In this paper we present the application of the homotopy analysis method for determining the free vibrations of the simply supported beam. The linear and nonlinear cases are considered.

The homotopy analysis gives the possibility to search for the solution of a wide range of problems described by means of the operator equations. The numerical examples are presented to confirm the exactness and fast convergence of the introduced method. The presented computational examples confirm the precision and the fast convergence of investigated method. In the linear case we knew the exact solutions, so we could compare with them the solutions obtained with the aid of homotopy analysis method. Differences between the obtained solutions were slight. However, the advantage of the examined method is that we receive here the approximate solution in the form of continuous function which can be used then in a further analysis or to perform various simulations.

W artykule przedstawiono homotopijną metodę analizy równań opisujących drgania swobodne układu o jednym stopniu swobody opisane równaniami liniowymi oraz nieliniowymi.

Homotopijna metoda analizy daje możliwość poszukiwania rozwiązania szerokiego zakresu problemów opisanych za pomocą równań operatorowych. W artykule przedstawiono przykłady liczbowe w celu potwierdzenia dokładności i szybkiej zbieżności metody. W przypadku liniowym znane było dokładne rozwiązanie, dzięki czemu można było porównać je z rozwiązaniami uzyskanymi za pomocą analizy homotopijnej. Różnice między otrzymanymi roztworami były niewielkie. Zaletą badanej metody jest jednak to, że otrzymano przybliżone rozwiązania w postaci ciągłych funkcji, które można wykorzystać w dalszych analizach, w tym do interpretacji wyników badań doświadczalnych.

Vibrations of the building structures can be divided into the forced ones and the free ones. The forced vibrations appear when the variable load acts on the structure. The free vibrations take place when the variable load does not act on the structure during the vibrations and the motion of the structure results from the initial conditions [

The vibrations registered while testing the real structures can be presented with the aid of continuous function. Such function is the polynomial of degree

Mathematical modeling of the building structures consists in deriving the differential equations including the linear and nonlinear parameters of the construction. With respect to the form of describing the vibrations of the real structures it is essential to get the solution of a mathematical model also in the form of continuous function. This function can be obtained by applying the homotopy analysis method. The homotopy analysis method has been developed in the 90s of the last century [

In the current paper we describe the application of the homotopy analysis method for determining the free vibrations of the beam in the linear and nonlinear case. The numerical examples are presented to confirm the exactness and fast convergence of the introduced method. In the linear case we know the exact solution, so we compare with it the solution obtained with the aid of homotopy analysis method. This case serves then to verify the exactness and convergence of the discussed method. In the nonlinear case the exact solution is unknown, therefore the approximate solution received by using the homotopy analysis method is compared with the approximate solution determined numerically by applying the Mathematica software [

We describe the concept of the homotopy analysis method by the example of its application for solving the ordinary differential equations of the second kind in the form

with the initial conditions

where _{1} is the linear operator, _{2} is the non-linear operator, _{a}_{a}

In the first step, we specify the homotopy operator ℋ as:

where _{} means the initial approximation of the solution of problem

Considering the equation ℋ(Φ

Substituting _{0}(t))=0. However, if we assume _{0} to

Taking the Maclaurin series of function Φ (with respect to parameter

where _{0}(

If the above series is convergent for

If we are not able to determine the sum of series in

In order to determine the form of function _{m}

where

and

In view of the above definition of operator

whereas for

Thus, the equation

whereas for

After selecting operator _{m}

As the operator

In the considered case we receive the simple form of equations by taking

Then we get for

and for

The last component in the above equation is nonlinear. We will be able to determine this nonlinear element by knowing the form of nonlinear operator _{2}.

As the initial approximation the best is to select a function satisfying the given initial conditions

But if one makes another choice of the linear operator (other than in the form of second order derivative), then one can connect the initial approximation with the selected linear operator and the given initial conditions. On can then determine _{0} as the solution of linear equation _{0}) = 0 with the same initial conditions as in the original equation.

In order to ensure the uniqueness of solution of equations

for _{n}

In this way the solution of considered problem is reduced to the solution of a sequence of differential equations

The proper selection of the convergence control parameter

And next, the optimum value of the convergence control parameter is obtained by determining the minimum of this squared residual.

Vibrations of the simply supported reinforced concrete beam of span ^{3}, is equal to

Such beam can be considered as the set of segments of length

Reduction of the beam with the continuous distribution of mass into the system with one degree of freedom a) system with the continuous distribution of mass

where

where _{m}

Velocity of every point of coordinate

where

By substituting

The kinetic energy described by equation

focused in the deflection, that is in the point in which the deflection of a beam is the largest (Figure

where

Equation describing the first free vibration of the reinforced concrete beam of span

where

Equation

where

We show now how the described above homotopy analysis method may be used for solving the equation _{0} = 140.9 rad/s and the initial conditions _{a}_{a}

If we take as the initial approximation the function satisfying the initial conditions

then in the first steps of the method we get

Figure _{n}

Oscillator of parameters

The squared residual _{9}

Values of errors in the reconstruction of the exact solution (Δ_{n}_{e}_{n}

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

Δ_{n} |
0.187 | 0.390 | 0.287 | 0.101 | 1.767·10^{-2} |

6 | 7 | 8 | 9 | 10 | |

Δ_{n} |
1.187·10^{-3} |
4.884·10^{-5} |
7.253·10^{-6} |
3.744·10^{-7} |
2.572·10^{-8} |

