**Chapter 1, An Inexpensive Vertical Seismometer for Classroom
Demonstrations: Theory of Operation**

**Introduction**

The low-cost vertical seismometer described in this document is based on the theory and design for a gravity meter. Figure 1 gives a simplified design of the instrument. This version has been found to be convenient for the direct observation of relatively large motions, the demonstration of the principles of operation of a seismometer, and for operation in the typical classroom environment. Electronic amplification for recording earthquakes in a quiet environment is under development. However, a simple light sensitive resistor used as recommended can provide sufficient signal to use in classroom experiments, such as determining gain, free period and damping ratio. In this chapter we explain the theory of its operation. In additional chapters we provide a description of a low-cost version of the vertical seismometer that is easy to construct, provide assembly instructions for this seismometer, and describe experiments appropriate for teaching the physics and operation of seismometers.

**Overview**

The vertical seismometer in its simplest form is a weight suspended by a spring. The gravitational attraction of the weight is canceled by the tension of the spring, thus holding the weight in position. When the ground moves, the support for the lever arm and spring also moves and the weight, because it has inertia, tries to remain still. The inertial force acting on the weight will cause the weight to change position relative to the frame, stretching the spring and creating a restoring force. The spring-mass systems set up in this form is the classical harmonic motion problem of physics. All such systems have a natural period of oscillation which when not damped is the called the free period. To prevent continuous oscillation, the motion of the mass is damped. If the free period is very long, the restoring force from the combined effect of moving the weight and stretching the spring is small. Then, the position of the mass relative to the frame (ground reference) is a measure of the movement of the ground for periods shorter than the free period of the system. A typical free period for a long period seismometer is 15 seconds, because the most interesting and largest earthquake waves have periods from 1 to 20 seconds.

A simple weight suspended by a spring would not be practical for seismic measurements. A weight suspended on a spring would have to be large and the spring would have to be very long in order to obtain a free period long enough for recording earthquakes. Instead, a clever use of geometry allows us to make the change in position of the mass change its force in proportion to the change in force applied by the stretched spring. In such a system, the effective restoring force of the spring can be made to behave as though it is very long, and hence give the desired long period for a significantly smaller instrument.

**Theory**

The Weight on the right in Figure 1 is supported by the extended spring on the left. The horizontal bar (lever arm) is free to rotate in the vertical plain about the hinge point. The hinge is designed to prevent horizontal movement of the mass. The period of motion is determined by the magnitude of the restoring torque applied to the lever when it is displaced from the equilibrium position. If the torque can be made independent of the angular rotation of the bar, as measured by the angle q, then the net restoring force becomes nearly zero and, hence, the period can be made very long. The torque on the bar due to the mass is given by the expression,

(1)

where: D is the effective length from the hinge to the center of mass.

M is the total effective mass of the suspended system

g is the acceleration of gravity

q is the angle between the lever arm and the vertical.

If the mass is large, and it should be to give the system energy for recording low-level signals, then the effective center of mass is near the weight attached to the lever arm. As the angle q changes from the vertical where vertical is p/2, the torque decreases. This model assumes the mass is placed on the horizontal bar. If the mass is displaced above the bar, it also acts as an inverted pendulum and decreases the restoring force, causing the period to be even longer. We will assume that the mass is on the bar and note that displacement of the mass above or below the bar will respectively decrease or increase the restoring torque and respectively increase or decrease the free period. We will also explain below that the restoring force and, hence, the period is adjusted by the position of the attachment point of the spring to the base plate. Modification of the spring attachment position can compensate for the position of the mass.

The torque imparted by the spring is proportional to its extension, x and to the spring constant, k. The torque imparted by the spring is then,

(2)

where d is the distance from the hinge to the connecting point of the spring and sina is the projection of the force onto the perpendicular to the bar.

The total physical length of the spring is the sum of the length of extension x and the length, n, that the spring would have if it could collapse to the length it would have with zero force applied. Hence, the length is given by,

(3)

The torque due the spring is found by substituting *x = R-n* into
equation 2, thus,

(4)

At this point we can introduce the law of sines for the triangle formed by the bar, spring, and near vertical line connecting the end of the spring and the hinge,

(5)

and use this to remove the variable length R in the torque. Because the net torque is zero when the rod is not moving, we can set the total torque to zero for the equilibrium position. Substituting from equation 2, equation 4, and equation 5 we obtain the following,

(6)

In this expression, if the length of the relaxed spring is zero, that is if n=0, then the expression simplifies to

(7)

which is now independent of q. In order to set n = 0, the spring needs to be a Azero length spring". The coils may prevent the spring from actually reaching the relaxed length. We now know that by adjustment of the level of the horizontal bar and the lengths, D, d and h, the instrument can be set to a very long period. If these adjustments make the coefficient positive, the system will become unstable (that is the mass will always move away from the zero position). The objective is to set the system to give a very slight restoring force and to create the conditions that maintain stability but still give a very long period.