Engineering Acoustics/Forced Oscillations(Simple Spring-Mass System)

Recap of Section 1.3
In the previous section, we discussed how adding a damping component (e. g. a dashpot) to an unforced, simple spring-mass system would affect the response of the system. In particular, we learned that adding the dashpot to the system changed the natural frequency of the system from  to a new damped natural frequency, and how this change made the response of the system change from a constant sinusoidal response to an exponentially-decaying sinusoid in which the system either had an under-damped, over-damped, or critically-damped response.

In this section, we will digress a bit by going back to the simple (undamped) oscillator system of the previous section, but this time, a constant force will be applied to this system, and we will investigate this system’s performance at low and high frequencies as well as at resonance. In particular, this section will start by introducing the characteristics of the spring and mass elements of a spring-mass system, introduce electrical analogs for both the spring and mass elements, learn how these elements combine to form the mechanical impedance system, and reveal how the impedance can describe the mechanical system’s overall response characteristics. Next, power dissipation of the forced, simple spring-mass system will be discussed in order to corroborate our use of electrical circuit analogs for the forced, simple spring-mass system. Finally, the characteristic responses of this system will be discussed, and a parameter called the amplification ratio (AR) will be introduced that will help in plotting the resonance of the forced, simple spring-mass system.

Forced Spring Element
Taking note of Figs. 1, we see that the equation of motion for a spring that has some constant, external force being exerted on it is...

$$\hat{F} = s_M \Delta \hat{x}\qquad (1.4.1) \,$$

where $$ s_M \,$$ is the mechanical stiffness of the spring.

Note that in Fig. 1(c), force $$\hat{F}$$ flows constantly (i.e. without decreasing) throughout a spring, but the velocity $$\hat{u}$$ of the spring decrease from $$\hat{u_1}$$ to $$\hat{u_2}$$ as the force flows through the spring. This concept is important to know because it will be used in subsequent sections.

In practice, the stiffness of the spring $$ s_M \,$$, also called the spring constant, is usually expressed as $$ C_M = \frac {1}{s_M}\, $$, or the mechanical compliance of the spring. Therefore, the spring is very stiff if $$ s_M \,$$ is large $$ \Rightarrow \; C_M $$ is small. Similarly, the spring is very loose or “bouncy” if $$ s_M \,$$ is small $$ \Rightarrow \; C_M $$  is large. Noting that force and velocity are analogous to voltage and current, respectively, in electrical systems, it turns out that the characteristics of a spring are analogous to the characteristics of a capacitor in relation to, and, so we can model the “reactiveness” of a spring similar to the reactance of a capacitor if we let $$ C = C_M \,$$ as shown in Fig. 2 below.

$$ Reactance\ of\ Capacitor:\ X_C = \frac{1}{j\omega C} \qquad (1.4.2a) $$

$$ Reactance\ of\ Spring:\ X_{MS} = \frac{1}{j\omega C_M} \qquad(1.4.2b) $$

Forced Mass Element
Taking note of Fig. 3, the equation for a mass that has constant, external force being exerted on it is...

$$ \hat{F} = M_M \hat{a} = M_M \hat{\dot{u}} = M_M \hat{\ddot{x}} \qquad (1.4.3)   $$

If the mass $$ M_M \, $$ can vary its value and is oscillating in a mechanical system at max amplitude $$ A_M \, $$ such that the input the system receives is constant at frequency $$ \omega \, $$, as $$ M_M \, $$ increases, the harder it will be for the system to move the mass at $$ \omega \, $$  at $$ A_M \, $$  until, eventually, the mass doesn’t oscillate at all. Another equivalently way to look at it is to let $$ \omega \, $$ vary and hold $$ M_M \, $$ constant. Similarly, as $$ \omega \, $$ increases, the harder it will be to get $$ M_M \, $$ to oscillate at $$ \omega \, $$  and keep the same amplitude $$ A_M \, $$  until, eventually, the mass doesn’t oscillate at all. Therefore, as $$ \omega \, $$ increases, the “reactiveness” of mass $$ M_M \, $$ decreases (i.e. $$ M_M \, $$  starts to move less and less). Recalling the analogous relationship of force/voltage and velocity/current, it turns out that the characteristics of a mass are analogous to an inductor. Therefore, we can model the “reactiveness” of a mass similar to the reactance of an inductor if we let $$ L = M_M \, $$ as shown in Fig. 4.

