University of California, Berkeley Department of Mechanical Engineering Spring Semester 2020 Instructor: T. Keaveny
CEC30/MEC85 – Introduction to Solid Mechanics Computer Assignment Nr. 1
Due Noon Friday, March 6th
Consider a system of ten particles Pi, i = 1, 2, . . . , 10, as in Figure 1(a). The cartesian components of the position vectors of the particles are given in Figure 1(b). Each particle Pi has forces Fij exerted on it by each of the other nine particles Pj and each such force lies along the line defined by particles Pi and Pj. Further, by convention, the force Fij is positive if it is attractive between the two particles and negative if it is repulsive. The magnitudes of the inter-particle forces are included in Table 1.
4 3 2 1 0
−1 −2 −3 −4
2
2
00
−2
−2
−4 yx
x
y
z
P1
0.1291
-3.0785
-1.3918
P2
-0.3952
-0.2864
-1.6352
P3
-1.4960
-3.5508
-3.2673
P4
-4.0495
2.1784
-4.1388
P5
-0.6633
1.6171
-1.0666
P6
2.0924
-0.6813
3.0437
P7
-3.8403
-0.5397
-4.8892
P8
-4.2192
0.0833
-2.6689
P9
-1.3075
0.2809
4.3385
P10
-4.6637
0.7288
-2.7320
Figure 1: (a) The system of particles. (b) The cartesian coordinates of the particles
For the given system of particles, we will use MATLAB to compute the resultant force acting on each particle Pi due to the other nine particles, as well as the moment of this force about the origin O with coordinates (0, 0, 0) and about the point T with coordinates (1, 2, 3). In addition, we want to confirm that the resultant force for the entire system, as well as the sum of the moments of all the forces about points O and T are equal to zero.
The MATLAB code particlesystem provided on bCourses was developed to per- form the above tasks. However, it is missing some steps that you will have to code
as comp1 CEC30/MEC85
z
Procedure
The script first creates the position vectors in a 10 × 3 matrix [Pia] and the inter- particle forces shown in Table 1 in a 10 × 10 matrix [Fij ].
Complete the following steps:
1. Compute the unit vector between particle pairs (Pi,Pj). This will be used by the code to calculate the three components of the resultant force at each particle, which is stored in the 10 × 3 matrix [F P iia].
2. Compute the 10 × 3 matrices [M Oia] and [M Tia], which contain the three components of the resultant moments of each particle about points O and T, respectively.
3. If each particle is not individually in equilibrium, determine the components of the external forces necessary to equilibrate each particle and store them in a 10×3 matrix [FEia].
4. Check the force equilibrium of the entire system. Is it satisfied?
5. Check the moment equilibrium of the entire system by considering both points
O and T. Is equilibrium satisfied?
The MATLAB code will also provide a plot of the resultant force at each particle.
To Submit
Submit on bCourses a PDF file (lastname firstname CA1) containing the com- pleted m-file and the outputs requested. Your final PDF file should be less than 5 pages for this assignment. A good way of presenting the code and results is by using the publish command, which can be found under ‘File’.
2
in the areas denoted by ‘% ADD CODE’.
1
2
3
4
5
6
7
8
9
10
1
0.0000
1.0534
-2.6365
-0.0617
0.6077
-0.0215
0.7822
-0.3871
0.4466
-0.0665
2
1.0534
0.0000
-0.2971
1.0213
0.1834
1.4618
-0.2388
-1.2029
-1.7173
0.2002
3
-2.6365
-0.2971
0.0000
0.0144
1.8676
0.3647
-1.3845
1.6571
0.2077
-0.8231
4
-0.0617
1.0213
0.0144
0.0000
1.5520
-0.4121
-0.1447
0.4662
0.1484
-1.9719
5
0.6077
0.1834
1.8676
1.5520
0.0000
1.0968
-2.1540
-1.5109
-0.8345
-1.5307
6
-0.0215
1.4618
0.3647
-0.4121
1.0968
0.0000
-1.6281
1.7112
1.1388
0.6163
7
0.7822
-0.2388
-1.3845
-0.1447
-2.1540
-1.6281
0.0000
-0.8910
1.5033
-1.9832
8
-0.3871
-1.2029
1.6571
0.4662
-1.5109
1.7112
-0.8910
0.0000
1.7642
0.5628
9
0.4466
-1.7173
0.2077
0.1484
-0.8345
1.1388
1.5033
1.7642
0.0000
-0.3741
10
-0.0665
0.2002
-0.8231
-1.9719
-1.5307
0.6163
-1.9832
0.5628
-0.3741
0.0000
as comp1
CEC30/MEC85
Table 1: Inter-particle force matrix [Fij].