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Free vibration of particles | Undamped vibrations

If the motion of a system is maintained by only restoring forces, the vibration of the particles in the system undergoes free vibration. If a periodic force is applied to the system, the vibration of the system is called forced vibration.

In the analysis of vibration of particles or system, if friction is considered then vibration is damped otherwise it is undamped vibrations. In this article, we will see more in detail about undamped vibration after understanding about some basic terms related to vibrations.

Undamped vibration
Undamped vibration

Fundamental Terms in Vibration

Time period(T): It is the time interval required for the system to complete a full cycle of motions. Its unit is seconds(s).

Frequency(f): It is the number of vibrations recorded per second. Its unit is s^(-1) or Hertz(Hz).

Amplitude(A): It is the maximum displacement of the system from its initial point of equilibrium. Its unit is m.

Undamped vibrations

Consider a spring-mass system in which a body of weight W is hanging from a spring with stiffness k. Let Δ be the initial downward displacement of the spring-mass system. At equilibrium position of the system, an initial displacement x0 and velocity v0 is assumed.

In the equilibrium position, weight W is balanced by the tension T in the spring.

Then we get W = T = kΔ.

After displacement x = x0 is given to the system, a net force F exists in the system.

Thus, F = W - k(Δ + x) = kΔ - kΔ - kx = -kx.

The negative sign indicates that net force acts against the direction in which the displacement of the spring-mass system occurs.

From Newton's second law of motion,

F = ma = md²x/dt² and F = -kx,

we get md²x/dt² = -kx

md²x/dt² + kx =0.

The equation can also written as d²x/dt² + (k/m)x = 0

This is a homogeneous linear differential equation of second order. The solution of this equation is given as

Here, x is a periodic function of t, k/m = w² where w is the natural circular frequency of the system.

The initial conditions for the spring-mass system are given as displacement x = x0, velocity v = v0 and time t = 0. From the above solution, x0 = A and v0 = Bw.

After the graphical analysis of the solutions, we obtain various expressions for acceleration, velocity, etc., of the spring-mass system.

  • Maximum displacement x = xm, where xm is the amplitude

  • Maximum velocity v = xm x w, where w - natural circular frequency of the system.

  • Maximum acceleration a = xm x w²

  • The time period T = 2𝜋/w

  • Frequency f = w/2𝜋

Now try this question out!

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