The Dynamics of a Pendulum and an Encouragement of Useless (?) Studies

I would like to explore what kind of academic disciplines one should acquire in university to create truly useful engineering systems, through the new idea of actively utilizing the nonlinearity inherent in mechanical systems.

1. Linear and Nonlinear

Most mechanical systems possess nonlinearity. For example, the restoring force of a pendulum is proportional to the sine of the angle, not the angle itself (i.e., it is nonlinear). Due to this nonlinearity, even a pendulum, the simplest of mechanical systems, can produce chaos, where future states are impossible to predict, when its pivot point is periodically excited. Let's examine what system nonlinearity is through the motion of the familiar pendulum and spring-mass system (Figure 1), which are repeatedly studied at various levels starting from elementary school. The equation of motion for a spring-mass system, ma=F (where m is the mass of the point mass, a is its acceleration, and F is the external force acting on it), is

What is particularly noteworthy here is that the coefficient of x on the right-hand side is negative. This means that in response to the displacement of the point mass, the spring exerts a force in the opposite direction (i.e., a restoring force: if the point mass pulls the spring in the positive direction, the spring pulls the point mass back in the opposite, or negative, direction). This relationship is represented by a downward-sloping graph, as shown in Figure 2.

On the other hand, the equation of motion for a pendulum, as shown in Figure 3, is as follows.

Let's compare this with that of the spring-mass system. In the spring-mass system, the external force (F=-kx) is proportional to the displacement x (such a force is called a linear force), whereas in the pendulum, the force, like F=-mgsinθ, is not proportional to the angle θ (such a force is called a nonlinear force) ^* 1.

2. Equilibrium Points and Stability

Next, let's derive the equilibrium points (in essence, the states of balance) of the pendulum from the equation of motion shown above. An equilibrium point is a state that does not change over time, so it is the state (angle) that satisfies the condition that "the first and second time derivatives are zero" for the equation of motion (in this case, when the angular acceleration on the left-hand side of the equation of motion is set to zero). That is,

The angles that satisfy this are the equilibrium points, and we can see that the pendulum has two of them (θ=0 and θ=π, which we will call Equilibrium Point 1 and Equilibrium Point 2). Unlike the spring-mass system, which has only one equilibrium point (x=0), the pendulum has multiple equilibrium points due to the nonlinearity of the external force. Of course, θ=0 is the position pointing in the direction of gravity, and θ=π is the inverted position.

In physics up to the high school level, the approximation sinθ ≈ θ is made under the condition that "the angle of the pendulum is assumed to be very small." (This is an approximation of sinθ that holds near θ=0, obtained by considering the Taylor expansion of sinθ around θ=0 up to the first-order term in θ). As a result, the external force becomes proportional to θ (see Figure 4), and the equation of motion (2) is

approximated as shown, becoming equivalent to equation (1). We can see that near θ=0, where |θ|≪1 and the approximation holds, the pendulum undergoes simple harmonic motion similar to a spring-mass system. However, if we had only considered this approximated (linearized, with nonlinearity ignored) equation of motion from the beginning, we would only find one type of equilibrium point (θ=0) and would be unable to discover that the inverted position (θ=π) is also an equilibrium point.

By the way, we know intuitively that the two equilibrium points, θ=0 and θ=π, have clearly different physical properties. That is, it is easy to keep a pendulum in its downward-pointing state, but very difficult to maintain it in the inverted position. Where does this difference come from? The concept of an "equilibrium point" mentioned above cannot answer this; the concept of "stability" is required. We could solve the pendulum's equation of motion (2) without approximation, but unfortunately, an exact solution cannot be written using elementary functions ^* 2. Therefore, we consider an approximate equation that holds near the inverted state. This time, we use the approximation for sinθ that holds near θ=π, which is the Taylor expansion of sinθ around θ=π:

With this, the pendulum's equation of motion (2) can be transformed into the following linearized equation near the inverted state, where |θ-π|≪1.

