Upload
etcenterrbru
View
1.257
Download
1
Embed Size (px)
DESCRIPTION
Citation preview
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
� Method of virtual work most suited for solving equilibrium problems involving a system of several connected rigid bodies
� Before applying the principle of virtual work to � Before applying the principle of virtual work to the systems, specify the number of degrees of freedom for the system and establish the coordinates that define the position of the system
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Degrees of Freedom� A system of connected bodies takes on a unique
shape that can be specified provided the position of a number of specific points on the system is knownknown
� Positions are defined using independent coordinates q, measured from fixed reference points
� For every coordinate established, the system will have a degree of freedom for displacement along the coordinate axis such that it is consistent with the constraining action of supports
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Degrees of Freedom
� An n degree of freedom system requires n independent coordinates to specify all the locations of all its members
ExampleExample
� Consider one degree of freedom system
where independent coordinate
q = θ is used to specify
location of two connecting
links and the block
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Degrees of Freedom
� Coordinate x could be used as the independent coordinate
� Since the block is constrained to move � Since the block is constrained to move within the slot, x is not independent of θ; rather, it can be related to θ using the cosine law b2 = a2 + x2 -2axcos θ
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Degrees of Freedom
� Consider double link arrangement as a 2 degrees of freedom system
� To specify the location of
each link, the coordinate each link, the coordinate
angles θ1 and θ2 must be
known since a rotation of
one link is independent of
a rotation of the other
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Principles of Virtual Work
� A system of connected rigid bodies is in equilibrium provided that the virtual work done by all the external forces and couples acting on the system is zero for each done by all the external forces and couples acting on the system is zero for each independent displacement of the system
δU = 0
� For a system with n degrees of freedom, it takes n independent coordinates to completely specify the location of the system
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Principles of Virtual Work
� For the system, it is possible to write n independent virtual work equations, one for every virtual displacement taken along each of the independent coordinate axes while the every virtual displacement taken along each of the independent coordinate axes while the remaining n-1 independent coordinates are held fixed
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Procedure for Analysis
Free Body Diagram
� Draw the FBD of the entire system of connected bodies and sketch the connected bodies and sketch the independent coordinate q
� Sketch the deflected position of the system on the FBD when the system undergoes a positive virtual displacement δq
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Procedure for AnalysisVirtual Displacements� Indicate position coordinates si, measured from a
fixed point on the FBD to each i number of active forces and couplesforces and couples
� Each coordinate system should be parallel to line of action of the active force to which it is directed, to calculate the virtual work along the coordinate axis
� Relate each of the position coordinates si to the independent coordinate q, then differentiate for virtual displacements δsi in terms of δq
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Procedure for AnalysisVirtual Work Equation� Write the virtual-work equation for the system
assuming that all the position coordinates siundergo positive virtual displacement δsiundergo positive virtual displacement δsi
� Using the relations for δsi, express the work of each active force and couple in the equation in terms of the single independent virtual displacement δq
� Factor out this common displacement from all the terms and solve for unknown force, couple, or equilibrium position, q
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Procedure for Analysis
Virtual Work Equation
� If the system contains n degrees of freedom, n independent coordinates qn must be specified
Follow the above procedure and let only one of n
� Follow the above procedure and let only one of the independent coordinate undergo a virtual displacement, while the remaining n – 1 coordinates are held fixed
� n virtual work equations can be written, one for each independent coordinate
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Example 11.1
Determine the angle θ for equilibrium of the
two-member linkage. Each member has a
mass of 10 kg.mass of 10 kg.
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
FBD
� One degree of freedom since location of both links may be specified by a single independent coordinatemay be specified by a single independent coordinate
� θ undergoes a positive (CW) virtual rotation δθ, only the active forces, F and the 2 9.81N weights do work
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
SolutionVirtual Displacements� Origin of the coordinates established at the fixed
pin support DLocation of F and W specified by position � Location of F and W specified by position coordinates xB and yw
� To determine work, note these coordinates are parallel to lines of action of their associated forces
� Express position coordinates in terms of independent coordinate θ and taking derivatives
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
It can be seen by the signs of these
mymy
mxmx
ww
BB
θδθδθ
θδθδθ
cos5.0)sin1(2
1
sin2)cos1(2
==
−==
� It can be seen by the signs of these equations that an increase in θ causes an increase in xB and an increase in yw
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
SolutionVirtual Work Equation� If the virtual displacements δxB and δyw were
both positive, then the forces W and F would do positive work since the forces and their positive work since the forces and their corresponding displacements would have the same sense
� For virtual work equation for displacement δθ, δU = 0; Wδyw + Wδyw + FδxB = 0
� Relating virtual displacements to common δθ, 98.1(0.5cosθ δθ) + 9.81(0.5cosθ δθ)
+ 25(-2sinθ δθ) = 0
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
SolutionVirtual Work Equation� δθ ≠ 0,
(98.1cosθ -50 sinθ) δθ = 0θ = tan-1(9.81/50) = 63.0°θ = tan-1(9.81/50) = 63.0°
� If problem solved using equations of equilibrium, dismember the links and apply 3 scalar equations to each link
� Principle of virtual work, by means of calculus, eliminated this task so that answer is obtained directly
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Example 11.2
Determine the angle θ required to maintain
equilibrium of the mechanism. Neglect the weight
of the links. The spring is un-sketched when θ = 0°, of the links. The spring is un-sketched when θ = 0°, and it maintains a horizontal position due to the
roller.
