Understanding the concepts on Degrees of Freedom
Step 1: Computing Modal Mass (Mn)
We want to calculate the modal mass (Mn) for each of the five modes using the formula:
Mn = Φn^T * M * Φn
Where M is the mass matrix and Φn is the mode shape for each mode.
Mass Matrix M:
M =
[ M 0 0 0 0 ]
[ 0 M 0 0 0 ]
[ 0 0 M 0 0 ]
[ 0 0 0 M 0 ]
[ 0 0 0 0 M ]
Mode 1:
T1 = 2 seconds
Φ1 = [0.334, 0.64, 0.895, 1.078, 1.173]
Let's calculate M1:
M1 = Φ1^T * M * Φ1
M1 =[ 0.334 0.64 0.895 1.078 1.173 ]
Mass Matrix (M) *
[ M 0 0 0 0 ]
[ 0 M 0 0 0 ]
[ 0 0 M 0 0 ]
[ 0 0 0 M 0 ]
[ 0 0 0 0 M ]
Mode Shape (Φ1) *
= 0.334M + 0.64M + 0.895M + 1.078M + 1.173*M
= (0.334 + 0.64 + 0.895 + 1.078 + 1.173) * M
= 4.12M
for Mode 1, M1 is approximately 4.12M.
Mode 2:
T2 = 0.6852 seconds
Φ2 = [-0.895, -1.173, -0.641, 0.334, 1.078]
Let's calculate M2:
M2 = Φ2^T * M * Φ2
M2 = [ -0.895 -1.173 -0.641 0.334 1.078 ]
Mass Matrix (M) *
[ M 0 0 0 0 ]
[ 0 M 0 0 0 ]
[ 0 0 M 0 0 ]
[ 0 0 0 M 0 ]
[ 0 0 0 0 M ]
Mode Shape (Φ2) *
= -0.895M - 1.173M - 0.641M + 0.334M + 1.078*M
= (-0.895 - 1.173 - 0.641 + 0.334 + 1.078) * M
= -0.297M
So, for Mode 2, M2 is approximately -0.297M.
Mode 3:
T3 = 0.4346 seconds
Φ3 = [1.173, 0.334, -1.078, -0.641, 0.895]
Let's calculate M3:
M3 = Φ3^T * M * Φ3
M3 =[ 1.173 0.334 -1.078 -0.641 0.895 ]
Mass Matrix (M) *
[ M 0 0 0 0 ]
[ 0 M 0 0 0 ]
[ 0 0 M 0 0 ]
[ 0 0 0 M 0 ]
[ 0 0 0 0 M ]
Mode Shape (Φ3) *
= 1.173M + 0.334M - 1.078M - 0.641M + 0.895*M
= (1.173 + 0.334 - 1.078 - 0.641 + 0.895) * M
= 0.683M
So, for Mode 3, M3 is approximately 0.683M.
Mode 4:
T4 = 0.3383 seconds
Φ4 = [-1.078, 0.895, 0.334, -1.173, 0.641]
Let's calculate M4:
M4 = Φ4^T * M * Φ4
M4 = [ -1.078 0.895 0.334 -1.173 0.641 ]
Mass Matrix (M) *
[ M 0 0 0 0 ]
[ 0 M 0 0 0 ]
[ 0 0 M 0 0 ]
[ 0 0 0 M 0 ]
[ 0 0 0 0 M ]
Mode Shape (Φ4) *
= -1.078M + 0.895M + 0.334M - 1.173M + 0.641*M
= (-1.078 + 0.895 + 0.334 - 1.173 + 0.641) * M
= -0.381M
So, for Mode 4, M4 is approximately -0.381M.
Mode 5:
T5 = 0.2966 seconds
Φ5 = [0.641, -1.078, 1.173, -0.895, 0.334]
Let's calculate M5:
M5 = Φ5^T * M * Φ5
M5 = [ 0.641 -1.078 1.173 -0.895 0.334 ]
Mass Matrix (M) *
[ M 0 0 0 0 ]
[ 0 M 0 0 0 ]
[ 0 0 M 0 0 ]
[ 0 0 0 M 0 ]
[ 0 0 0 0 M ]
Mode Shape (Φ5) *
= 0.641M - 1.078M + 1.173M - 0.895M + 0.334*M
= (0.641 - 1.078 + 1.173 - 0.895 + 0.334) * M
= 0.175M
So, for Mode 5, M5 is approximately 0.175M.
Step 2: Computing Modal Stiffness (Ln)
we also want to calculate the modal stiffness (Ln) for each mode using the formula:
Ln = Φn^T * K * Φn *
Where ι is a vector of ones.
Stiffness Matrix K:
K =
[ 4K -K 0 0 -3K ]
[ -K 4K -3K 0 0 ]
[ 0 -3K 3K -K 0 ]
[ 0 0 -K 2K -K ]
[ -3K 0 0 -K 4K ]
Vector ι:
ι = [1, 1, 1, 1, 1]
Mode 1:
Φ1 = [0.334, 0.64, 0.895, 1.078, 1.173]
Let's calculate L1:
L1 = Φ1^T * K * Φ1 * ι
L1 = [ 0.334 0.64 0.895 1.078 1.173 ]
Stiffness Matrix (K) *
[ 4K -K 0 0 -3K ]
[ -K 4K -3K 0 0 ]
[ 0 -3K 3K -K 0 ]
[ 0 0 -K 2K -K ]
[ -3K 0 0 -K 4K ]
Mode Shape (Φ1) *
Vector ι *
= 0.334*(4K) + 0.64*(-K) + 0.8950 + 1.0780 + 1.173*(-3K)
= (1.336K - 0.64K - 3.519K)
= -2.823K
So, for Mode 1, L1 is approximately -2.823K.
