# Dynamics of a trolley rails

Topics: Hooke's law, Velocity, Friction Pages: 7 (2958 words) Published: October 28, 2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MECHANICAL ENGINEERING
Part I APPLIED DYNAMICS LAB (full report required)
DYNAMICS OF A TROLLEY ON RAILS
Based on document MGS/ID137/203 by Dr MG Sainsbury
Modified in Feb 2007 & Jan 2008 by Dr L Huang and Mr WS Sze
1. Objectives
(1) To measure the velocity and acceleration of a trolley as it descends on inclined track and is stopped by a nonlinear elastic arrester system.
(2) To compare the measured acceleration, velocity and displacement with theoretical predictions of various methods and discuss the physics learned. 2. Apparatus
The test setup comprises a short inclined railway track with a trolley that rolls down the rails and is stopped by a transversely mounted elastic cord at the bottom, as shown below:

Although the velocity and acceleration can be estimated approximately from measurements with a ruler and a stopwatch, it is more accurate to collect data automatically from a sensor. In this experiment, a magnetic sensor mounted underneath the trolley does the job. The sensor contains a small magnet surrounded by a coil. As the sensor moves over each edge of the steel sleeper, the field around the magnet suddenly changes, and this induces a sudden voltage across the coil. It generates two pulses every time it passes over one steel sleeper, one positive and one negative. Since all the sleepers are equi-spaced, a measurement of the time interval between successive pulses allows instantaneous velocities to be calculated.

The pulses could be viewed on an oscilloscope, but the screen would only be wide enough to look at 1 of 7

several pulses. To avoid this problem, and to analyze the results more thoroughly, the pulses are sampled as thousands of instantaneous voltages using an analog-to-digital converter (ADC) card in a computer. The information is stored in a file that can then be read by a specially written computer program called “ScrollGraph”. This program has a Windows® graphical user interface (GUI) and includes built-in help on how to use it. It has the special feature of allowing you to scroll along the graph and examine the pulses in detail.

The transversely mounted elastic cord is almost tension-free when first mounted, but the tension builds up as soon as the trolley begins to deflect it. The system is highly nonlinear unless a very high initial tension is used. A pulley, hanger and weights are provided to facilitate measurement of the changing “stiffness”, and an equivalent linear spring coefficient would be used in kinematic analysis. 3. Theory

(A) Pulse Analysis for Velocities
When the digitized pulses are displayed in “ScrollGraph” and the zoom facility is activated, the pulses look similar to those in the figure below. Provided the time scale has been set to seconds,

the data-readout cursor allows the pulse times t0, t1, t2, … to be measured directly off the record. The instantaneous velocity v(t) at a time midway between each pair of pulses is estimated according to the formulae shown above. In fact, a more accurate displacement trace s(t) can be obtained by locating the peak positions of the positive and negative pulses. If the pulse times are entered into an Excel spreadsheet, the velocities and associated times may be calculated using Excel formulae. Alternatively, the data file containing the pulses may be read and analysed in one go using a specially written C++ program. “ScrollGraph” actually includes such a facility. (B) Theoretical Dynamic Analysis

The motion of the trolley comprises two distinct phases, as indicated in the figure below. In phase (1), from A to B, the trolley moves freely under gravity and rail friction, whereas in phase (2) it moves under the action of the restoring force imposed by the elastic. The solution for phase (1) is obtained using Newton’s Second Law taking into account the gravitational force as well as the friction. In this test, the friction coefficient is measured by the time it takes for the trolley...