Free vibration means that after the external force makes the ball of the spring vibrator and the ball of the simple pendulum deviate from the equilibrium position, they will vibrate under the force of the internal elasticity or gravity of the system, and no external force is required to push them.
doVibrationThe object leaves under the action of external forcebalanceAfter the position, you can press it by yourselfnatural frequencyVibration without external force is called free vibration. The ideal free vibration is called nonedampingFree vibration. The period of free vibration is called natural period, and the frequency of free vibration is called natural frequency. They are determined by the conditions of the vibration system itself and are independent of the amplitude
free vibration
Difference from forced vibration:
free vibration
free vibration
Forced vibration is also called forced vibration. Under the continuous action of external periodic force, the vibration of the vibration system is called forced vibration. This "external periodic force" is called driving force (or forced force). When the forced vibration of an object reaches a stable state, its vibration frequency is the same as the frequency of the driving force, but not the natural frequency of the object
The free vibration cannatural frequencyVibration without external force
Formula derivation
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The vibration of a mechanical system near its equilibrium position without any other excitation after initial disturbance.Since the medium damping and internal friction are regarded as belonging to the vibration system, the free vibration also includes the vibration with damping force.The simplest free vibration is simple harmonic vibration.The second is the single degree of freedom linear vibration with damping force (see linear vibration).For the free vibration with multiple degrees of freedom, since the vibration process occurs near the stable equilibrium position of the system, if the equilibrium position is taken as the origin of the generalized coordinates, then the kinetic energy of the systemTAnd potential energyVIt can be approximated as:
WhereqIs a generalized coordinate;mIs the quality;kIs the stiffness.There is also a dissipative force acting on the system similar to the damping force.The motion equation of this mechanical system is:
，（j=1，2，…，n） （1）
WhereFIs Rayleigh dissipation function,;L=T－VIs a Lagrange function.
For conservative systems,F=0, Equation (1) becomes the Lagrange equation of holonomic conservative system:
。（j=1，2，…，n）
Applying the above formula to the free linear vibration of the multi degree of freedom conservative system, the vibration equation can be obtained:
， （2）
Where
They are mass matrix, stiffness matrix and generalized displacement vector.
The vibration characteristic of this conservative system is formed by the simple harmonic vibration of each generalized displacement.The main vibration can be set as:q=usin（ωt+
）， （3）
Where, is called the main mode vector;qanduCan be regarded as column matrix.Substitute equation (3) into equation (2) and reduce sin (ω t+
）, then:
The above equation is called the eigenvector equation and the eigenvector matrix.The conditions for the non-zero solution of equation (4) are:
Equation (5) is called characteristic equation;It can be solved from equation (5)nPieces（i=1，2，…，n）。After substituting into equation (4), the correspondingn.Natural frequency (dominant frequency), or characteristic value;It is called natural mode (main mode) or eigenvector.WhenKandMbynOrder real symmetric matrix, andMPositive timing, presencenReal eigenvalues and correspondingnThe special solution of equation (2) can be written as:
Neutralization is an undetermined constant, which is determined by the initial conditions.For example, knownt=0, there are:
