Teoria - Dynamics of Mechanical Systems

Dynamics of Lumped Parameter Systems

mass, inertia, elasticity and damper divided

Equations of motion for a mechanical system

Let us consider a mechanical system having n degree of freedom (d.o.f.)

• The equations of motion are a set of 2^nd order differential equations in m unknowns represented by the system’s coordinate x.
• With a standard mathematical treatment it is possible to end up into a set of 2n 1^st order differential equations.

1.1. Newtonian Dynamics

F = m · ȧ approach based on “Newton’s second law”

1. All forces applied to the point
2. Acceleration of the point mass

F = m · ẍ

M₀ = J₀ · ω̇

• Newton’s second law for a single rigid body in a planar motion
• ẍ: acceleration of the centre of mass (C.O.M.) of the body
• M₀: moment of all forces acting on the body referred to a point O called the “centre”
• J₀: moment of inertia of the body w.r.t. the same point O
• ω̇: angular velocity of the body

Example: 1 d.o.f. system

∑F = m · ẍ → F - Kx - cẋ = mẍ

mẍẍ + cẋ + Kx = F

2^nd order differential equation (equation of motion of the system)

1.2. D'Alembert’s principle

F_in = -m · a Inertia force

M_in = - Jω̇ Moment of inertia forces

Example:

F_ed: kx mẍ

F_td: cẋ R

F_in: mẍ

F - kx - ci - mẍ = φ

mẍ + cẋ + Kx = F

we end up with the same equation of motion

1.3. The principle of virtual work

It represents the movement of the system in the form of equilibrium equations.

∑δW = ∑δW_e = φ for any virtual displacement

Under the assumption of non-dissipative constraints (constraint forces generate zero work, frictionless constraints and rolling without sliding condition)

• The equation brings a set of m independent equations of motion (m d.o.f.)
• (This equation not include any unknown constraint force)

Example:

Fδx - (kxcxẋmẍ) δx = φ

must hold for any value of virtual displacement

mẍ + cẋ + Kx = F

same equation again

Lagrange Equations

A different form of the principle of virtual work by using a special expression for the virtual work. (of some forces: conservative, dissipative and inertial, Lagrange equation valid as the d.o.f.)

1 D.O.F. EQUATION: d/dt (∂T/∂_ẋ) - ∂T/∂x + ∂D/∂_ẋ + ∂V/∂x = Q

• x: single independent coordinate
• T: kinetic energy
• D: dissipation function
• V: potential energy
• Q: Lagrangian component of all forces apart from conservative, dissipative and inertial ones. (elastic, viscous, inertial)

Q = δW/δx = δL/δx rate of virtual work and the infinitesimal variation of the independent variable x

EXAMPLE: T = 1/2 mv̇² = 1/2 m ẋ² V = 1/2 kx² = 1/2 k x² D = 1/2 c_ẋ² = 1/2 c ẋ² Q = F δx/δx = F

m ẍ - ϕ + cẋ + kx = F → SAME EQUATION AGAIN

1.5. Example with many constraints

In this case, the use of the principle of virtual work or of Lagrange Equations provides substantial benefits when the system addressed is composed by a relatively large number of bodies, interconnected by constraints. Lagrange Equations provide a general procedure to write the equations of motion, that can be easily applied to a wide variety of systems.

SLIDER-CRANK MECHANISM α → independent coordinate l >> r => x₁ = r cos(α) ; ẋ₁ = -r sin(α) α̇ ẍ₁ = -r sin(α) α̈ - r cos(α) α̇²

D'Alembert Principle:

This method brings us to a set of 3 unknowns in 3 equations (1 for motion, 2 for constraint forces)

Lagrange Equations: T = 1/2 J α̇² + 1/2 m ẋ₂² = 1/2 (J₀ + mr² sin²(α)) α̇² D = ϕ V = ϕ δW = M δα + F Sx = (M - Fr sin(α)) δα => Q = δW/δα = M - Fr sin(α) Applying the Lagrange equation w.r.t. α

(J₀ + mr² sin²(α)) α̈ + (mr² sin(α) cos(α) α̇²) α̈ = M - Fr sin(α) → NON LINEAR 2^nd ORDER DIFFERENTIAL EQUATION