$$ Reactance\ of\ Inductor:\ X_L = j\omega L \qquad (1.4.4a) $$

$$ Reactance\ of\ Mass:\ X_{MM} = j\omega L_M \qquad(1.4.4b) $$

Mechanical Impedance of Spring-Mass System
As mentioned twice before, force is analogous to voltage and velocity is analogous to current. Because of these relationships, this implies that the mechanical impedance for the forced, simple spring-mass system can be expressed as follows:

$$ \hat{Z_M} = \frac{\hat{F}}{\hat{u}} \qquad (1.4.5) \,$$

In general, an undamped, spring-mass system can either be “spring-like” or “mass-like”. “Spring-like” systems can be characterized as being “bouncy” and they tend to grossly overshoot their target operating level(s) when an input is introduced to the system. These type of systems relatively take a long time to reach steady-state status. Conversely, “mass-like” can be characterized as being “lethargic” and they tend to not reach their desired operating level(s) for a given input to the system...even at steady-state! In terms of complex force and velocity, we say that “ force LEADS velocity” in mass-like systems and “velocity LEADS force” in spring-like systems (or equivalently “ force LAGS velocity” in mass-like systems and “velocity LAGS force” in spring-like systems). Figs. 5 shows this relationship graphically.

Power Transfer of a Simple Spring-Mass System
From electrical circuit theory, the average complex power $$ P_E \,$$ dissipated in a system is expressed as ...

$$ P_E = \frac{1}{2} \mathbf{Re}\left\{ \hat{V}\hat{I^*} \right\} \qquad (1.4.6)\,$$

where $$ \hat{V}\,$$ and $$ \hat{I^*}\;$$ represent the (time-invariant) complex voltage and complex conjugate current, respectively. Analogously, we can express the net power dissipation of the mechanical system $$ \hat{P}_E\,$$ in general along with the power dissipation of a spring-like system $$ \hat{P}_{MS}\,$$ or mass-like system $$\hat{P}_{MM}\,$$ as...

$$ \hat{P}_E = \frac{1}{2} \mathbf{Re}\left\{ \hat{F}\hat{u^*} \right\} \qquad \qquad \qquad (1.4.7a)\,$$

$$ \hat{P}_{MS} = \frac{1}{2} \mathbf{Re}\left\{\hat{F} \left( \frac{j\hat{F}\omega}{s_M} \right)^* \right\} \qquad (1.4.7b) $$

$$ \hat{P}_{MM} = \frac{1}{2} \mathbf{Re}\left\{\hat{F} \left( \frac{\hat{F}}{j \omega M_M} \right)^* \right\} \qquad (1.4.7c) $$

In equations 1.4.7, we see that the product of complex force and velocity are purely imaginary. Since reactive elements, or commonly called, lossless elements, cannot dissipate energy, this implies that the net power dissipation of the system is zero. This means that in our simple spring-mass system, power can only be (fully) transferred back and forth between the spring and the mass. But this is precisely what a simple spring-mass system does. Therefore, by evaluating the power dissipation, this corroborates the notion of using electrical circuit elements to model mechanical elements in our spring-mass system.

Responses For Forced, Simple Spring-Mass System
Fig. 6 below illustrates a simple spring-mass system with a force exerted on the mass.

This system has response characteristics similar to that of the undamped oscillator system, with the only difference being that at steady-state, the system oscillates at the constant force magnitude and frequency versus exponentially decaying to zero in the unforced case. Recalling equations 1.4.2b and 1.4.4b, letting  be the natural (resonant) frequency of the spring-mass system, and letting $$ \omega_n \,$$ be frequency of the input received by the system, the characteristic responses of the forced spring-mass systems are presented graphically in Figs. 7 below.

$$ \mathbf{Figs.7}\,$$

Amplification Ratio
The amplification ratio is a useful parameter that allows us to plot the frequency of the spring-mass system with the purports of revealing the resonant freq of the system solely based on the force experienced by each, the spring and mass elements of the system. In particular, AR is the magnitude of the ratio of the complex force experienced by the spring and the complex force experienced by the mass, i.e.

$$ \mathbf{AR} = \left| \frac{s_M \hat{x}}{M_M \hat{a}} \right| = \left| \frac{s_M \hat{x}}{M_M \hat{\dot{u}}} \right| = \left| \frac{s_M \hat{x}}{M_M \hat{\ddot{x}}} \right| \qquad (1.4.8) \,$$

If we let $$ \zeta = \frac{\omega}{\omega_n} $$, be the frequency ratio, it turns out that AR can also be expressed as...

$$ \mathbf{AR} = \frac{1}{1 - \zeta^2} \qquad (1.4.9)\,$$.

AR will be at its maximum when $$ \left| X_{MS} \right| = \left| X_{MM} \right| \,$$. This happens precisely when $$ \zeta^2 = 1 \,$$. An example of an AR plot is shown below in Fig 8.

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