Abstract
Periodic oscillations of a thin rod, lying on a nonlinear elastic foundation.
1 Periodic Oscillations of a Thin Rod, Lying on a Nonlinear Elastic Foundation
The equation of bending vibrations of a thin rod lying on a nonlinear elastic foundation can be written as
These equations differ from Eq. (1.51) by the presence of a member \( f(u) \), which takes the action of a nonlinear elastic foundation into account. Moreover, we introduce the forces of viscous friction \( g\dot{u} \) in consideration of this. The notation is exactly the same as specified in Chap. 1. Considering the coefficient of bending elasticity \( {\upbeta} = {\upbeta} (x) \ge ({\upbeta} (x))_{\hbox{min} } > 0 \) and assuming the external distributed moment as \( k = k(x,t) = 0 \), let us move on to a single equation with respect to displacement
where
l is the length of a rod; and T is the oscillation period.
The periodicity conditions should be added to Eq. (8.1)
We assume that
We further assume that the function \( f(u)\,\, = f(x,t) \) is a periodic function of time t with the periodic function \( u\,(x,t) \) in a domain, which will be designated below when presenting a method for solving equations of the type (8.2). For simplicity, let us also assume that its mean (static) component is equal to zero.
Furthermore, the first and second derivatives of the function \( A(u) \) with respect to the function u will be required. We introduce \( \Delta = \Delta (x , t) \) as the variation of function \( u = u (x , t). \) We consider this function is also twice differentiable with respect to coordinate and time, and satisfies both the boundary conditions and periodicity conditions of the form (8.5). Let us use general formulas for the differential and derivative of the nth order [7]
In accordance with formulas (8.6), instead of u we substitute \( u + {\upalpha} \Delta \) into formula (8.3), differentiate the obtained result with respect to \( {\upalpha} \) and then, setting the value \( {\upalpha} \) equal to 0, we get
and
Let us compose the functional as
The functions \( u = u (x , t) \) and \( \dot{u} = \dot{u}(x,t) \) satisfy the periodicity conditions (8.5). Therefore, there will be
Moreover, as already mentioned, we consider that the function \( f = f (u) \) is also a periodic function of time at the periodic function \( u = u(x,t) \) and that the mean (static) component of this function is equal to zero. Then
As a result, we obtain
Let us consider an oscillatory process. For simplicity we also suppose that the static component of the displacement is equal to zero and represent a vibrating (variable) component of displacement in the form of a Fourier series expansion
and
where \( {\uplambda} = \frac{{2{\uppi} }}{T} \).
Hence, it follows that \( |\dot{u}|| \ge {\uplambda} ||u|| \), where \( ||u||^{2} = \int\nolimits_{0}^{T} {\int\nolimits_{0}^{l} {u^{2} } {\text{d}}x} {\text{d}}t \).
On the other hand, according to the Schwarz inequality \( (A^{\prime } u, \dot{u}) \le ||A^{\prime } u||\, ||\dot{u}|| \) and taking formula (8.10) into account, we finally come to the important estimate
This inequality means that there exists an inverse bounded linear operator
The function \( f(u) \) is considered to be sufficiently smooth so as to satisfy
Estimates of the form (8.12) and (8.13) are used in the theorem on the convergence of the iterative Newton-Kantorovich method.
2 The Newton-Kantorovich Method for Solving Nonlinear Operator Equations
Let us consider a nonlinear operator equation
which is the same as Eq. (8.2). The equation is called “functional” since both the argument and value of the operation \( A(u) \) are functions representing those of time and coordinates in the case considered above. Let us suppose the domain of definition and domain of values of the operator A are Hilbert spaces to which the notion of scalar products of elements of the space has been introduced. In the problem considered in Sect. 8.1 the elements of a Hilbert space are the functions \( u\,(x,t) \), and the scalar product of two such functions can be calculated as the integral of their product over the coordinates and time.
The squared norm of the Hilbert space element is defined as the scalar product of this element with itself. Other properties of the operation \( A(u) \) will be discussed below.
Let us denote the exact solution of Eq. (8.14) by \( u = u_{ *} \). Suppose also that \( u = u_{ 0} \) is a function close to the solution (one of the solutions) of this equation in a certain sense. The iterative Newton-Kantorovich method of solution of a nonlinear equation (8.14) is based on approaching linear approximation of the nonlinear operation \( A(u) \) in a bounded domain containing \( u_{ 0} \). Let us represent
approximately and put
instead of (8.14).
