Pinned bar

Reference: W. G. McLean, E. W. Nelson, C. L. Best, Schaum’s Outline of Theory and Problems Reference: of Engineering Mechanics, Statics and Dynamics, McGraw-Hill Book Co., Inc., New York, NY, 1978, p. 336. Analysis: Explicit dynamics, bilaterally constrained motion. Purpose: Examine the accuracy of an analysis involving spherical joints. Summary: A homogeneous bar, pinned at a distance a from one end, with total length $L$, is subjected to gravity loading and released from rest at an angle $\theta =30\mathrm{deg}$ from the vertical. The rotational speed when it passes through $\theta =0\mathrm{deg}$ is calculated and compared to an analytical expression.

The length of the spatial angular velocity vector at $\theta =0\mathrm{deg}$ reads

$$\left|\omega \right|=\sqrt{\frac{0.402g_3(L-2a)}{L^2-3La+3a^2}}$$

provided the bar was released at $\theta =30\mathrm{deg}$.

Fig. 15 The pinned bar in the initial configuration.

Input parameters

Density $\left(kg/m^3\right)$	$\rho =1$
Square cross-section $\left(m^2\right)$	$b\times b=0.1\times 0.1$
Length $\left(m\right)$	$L=1$
Distance to joints $\left(m\right)$	$a=0.25$
Gravity acceleration $\left(m/s^2\right)$	$\mathbf{g}=[0,0,-9.8]$

Results

The time step used in the analysis is $h=2^{-8}$. The hinge is modeled by a pair of spherical joints. The computations are terminated for the first n such that $\theta \left(nh\right)\le 0$ (interpolation of the results to the exact point $\theta \left(t\right)=0$ is omitted). The table below summarizes the results

	Target	Solfec-1.0	Ratio
Length of angular velocity when $\theta =0\mathrm{deg}\left(rad/s\right)$	2.121	2.116	0.997

Fig. 16 Animation of the pinned bar (reload page or click on image to restart).