1.1. Linearization of the inertial terms

Expanding in series of power the generalized mass, m(x):

m(x) ≜ m(0) + _dm/_dx |₀ x and substituting in m(x) ẍ

⇒ m(0) ẍ + _dm/_dx |₀ x ẍ ≜ m(0) ẍ

Neglecting residual terms of second order

Generalized mass evaluated around the constant equilibrium position

NB: m(b) = Σ_i=1⁵ (M_si m_si(o) + Ms_i ms_i(o) + Ss_i ms_e(o))

It may be obtained a priori by replacing the equilibrium position in the Jacobian functions

1.2. Linearization of the dissipative terms

Expanding in series of power the c(x):

c(x) ≜ c(0) + _2c/_2x |₀ x and substituting in c(x) ẋ

⇒ c(0) ẋ + _2c/_2x x ≜ c(0) ẋ

Same as inertia terms, replace the steady state position in the c(x) function.

1.3. Linearization of the terms related to conservative forces

V(x) ≅ V(0) + _∂V/_∂x |₀ x + _2∂V/_2∂x |₀ |x² V(0) + ₁/₂ K(0) x² ⇒ _∂V/_∂x = Q_c

V ≜ V_el + V_g

❍ V_el = ₁/₂ K_s ΔL_s(x)

with k_s stiffness of the s-th spring and ΔL_s is the variation of length (also undeformed state that may be different from equilib.)

Q_c,_el = - _∂Vel /_∂x

⇒ so we need to expand V_el in series power up to second order terms

ΔL_s(x) = ΔL_s(0) + _∂ΔL/_∂x |₀ x + ₁/_{2 2∂ΔL}/_2∂x |₀ x² - ΔL ≜ 0 + λks(0) x + ₁/₂ Wks(0) x²

Substituting it in V(x) and neglecting all powers greater than 2:

V_el(x) ≈ ₁/₂ k_s ΔL_s x + k_s λ_ks(0) x + ₁/₂ k_s λ_ks(x) x² + ₁/₂ k_s ΔL_se Wks(0) x² ≜ ₁/₂ Ke_R (0)

Ke_TOT = K(0)

Ke_R = φ

ΔL_se = φ undeformed spring in the equilibrium position

width linear dL second derivative becomes φ ± φ

+ the Hessian is zero for all the values of x

Lesson 2

Solution of the equations of motion for linear 1 D.O.F and multi-D.O.F systems

We consider the equation of motion for a linear (or linearised) 1 d.o.f. system and we know that it takes the form of a linear, 2^nd order differential equation with constant coefficients:

mẍ + cẋ + kx = F(t)

We assume m,c,k to be positive valued. The solution takes the form

x(t) = x_g(t) + x_p(t)

Particular solution -> with also the forcing effect (forced motion) added to the system

General solution (free motion) if it is an homogeneous equation

1. 1 D.o.F. System

1.1 Free motion of the 1 d.o.f. system

We consider the homogeneous differential equation:

mẍ + cẋ + kx = ϕ

The x_g(t) solution is a combination of 2 linearly independent solutions having the form:

x_g(t) = X₀ e^λt

ẍ_g(t) = λ² X₀ e^λt

λ always positive

ẋ_g(t) = λX₀ e^λt

X₀ = ϕ trivial solution

λ²m + cλ + k λ_1,2 = ϕ , λ_1,2 = -^c⁄_2m ± √(^{(c/2m)2 - k}⁄_m)

Case 1

c = ϕ undamped system (pure idealization impossible in nature)

λ_1,2 = ± iω with ω = √^k⁄_m

Complex conjugate

Constant and complex-valued

x_g(t) = C₁ e^-ωit + C₂ e^+ωit

C₁ = A + iB⁄2

C₂ = A - iB⁄2

Alternatively

x_g(t) = A cos(ωt) + B sen(ωt)

C cos(ωt + ϕ_p) with C = √A² + B²

A = C cos(ϕ)

-ic i = 1

C = A cos(ωt) + B sen(ωt)

B = -Csen(ϕ)

-ic i = 1

ϕ_p = -tan^-1(^B⁄_A)

The motion is harmonic with circular frequency ω not depending upon the initial conditions applied and it is called natural frequency of the undamped 1 d.o.f. system.