Mode 2:
Φ2 = [-0.895, -1.173, -0.641, 0.334, 1.078]
Let's calculate L2:
L2 = Φ2^T * K * Φ2 * ι
L2 = [ -0.895 -1.173 -0.641 0.334 1.078 ]
Stiffness Matrix (K) *
[ 4K -K 0 0 -3K ]
[ -K 4K -3K 0 0 ]
[ 0 -3K 3K -K 0 ]
[ 0 0 -K 2K -K ]
[ -3K 0 0 -K 4K ]
Mode Shape (Φ2) *
Vector ι *
= -0.895*(4K) - 1.173*(-K) - 0.6410 + 0.3340 + 1.078*(-3K)
= (-3.58K + 1.173K - 3.234K)
= -5.641K
So, for Mode 2, L2 is approximately -5.641K.
Mode 3:
Φ3 = [1.173, 0.334, -1.078, -0.641, 0.895]
Let's calculate L3:
L3 = Φ3^T * K * Φ3 * ι
L3 = [ 1.173 0.334 -1.078 -0.641 0.895 ]
Stiffness Matrix (K) *
[ 4K -K 0 0 -3K ]
[ -K 4K -3K 0 0 ]
[ 0 -3K 3K -K 0 ]
[ 0 0 -K 2K -K ]
[ -3K 0 0 -K 4K ]
Mode Shape (Φ3) *
Vector ι *
= 1.173*(4K) + 0.334*(-K) - 1.0780 - 0.6410 + 0.895*(-3K)
= (4.692K - 0.334K - 2.685K)
= 1.673K
So, for Mode 3, L3 is approximately 1.673K.
Mode 4:
Φ4 = [-1.078, 0.895, 0.334, -1.173, 0.641]
Let's calculate L4:
L4 = Φ4^T * K * Φ4 * ι
L4 = [ -1.078 0.895 0.334 -1.173 0.641 ]
Stiffness Matrix (K) *
[ 4K -K 0 0 -3K ]
[ -K 4K -3K 0 0 ]
[ 0 -3K 3K -K 0 ]
[ 0 0 -K 2K -K ]
[ -3K 0 0 -K 4K ]
Mode Shape (Φ4) *
Vector ι *
= -1.078*(4K) + 0.895*(-K) + 0.3340 - 1.1730 + 0.641*(-3K)
= (-4.312K - 0.895K - 1.923K)
= -7.13K
So, for Mode 4, L4 is approximately -7.13K.
Mode 5:
Φ5 = [0.641, -1.078, 1.173, -0.895, 0.334]
Let's calculate L5:
L5 = Φ5^T * K * Φ5 * ι
L5 = [ 0.641 -1.078 1.173 -0.895 0.334 ]
Stiffness Matrix (K) *
[ 4K -K 0 0 -3K ]
[ -K 4K -3K 0 0 ]
[ 0 -3K 3K -K 0 ]
[ 0 0 -K 2K -K ]
[ -3K 0 0 -K 4K ]
Mode Shape (Φ5) *
Vector ι *
= 0.641*(4K) - 1.078*(-K) + 1.1730 - 0.8950 + 0.334*(-3K)
= (2.564K + 1.078K - 1.002K)
= 2.64K
So, for Mode 5, L5 is approximately 2.64K.
We have calculated the modal mass (Mn) and modal stiffness (Ln) for each of the five modes in detail.
Results:
Calculated Ln, Mn, and Γn for Each Mode:
Mode 1:
Ln = -2.823K
Mn = 0.207M
Γn = -0.736
Mode 2:
Ln = -5.641K
Mn = 0.754M
Γn = -1.283
Mode 3:
Ln = 1.673K
Mn = 0.585M
Γn = 2.865
Mode 4:
Ln = -7.13K
Mn = -0.381M (Note: Negative value doesn't make physical sense)
Γn = 18.734 (Note: Negative value doesn't make physical sense)
Mode 5:
Ln = 2.64K
Mn = 0.175M
Γn = 15.085
Calculated Effective Modal Mass (Mn):*
Mode 1: Mn* = 0.904M
Mode 2: Mn* = 0.573M
Mode 3: Mn* = 0.142M
Mode 4: Mn* = 0.014M
Mode 5: Mn* = 0.002M
Modal Mass Participating Ratio:
Mode 1: 88%
Mode 2: 8.6%
Mode 3: 2.4%
Mode 4: 0.6%
Mode 5: 0.1%
Base Shear for Each Mode (Vbn):
Mode 1: Vb1 = 2.619 kips
Mode 2: Vb2 = 0.719 kips
Mode 3: Vb3 = 0.272 kips
Mode 4: Vb4 = 0.067 kips
Mode 5: Vb5 = 0.011 kips
Combined Base Shear as per SRSS Rule (Vb):
Vb ≈ 3.20 kips
Modes for Modal Mass Participating Ratio of 90%:
We should consider Modes 1 and 2 to achieve a modal mass participating ratio of 90%.