Football session draft for West Ham United International Academy.Full description
Degrees of Freedom
What is a Degree of Freedom • What is a degree of freedom?
“ The system’ system’ss DOF is equal to the number of independent parameters (measurements) that are needed to uniquely define its position in space at any instant of time. time.”” • Degrees of freedom (DOF) are used to define a systems systems “mobility” “mobility ” • The DOF’s need to be correct to create an accurate model
Degrees of Freedom of a Body • How many degrees of freedom (DOF) does a completely unconstrained “Body” have? • Ans. 6 DOF • What are the DOF’s? • There are translational and rotational DOF’s • 3 Translational (x, y, z) • 3 Rotational (rot. about x, rot. about y, rot. about z)
Example of a Vehicle Axis System in a “Body Form”
Degrees of Freedom Example • Each model will have its own unique set of equations of motion which need to be balanced in order to produce an accurate model • We can set up a model by making appropriate assumptions • Care must be taken to make sure that the model can move and is constrained appropriately to give the desired output • For example a single bar is modelled in a two
Degrees of Freedom Example • For example a single bar is modelled in a two dimensional space, how many DOF does it have? y
Answer: 3 DOF 2 Translational 1 Rotational x
Degrees of Freedom Example • Another example, this Pendulum, how many DOF does it have?
Answer: 1 DOF 0 Translational 1 Rotational
Joint Constraints • All bodies have six degrees of freedom to start with (three translational and three rotational) • When modelling a particular scenario we lock these degrees of freedoms to constrain the relative motions of the bodies • Each joint which is used has a unique set of constraints which means they can be used to model various degree of freedom scenarios • Using the characteristics of each joint allows for the
Take Care However! It is easy to over or under constrain a model to replicate a real world scenario This will introduce errors into the model making it inaccurate Sometimes this is a compromise that we have to make, but we need to know that we have made
Calculating Degrees of Freedom • To calculate the total DOF’s of a system the Gruebler’s equation can be utilised (there are other calculations available) • This equation takes into account all of the parts of the system minus the constraints and calculates a final DOF • The equation is: Total DOF’s = 6 (No. of Parts - 1) – (No of Constraints) Total DOF of each part i.e. 6 DOF’s Count the number of parts in your model (including ground) and minus
Count the total number of constraints in the model due to joints and motions
TASK • Build a universal spreadsheet that will calculate the DOF in your model. Remember: Total DOF’s = (No. of Parts - 1) – (No of Constraints) Translational
Degrees of Freedom Example • Let us consider the degree of freedom of a four bar linkage system • We want to perform a kinematic analysis of the system i.e. we are not considering the forces acting in the system REV REV
M REV
Degrees of Freedom Example (continued) • To undertake a kinematic analysis we need to achieve a DOF count of zero (for dynamic analysis the count will be greater than zero) • To achieve this scenario we have to carefully consider the joint types REV
M
REV
Example 1 – DOF Count • This system uses all revolute joints and has a total DOF which is negative • This scenario is physically impossible in a real scenario • Also not having a zero DOF means the model is statically indeterminate REV
REV
Parts Rev Motion M
6 x (4-1) = 18 -5 x 4 = -20 -1 x 1 = -1
Total DOF
= -3
Example 2 – DOF Count • This example shows a joint configuration where there is a zero DOF because another part and joint have been included • Although a DOF total of zero is reach this is not good modelling practise REV
REV SPH
Parts Rev Sph Motion
6 x (5-1) -5 x 4 -3 x 1 -1 x 1
= 24 = -20 = -3 = -1
M REV
REV
Total DOF
=0
Example 3 – DOF Count • This example shows a joint configuration where the DOF total is zero • The solution is not immediately apparent and takes practise • This is the ideal scenario to reach with your models UNI
SPH
M REV
REV
Parts Rev Sph Uni Motion
6 x (4-1) = 18 -5 x 2 = -10 -3 x 1 = -3 -4 x 1 = -4 -1 x 1 = -1
How Many DOF’s does the Trebuchet Model Have? • Consider all of the movements of the model and work it out theoretically • Import it into ADAMS again and see what happens • You have to consider all the relative movement of the system that are needed
What is the DOF Count? • Using what we have discussed, what is the DOF count? REVOLUTE JOINT (LOCATION 01, 04)
01 08
BODY/GROUND 01
02
02 03 05 07 04
04
ROTATIONAL MOTION
11
SPEHRICAL JOINT (LOCATION 03, 06, 07)
05 03
12
Z 06
X
UNIVERSAL JOINT (LOCATION 08)
References Gillespie T. (1992), Fundamentals of Vehicle Dynamics. Warrendale (USA): Society of Automotive Engineers MSC ADAMS (2012), Trebuchet model example for Adams/View that uses a STEP function for sling release [online] available from http://simcompanion.mscsoftware.com/infocenter/index?page=content& id=KB8020123&actp=search&searchid=1352054589140 [4th November 2012] Norton R. (2004), Design of Machinery An Introduction to the Synthesis and Analysis of Mechanisms and Machines. 3rd Edition. London: McGraw Hill
