With the use of sketches, free body diagram and equations where necessary, explain the following terms used in the study of mechanical vibration: Single Degree of freedom. Double degree of freedom. Resonance. Damped free vibration. Forced vibration with harmonic excitation. Solve these questions showing the free body diagrams, please.

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Here are the explanations for the terms used in mechanical vibration:

  1. Single Degree of Freedom (SDOF)

    • Explanation: A single degree of freedom system is one whose motion can be completely described by a single independent coordinate. This means only one variable is needed to specify the position of all parts of the system at any given time.
    • Sketch: Imagine a block of mass mm attached to one end of a spring with stiffness kk, with the other end of the spring fixed to a rigid wall. The block rests on a frictionless horizontal surface. The only motion allowed is horizontal translation.
    • Free Body Diagram (FBD):
      • A block of mass mm.
      • A spring force Fs=−kxF_s = -kx acting on the block, opposing displacement xx.
      • Gravitational force mgmg acting downwards.
      • Normal force NN acting upwards, balancing gravity.
    • Equation of Motion (undamped): mx¨+kx=0m\ddot{x} + kx = 0 where mm is mass, kk is spring stiffness, and xx is displacement.

  2. Double Degree of Freedom (DDOF)

    • Explanation: A double degree of freedom system requires two independent coordinates to completely describe its motion. This typically involves two masses or two distinct modes of motion that are coupled.
    • Sketch: Imagine two blocks, m1m_1 and m2m_2, connected in series by springs k1k_1 and k2k_2. Block m1m_1 is attached to a fixed wall by spring k1k_1, and block m2m_2 is attached to m1m_1 by spring k2k_2. Both blocks rest on a frictionless horizontal surface.
    • Free Body Diagram (FBD):
      • For mass m1m_1:
        • Spring force Fs1=−k1x1F_{s1} = -k_1x_1 from spring k1k_1.
        • Spring force Fs2=k2(x2−x1)F_{s2} = k_2(x_2 - x_1) from spring k2k_2.
        • Gravitational force m1gm_1g and normal force N1N_1.
      • For mass m2m_2:
        • Spring force Fs2=−k2(x2−x1)F_{s2} = -k_2(x_2 - x_1) from spring k2k_2.
        • Gravitational force m2gm_2g and normal force N2N_2.
    • Equations of Motion (undamped): m1x¨1+k1x1−k2(x2−x1)=0m2x¨2+k2(x2−x1)=0\begin{align*} m_1\ddot{x}_1 + k_1x_1 - k_2(x_2 - x_1) &= 0 \\ m_2\ddot{x}_2 + k_2(x_2 - x_1) &= 0 \end{align*} where m1,m2m_1, m_2 are masses, k1,k2k_1, k_2 are spring stiffnesses, and x1,x2x_1, x_2 are displacements.

  3. Resonance

    • Explanation: Resonance is a phenomenon that occurs when the frequency of an external exciting force matches or is very close to the natural frequency of a system. This leads to a significant increase in the amplitude of vibration, potentially causing large oscillations and structural failure.
    • Sketch: A graph showing the amplitude of vibration on the y-axis versus the excitation frequency on the x-axis. The graph would show a sharp peak in amplitude at a specific frequency, which is the natural frequency (ωn\omega_n) of the system.
    • Free Body Diagram (FBD): No specific FBD is needed for the concept of resonance itself, but it occurs in systems undergoing forced vibration (see point 5).
    • Equation (Amplitude for undamped forced vibration): X=F0/k1−(ωωn)2X = \frac{F_0/k}{1 - \left(\frac{\omega}{\omega_n}\right)^2} where XX is the amplitude, F0F_0 is the amplitude of the exciting force, kk is stiffness, ω\omega is the excitation frequency, and ωn\omega_n is the natural frequency. Resonance occurs when ω≈ωn\omega \approx \omega_n, causing the denominator to approach zero and XX to become very large.

  4. Damped Free Vibration

    • Explanation: Damped free vibration describes the oscillatory motion of a system that gradually decreases in amplitude over time due to energy dissipation mechanisms (damping), without any external driving force. The system eventually returns to its equilibrium position.
    • Sketch: A graph of displacement versus time, showing a decaying sinusoidal wave. The peaks of the oscillations would progressively get smaller.
    • Free Body Diagram (FBD):
      • A block of mass mm.
      • A spring force Fs=−kxF_s = -kx opposing displacement xx.
      • A damping force Fd=−cx˙F_d = -c\dot{x} opposing velocity x˙\dot{x}, where cc is the damping coefficient.
      • Gravitational force mgmg and normal force NN.
    • Equation of Motion: mx¨+cx˙+kx=0m\ddot{x} + c\dot{x} + kx = 0 where mm is mass, cc is damping coefficient, kk is spring stiffness, xx is displacement, and x˙\dot{x} is velocity.

  5. Forced Vibration with Harmonic Excitation

    • Explanation: Forced vibration with harmonic excitation occurs when a system is subjected to a continuous, periodic external force that varies sinusoidally with time. The system will oscillate at the frequency of the applied force, and its amplitude will depend on the force's magnitude, the system's properties, and the proximity of the excitation frequency to the natural frequency.
    • Sketch: A mass-spring-damper system with an external force F(t)=F0sin⁡(ωt)F(t) = F_0 \sin(\omega t) applied to the mass.
    • Free Body Diagram (FBD):
      • A block of mass mm.
      • A spring force Fs=−kxF_s = -kx opposing displacement xx.
      • A damping force Fd=−cx˙F_d = -c\dot{x} opposing velocity x˙\dot{x}.
      • An external harmonic force F(t)=F0sin⁡(ωt)F(t) = F_0 \sin(\omega t) acting on the mass.
      • Gravitational force mgmg and normal force NN.
    • Equation of Motion: mx¨+cx˙+kx=F0sin⁡(ωt)m\ddot{x} + c\dot{x} + kx = F_0 \sin(\omega t) where mm is mass, cc is damping coefficient, kk is spring stiffness, xx is displacement, x˙\dot{x} is velocity, F0F_0 is the amplitude of the exciting force, and ω\omega is the excitation frequency.
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