Mdof systems

MDOF system dynamics

n degrees of freedom → n independent variables q_i to describe system conf. {q_i} generalized coordinate vector.

M ODE, scalar → matrix form.

Vibrational modes

• Natural freq.
• Modal damping factor
• Mode shape

Description of the way the system vibrates at each natural frequency: it represents relative displacement of all parts of mech. system for a particular mode.

Mode shapes

"Normal" modes, Ψ ∈ ℝ^m.

"Complex" modes, Ψ ∈ ℂ^m.

Damping distribution determines whether modes are normal or complex.

• Undamped system, very lightly damped or proportionally damped exhibit normal modes.
• System with much localized damping exhibit complex modes (ex. cars with shock absorbers).

Linear and natural mech. system

Kinetic energy can be expressed as a quadratic form of velocities.

MDOF system dynamics

Degrees of freedom — independent variables q_i to describe system conf. {q_i} generalized coordinate vector.

ODE, scalar → matrix form.

Vibrational modes

• Natural freq.
• Modal damping factor
• Mode shape → eigenmode u⁽ⁱ⁾.

Description of the way the system vibrates at each natural frequency: it represents relative displacement of all parts of mech. system for a particular mode.

Mode shapes

Normal modes, Ψ⁽ⁱ⁾ ∈ Rⁿ.

Complex modes, Ψ⁽ⁱ⁾ ∈ Cⁿ.

Damping distribution determines whether modes are normal or complex.

• Undamped system, very lightly damped or proportionally damped exhibit normal modes.
• Syst. with much localized damping exhibit complex modes (i.e., cars with shock absorbers).

Linear and natural mech. system

Kinetic energy can be expressed as a quadratic form of velocities.

1) n = 2

[ q₁ ] [ x₁ ] [ y₁ ] [ q₂ ] [ x₂ ] [ y₂ ]

Different sets of coordinates are available, we choose the first { q_i }.

( [ q₁ ] ) ( [ y₁ - y₂ ] ) [ R₁ ]

( [ q₂ ] ) ( [ y₂ - y₁ ] )

Relative displ. of M_i w.r.t. M₂.

• Generalized coord. vector { q }.
• TBD... (check theory book).
• EOM. Dynamic balance of each mass (Newt. 2^nd law).

Rearrange equations -> grouping: ( L/2m x₁. + x₂· x₁· x₂ - x₂ + x₁ + x₂ )

Conversion into a matrix form

Particular cases:

• Unforced: {F} = {0} HOF UNAMPED FREE RESPONSE.
• Undamped: [C] = [0] [M] { q_i''} + [K] { q_j} = 0.

Properties of [M] and [K] matrices

Constant and real coff.

Symmetric?

We can always get sym. matrices for a natural mech. system with Lagrange's approach, always sym.

Neutral "S" we can get non-sym. mech.

Positive definite and positive semi-definite (P.s.d.).

A square matrix [A] is P.D. if ∀ {z} ≠ {0} ∈ Rⁿ and {z}_i ≠ {0}, {z}^T[A]{z} > 0 strictly positive scalar quantity.

A square matrix [B] is PS-D if ∀ {z} ≠ {0} ∈ R, {z}^T[B]{z} ≥ 0.

[M] is P.D. and [K] is P.S.D.

Example: Consider a system of N masses connected by springs

Kinetic energy: T = (1/2) ∑ m_iv_i².

Absolute velocity of m_i.

T = 1/2 {V}^T[m]{V} > 0 ∀ {V} ≠ {0}.

Matrix expression quadratic form of velocities.

{V} = [v₁ ... v_i ... v_N]

[m] = diag (m₁, m₂, ..., m_N)

Potential elastic energy: V = (1/2) ∑_{i=1 to N-1} k_i(x_i,j)² (x is Spring change).

V = 1/2 {x}^T[K]{x} ≥ 0 if the system allows rigid body motion.

∀ {x} ≠ {0}

{x} = [x₁ ... x_i ... x_N]

2-DOF example

{x} = { x₁ x₂ }

{v} = { v₁ v₂ }

u = x₁ - x₂ → V ≠ 0     if    {x₁ x₂} ≠ {x x}

i.e. when x₁ = x₂ = x → u = x₁ - x₂ = 0.

Translation of the system w/o spring deflections.

T > 0 ; V ≠ 0.

v = { v₁ v₂ }

[_m1     0]         [x₁]

[0     _m2]    [x₂]     [K     -K]    [x₁]    {0}

[x₂]    -K     K    [x₂] = {0}

V = ¹/₂ K (x₁ - x₂)² = ¹/₂ ¹/₂ [x₁ x₂]^T K [x₁ x₂] = ¹/₂    [x₁ x₂]    [Kx₁ - Kx₂] [Kx₂ - Kx₁]    = K [(x₁ - x₂)² - x₁² - x₂² + 2 x₁x₂] = ¹/₂ [ (x₁ - x₂)² ]

Sylvester criterion (...)

Example (theory back).

Eigen value problem (EVB) (...)