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The piecewise full decoupling method is a new developed numerical procedure of explicit integration based on piecing together local linear solutions. The method is applied for solving piecewise linear dynamic systems under periodic excitations. Close agreement is found between obtained results and published findings of a harmonic balance method and a finite element method in time domain. Problem formulationIn many vibrating systems exist gaps and clearances as the result of manufacturing errors or mechanical failures. Generally, such systems can be modelled as n + 1-degree-of-freedom semi-definite system with clearances. The mechanical model consists of n + 1 mass elements, n linear viscous dampers, m ≤ n piecewise linear stiffness elements and n − m linear springs. The equation of motion, in nondimensional form, yieldswhere q is the relative displacement vector, h(q) is the displacement vector of stiffness elements with piecewise linear and linear terms while Z and Ω are the damping and stiffness matrices, respectively. Furthermore, f 0 and f a are the amplitude vectors of mean and alternating load and η denotes a nondimensional excitation frequency. Piecewise linear stiffness elements are assumed by the piecewise linear displacement function as follows The piecewise full decoupling methodExact solutions of piecewise linear equations of motion are very rare and almost all of the methods for their solving are only approximate. The most commonly used solution methods are the classical numerical time integrations (RungeKutta, etc.), the harmonic balance method, the incremental harmonic balance method and the finite element in time method. The piecewise full decoupling method is new developed numerical procedure based on the substitution of the equation (1) with a set of linear equations of motion, defined inside each of the stage stiffness region:In the equation (3), Π jΩ denotes the local stiffness matrix while p jΩ is the vector of the breakpoints. The mechanical system starts from an initial position described with one of the local equations of motion. When the system changes a stage stiffness region, the system is represented with the new local equation of motion. The determination of times of flight in each stage stiffness region can be done only numerically. Local equations of motion are solved by applying the state-space formulation. By employing the state vector y = [y y ] T , the equation (3) can be transformed into the first-order differential equation, with the state matrix A of the formThe matrix A (4) is, in general, a real nonsymmetric matrix and its eigenvalues can be calculated using one of numerical routines for the nonsymmetric eigenvalue problem. Obtained eigenvalues enable a transformation of the state variable y to the normal coordinate zwhere V is the matrix of eigenvectors. The coordinate transformation (5) where Λ = diag(λ k ), k = 1(1)2n is the matrix of eigenvalues and g is the excitation vector. The uncoupled equation of motion (6) has well-known analytical solutions. Described solution...

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