Sound is caused by vibrating objects or media. In this section, we study somewhat overly simplified, ideal structures to gain an understanding of the fundamental concepts of vibrating systems. The analysis of systems in terms of finite collections of masses, springs, and friction sources is referred to as lumped modeling
and the resulting approximate system responses are called ``lumped characterizations''.
- The Ideal Dashpot:
An ideal dashpot or damper.
- The force necessary to overcome a mechanical dashpot or resistance is typically approximated as being proportional to velocity:
- The dashpot impedance is simply given by , the frequency-independent damping constant.
- The electrical correlate of a mechanical dashpot is a resistor, characterized by v(t) = R i(t). In an analog equivalent circuit, a dashpot can be represented by a resistor .
- The Ideal Mass:
An ideal mass.
- An ideal mass is assumed to move on a friction-less surface or guide-rod.
- The ideal mass is completely rigid.
- By Newton's Second Law:
- Making use of the Laplace transform and the differentiation theorem:
F(s) = m[sV(s) - v(0)] = m[s2X(s) - sx0 - v0]
- Assuming zero initial conditions, the mass impedance is given by:
Zm(s) = F(s)/V(s) = m s.
- If we characterize our mass in terms of an applied force (input) and a resulting velocity (output), the transfer function is given by:
which is the admittance. The impulse response of the mass is then found as the inverse Laplace transform of the transfer function:
where u(t) is the unit step function. Applying an input corresponds to transferring a unit of momentum to the mass at time 0, with a resulting velocity v(t) = 1/m.
- The frequency response of the mass system can be determined by evaluating the transfer function for (along the axis in the s-plane):
. This is the frequency response of an integrator. The magnitude response rolls off by -6 dB per octave and the phase shift is radians for all frequencies.
- The electrical correlate of mechanical mass is an inductor, characterized by
v(t) = L di/dt. In an analog equivalent circuit, a mass can be represented using an inductor with value L = m.
- The Ideal Spring:
An ideal spring.
- The ideal spring has no mass or internal damping.
- By Hooke's Law:
for x(0) = 0 (valid for small, non-distorting displacements)
- The spring's equilibrium position is given by x=0.
- A positive value of x produces a negative restoring force.
- The spring constant k can also be referred to as the spring stiffness.
- Making use of the Laplace transform and the integration theorem:
F(s) = k V(s) / s
- For x(0) = 0, the spring impedance is given by:
Zk(s) = k / s.
- If we characterize the spring in terms of an applied force (input) and a resulting velocity (output), the transfer function is given by:
- The frequency response of the spring is given by:
. This is the frequency response of a differentiator (an ideal spring differentiates force , divided by k, to produce an output velocity). The magnitude response grows by +6 dB per octave and the phase shift is radians for all frequencies.
- The electrical correlate of a mechanical spring is a capacitor, characterized by
i(t) = C dv/dt. In an analog equivalent circuit, a spring can be represented by a capacitor of value C = 1/k.
In this section, we analyze the behaviour of particular combinations of masses, springs, and dashpots when given an initial displacement and/or velocity. We will consider forced oscillations in a subsequent section.
An ideal mass-spring system.
- The series combination of an ideal mass and spring shown in Fig. 4 is characterized by the system equation:
- This second-order homogeneous differential equation has solutions of the form
representing periodic motion.
is the characteristic (or natural) angular frequency of the system.
- A and are determined by the initial displacement and velocity.
- There are no losses in the system, so it will oscillate forever.
- This system can be analyzed using the Laplace transform
with non-zero initial conditions as follows:
. From a table of Laplace transform pairs, the solution is found as:
A physical system without losses is rare (if not impossible). The introduction of a mechanical dashpot in the mass-spring system provides damping and causes the resulting vibrations to decay over time.
An ideal mass-spring-damper system.
- Longitudinal Motion (along x-axis):
A one-mass, two-spring system: Longitudinal motion.
- The net restoring force on the mass:
fx(t) = -2 k x(t)
- System natural frequency:
- Longitudinal Motion (along x-axis):
A two-mass, three-spring system: Longitudinal motion.
- Natural frequencies:
An infinite mass-spring system.
- Each additional mass-spring combination adds another natural mode of vibration per axis of motion.
- As the number of masses and springs increases, the system begins to resemble a uniform string (assuming all masses and all springs are roughly equal in value). Eventually, it becomes more convenient to consider the mass and compression characteristics of the system to be uniformly distributed along its length (as we did in deriving the wave equation).
- We are typically interested in the behavior of systems when driven by an external force f(t).
- For a sinusoidal driving force, the resulting response has a transient component and a steady-state term. The transient component, which involves motion at the natural frequency of the system, decays away at a rate proportional to the damping in the system.
- The mechanical impedance,
Z(s) = F(s) / V(s), (evaluated at ) of the system characterizes its steady-state response, after its initial transient behavior has decayed away.
- Components connected in series share a common velocity and their impedances add (i.e.,
Z(s) = Z1(s) + Z2(s)).
- Components connected in parallel share a common driving force but can develop different velocities. Admittances add in parallel, while a parallel combination of impedances is given by
- When a mass-spring-damper system is driven by an external force, the system equation is
- The driven mass-spring-damper system can be described using the Laplace Transform as (assuming v(0) = 0 and x(0) = 0):
and its impedance determined as
This impedance expression, Z(s), can be evaluating for , as shown in Fig. 10 for three different damping constants.
Impedance of a mass-spring-damper system.
- Mechanical driving-point impedance minima indicate the frequencies at which a sinusoidal driving force of constant amplitude will produce the greatest mass velocity (and displacement).
- The Laplace Transform for the mass-spring-damper system can also be expressed in terms of its displacement (assuming x(0) = 0 and v(0) = 0):
The steady-state force-to-displacement response is then given by:
and is plotted in Fig. 11 for three damping constants. Resonance occurs when a system is driven near its natural frequency.
Mass displacement vs. driving frequency for mass-spring-damper system.
- Below resonance, the system is said to be ``stiffness dominated''.
- Above resonance, the system is said to be ``mass dominated'' and the resulting displacement approaches zero with increasing frequency.
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