Natural Frequency Of A Pendulum
Problem Description There is a model clock that is equanimous of an escapement wheel and a swinging pendulum.The timing of the clock depends on the natural frequency of the pendulum, and the escapement cycle provides energy to overcome frictional losses and keep the pendulum aquiver. Objective
To find the natural frequency of the pendulum using information obtained from CAD and actual
measurements. In this assay we consider the whole body of the pendulum, and the rotational inertia
that affects it. This analysis is more than in depth that the Point Mass Pendulum Analysis, just the results
are accurate. Assumptions
i. Friction tin be neglected.
2. The maximum bending of movea, is relatively small-scale.
iii. The pendulum swings freely. (nosotros practise not consider the effect of the escapement wheel) Analysis
Gratis Trunk Diagram
Fp = tension force in pendulum
Lcom = constructive length of pendulum
mg = gravitational force on pendulum
a = angle between Fp and mg
Bones Equations
Click to see derivation of equations
Luckily this integral of even a complex shape tin can be done hands by our CAD package, but we must make sure to specify that the point nearly which the moment of inertia is being calculated is the pivot of the pendulum.
Calculating the Heart of Mass of the Pendulum
Make sure that the Sketch Origin in Inventor is at the pivot indicate of the pendulum.
Finding the Rotational Inertia of the Pendulum
In the previous analysis, it is assumed that all of the mass of the pendulum is concentrated at the center of mass. However, a more accurate assay can be performed that includes the effect of the rotational inertia (I) of the pendulum. The rotational inertia (I) of a body is the quantity that tells us how the mass of a rotating body is distributed almost its axis of rotation.
Like to the center of mass analysis, the rotational inertia analysis can exist cleaved up into 2 parts: The acrylic pendulum and the bolts.
Step 1: Finding the Rotational Inertia of the Acrylic
This process again requires values from Inventor. The start of the procedure has been copied for your convenience.
Create a new sketch on the front face of the pendulum. Click on the "Project Geometry" tool, and select the front face of the pendulum. Click Tools on the Ribbon, and and so click the drop down arrow under "Measure." Click on "Region Backdrop."
The Region Properties dialog box will pop up. Click on "Click to Add" in the top left of the box, and then click on the front face of the pendulum. Click on the "Calculate" button, and and so mass properties about the pendulum will appear. The moment of inertia values are generated from the 2nd profile of the sketch of the pendulum. To summate the rotational inertia, utilise the equation beneath. Annotation: The units of the moments of inertia are given every bit in^4. 
rotational moment of inertia of acrylic: Iair conditioning = p t Ipolar
t => thickness of pendulum
I polar => polar moments of inertia with respect to Sketch Origin
Step 2: Finding the Rotational Inertia of the Pendulum with Bolts
To find the rotational inertia of the pendulum with bolts, the bodily altitude from each bolt to the pivot is needed.
1. Create a new sketch on the front face of your pendulum.
2. Select the "Project Geometry" tool from the Ribbon
iii. Click anywhere on the front confront of your pendulum sketch. The shape of the pendulum should now highlight in yellow. This tool is used to bring basic the geometry of a shape into the current sketch.
4. At present, click on the "Tools" tab of the Ribbon. Select the "Distance" tool from the Measure box. At present, click the center of the pivot pigsty, followed by the eye of a bolt pigsty. A blue line is drawn, and a distance box displays relevant information almost the selection. The value displayed in the top box is the actual altitude between the two selected points, and should be used for calculations in this section. Echo this process for all bolts.
The final fix of measurements may exist similar to this:
(Note: If you lot desire to include dimensions that await like this, use the Dimension tool and correct click after bespeak selection to choose "Aligned." Google "Inventor Dimension Aligned" if you need more assist.)
Tape the distances from each bolt to the pivot. Nosotros assume that each bolt is a point mass. Thus the rotational inertia of each bolt on the pendulum can be calculated by:
Ibolt1=mrone 2
I bolt2 =mr2 two
Full Bolt Inertia: I bolts= I bolt1 + I bolt2 +...+ I bolt8
Footstep 3: Combine the Acrylic and Bolts Moment of Inertia
Pace 4: Rigid Body Pendulum Frequency Calculations
Using the values obtained above, solve the frequency of the pendulum using rigid trunk mechanics.
The total time that the clock will run for depends on three things.
1. The flow of the pendulum which is the corporeality of time that it takes for the pendulum to do ane oscillation or cycle.
2. The number of cycles the pendulum can do per rotation of the escapement wheel. What this basically boils down to is the number of teeth on the escapement wheel.
3. The number of rotations the escapement cycle performs.
total time =
Comparison of Calculated Time to Bodily Time
What were the results? Compare the results to the actual time the pendulum runs and see if at that place is a discrepancy. If and then, why?
Natural Frequency Of A Pendulum,
Source: http://mae3.eng.ucsd.edu/clock-project/clock-timing-information/natural-frequency-of-a-rigid-body-pendulum
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