Penalty function method for FEM dynamic analysis of trusses with nonlinear multi-node constraints under harmonic load

. This work introduces the establishment of the FEM model for investigating the dynamic behavior of trusses, subjected to multi-node constraints, under harmonic excitation based on utilizing the method of Penalty function. The technique for treating the nonlinear relationship of multi-node constraints is developed by using the Penalty function to extremum the Hamiltonian augmented potential energy function and to convert the multi-node constrained relation. The method of Newmark integration is utilized to convert dynamic differential equations of a truss system having multi-node boundary constraints subjected to a time-dependent harmonic force to incremental equations. The incremental-iterative algorithm had been developed based on the Newton-Raphson method, used for solving the dynamic incremental equation of motion for the truss system. Utilizing the constructed algorithm, the Matlab program is written to compute the investigated truss system's internal force-time response and displacement-time response subjected to harmonic load. The advice for choosing the weight value of the Penalty function is proposed based on analyzing numerical results.


Introduction
Observing the nonlinear structural dynamic response has attracted attention during several past decades.In most of the past research works, the mathematical technique for solving the structural dynamic problems closely deals with utilizing FEM [1][2][3][4][5][6].The finite element simulation is extensively employed for conducting nonlinear analysis of structures, which undergo a time duration such as vibration.The step of establishing an appropriate FE mathematical model is based on selected properties of geometry and material, loading, and boundary condition.In building the FE model for dynamic analysis of the structures having dependent displacement boundary constraints such as multi-node constraints, the additional step for imposing the nonlinear multi-node boundary relation is required by utilizing the mathematical optimization technique [7][8][9][10].This research is concerned with building the solution technique for the vibration solution of truss systems with nonlinear multi-node boundary conditions.The proposed solution scheme in this work is established by going through two steps, including converting the nonlinear boundary constraints into the FE dynamic equilibrium equation and establishing the mathematical algorithm, used for solving nonlinear equations with time-dependent variables.The FE system's dynamic equilibrium equation is developed according to the Principle of Least Action.For treating the dependent multi-node constraints, this research introduces the idea of the author's proceeding works [11][12] to use the method based on penalty function to convert a constrained problem due to nonlinear multi-node constraints into an unconstrained problem.The incremental-iterative algorithm is built based on utilizing the Newmark integration method [13] to convert the time-dependent differential equations of a truss system having multi-node boundary conditions to incremental equations combining the Newton-Raphson method [14] to solve the dynamic increment equilibrium equation of the truss system.The established incremental-iterative algorithm is founded for writing the calculation program for computing the internal force-time response and the displacement-time response of trusses having nonlinear multi-node boundary conditions under harmonic loading.Analyzing numerical results, the advice for choosing the weight value of the Penalty function is proposed.

Problem formulation
The FEM governing equation of the truss system subjected to harmonic load is built by implementing Hamilton's Principle of Least Action for the n th -DOF plan truss.The boundary conditions of the plan truss are multi-node nonlinear constraints as shown in Fig. 1, described as , uu are nodal velocity vector of and acceleration; ,, M C K are mass matrix,  The dynamic equilibrium equation of the truss system, having "m" nonlinear multi-node boundary relations ()  g u 0 , is established by minimizing the Hamiltonian function augmented potential satisfied these boundary relations, described following For incorporating the nonlinear boundary equations building into the dynamic equilibrium equation based on the method of Penalty function [3,4,7,8,9], the first step is penalizing the objective function   , Qw u with the penalty element Equation ( 6) is becoming as follows The incorporated dynamic equilibrium equation can be get by minimizing the objective function The minimization of objective function is realized by satisfying the Euler condition of necessity and sufficiency as expressed in the equation ( 10 7) into (10), getting the equation Identifying right hand components in equations (1-3) with components in equations (11), having  (14) where: ,  uu are vectors of nodal incremental displacement, acceleration, velocity.
Designating the matrix built from the nonlinear multi-node boundary constraints )) The incremental equation ( 14) can be expressed in the compact form Algorithm for finding the solution of nonlinear time-variated equation is established by employing the Newmark integration technique.Utilizing the Newmark method [1,2], the incremental form of the equation of motion can be described as follows -expanded tangent stiffness matrix, included penalty element.

Solving algorithm
For solving the time-variated incremental equation ( 18), the author proposed employing the Newton-Raphson technique to establish the incremental-iterative algorithm as shown in Fig. 2.

Fig. 2. Block diagram of incremental-iterative algorithm based on Newton-Raphson technique 4 Numerical investigation
Investigate the dynamic behaviour of truss system with nonlinear multi-node boundary constraints subjected harmonic load (shown in fig.3).Truss elements (1), ( 2), ( 3), ( 4) and ( 5) are made of the same material and have the same geometrical and material properties.The parameters of loaded truss system are given as below.The convergent speed of the solution depends on choosing the value of the penalty element, it will be faster according to increasing the weight value.The optimized choice can be reached in the case of choosing an initial weight value approximated (or about) with an element stiffness value.

Conclusions
In this paper, the mathematical algorithm has been established for seeking the FEM dynamic solution of a truss system having nonlinear multi-node boundary conditions subjected to harmonic loading.The Penalty function method has been effectively employed to extremum the Hamiltonian augmented potential energy function of the truss system under harmonic load, used to convert the nonlinear multi-node boundary constraints for building the incorporated nonlinear dynamic equilibrium equation.The incremental-iterative algorithm for solving the nonlinear time-dependent problem has been successfully constructed using the combination of the Newmark integration and the Newton-Raphson technique.The numerical results show the convergence and effeteness of the proposed model and algorithm for analyzing the dynamic behavior of truss systems having nonlinear multi-node boundary relations under the action of harmonic loading.

Fig. 1 .
Fig. 1.Observed truss model For building the augmented potential function, computing T -the kinetic energy and Vstrain energy of the truss system by equations (1) & (2), the total work of the truss system is defined by summing work done by all forces, included damping and harmonic forces, expressed in equation (3) 1 2 () 

E3S
Web of Conferences 458, 07009 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345807009damping matrix, stiffness matrix of FE truss system; of external forces.Implementing Hamilton's Principle, between two points in a time interval   01 , tt, the actual path of motion is realized while the Hamiltonian function, in equation (4), is reaching the extremum.
second-order differential dynamic equilibrium equation, incorporated nonlinear boundary condition due to the multi-node constraints ( for finding the solution FEM nonlinear equation is established by employing the iterative load-increment technique.The dynamic incremental equation is built by employing the Taylor series to expand the function and keep the linear terms of increments ,, case of average acceleration method.By replacing the right sides of equations (16) into equation (15), having equation (17) can be expressed in shortened form as ()   K u u P (18) E3S Web of Conferences 458, 07009 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345807009

u u u g0 Fig. 3 .
Fig. 3. Investigated truss system having nonlinear multi-node boundary conditions under harmonic load For solving the dynamic problem of investigated truss system, the Matlab calculation program is coded utilizing the established iterative-incremental algorithm.The Newmark's average acceleration choice 0.5; 0.25  

Fig. 6 .
Fig. 6.Normal force-time relation 1 ()  Nt with different weight values Comments:The convergent speed of the solution depends on choosing the value of the penalty element, it will be faster according to increasing the weight value.The optimized choice can be reached in the case of choosing an initial weight value approximated (or about) with an element stiffness value.