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MATHEMATICS 1170

LAB #7MODELS OF RESONANCE IN PLANT STEMS

Today we are going to build models of plants using oak dowels and playdo. The playdowill be used to vary the mass at the top of the beam. These models will best representcorn, wheat, and other crops that consist of a shoot and a mass on the end.

If we deflect the beam and release it, it will oscillate until it comes to rest. If we neglectfriction (damping), the equation describing the motion of the TOP of the beam is asfollows:

)cos()sin()( t At v

t p nnn

o

+=

Where vo is the initial velocity, A is the initial amplitude, and n is the natural frequency.**Note that the natural frequency is the frequency of vibration when the beam isdeflected and release.

Today we will measure the natural frequency of beams of different stiffnesses, lengths,and masses. We will measure the length and stiffness of the beams, and calculate themass of the playdo based on the measured natural frequency.

Resonance Trees often uproot or snap when strong winds gust at their natural frequency. In order toselect better crops, agricultural engineers are interested in determining the parameters thatinfluence the natural frequency of crops. Storm winds typically gust at a certain range offrequencies (0.2 2 Hz or so). If the natural frequencies of crops and other plants areoutside of this range, they are less likely to uproot or snap.

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Natural frequency

A very simple model of the natural frequency can be made by modeling the plant as aspring-mass system. The natural frequency of the plant can be calculated as:

mk

n =

Let T be the period of the oscillation (the time in seconds it takes the plant to sway backand forth once). The natural frequency can be calculated from this value as follows:

T n

2

=

Knowing the natural frequency and the effective stiffness (k) the effective mass (m) canbe calculated using the first equation.

The Effective Stiffness, kThe effective stiffness is proportional to the resistance of the beam (or trunk or stem) tobending. It can be calculated using the following equation:

3

3 L EI

k =

where L describes the length of the beam. E is the elastic modulus which is a property ofthe material. The beams we are using in class are made of oak. E is given in the tablebelow for a couple of materials. I is the second moment of area (also called the second

moment of inertia). Elastic Modulus (E)In solid mechanics, the elastic modulus (E) is a measure of the stiffness of a givenmaterial.

Material Elastic Modulus (psi, pounds per square inch)*Oak 1,600,000Ash 1,450,400Douglas Fir 1,740,480Steel 29,008,000

Aluminum 10,100,000

Second Moment of Area (I)The second moment of area of a shape is a property which measures the efficiency of thatshape with respect to its resistance to bending.

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For a circular cylinder:

4

4r I

=

Effective mass, m

Let the mass of the beam be mb and the mass at the end of the beam be me. Then theeffective mass of the total system is given by the following equation:

eb mmm += )140 / 33(This is the same m as appears in the equation for the natural frequency.

Lab ExerciseWe are going to build various models of plants of different lengths, diameters, andmasses. For each model, record the length of the beam, the radius of the beam, and thediameter of the mass of playdo at the end. We are then going to measure the period ofoscillation of each model and use this data to estimate its effective mass. Please fill in theattached chart and make the appropriate calculations.

Questions to Answer

1. As crops and trees were bred to produce larger fruits, seeds, and vegetables,agriculturalists noticed that these plants became more susceptible to windthrow(uprooting during storms) and snapping. What effect does increasing the mass of thecrops have on the natural frequency? Why might this cause the plants to uproot and snapmore frequently?

2. Pick one of the models built in class. Assuming that the initial velocity is zero, theequation of motion can be written as:

)cos()( t mk

At p =

Calculate the p(t) and p(t). The force acting on the stem (or beam) can be estimated asF = m*a = m*p(t). Double the mass of the model. What happens to the force? Doublethe stiffness of the beam. What happens to this force?

3. In order to reduce the number of crops that uproot or snap, agriculturalists have bred

for increased effective stiffness in the stems. What are three things that could be changedin the plant to increase the effective stiffness?