Mechanical Vibrations

The vibrations of mechanical systems can sometimes be analyzed in terms of a finite collection of masses, springs, and damping objects. This approach is referred to as lumped modeling and the resulting approximate system response is called a “lumped characterization”.

Figure 11: An ideal mass-spring-damper system.
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Figure 11 represents a system consisting of an ideal mass, spring, and damper connected “in series” (each element has the same velocity or displacement).
An ideal mass (represented in Fig. 12) is assumed to move on a friction-less surface or guide-rod and to be completely rigid.

Figure 12: An ideal mass.
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From Newton's Second Law: $f(t) = m a(t) = m \frac{dv}{dt} = m \frac{d^{2}x}{dt^{2}}$
An ideal spring (represented in Fig. 13) has no mass or internal damping and is characterized by a spring constant (or stiffness constant) .

Figure 13: An ideal spring.
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By Hooke's Law: $f(t) = -k x(t) = k \int_{0}^{t} v(\tau) d\tau$ for (valid for small, non-distorting displacements)
The spring's equilibrium position is given by . A positive value of produces a negative restoring force.
The force necessary to overcome a mechanical dashpot or resistance (diagrammed in Fig. 14) is typically approximated as being proportional to velocity: $f(t) = \mu v(t) = \mu \frac{dx}{dt}$

Figure 14: An ideal dashpot or damper.
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In the series combination of an ideal mass, spring, and damper (Fig. 11), all of the forces must sum to zero (assuming no external force is applied). Thus, this system can be characterized by the equation: $m \frac{d^{2}x}{dt^{2}} + \mu \frac{dx}{dt} + k x = 0$
This second-order homogeneous differential equation has solutions of the form $x = e^{-\alpha t} A \cos(\omega_{d}t + \phi)$ , where $\alpha = \mu/(2m)$ is a decay constant and $\omega_{d} = \sqrt{\omega_{0}^{2} - \alpha^{2}}$ is the characteristic (or natural) angular frequency of the system.

Figure 15: A decaying sinusoid.
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and $\phi$ are determined by the initial displacement and velocity.
The natural frequency $\omega_{d}$ is lower than that of the mass-spring system (without damping) ( $\omega_{0} = \sqrt{k / m}$ ).