This paper developes the multi-parameter equivalent linearization (mpEL) investigating the replacement of the original nonlinear part \(A(x)\) with the linear function \(k_{d} B(x)\) where \(A(x)\) can be expressed as an algebraic sum of nonlinear functions \(A_{i} (x),\,i = 1,2,...,N\) . When many pairs \((A_{i} ,B),\,i = 1,2,...,N\) are considered together there can be a case where there exist two functions \(A_{i}\) and \(A_{j} ,\,i \ne j\) such that \(A_{i} ,B\) are on the same side while \(A_{j} ,B\) are on different sides. Therefore, DEL is first extended to the pair \((A,B)\) whose correlation coefficient r is a real number and the proposed weighting coefficient p depends not only on the absolute value of \(\sqrt {r^{2} }\) but also on the sign of r. Further, considering \(A(x)\) as the sum of \(A_{i} (x)\) , each term \(A_{i} (x)\) is replaced by linear function \(k_{is} B(x)\) using spEL. An alternative to spEL, the proposed mpEL consider the sum of equivalent linearization coefficients \(k_{is}\) as an equivalent linearization coefficient \(k_{m}\) when replacing \(A(x)\) with \(k_{m} B(x)\) . It turns out that for an algebraic sum of nonlinear functions, DEL has two approaches, spEL or mpEL. Using spEL, the sum of nonlinear functions is considered as the entire nonlinear function while mpEL requires applying spEL to each nonlinear term. These two approaches lead to two different equivalent linearization coefficients for the original nonlinear function. The accuracy of spEL and mpEL is tested for nonlinear free vibration frequency analysis. A generalized nonlinearity measure is built based on the difference in direction and magnitude of the nonlinear function compared to the linear function. For tested nonlinear systems, it is obtained that among the solutions obtained from several approximate methods, spEL provides the lowest maximal errors within weak and weak moderate nonlinearities, and mpEL mainly provides the lowest maximal errors within moderate strong and strong nonlinearities.
