# 统计代写|主成分分析代写Principal Component Analysis代考|Mathematical Notes: Free-Vibration Response of a Two-Degree-of-Freedom System

This passage describes the free-vibration response of a two-degree-of-freedom mechanical system consisting of two spring-mass elements arranged in series, subject to gravity but without any additional external forcing. The system is initially disturbed and allowed to oscillate freely thereafter.

System Stiffness Matrix ([K]):

The stiffness matrix for the system is given by Equation 10.30, which reflects the connectivity and relative stiffness between the nodes. Mass Matrix ([M]):

The mass matrix, as shown in Equation 10.31, is diagonal, reflecting the fact that the masses are concentrated at the nodes and accounting for their inertia. Equations of Motion (10.32):

The equations of motion combine the mass and stiffness matrices with the acceleration vector and displacement vector to account for the gravitational forces and reactions. Free-Vibration Response Equations (10.34):

After enforcing the constraint U1=0 and neglecting gravitational forces in the dynamic equilibrium condition, the system reduces to a homogeneous system of two second-order ODEs. Assumed Harmonic Solutions:

Assuming harmonic motion for the displacements of the masses at nodes 2 and 3, the second derivatives are substituted back into the free-vibration equations leading to a homogeneous algebraic system (Equation 10.37). Natural Frequencies (10.39):

Solving the characteristic equation derived from the homogeneous algebraic system yields two natural circular frequencies, Ω₁ and Ω₂, which are the eigenvalues of the system. Amplitude Ratios:

For each mode of vibration (fundamental and higher), the ratio of the amplitudes of the masses’ oscillations is derived from the coefficients in the algebraic equations. Displacement Equations (10.45):

The final solution for the free-vibration response combines both natural modes of vibration with unknown amplitude constants and phase angles. For the specific numerical example provided, with k=40 lb/in and mg=20 lb:

Natural frequencies are calculated to be approximately 27.8 rad/sec (fundamental) and 68.1 rad/sec (higher mode). Initial conditions are given for displacements and velocities at t=0. Due to the complexity introduced by trigonometric functions, solving for the amplitudes and phase angles requires manipulation and substitution. It is noted that the phase angles turn out to be π/2, and the amplitudes are calculated accordingly. The final free-vibration responses for masses at nodes 2 and 3 are described as combinations of cosine functions at the natural frequencies with their respective amplitudes. This illustrates the concept of modal superposition, where each mode contributes independently to the overall motion of the system.

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