We study undamped harmonic motion as an application of second order linear differential equations.
We consider the motion of an object of mass , suspended from a spring of negligible mass. We say that the spring–mass system is in equilibrium when the object is at rest and the forces acting on it sum to zero. The position of the object in this case is the equilibrium position. We define to be the displacement of the object from its equilibrium position measured positive upward.
Our model accounts for the following kinds of forces acting on the object:
- The force , due to gravity.
- A force exerted by the spring resisting change in its length. The natural length of the spring is its length with no mass attached. We assume that the spring obeys Hooke’s law: If the length of the spring is changed by an amount from its natural length, then the spring exerts a force , where is a positive number called the spring constant. If the spring is stretched then and , so the spring force is upward, while if the spring is compressed then and , so the spring force is downward.
- A damping force that resists the motion with a force proportional to
the velocity of the object. It may be due to air resistance or friction in
the spring. However, a convenient way to visualize a damping force is to
assume that the object is rigidly attached to a piston with negligible mass
immersed in a cylinder (called a dashpot) filled with a viscous liquid
As the piston moves, the liquid exerts a damping force. We say that the motion is undamped if , or damped if .
- An external force , other than the force due to gravity, that may vary with , but is independent of displacement and velocity. We say that the motion is free if , or forced if .
From Newton’s second law of motion,We must now relate to . In the absence of external forces the object stretches the spring by an amount to assume its equilibrium position, as shown in the figure below.
Since the sum of the forces acting on the object is then zero, Hooke’s Law implies that . If the object is displaced units from its equilibrium position, the total change in the length of the spring is , so Hooke’s law implies that Substituting this into (eq:6.1.1) yields Since this can be written asWe call this the equation of motion.
Throughout the rest of this section we’ll consider spring–mass systems without damping; that is, . We’ll consider systems with damping in the next section.
We first consider the case where the motion is also free; that is, =0. We begin with an example.
- Set up the equation of motion and find its general solution.
- Find the displacement of the object for if it’s initially displaced 18 inches above equilibrium and given a downward velocity of 3 ft/s.
We’ll now consider the equation where and are arbitrary positive numbers. Dividing through by and defining yields The general solution of this equation isWe can rewrite this in a more useful form by defining and Substituting from (eq:6.1.9) into (eq:6.1.7) and applying the identity yields
Given and , we can compute from (eq:6.1.8). From (eq:6.1.8) and (eq:6.1.9), we see that is related to and by There are infinitely many angles , differing by integer multiples of , that satisfy these equations. We will always choose so that .
The motion described by (eq:6.1.7) or (eq:6.1.10) is simple harmonic motion. We see from either of these equations that the motion is periodic, with period This is the time required for the object to complete one full cycle of oscillation (for example, to move from its highest position to its lowest position and back to its highest position). Since the highest and lowest positions of the object are and , we say that is the amplitude of the oscillation. The angle in (eq:6.1.10) is the phase angle. It’s measured in radians. Equation (eq:6.1.10) is the amplitude–phase form of the displacement. If is in seconds then is in radians per second (rad/s); it’s the frequency of the motion. It is also called the natural frequency of the spring–mass system without damping.
In many mechanical problems a device is subjected to periodic external forces. For example, soldiers marching in cadence on a bridge cause periodic disturbances in the bridge, and the engines of a propeller driven aircraft cause periodic disturbances in its wings. In the absence of sufficient damping forces, such disturbances – even if small in magnitude – can cause structural breakdown if they are at certain critical frequencies. To illustrate, this we’ll consider the motion of an object in a spring–mass system without damping, subject to an external force where is a constant. In this case the equation of motion (eq:6.1.2) is which we rewrite aswith . We’ll see from the next two examples that the solutions of (eq:6.1.13) with behave very differently from the solutions with .
It is revealing to write this in a different form. We start with the trigonometric identities
From (eq:6.1.20) we can regard as a sinusoidal variation with frequency and variable amplitude . In the figure below, the dashed curve above the axis is , the dashed curve below the axis is , and the displacement appears as an oscillation bounded by them. The oscillation of for on an interval between successive zeros of is called a beat.
You can see from (eq:6.1.20) and (eq:6.1.21) that moreover, if is sufficiently large compared with , then assumes values close to (perhaps equal to) this upper bound during each beat. However, the oscillation remains bounded for all . (This assumes that the spring can withstand deflections of this size and continue to obey Hooke’s law.) The next example shows that this isn’t so if .
This phenomenon of unbounded displacements of a spring–mass system in response to a periodic forcing function at its natural frequency is called resonance. More complicated mechanical structures can also exhibit resonance–like phenomena. For example, rhythmic oscillations of a suspension bridge by wind forces or of an airplane wing by periodic vibrations of reciprocating engines can cause damage or even failure if the frequencies of the disturbances are close to critical frequencies determined by the parameters of the mechanical system in question.
Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)