Chapter 12  Fourier Series and the Laplace Transform

12.3  Vibrations in Mechanical Systems

    Consider a spring that resists compression as well as extension, that is suspended vertically from a fixed support, and a body of mass  m  that is attached at the lower end of the spring.  We make the assumption that the mass  m  is much larger than the mass of the spring so that we can neglect the mass of the spring.  If there is no motion then the system is in static equilibrium, as illustrated in Figure 12.17(a).  If the mass is pulled down further and released, then it will undergo an oscillatory motion.

                         Figure 12.17.  The spring-mass system.

     We will determine the Simple Harmonic Motion of this mechanical system by considering the forces acting on the mass during the motion.  For details see Section 12.3 in the textbook.  

    If there is no friction to slow the motion of the mass, then we say that the system is undamped.  We determine the motion of this mechanical system by considering the forces acting on the mass during the motion.  Doing so leads to a differential equation relating the displacement as a function of time.  The most obvious force is that of  gravitational attraction acting on the mass m and given by

            ,

where g is the acceleration of gravity.

    The next force to be considered is the spring force acting on the mass and directed upward if the spring is stretched and downward if it is compressed.  It obeys Hooke's law,

            ,

where s is the amount the spring is stretched when and is the amount it is compressed when .

    When the system is in static equilibrium and the spring is stretched by the amount , the resultant of the spring force and the gravitational force is zero, which is expressed by the equation

            .

We let denote the displacement from static equilibrium with the positive s direction pointed downward, as indicated in Figure 12.17(b), and write the spring force as

            .

The resultant force is

(12.16)            .

We obtain the differential equation for motion by using Newton's second law, which states that the resultant of the forces acting on the mass at any instant satisfies

(12.17)            .

The distance from equilibrium at time t is measured by , so the acceleration a is given by  .  Applying Equations (12.16) and (12.17) yields

            .

Hence the undamped mechanical system is governed by the linear differential equation

            ,  

and the general solution to this D. E. is known to be  .  

12.3.1  Damped Systems

     If we consider frictional forces that slow the motion of the mass, then we say that the system is damped.  To help visualize this situation, we connect a dashpot to the mass, as indicated in Figure 12.18.  For small velocities we assume that the frictional force is proportional to the velocity; that is,

            .

                              Figure 12.18.  The spring-mass-dashpot system.

    The damping constant c must be positive, for if  ,  then the mass is moving downward and hence must point upward, which requires that be negative.  The result of the three forces acting on the mass is given by

            .  

Hence the Damped Mechanical Motion is governed by the linear differential equation

            .  

For details see Section 12.3 in the textbook.

12.3.2  Forced Vibrations

    The vibrations discussed earlier are called free vibrations because all the forces that affect the motion of the system are internal to the system.  We extend our analysis to cover the case in which an external force acts on the mass, as depicted in Figure 12.19.  Such a force might occur from vibrations of the support to which the top of the spring is attached or from the effect of a magnetic field on a mass made of iron.  As before, we sum the forces , , , and and set this sum equal to the resultant force , obtaining  

            .   

Therefore the forced motion of the mechanical system satisfies the nonhomogenous linear differential equation

(12.18)            .  

The function is called the input, or driving force, and the solution is called the output, or response.  Of particular interest are periodic inputs that can be represented by Fourier series.

                              Figure 12.19.  The spring-mass-dashpot system with an external force.

    For damped mechanical systems that are driven by a periodic input  F(t),  the general solution involves a transient part that vanishes as  ,  and a steady state part that is periodic.  We find the transient part of the solution    is found by solving the homogeneous differential equation

            .  

This homogeneous equation has the characteristic equation  ,  and it's roots are  .  The constants  m, c, and k  are all positive, and there are three cases to consider.

Case 1.  If  , then the roots are real and distinct, and since  , it follows that the roots   are negative real numbers.  Thus for this case we have

            ,        and  .

Case 2.  If  , then the roots are real and equal, ,  where   is a negative real number, and for this case we have

            ,          and  .

Case 3.  If  , then the roots are complex conjugates,  ,  where    are positive real number, and it follows that

            ,       and  .

In all three cases, the homogeneous solution    decays to  0  as  .  

    We obtain the steady state solution    by representing    by its Fourier series, substituting    into the nonhomogeneous differential equation and solving the resulting system for the Fourier coefficients of  .  The general solution to the nonhomogeneous linear differential equation is  

            .

More Details for Mechanical Systems

Example 12.4.  Find the general solution to  ,  where is given by the Fourier series  .  

Solution.

First, we solve    for the transient solution. The characteristic equation is    , which has a double root .  Hence  

            .  

We obtain the steady state solution by assuming that    has the Fourier series representation  

            ,  
            
and that and can be obtained by termwise differentiation:

            ,  
            
            ,  

Substituting these expansions into the differential equation results in

            

Equating the coefficients with the given series for , we find that  ,  and that

            
        
                for all n.

Solving this linear system for and , we get

            

            

And with a little algebra, they can be written as follows

            

            

The general solution is

          

    For illustration purposes, consider the initial conditions and .  If the trigonometric polynomial is used to form an approximation to the solution then the graph of the approximate solution (in blue) is given in the figure below.
                                        

Explore Solution 12.4.

Extra Example 1.  Find the general solution to  ,  where is given by the Fourier series  .  

Explore Extra Solution 1.

Extra Example 2.  Find the general solution to  ,  where is given by the Fourier series  .  

Explore Extra Solution 2.