Why do we choose the fastest point as a reference to measure the simple pendulum period by a stopwatch?

Since the vibration of a single pendulum is very periodic, measuring its period is a very interesting topic. When students are asked to measure the period of a single vibration, they start and stop the stopwatch when the pendulum reaches the top point as a reference point. In this paper, we try to show that the error can be reduced more by using the equilibrium point, that is, the bottom position as the reference rather than the top position. We think it would be beneficial for students to measure the period in both cases, compare the errors, and think about the reasons for error differences. Students believe that the moment the pendulum moved slowly and nearly stopped at the top was more likely to measure the time to be more accurate. The reason for this is that they think the pendulum will move so fast when it passes the bottom point that they will not be able to start or stop the stopwatch accurately at the instant passing the lowest point. We are also able to obtain the error of the position measurement by using the recorded video of a simple pendulum.

How is the period of a simple pendulum growing with increasing amplitude?

For the period T(α) of a simple pendulum with the length L and the amplitude (the initial elongation) α ∈ (0, π), a strictly increasing sequence Tn (α) is constructed such that the relations T 1 ( α ) = 2 L g π − 2 + 1 ϵ ln 1 + ϵ 1 − ϵ + π 4 − 2 3 ϵ 2 , T n + 1 ( α ) = T n ( α ) + 2 L g π w n + 1 2 − 2 2 n + 3 ϵ 2 n + 2 , $$\begin{array}{c} \displaystyle T_1(\alpha)=2\sqrt{\frac{L}{g}}\left[\pi-2+\frac{1}{\epsilon} \ln\left(\frac{1+\epsilon}{1-\epsilon}\right)+\left(\frac{\pi}{4}-\frac{2}{3}\right)\epsilon^2\right],\\ \displaystyle T_{n+1}(\alpha)=T_n(\alpha)+2\sqrt{\frac{L}{g}}\left(\pi w_{n+1}^2 - \frac{2}{2n+3}\right)\epsilon^{2n+2}, \end{array}$$ and 0 < T ( α ) − T n ( α ) T ( α ) < 2 ϵ 2 n + 2 π ( 2 n + 1 ) , $$\begin{array}{} \displaystyle 0 \lt \frac{T(\alpha)-T_n(\alpha)}{T(\alpha)} \lt \frac{2\epsilon^{2n+2}}{\pi(2n+1)}\,, \end{array}$$ holds true, for α ∈ (0, π), n ∈ ℕ, w n := ∏ k = 1 n 2 k − 1 2 k $\begin{array}{} \displaystyle w_n:=\prod_{k=1}^n\frac{2k-1}{2k} \end{array}$ (the nth Wallis’ ratio) and ϵ = sin(α/2).

Modeling and Simulation of a Human Knee Exoskeleton's Assistive Strategies and Interaction

Exoskeletons are increasingly used in rehabilitation and daily life in patients with motor disorders after neurological injuries. In this paper, a realistic human knee exoskeleton model based on a physical system was generated, a human–machine system was created in a musculoskeletal modeling software, and human–machine interactions based on different assistive strategies were simulated. The developed human–machine system makes it possible to compute torques, muscle impulse, contact forces, and interactive forces involved in simulated movements. Assistive strategies modeled as a rotational actuator, a simple pendulum model, and a damped pendulum model were applied to the knee exoskeleton during simulated normal and fast gait. We found that the rotational actuator–based assistive controller could reduce the user's required physiological knee extensor torque and muscle impulse by a small amount, which suggests that joint rotational direction should be considered when developing an assistive strategy. Compared to the simple pendulum model, the damped pendulum model based controller made little difference during swing, but further decreased the user's required knee flexor torque during late stance. The trade-off that we identified between interaction forces and physiological torque, of which muscle impulse is the main contributor, should be considered when designing controllers for a physical exoskeleton system. Detailed information at joint and muscle levels provided in this human–machine system can contribute to the controller design optimization of assistive exoskeletons for rehabilitation and movement assistance.

Self-Sustained Oscillation of a Photothermal-Responsive Pendulum under Steady Illumination

Self-sustained oscillation has the advantages of harvesting energy from the environment and self-control, and thus, the development of new self-oscillating systems can greatly expand its applications in active machines. In this paper, based on conventional photothermal shrinkable material or photothermal expansive material, a simple pendulum is proposed. The light-powered self-sustained oscillation of the simple pendulum is theoretically studied by establishing a dynamic model of the photothermal-responsive pendulum. The results show that there are two motion modes of the simple pendulum, which are the static mode and the oscillation mode. Based on the photothermal-responsive model, this paper elucidates the mechanism of the self-excited oscillation. The condition for triggering self-excited oscillation is further studied. In addition, the influence of the system parameters on the amplitude and frequency is also obtained. This study may have potential applications in energy harvesting, signal monitoring, and soft machines.