### Overview

This tutorial is part of the National Instruments Measurement Fundamentals series. Each tutorial in this series will teach you a specific topic of common measurement applications by explaining theoretical concepts and providing practical examples. There are several physical processes that can be used to develop a sensor to measure acceleration. In applications that involve flight, such as aircraft and satellites, accelerometers are based on properties of rotating masses. In the industrial world, however, the most common design is based on a combination of Newton's law of mass acceleration and Hooke's law of spring action.

### Table of Contents

## Spring-Mass System

Newton's law simply states that if a mass, *m*, is undergoing an acceleration, *a*, then there must be a force *F* acting on the mass and given by *F* = *ma.* Hooke's law states that if a spring of spring constant *k* is stretched (extended) from its equilibrium position for a distance D*x***,** then there must be a force acting on the spring given by *F* = *kDx.*

**FIGURE 5.23 The basic spring-mass system accelerometer.**

In Figure 5.23a we have a mass that is free to slide on a base. The mass is connected to the base by a spring that is in its unextended state and exerts no force on the mass. In Figure 5.23b, the whole assembly is accelerated to the left, as shown. Now the spring extends in order to provide the force necessary to accelerate the mass. This condition is described by equating Newton's and Hooke's laws:

*ma = kDx*

**(5.25)**

where *k =* spring constant in N/m

Dx = spring extension in m

*m =* mass in kg

*a* *=* acceleration in m/s^{2}

Equation (5.25) allows the measurement of acceleration to be reduced to a measurement of spring extension (linear displacement) because

If the acceleration is reversed, the same physical argument would apply, except that the spring is compressed instead of extended. Equation (5.26) still describes the relationship between spring displacement and acceleration.

The *spring-mass principle* applies to many common accelerometer designs. The mass that converts the acceleration to spring displacement is referred to as the *test mass* or *seismic mass.* We see, then, that acceleration measurement reduces to linear displacement measurement; most designs differ in how this displacement measurement is made.

## Natural Frequency and Damping

On closer examination of the simple principle just described, we find another characteristic of spring-mass systems that complicates the analysis. In particular, a system consisting of a spring and attached mass always exhibits oscillations at some characteristic *natural frequency.* Experience tells us that if we pull a mass back and then release it (in the absence of acceleration), it will be pulled back by the spring, overshoot the equilibrium, and oscillate back and forth. Only friction associated with the mass and base eventually brings the mass to rest. Any displacement measuring system will respond to this oscillation as if an actual acceleration occurs. This natural frequency is given by

where *f _{N}*

*=*natural frequency in Hz

*k =*spring constant in N/m

*m*= seismic mass in kg

The friction that eventually brings the mass to rest is defined by a

*damping coefficient ,*which has the units of s

^{-1}. In general, the effect of oscillation is called

*transient response,*described by a periodic damped signal, as shown in Figure 5.24, whose equation is

*X*X

_{T}(t) =*e*

_{o}^{-µt}sin(2p

*f*

_{N}

*t*)

**(5.28)**

where *Xr(t) =* transient mass position

*X _{o}*

*=*peak position, initially

*µ*= damping coefficient

*f*= natural frequency

_{N}The parameters, natural frequency, and damping coefficient in Equation (5.28) have a profound effect on the application of accelerometers.

## Vibration Effects

The effect of natural frequency and damping on the behavior of spring-mass accelerometers is best described in terms of an applied vibration. If the spring-mass system is exposed to a vibration, then the resultant acceleration of the base is given by Equation (5.23)

*a(t) = -w*

^{2}x_{o}*sin wt*

If this is used in Equation (5.25), we can show that the mass motion is given by

where all terms were previously denned and *w* = 2p*f*, with/the applied frequency.

**FIGURE 5.24 A spring-mass system exhibits a natural oscillation with damping as response to an impulse input.**

**FIGURE 5.25 A spring-mass accelerometer has been attached to a table which is exhibiting vibration. The table peak motion is x**

_{o}**and the mass motion is**

**D**

*x*.

To make the predictions of Equation (5.29) clear, consider the situation presented in Figure 5.25. Our model spring-mass accelerometer has been fixed to a table that is vibrating. The *x*_{o} in Equation (5.29) is the peak amplitude of the table vibration, and *Dx* is the vibration of the seismic mass within the accelerometer. Thus, Equation (5.29) predicts that the seismic-mass vibration peak amplitude varies as the vibration frequency squared, but linearly with the table-vibration amplitude. However, this result was obtained without consideration of the spring-mass system natural vibration. When this is taken into account, something quite different occurs.

Figure 5.26a shows the actual seismic-mass vibration peak amplitude versus table-vibration frequency compared with the simple frequency squared prediction.You can see that there is a resonance effect when the table frequency equals the natural frequency of the accelerometer, that is, the value of *Dx* goes through a peak. The amplitude of the resonant peak is determined by the amount of damping. The seismic-mass vibration is described by Equation (5.29) only up to about *f _{N}/2.5.*

Figure 5.26b shows two effects. The first is that the actual seismic-mass motion is limited by the physical size of the accelerometer. It will hit "stops" built into the assembly that limit its motion during resonance. The figure also shows that for frequencies well above the natural frequency, the motion of the mass is proportional to the table peak motion,

*x*but not to the frequency. Thus, it has become a displacement sensor. To summarize:

_{o},1.

*f*<

*f*

_{N}*-*For an applied frequency less than the natural frequency, the natural frequency has little effect on the basic spring-mass response given by Equations (5.25) and (5.29). A rule of thumb states that a safe maximum applied frequency is

*f*< 1

*/2.5f*

_{N}.*2. f > f*

_{N}*-*For an applied frequency much larger than the natural frequency, the accelerometer output is independent of the applied frequency. As shown in Figure 5.26b, the accelerometer becomes a measure of

*vibration displacement x*of Equation (5.20) under these circumstances. It is interesting to note that the seismic mass is stationary in space in this case, and the housing, which is driven by the vibration, moves about the mass. A general rule sets

_{o}*f*> 2.5

*f*for this case.

_{N}Generally, accelerometers are not used near the resonance at their natural frequency because of high nonlinearities in output.

**FIGURE 5.26 In (a) the actual response of a spring-mass system to vibration is compared to the simple**

**w**

^{2}**prediction In (b) the effect of various table peak motion is shown**

**EXAMPLE 5.14**

An accelerometer has a seismic mass of 0.05 kg and a spring constant of 3.0 X 10^{3} N/m Maximum mass displacement is ±0 02 m (before the mass hits the stops). Calculate **(a)** the maximum measurable acceleration in **g,** and **(b)** the natural frequency.

*Solution*

We find the maximum acceleration when the maximum displacement occurs, from Equation (5.26).

**a.**

or because

**b.** The natural frequency is given by Equation (5.27).

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For more tutorials, return to the NI Measurement Fundamentals Main Page.

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__Types of Motion__

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