AAE203F2010LEC15.pdf

AAE 203: Aeromechanics I Statics of Bodies Dengfeng Sun School of Aeronautics and Astronautics Engineering November 5, 2010 Reference: AAE 203 Notes, by Prof. Martin Corless. Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 1 / 15 Force systems A force system is just a bunch of forces, ¯F1 , ¯F1 , . . . , ¯FN. The resultant of a force system is the sum of all its forces and is given by: summationdisplay¯ F := Nsummationdisplay j=1 ¯Fj = ¯F1 + ¯F2 + . . . + ¯FN Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 2 / 15 Moment resultant The moment resultant of a force system about a point Q is the sum of the moments of all its forces about Q and is given by: summationdisplay ¯ MQ := Nsummationdisplay j=1 ¯rj null ¯Fj where ¯rj is a vector from Q to a point on the line of action of ¯Fj. Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 3 / 15 Fact For any two points Q and Qnull, summationtext ¯MQ =summationtext ¯MQnull +¯r null?¯F where ¯r = ¯rQQnull (the vector from Q to Qnull). Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 4 / 15 Couples and torques A couple is a pair of forces which have equal magnitude but opposite direction. If (¯F1, ¯F2) is a couple, then, ¯F2 = null¯F1 . null A couple has zero resultant. null The moment resultant of a couple about every point is the same. The torque ¯T of a couple is its moment resultant about any point. Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 5 / 15 Equivalent force systems Two force systems are equivalent if they have the same resultant and the same moment resultant about some point. null The system of internal forces in any body is equivalent to a zero force. null If a body is in static equilibrium, its system of external forces is equivalent to a zero force. null If two force systems are equivalent, then they have the same moment about every point. This follows from the relationship,summationtext ¯MQ =summationtext ¯MQnull +¯r null?¯F. Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 6 / 15 Example 2 m 2 N 3N 5 N 6 N m 5 N 4 N m 5 N 6/5 m 4/5 m Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 7 / 15 A force and a couple Every force system (regardless of complexity) is equivalent to a force and a couple. Choose any point Q and let ¯F = ?¯F and ¯T = ? ¯MQ It can readily seen that the new force system consisting of a force ¯F placed at Q and a couple of torque ¯T is equivalent to the original force system. Note that Q can be any point. Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 8 / 15 Other equivalent force systems null Force systems which are equivalent to a couple Suppose ?¯F = ¯0 Then the original force system is equivalent to a couple of torque ¯T = ? ¯MQ. Since ?¯F = ¯0, this torque is independent of the point Q. null Force systems which are equivalent to single force Suppose there is a point Q such that ? ¯MQ = ¯0 Then the original force system is equivalent to a single force ¯F = ?¯F whose point of application is Q. Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 9 / 15 Some properties null The point of application of a single equivalent force is not unique. One can ?frame? a single equivalent force along its line of application and obtain another equivalent force with a different point of application. null Suppose one has a force system which is equivalent to single force ¯F placed at a point Q. Let ¯T =summationtext ¯MO where O is any point. Then it is necessary that ¯T is perpendicular to ¯F. null Given any two mutually perpendicular vectors ¯F and ¯T with ¯F null= 0, one can always find a vector ¯r such that holds. One such vector is given by ¯r = ¯F null ¯T F2 . (1) Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 10 / 15 Parallel force systems If all forces are parallel to ?e3, ¯Fj = Fj ?e3 and ?¯F = F?e3 where F := Nsummationdisplay i=1 Fj Let O be the origin of the reference frame and let rj be the vector from O to the point of application of Fj; then rj can be expressed as ¯rj = xj ?e1 + yj ?e2 + zj ?e3 Hence, ¯rj null ¯Fj = (yjFj)?e1 null(xjFj)?e2 and ? ¯MO = T1 ?e1+T2 ?e2 where T1 := Nsummationdisplay i=1 yjFj and T2 := null Nsummationdisplay i=1 xjFj Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 11 / 15 Parallel force systems cont?d If the vector from O to the point of application of ¯F is ¯r? = x? ?e1 + y? ?e2 + z? ?e3 ¯r? null ¯F = (y? Nsummationdisplay i=1 Fj)?e1 null(x? Nsummationdisplay i=1 Fj)?e2 Since ¯r? null ¯F = ? ¯MO, we obtain y? Nsummationdisplay i=1 Fj = Nsummationdisplay i=1 yjFj x? Nsummationdisplay i=1 Fj = Nsummationdisplay i=1 xjFj x? = summationtextN i=1 x jFj F y? = summationtextN i=1 yjFj F Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 12 / 15 Distributed force systems null In distributed force systems, forces do not act a single point, but act over a region of space. null body forces null surface forces Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 13 / 15 Body forces A body force acts over a three-dimensional region of space. The main example of a body force is the gravitational attraction of one body on another. null The gravitational force is exerted by one body (the offending body) on another body (the suffering body). null It is a force system which is distributed throughout the entire suffering body. Every particle of the suffering body is subject to the gravitational attraction of the offending body. null When the suffering body is near the surface of the offending body and its dimensions are insignificant compared to the offending body, then, the gravitational force system is a parallel force system. It is equivalent to a single force ¯W placed at the mass center of the suffering body. For bodies of uniform mass density, the mass center is at the geometric center. Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 14 / 15 Surface forces A surface force acts over a two-dimensional region of space. null Hydrostatic force: the forces exerted on a body when it is immersed in water. null Archimedes principle null center of buoyancy null Aerodynamic forces: the forces on a body moving relative to the air. null lift null drag null pitching moment null center of pressure null Connection forces: the force exerted by one body on another at a connection between the two bodies. It is a distributed force system, usually represented by an equivalent force system consisting of a single force and a couple of torque. Dengfeng Sun (dsun@purdue.edu) AAE 203: Aeromechanics (Fall 2010) November 5, 2010 15 / 15