A Novel Method for Deriving High-Order Frequency Equations with Applications to Cantilever Beam Vibrations

Abstract Purpose The determination of natural frequencies in structural systems is governed by the order of the underlying partial differential equation that describes the system behavior. In the specific case of beam transverse vibration, the equation of motion is represented by a fourth order differential equation. Consequently, four distinct frequency equations are obtained by imposing appropriate boundary conditions. To calculate the natural frequencies in such scenarios, it becomes necessary to solve a determinant equation of size $$4\times 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>×</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . However, in more complex situations involving the study of coupled longitudinal and transverse vibration, where the frequency equations surpass the dimensions of $$4\times 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>×</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , it is customary to express these equations in matrix form and employ computational programs to solve them. The current manuscript introduces novel systematic methodologies for the expansion of determinant matrices of dimensions $$4\times 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>×</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , $$6\times6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>6</mml:mn> <mml:mo>×</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> , and $$8\times8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8</mml:mn> <mml:mo>×</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> . Methods This paper introduces a set of procedures that facilitate the derivation of exact closed-form solutions for frequency equations of dimensions $$6\times6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>6</mml:mn> <mml:mo>×</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> and $$8\times8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8</mml:mn> <mml:mo>×</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> . These procedures are based on established numerical procedures applied to the constituent elements of the matrices. By employing these procedures, accurate and efficient determination of natural frequencies in coupled longitudinal and transverse vibration systems can be achieved. Results Numerical examples and practical applications were used to compute the determinants of matrices ranging from $$4\times4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>×</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> to $$8\times8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8</mml:mn> <mml:mo>×</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> , validating the proposed analysis. These examples included lateral vibration problems. The results were consistent with both the traditional numerical matrix expansion method and the literature, matching up to the third decimal place. Conclusion The proposed analysis is effective for the symbolic expansion of determinants for matrices of sizes $$4\times4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>×</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and $$8\times8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8</mml:mn> <mml:mo>×</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> . Its utility extends beyond vibration analysis, with potential applications in various fields, including mechanical, structural, and aerospace engineering.