Considering \( u_{ 0} \) as the initial zero approximation of the iterative process, we will take the solution of the linear equation (8.15) for its first approximation \( u_{ 1} \) to be
Similarly, for the nth step there will be
or
This is the theorem of Kantorovich [6].
Let us suppose the following conditions are fulfilled:
-
(1)
For the element \( u_{ 0} \) (an initial approximation) the operation \( A^{\prime}(u_{ 0} ) \) is the inverse of it (i.e. \( G_{0} = \left( {A^{\prime}(u_{ 0} )} \right)^{ - 1} \)), and an estimate of its norm can be given as
$$ ||G_{0} || \le B_{ 0} . $$(8.18) -
(2)
The element \( u_{ 0} \) approximately satisfies Eq. (8.14) such that
$$ ||G_{0} A(u_{ 0} )|| \le {\upeta}_{0} . $$(8.19) -
(3)
The second derivative \( A^{\prime\prime}(u) \) is bounded
$$ ||A^{\prime\prime}(u)|| \le K $$(8.20)
in the domain defined by the inequality
-
(4)
The constants \( B_{0 } , {\upeta}_{0 } , K \) satisfy the inequality
Then Eq. (8.14) has the solution \( u_{ *} \), which is close to \( u_{ 0} \) in the domain defined with the same inequality (8.21), and the iterative process (8.16) or (8.17) converges to solution of Eq. (8.14).
Note that, in the problem of the bending vibrations of a thin rod on a nonlinear elastic foundation, we have proved that conditions (8.18), (8.20) are satisfied under both certain properties of the elastic foundation and the mandatory presence of viscous friction forces. Moreover, conditions (8.19), (8.21) and (8.22) have to be fulfilled for the proof of existence and convergence.
3 Iterative Gradient Method for Solving Operator Equations
The iterative gradient method can also be used to solve the nonlinear operator equation
where the derivatives of the operators \( A^{\prime} \) and \( A^{\prime\prime} \) must satisfy conditions (8.12), (8.13). However, the operator \( A^{\prime} \) is also required to be bounded
The domain of definition of the operator \( A^{\prime} \), where this inequality needs to be fulfilled, is specified in the theorem of the convergence of the iterative gradient process. The integral equations and systems of algebraic equations satisfy the requirement (8.24). The iterative gradient method can also be used to solve differential equations approximately as long as, with the help of the projection procedure, the differential equation can first be approximately replaced by the system of algebraic equations.
The successive approximations can be calculated by the formula
where the “step length” \( \upvarepsilon_{n} \) is a real number; and the gradient of the functional is \( z_{n} \). Hence
The gradient of the functional \( l(u) \) at the point u can be determined by the formula
Let us choose an element of the domain of definition of the functional (8.26) as z, an element that imparts the maximum value to the derivative (8.27) subject to the normalization condition
Clearly, such an element is precisely the gradient itself
so, we put
in formula (8.25).
Let us calculate \( {\text{grad}}\,l(u) \) for the functional (8.26). The derivative (8.27) of this functional takes the form
and its maximum value subject to the normalization condition (8.28) is obviously attained at
Accordingly, we put
in formula (8.25).
Let us choose the coefficient \( \upvarepsilon_{n} \) from the condition of error norm minimum at each step of the iterative process
The squared error norm is
Obviously, the first derivative of this quantity with respect to \( \upvarepsilon_{n} \) vanishes at
while the second derivative, equal to \( 2(z_{n} ,z_{n} ) \), is positive. Consequently, the error minimum condition (8.31) can be carried out at \( \upvarepsilon_{n} \) of (8.32).
Let us substitute \( z_{n} \) from (8.30) into formula (8.32)
However, if the element \( u_{n} \) is close to the solution \( u_{*} \), then there will be
and the formula for the coefficient \( \upvarepsilon_{n} \), which allows the minimum condition of the error norm (8.31) to be approximately fulfilled, takes the form
When the operator of Eq. (8.23) is linear, the approximate equality (8.33) becomes exact
and the error minimum condition (8.31) is also satisfied accurately. Combining formulas (8.25), (8.30), (8.34), we obtain the iterative algorithm (iterative “method with minimal errors”)
Convergence of the iterative procedure (8.36) is investigated in greater depth in [3, 5].
