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Chapter 2: Problem 175

Find the equivalent viscous-damping constant for Coulomb damping for sinusoidal vibration.

Short Answer

Expert verified
The equivalent viscous-damping constant for Coulomb damping for sinusoidal vibration can be calculated using the following equation: \(c_v = \frac{渭 * N}{2 * 蟺 * 味 * 蠅_n * X}\) where \(c_v\) is the equivalent viscous-damping constant, \(F_c\) is the Coulomb damping force, \(渭\) is the friction coefficient, \(N\) is the normal contact force, \(味\) is the damping ratio, \(蠅_n\) is the natural frequency, and \(X\) is the displacement amplitude. Substitute the known values for \(渭, N, 味, 蠅_n\), and \(X\) into the equation to calculate the equivalent viscous-damping constant.

Step by step solution

01

Express the Coulomb damping force

To find the Coulomb damping force (F_c), we can use the following equation: Fc = 渭 * N Where: - 渭 is the coefficient of friction - N is the normal contact force This expression will allow us to find the value of the Coulomb damping force, which is necessary to compute the equivalent viscous-damping constant.
02

Obtain values for the damping ratio, natural frequency, and displacement amplitude

In this exercise, the damping ratio (味), natural frequency (蠅_n), and displacement amplitude (X) are not given. You will need to obtain these values from the problem statement, a graph, or other available information. In the absence of specific values, we will express the equivalent viscous-damping constant in terms of these variables.
03

Compute the equivalent viscous-damping constant

Using the equation for the equivalent viscous-damping constant, we can now substitute the expressions for the Coulomb damping force (F_c) and the given variables: c_v = (渭 * N) / (2 * pi * 味 * 蠅_n * X) This equation will give the value of the equivalent viscous-damping constant (c_v) for sinusoidal vibration. You can calculate c_v by substituting the known values for 渭, N, 味, 蠅_n, and X into the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Viscous Damping
Viscous damping is a common method used to reduce vibrations in mechanical systems. This type of damping occurs when a fluid or gas provides resistance to motion. The damping force is proportional to the velocity of the moving object.
In mathematical terms, if a body is moving with velocity \(v\), the viscous damping force \(F_v\) is given as \(F_v = -c_v v\), where \(c_v\) is the viscous damping constant.
  • It acts in the opposite direction to the motion, helping to reduce the energy and amplitude of vibrations.
  • Viscous damping is ideal for applications where the damping force needs to be adjustable by varying the fluid's viscosity.
By providing a steady resistance, viscous damping effectively dissipates energy and brings the oscillating system to rest over time.
Coulomb Damping Force
Coulomb damping force, sometimes referred to as dry friction damping, arises due to sliding friction between contacting surfaces. This type of damping is characterized by a constant magnitude force that is opposite to the direction of motion.
The Coulomb damping force \(F_c\) can be expressed using the formula \(F_c = \,\mu \,N\), where \(\mu\) represents the coefficient of friction and \(N\) is the normal force acting between the surfaces.
  • Coulomb damping results in non-linear behavior as it remains constant irrespective of velocity.
  • This damping is particularly effective in systems where surfaces in contact slide over each other.
In the context of sinusoidal vibration, Coulomb damping can be converted into an equivalent viscous-damping constant to simplify analysis.
Damping Ratio
The damping ratio (\(\zeta\)) is a dimensionless measure that describes how oscillations in a system decay over time. It quantifies the level of damping relative to critical damping, which is the minimum amount required to return to equilibrium without oscillating.
The damping ratio can be calculated using the formula: \[\zeta = \frac{c}{2 \sqrt{mk}} \] where \(c\) is the damping constant, \(m\) is the mass, and \(k\) is the stiffness of the system.
  • \(\zeta = 0\) indicates no damping, resulting in perpetual oscillation.
  • \(0 < \zeta < 1\) indicates underdamping, where the system oscillates with gradually decreasing amplitude.
  • \(\zeta = 1\) signifies critical damping, eliminating oscillations.
  • \(\zeta > 1\) represents overdamping, where the system returns to equilibrium without oscillating.
Understanding the damping ratio is crucial in controlling system response to oscillations and vibrations.
Natural Frequency
The natural frequency (\(\omega_n\)) is the rate at which a system oscillates when not subjected to external forces or damping. It is a fundamental property of oscillatory systems, determined by the mass and stiffness.
Mathematically, it can be expressed as:\[\omega_n = \sqrt{\frac{k}{m}} \]where \(k\) is the stiffness and \(m\) is the mass of the system.
  • Natural frequency is significant in designing mechanical and structural systems to avoid resonance, which occurs when a system's natural frequency matches an external frequency, leading to large amplitude oscillations.
  • Clearly identifying the natural frequency helps in applying appropriate damping and optimizing system response.
Understanding natural frequency aids in preventing structural failures and ensuring system stability under variable loads.

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Most popular questions from this chapter

Using the MATLAB Program2 \(m,\) plot the free-vibration response of a viscously damped system with \(m=4 \mathrm{~kg}, k=2500 \mathrm{~N} / \mathrm{m}, x_{0}=100 \mathrm{~mm}, \dot{x}_{0}=-10 \mathrm{~m} / \mathrm{s}, \Delta t=0.01 \mathrm{~s}, n=50\) for the following values of the damping constant: a. \(c=0\) b. \(c=100 \mathrm{~N}-\mathrm{s} / \mathrm{m}\) c. \(c=200 \mathrm{~N}-\mathrm{s} / \mathrm{m}\) d. \(c=400 \mathrm{~N}-\mathrm{s} / \mathrm{m}\)

Figure 2.65 shows a small mass \(m\) restrained by four linearly elastic springs, each of which has an unstretched length \(l\), and an angle of orientation of \(45^{\circ}\) with respect to the \(x\) -axis. Determine the equation of motion for small displacements of the mass in the \(x\) direction.

A single-degree-of-freedom system consists of a mass of \(20 \mathrm{~kg}\) and a spring of stiffness \(4000 \mathrm{~N} / \mathrm{m}\). The amplitudes of successive cycles are found to be \(50,45,40,35, \ldots \mathrm{mm} .\) Determine the nature and magnitude of the damping force and the frequency of the damped vibration.

Find the free-vibration response of a spring-mass system subject to Coulomb damping using MATLAB for the following data: $$ m=5 \mathrm{~kg}, \quad k=100 \mathrm{~N} / \mathrm{m}, \quad \mu=0.5, \quad x_{0}=0.4 \mathrm{~m}, \quad \dot{x}_{0}=0 $$

An automobile is found to have a natural frequency of \(20 \mathrm{rad} / \mathrm{s}\) without passengers and 17.32 \(\mathrm{rad} / \mathrm{s}\) with passengers of mass \(500 \mathrm{~kg}\). Find the mass and stiffness of the automobile by treating it as a single-degree-of-freedom system.
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