11 | 12 | 13 | 14 | 15 | |

Δ_{n} |
3.130·10^{-9} |
8.340·10^{-11} |
1.165·1^{-11} |
1.101·10^{-12} |
3.706·10^{-14} |

where _{e}_{n}^{–2}, for ^{–8} and for ^{–14}. Next the error decreases a little bit slower, so for ^{–16}. In Figure _{e}_{n}

Distribution of error (|_{e}_{n}

In the second example we execute the calculations for the changed initial conditions by taking _{a}_{a}_{0} is not changed (_{0}=140.9 rad/s). In this case we also know the exact solution given by the function

As the initial approximation we take the function fulfilling the initial conditions

Then we obtain successively

Figure

The squared residual _{9}

Numerically computed optimum value of the convergence control parameter is equal to –0.9580711. In Table _{n}

Values of errors in the reconstruction of the exact solution (Δ_{n}_{e}_{n}

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

Δ_{n} |
0.236 | 0.456 | 0.333 | 0.122 | 2.388·10^{-2} |

6 | 7 | 8 | 9 | 10 | |

Δ_{n} |
2.283·10^{-3} |
4.017·10^{-5} |
8.665·10^{-6} |
8.046·10^{-8} |
3.744·10^{-8} |

11 | 12 | 13 | 14 | 15 | |

Δ_{n} |
7.420·10^{-10} |
1.238·10^{-10} |
7.527·10^{-12} |
1.437·10^{-13} |
2.802·10^{-14} |

Distribution of error (|_{e}_{n}

The initial condition

of the free vibrations can be achieved by dropping on the beam from height _{1} (the beater – Figure _{1} = 100 kg and

Impact of plastic character implying the initial conditions _{a}

where _{a}

After the impact of the plastic character the motion of the oscillator is executed according to the equation

the solution of which, after taking into account the initial conditions

is given by the function

where

Now we deal with the solution of equation _{1} = 100 kg, ^{2}, and the initial conditions _{a}_{a}

Numerically determined optimum value of the convergence control parameter is equal to –0.77670315. Table _{n}

Values of errors in the reconstruction of the exact solution (Δ_{n}_{e}_{n}

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

Δ_{n} |
3.367·10^{-2} |
2.084·10^{-2} |
1.741·10^{-3} |
1.374·10^{-3} |
4.188·10^{-4} |

6 | 7 | 8 | 9 | 10 | |

Δ_{n} |
5.135·10^{-5} |
1.785·10^{-5} |
9.227·10^{-5} |
2.483·10^{-6} |
4.230·10^{-7} |

11 | 12 | 13 | 14 | 15 | |

Δ_{n} |
6.678·10^{-8} |
2.612·10^{-8} |
1.181·10^{-8} |
3.571·10^{-9} |
8.398·10^{-10} |

Distribution of error (|_{e}_{n}

Nonlinearity in the mechanical dynamic system can occur in result of nonlinearity of the elastic force

The plus sign preceding

with the initial conditions

Referring to the previous example when we assumed the mass ^{9} with the amplitude of variable

The free vibrations with the damping include additionally one more term by the first derivative denoted usually as

In our case the physically real value of

Proceeding with the computational example we discuss the solution of equation _{a}_{0} the function satisfying the initial conditions

we get, by using the homotopy analysis method, the function

The optimum value of the convergence control parameter is equal to –0.7227439. The plot of the squared residual for

The squared residual _{20}

Plot of the residual _{n}

for _{n}

Figure

Comparison of the approximate solutions (solid line – solution _{20}

Maximal absolute differences (Δ_{n}

1 | 3 | 5 | |
---|---|---|---|

Δ_{n} |
3.35·10^{-2} |
4.98·10^{-3} |
7.12·10^{-5} |

10 | 15 | 20 | |

Δ_{n} |
1.40·10^{-6} |
2.99·10^{-8} |
3.71·10^{-9} |

In the next example we solve the equation ^{3} kg/s and _{a}_{0}, that is

we get successively the functions

Optimum value of the convergence control parameter is equal to –0.686325. The plot of the squared residual for

The squared residual _{20}

Plot of the residual

for _{n}

Figure ^{–4}, for ^{–6}, for ^{–8}, whereas for ^{–9}.

Comparison of the approximate solutions (solid line – solution _{20}

In the paper we have presented the application of the homotopy analysis method for determining the free vibrations of the reinforced concrete beam in the linear and nonlinear case. In the investigated method the series is created, the successive terms of which are derived by calculating the proper integrals obtained from the previous terms. The formed series is in general quickly convergent. For the optimal selection of the convergence control parameter computation of just several first terms of the series ensures a very good approximation of the sought solution. Performed calculations show that the proposed method is effective in solving the problems under consideration.

The presented computational examples confirm the precision and the fast convergence of investigated method. In the linear case we knew the exact solutions, so we could compare with them the solutions obtained with the aid of homotopy analysis method. This case served then to verify the exactness and convergence of the discussed method. In the nonlinear case, when the exact solution was unknown, we compared the approximate solutions received by using the homotopy analysis method with the approximate solutions determined numerically by applying the Mathematica software. Differences between the obtained solutions were slight. However, the advantage of the examined method is that we receive here the approximate solution in the form of continuous function which can be used then in a further analysis or to perform various simulations. Whereas when we apply the numerical methods, we get the discrete set of values which may be potentially used later for approximating the appropriate functions.

The executed calculations indicate that it is possible to obtain the mathematical solution of the model describing the free vibrations of the reinforced concrete beam in the form of polynomial. The obtained results can be compared with the results of experimental research also approximated by polynomials. Such comparison will give the possibility to verify the taken model of structure.