When the iterative algorithm (8.36) is applied to solve a linear operator equation, convergence of the process can be proved making the most general assumption of existence of a solution, even though it is not the only one [4]. Which of the possible solutions turns out to be the limit of successive approximations can be determined by the choice of the initial approximation.
An iterative formula of the gradient method, as in the Newton-Kantorovich method, presupposes knowledge of the first derivative \( A^{\prime} \). The advantage of the gradient method rests in use by the adjoint operator \( (A^{\prime} )^{*} \) of the multiplication operation in an iterative algorithm, which is much simpler than calculation of the inverse operation \( G = (A^{\prime} )^{ - 1} \), as in the Newton-Kantorovich method. The disadvantage of the gradient method is its slow convergence. However, the increasing performance of computers will promote wider application of the gradient method.
4 Nonlinear Vibrations, Close to Periodic
Close-to-periodic linear oscillations of a lumped mass attached to a spring were examined in Sect. 5.6. The approach presented there can also be extended to the case of nonlinear oscillations. We use the same designations, assumptions and some algorithms given in that section.
The equation of nonlinear oscillations of a lumped mass on a spring will be written as
where the term \( {\upgamma} \left( {{\uptau } , \frac{{{\text{d}}u}}{{{\text{d}}t}}} \right) \) takes a nonlinear frictional force acting on the oscillating mass into account, while the term \( c ({\uptau } , u) \) represents the nonlinear elastic force of the spring. All terms in the equation can be divided by the mass amount.
We still assume that the external force \( f({\uptau } , t) \) at a fixed slow time \( {\uptau } \) is a periodic function of a fast time t with the frequency \( {\upomega} ({\uptau } ) \) and period \( T({\uptau } ) = \frac{{2{\uppi} }}{{{\upomega} ({\uptau } )}} \). Putting the arguments of the functions in Eq. (8.37) in brackets indicates the nonlinear dependence of these functions of the arguments.
Taking the variable \( {\uptheta} (t) \) from formula (5.41) we get \( \frac{\text{d}}{{{\text{d}}t}}\, = \,\frac{\text{d}}{{{\text{d}} {\uptheta} }}{\upomega} \). Here Eq. (8.37) can be replaced by the system of equations
An approximate solution to this system of equations at \( {\uptau } \ge 0 \), as in Sect. 5.6, can be sought in the form
We give the projection conditions as
We also rewrite formulas (5.48) from Sect. 5.6 as
Substituting the series (8.39) into the projection conditions (8.40) and using formulas (8.41), we obtain equations that can determine the coefficients of the series (8.39)
where the functions \( v ({\uptau } , {\uptheta} ) \) and \( u({\uptau } , {\uptheta} ) \) are represented by the series (8.39). When calculating the integral (8.43), the slow time \( {\uptau } \) can be regarded as a parameter.
Let us now suppose that when \( {\uptau } = 0 \) there is a purely periodic motion at \( {\upgamma} (0,{\upnu} ) \), \( c (0, u ) \), \( f(0, {\uptheta} ) \) with a constant frequency \( {\upomega} \,(0) \). Entering these parameter values for external force and oscillation frequency in formulas (8.42), (8.43) and using the Runge-Kutta method, we get a movement which at the limit tends to the periodic. Representing the periodic motion obtained as a series
we are able to find the values \( u^{m} (0) \) and \( {\upnu}^{m} (0) \), \( m = - M, \ldots , M, \) that should be added to the system of Eq. (8.42) as the initial conditions. Hence, the initial value problem (8.39), (8.42) can again be solved by the Runge-Kutta method, but only for the slowly varying parameters of the oscillating system.
The projection method for solving the problems of vibrations that are close to periodic, as outlined in Chap. 5 and earlier in this chapter, is an alternative to the asymptotic methods that introduce the concepts of fast and slow time. However, the solution using the projection method can be represented as a series expansion in a small parameter, which is usually limited to one or two approximations [1, 2, 8].
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Fridman, V. (2018). Nonlinear Periodic Oscillations. In: Theory of Elastic Oscillations. Foundations of Engineering Mechanics. Springer, Singapore. https://doi.org/10.1007/978-981-10-4786-2_8
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