In daily analysis, we often confuse natural frequency and resonance frequency and think that they are the same thing.
In fact, this is not strict.
Natural frequency is the performance of natural structural characteristics, while resonant frequency is the performance of structural response under external forces.
Single degree of freedom spring system free vibration
A single degree of freedom system is a system where position can be fully determined by only one generalized coordinate at a given time. In simpler terms, the force acting on a body occurs in only one direction. The movement of the doll in the figure below can be represented as a mass-spring system.
A simplified model of the spring mass system can be shown in the following figure.
The static equilibrium position of the pickup block is considered the origin of the coordinate and is considered positive when it moves vertically downward along the direction of spring deformation. The distance between the block and the equilibrium position can be represented as x, and the block's differential equation of motion can be expressed as:
Where m is the mass of the block, k is the stiffness of the spring, c is the viscosity coefficient, 2n=c/m is the damping attenuation coefficient and when the damping coefficient is zero, it corresponds to the undamped vibrating system.
Pn natural frequency:
The natural frequency depends only on mass and stiffness and is not affected by factors such as damping. Structural boundary connections, material properties, shape, and other factors can impact natural frequency, but these influences are reflected in stiffness and mass and are not the ultimate determining factors.
Forced vibration of the spring system under harmonic excitation
Free vibration is the vibration of the system without external excitation, and the motion track is related to the initial state and natural characteristics.
Forced vibration refers to the vibration generated by the system under external excitation.
External excitation is generally a periodic or aperiodic function of time, among which simple harmonic excitation is the simplest.
Let the simple harmonic exciting force be:
Where, H is the amplitude of the exciting force, ω is the angular frequency of the exciting force.
When the block deviates from the equilibrium position by a distance x, the differential equation of motion of the block is
Where, h=H/m, the above equation is the forced vibration differential equation of a single degree of freedom with viscous damping, which is a second-order inhomogeneous linear ordinary differential equation of constant coefficient.
The above equation is completely consistent with the capacitive load voltage response expression we learned in circuit theory, which is a second-order constant coefficient inhomogeneous linear ordinary differential equation.
The damping in the circuit depends on the resistance, as the resistance only consumes and does not store energy.
Under simple harmonic excitation, the total solution of the differential equation of motion of a damped system is defined as:
Where, x1
