right) = - \frac{2}{n}\mathop \sum \limits_{k = 0}^{{k = \frac{n}{2} - 1}} \left\{ {P_{k} \cos \left( {\frac{2k + 1}{n}\pi } \right) - Q_{k} \sin \left( {\frac{2k + 1}{n}\pi } \right)} \right\},$$

for n positive even number and

$$F_{1} \left( x \right) = \frac{1}{n}\ln \left( {1 + x} \right) - \frac{2}{n}\mathop \sum \limits_{k = 0}^{{k = \frac{n - 3}{2}}} \left\{ {P_{k} \cos \left( {\frac{2k + 1}{n}\pi } \right) - Q_{k} \sin \left( {\frac{2k + 1}{n}\pi } \right)} \right\}$$

For n even, where

$$P_{k} = \frac{1}{2}\ln \left( {x^{2} - 2x\cos \left( {\frac{2k + 1}{n}\pi } \right) + 1} \right)\;{\text{and}}\;Q_{k} = \tan^{ - 1} \left[ {\frac{{x\sin \left( {\frac{2k + 1}{n}} \right)\pi }}{{1 - x\cos \left( {\frac{2k + 1}{n}} \right)\pi }}} \right].$$

$$F_{2} \left( x \right) = \frac{1}{n}\ln \frac{1 + x}{1 - x} - \frac{2}{n}\mathop \sum \limits_{k = 1}^{{k = \frac{n}{2} - 1}} \left\{ {P_{k} \cos \frac{2k}{n}\pi - Q_{k} \sin \frac{2k}{n}\pi } \right\}$$

for n even where

$$P_{k} = \frac{1}{2}\ln \left( {x^{2} + 2x\cos \left( {\frac{2k + 1}{n}\pi } \right) + 1} \right)\;{\text{and}}\;Q_{k} = \tan^{ - 1} \left[ {\frac{{x + \cos \left( {\frac{2k + 1}{n}\pi } \right)}}{{\sin \left( {\frac{2k + 1}{n}\pi } \right)}}} \right]$$

and

$$F_{2} \left( x \right) = - \frac{1}{n}\ln \left( {1 - x} \right) + \frac{2}{n}\mathop \sum \limits_{k = 0}^{{k = \frac{n - 3}{2}}} \left\{ {P_{k} \cos \left( {\frac{2k + 1}{n}\pi } \right) + Q_{k} \sin \left( {\frac{2k + 1}{n}\pi } \right)} \right\}$$

for n odd where

$$P_{k} = \frac{1}{2}\ln \left( {x^{2} - 2x\cos \frac{2k}{n}\pi + 1} \right)\;{\text{and}}\;Q_{k} = \tan^{ - 1} \left[ {\frac{{x - \cos \frac{2k}{n}\pi }}{{\sin \frac{2k}{n}\pi }}} \right].$$