Here, we have set θ-π=θ ₁ . In other words, θ ₁ is the angle from the inverted state. If we represent the external force shown on the right-hand side in the same way as in Figure 2, we get the upward-sloping graph in Figure 5. In terms of a spring, this would be like a spring that, when pulled in the positive direction by a point mass, pulls the point mass back in the same (positive) direction (though such a spring does not actually exist). Therefore, the force represented by the right-hand side of equation (4) is not a force that returns the angle from the inverted state, θ ₁ , to 0 (back to the inverted state), but rather a force that increases the magnitude of θ ₁ . We can see that a force is acting on the pendulum that tends to move it away from the inverted state. Such a state is called "unstable." ^* 1 This means that the inverted state is an unstable equilibrium point.

3. Utilization of Nonlinear Phenomena

In general, mechanical systems have even more complex nonlinearity than a pendulum (with multiple equilibrium points of varying stability) and can potentially exhibit unpredictable nonlinear behavior. To ensure safety and reliability, highly rigid (and consequently heavy) mechanical systems have traditionally been used within a narrow operating range (for a pendulum, the narrow range where the approximation sinθ ≈ θ holds) to prevent nonlinear behavior from occurring. However, recent expectations for higher performance and functionality in mechanical systems demand increased flexibility, lighter weight, and an expanded operating range into areas where nonlinear behavior occurs (for a pendulum, the range where sinθ ≈ θ does not hold). How to deal with this no-longer-negligible system nonlinearity while continuing to ensure safety and reliability has become a critical issue.

In this context, contrary to conventional thinking ^* 3, attempts to enable previously impossible motion control by actively utilizing the complexity and diversity of nonlinear phenomena arising from a system's inherent nonlinearity (such as the existence of multiple equilibrium points and their different stabilities in a pendulum) are gaining attention. Let me introduce a few examples*4. A manipulator with fewer actuators than degrees of freedom is called an underactuated manipulator; by inducing and actively utilizing nonlinear phenomena, motion control becomes possible without feedback control http://youtu.be/tP88f-SwO_E . Other examples where the active use of nonlinear phenomena has made things possible for the first time include a method for stabilizing the resonant phenomenon known as hunting oscillation, which occurs in high-speed railway vehicles, using the gyroscopic effect http://youtu.be/hGlZ8mWGkuM , and the nanometer-scale amplitude control of microcantilevers used in Atomic Force Microscopes (AFM), which are indispensable for observing biological samples.

4. An Encouragement of Useless (?) Studies in Engineering

One often hears from students, especially those aspiring to study engineering, "I don't see how the math and physics I'm learning in my first year will be useful." To actively utilize nonlinearity as described above, it is necessary to freely master cutting-edge mathematics and physics as tools—disciplines that lie beyond the studies often thought to be useless ^* 5. What mathematics and physics provide is the truth of things, and for engineering, they are in a sense far more useful tools than computers. And because they are truths, they never go out of style and provide a universal way of thinking that is valid in any era.

Breakthroughs for creating new engineering systems that accurately capture the needs of a rapidly changing era will likely not emerge from simply grafting onto existing engineering concepts and methods. It is necessary to return to fundamental academic disciplines such as mathematics and physics, which present us with truth, and master their cutting-edge ideas as tools. I hope that students will acquire a solid foundation in engineering based on these fundamental studies, which at first glance may seem to have no practical use. I want them to conduct the kind of basic engineering research that looks 10 or 20 years into the future—something possible precisely because universities, unlike corporations, do not need to turn a profit—and then go out into society to create engineering systems that are truly useful to humanity.

^* 1 For a rigorous definition, please refer to specialized books. For example, V.I. Arnold, Mathematical Methods of Classical Mechanics, 1988, Springer

^* 2 It can be solved analytically using elliptic functions.

^* 3 Following the conventional approach, the focus would be on how to suppress the complex phenomena caused by the system's inherent nonlinearity.

^* 4 Although only the experimental results are shown in the videos, the applied control methods were devised by establishing the equations of motion (nonlinear ordinary (or partial) differential equations) and performing nonlinear analysis on them.

^* 5 Of course, they are useful.