For a dynamical system governed by the equation,

$\ddot{x}(t)+2\zeta {\omega }_n\dot{x}(t)+{\omega }_n^{2}x(t)=0$

the damping ratio ζ is equal to $\frac{1}{2\pi }{\log }_{e}2$. The displacement x of this system is measured during a hammer test. A displacement peak in the positive displacement direction is measured to be 4 mm. Neglecting higher powers (> 1) of the damping ratio, the displacement at the next peak in the positive direction will be _______ mm (in integer).

The correct answer is 2.

Step 1: Understand the system dynamics
The given equation of motion for a free vibration, underdamped single degree of freedom system is:
$$\ddot{x}(t)+2\zeta {\omega }_n\dot{x}(t)+{\omega }_n^{2}x(t)=0$$ Where:
- x(t) is the displacement,
- ζ is the damping ratio, and
- ω_n is the natural frequency.
The damping ratio ζ is given as:
$$\zeta =\frac{1}{2\pi }{\log }_e(2)$$

Step 2: Calculate the logarithmic decrement (δ)
The logarithmic decrement δ represents the rate at which the amplitude of free damping vibrations decreases. It is defined in terms of the damping ratio ζ as:
$$\delta =\frac{2\pi \zeta }{\sqrt{1-{\zeta }^2}}$$ The problem states to neglect higher powers (> 1) of the damping ratio ζ. Therefore, we can approximate the denominator:
$$\sqrt{1-{\zeta }^2}\approx 1$$ Using this approximation, the logarithmic decrement simplifies to:
$$\delta \approx 2\pi \zeta$$ Substituting the given value of ζ into our simplified formula for δ:
$$\delta \approx 2\pi (\frac{1}{2\pi }{\log }_e(2))={\log }_e(2)$$ Thus, we find:
$$\delta \approx 0.693$$

Step 3: Relate successive peak displacements to the logarithmic decrement
The ratio of any two consecutive peak displacements in the same direction (e.g., the positive direction) is related to the logarithmic decrement by:
$$\frac{x_0}{x_1}=e^{\delta }$$ Where:
- x₀ is the initial peak displacement (4 mm),
- x₁ is the displacement at the next consecutive peak in the positive direction.
Using our calculated value for δ:
$$\frac{x_0}{x_1}=e^{{\log }_e(2)}=2$$

Step 4: Solve for the next peak displacement (x₁)
Substituting x₀ = 4 mm into the relation:
$$\frac{4}{x_1}=2$$ Solving for x₁ yields:
$$x_1=\frac{4}{2}=2\text{mm}$$ Therefore, the displacement at the next peak in the positive direction is 2 mm.