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
SolutionFBD� One degree of freedom since location of both
links may be specified by a single independent coordinatecoordinate
� θ undergoes a positive (CW) virtual rotation δθ, links AB and EC rotates by the same amount since they have the same length and link BC only translates
� Since a couple moment does work only when it rotates, work done by M2 = 0
� Reactive forces at A and E does no work
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
FBD
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
Virtual Displacements
� Position coordinates xB and xD are parallel to lines of action of P and F and these lines of action of P and Fs and these coordinates locate the forces with respect to fixed points A and E
mxmx
mxmx
DD
BB
θδθδθ
θδθδθ
cos2.0sin2.0
cos4.0sin4.0
==
==
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
Virtual Work Equation
� For positive virtual displacements Fs is opposite to δxD and hence does negative work
δU = 0; M δθ + Pδx - F δx = 0 D
δU = 0; M1δθ + PδxB - FsδxD = 0
� Relating virtual displacements to common δθ,
0.5 δθ + 2(0.4cosθ δθ) - Fs(0.2cosθ δθ) = 0
(0.5 + 0.8cosθ – 0.2Fscosθ) δθ = 0
� For arbitrary angle θ, spring is sketched a distance of xD = (0.2sinθ)m
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
Virtual Work Equation
� Therefore, Fs = 60N/m(0.2sinθ)m = (12sinθ)N
� δθ ≠ 0, � δθ ≠ 0,
0.5 + 0.8cosθ – 0.2Fscosθ = 0
� Since sin2θ = 2sinθcosθ
1 = 2.4sin2θ – 1.6cosθ
� By trial and Error,
θ = 36.3°
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Example 11.3
Determine the horizontal force Cx that the pin
at C must exert on BC in order to hold the
mechanism in equilibrium when θ = 45°. Neglect mechanism in equilibrium when θ = 45°. Neglect
the weight of the members.
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
SolutionFBD� Cx obtained by releasing the pin constraint at
C in the x direction and allowing the frame to be displaced in this directionbe displaced in this direction
� One degree of freedom since location of both links may be specified by a single independent coordinate
� θ undergoes a positive virtual displacement δθ, only Cx and the 200N force do work
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
FBD
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
Virtual Displacements
� Forces Cx and 200N are located from the fixed origin A using position coordinates yBfixed origin A using position coordinates yB
and xC
� Using cosine rule,
� Thus,
( ) ( ) ( )
θδθδ
θ
θδθθδ
θ
cos6.0
sin6.0
sin2.1cos2.1200
cos6.026.07.0 222
=
=
+−+=
−+=
B
B
CCC
CC
y
y
xxx
xx
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
SolutionVirtual Work Equation� yB and xC undergo positive virtual displacements
δyB and δxC
� Cs and 200N do negative work since they act in � Cs and 200N do negative work since they act in opposite sense to δyB and δxC
δU = 0; -200δyB + CxδxC = 0 � Relating virtual displacements to common δθ,
-200(0.6cosθδθ) – Cx[(1.2xCsinθ)/(1.2cosθ – 2xC)]δθ = 0
Cx = [-120cosθ(1.2cosθ – 2xC)]/(1.2xCsinθ)� At required equilibrium position, θ = 45°
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
(xC)2 – 1.2cos45°xC – 0.13 = 0
� Solving for positive root,
xC = 0.981mxC = 0.981m
� Thus,
Cx = 114N
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Example 11.4
Determine the equilibrium position of the
two-bar linkage. Neglect the weight of the two-bar linkage. Neglect the weight of the
links.
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
FBD
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
� Two degrees of freedom since the independent coordinates θ1 and θ2 must be known to locate the position of both links
Position vector x , measured from the fixed point � Position vector xB, measured from the fixed point O, is used to specify the location of P
� If θ1 is held fixed and θ2 varies by an amount δ θ2, for the virtual work equation,
[δU = 0]θ1 ; P(δxB)θ1 - M δθ2 = 0
where P and M represent magnitudes of applied force and couple moment acting on link AB
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution� Position coordinate xB related to independent
coordinates θ1 and θ2
xB = lsinθ1 +lsinθ2
To obtain variation of δx with respect to δθ� To obtain variation of δxB with respect to δθ2
(δxB/δθ2) = l cosθ2
(δxB)θ2 = l cosθ2 δθ2
� Therefore, (Pl cosθ2 – M) δθ2 = 0
� δθ2 ≠ 0, θ2 = cos-1(M/Pl)
11.3 Principle of Virtual Work for a System of Connected Rigid Bodies11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Solution
� To obtain variation of δxB with respect to θ1
(δxB/δθ1) = l cosθ1
(δxB)θ1 = l cosθ1 δθ1(δxB)θ1 = l cosθ1 δθ1
� Therefore,
(Pl cosθ1 – M) δθ1 = 0
� δθ1 ≠ 0, θ1 = cos-1(M/Pl)