Chapter 2: Vibration of Mechanical Systems

This chapter focuses on calculating the impulse response and frequency response function (FRF) of vibrating systems, including single-degree-of-freedom (SDOF), multi-degree-of-freedom (MDOF), and continuous systems.

1. Single Degree-of-Freedom (1-DOF) System

A 1-DOF system consists of a mass \(m\), damper \(c\), and spring \(k\) subjected to an external force \(f(t)\). The position \(x\) of the mass is zero at equilibrium.

k c m x(t) f(t)

Figure 1: Schema of a single degree-of-freedom (1-DOF) oscillator.

The governing equation is:

\[ m \ddot{x}(t) + c \dot{x}(t) + k x(t) = f(t) \]

Linear Differential Equation of Order Two

The governing equation is a linear differential equation of second order with constant coefficients. The general theory of differential equations states that the set of solutions of the homogeneous differential equation

\[ \ddot{x}(t) + 2 \xi \omega_0 \dot{x}(t) + \omega_0^2 x(t) = 0 \quad \mathrm{for} \: t\in\mathbb{R} \]

is a linear space of dimension 2. The problem then reduces to find two linearly independent solutions. We search for solutions of the type \(e^{\lambda t}\). Substituting into the differential equation shows that \(\lambda\) is a complex root of the characteristic equation

\[ \lambda^2 + 2 \xi \omega_0 \lambda + \omega_0^2 = 0 \]

When \(\xi > 1\), the characteristic polynomial admits exactly two different real roots \((-\xi \pm \sqrt{\xi^2-1}) \omega_0 \), the set of real solutions is

\[ x_0(t) = A_1 e^{(-\xi + \sqrt{\xi^2-1}) \omega_0 t} + A_2 e^{(-\xi - \sqrt{\xi^2-1}) \omega_0 t}, \quad A_1, A_2\in\mathbb{R} \]

When \(\xi = 1\), the characteristic polynomial admits exactly one real root \(-\omega_0\), the set of solutions is

\[ x_0(t) = A_1 e^{- \omega_0 t} + A_2 t e^{-\omega_0 t}, \quad A_1, A_2\in\mathbb{R} \]

When \(\xi < 1\), the characteristic equation admits exactly two different complex solutions \((-\xi \pm i \sqrt{1-\xi^2}) \omega_0 \), the set of real solutions is

\[ x_0(t) = A_1 e^{- \xi \omega_0 t} \sin( \sqrt{1-\xi^2} \omega_0 t) + A_2 e^{- \xi \omega_0 t} \cos(\sqrt{1-\xi^2}\omega_0 t), \quad A_1, A_2\in\mathbb{R} \]

The set of solutions of the inhomogeneous differential equation

\[ \ddot{x}(t) + 2 \xi \omega_0 \dot{x}(t) + \omega_0^2 x(t) = f(t)/m \quad \mathrm{for} \: t\in\mathbb{R} \]

is an affine space of dimension 2 whose linear space is the set of solutions of the homogeneous differential equation. This means that any solution is of the type

\[ x(t) = x_0(t) + x_1(t) \]

where \(x_0(t)\) is any solution of the homogeneous equation given above and \(x_1(t)\) is a particular solution of the inhomogeneous equation. So if we know a particular solution with \(f(t)\) on the right-hand side, we know all solutions. The problem therefore reduces to first, find a particular solution and second, determine the constants \(A_1\) and \(A_2\) that appear on the general solution.

Cauchy's Problem

The initial value problem is defined by the inhomogeneous differential equation with initial conditions:

\[ \left\{\begin{array}{ll} \ddot{x} + 2 \xi \omega_0 \dot{x} + \omega_0^2 x = \frac{f(t)}{m} & \mathrm{for} \: t\geq 0\\ x(0) = x_0, \quad \dot{x}(0) = v_0 \end{array}\right. \]

The Cauchy-Lipschitz theorem implies that it exists a unique solution when \(f(t)\) is continuously differentiable.

Impulse Response

The impulse response \(h(t)\) is the response to a Dirac delta force \(f(t) = \delta(t)\). It satisfies:

\[ \left\{\begin{array}{ll} \ddot{h}(t) + 2\xi\omega_0 \dot{h}(t) + \omega_0^2 h(t) = \delta(t)/m \\ h(t) = 0 \text{ for } t < 0 \end{array}\right. \]

The impulse response depends on the damping ratio \(\xi\):

  • Underdamped (\(\xi < 1\)): \[ h(t) = \frac{Y(t)}{m \omega_0 \sqrt{1 - \xi^2}} e^{-\xi \omega_0 t} \sin(\omega_0 \sqrt{1 - \xi^2} t) \]
  • Critically damped (\(\xi = 1\)): \[ h(t) = \frac{t Y(t)}{m} e^{-\omega_0 t} \]
  • Overdamped (\(\xi > 1\)): \[ h(t) = \frac{Y(t)}{m \omega_0 \sqrt{\xi^2 - 1}} e^{-\xi \omega_0 t} \sinh(\omega_0 \sqrt{\xi^2 - 1} t) \]

where \(Y(t)=1\) if \(t\geq 0\) and \(0\) otherwise (Heaviside step function).

Frequency Response Function (FRF)

The frequency response function is the complex function \(H(\omega)\) so that \(H(\omega)e^{i\omega t}\) is the unique solution of this type to the inhomogeneous differential equation with a harmonic force \(f(t) = e^{i \omega t}\). It is given by:

\[ H(\omega) = \frac{1}{m \omega_0^2 \left[ 1 + 2i \xi \frac{\omega}{\omega_0} - \left( \frac{\omega}{\omega_0} \right)^2 \right]} \]

The frequency response function and the impulse response form a Fourier pair:

\[ H(\omega) = \int_{-\infty}^{\infty} h(t) e^{-i \omega t} \, dt, \quad h(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} H(\omega) e^{i \omega t} \, d\omega \]

Plot of Impulse Response & Frequency Response Function

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Solution to Cauchy's Problem

The solution to the initial value problem is:

\[ x(t) = \int_0^t h(t - \tau) f(\tau) \, d\tau + (m v_0 + c x_0) h(t) + m x_0 \dot{h}(t) \]

For a causal system (\(f(t) = 0\) for \(t < 0\)), the steady-state response is:

\[ x(t) = \int_{-\infty}^{\infty} h(t - \tau) f(\tau) \, d\tau \]

2. N Degrees-of-Freedom (N-DOF) System

A multiple degree-of-freedom (N-DOF) oscillator is described by the vector of displacements \(X(t) = \begin{pmatrix} x_1(t) \\ \vdots \\ x_n(t) \end{pmatrix}\) and the vector of forces \(F(t) = \begin{pmatrix} f_1(t) \\ \vdots \\ f_n(t) \end{pmatrix}\). When \(F=0\), we set \(X=0\).

x2 x1 x4 x3 f3 f1 f2 f4

Figure 2: Schema of a multiple degree-of-freedom (N-DOF) oscillator.

The governing equation is:

\[ M \ddot{X}(t) + C \dot{X}(t) + K X(t) = F(t) \]

where \(M\), \(C\), and \(K\) are the mass, damping, and stiffness matrices, respectively.

Mass and Stiffness Matrices

The mass end stiffness matrices always verify the following properties:

  • Symmetry: \(M^T = M\), \(K^T = K\).
  • Positive definiteness: \(X^T M X > 0\) for all \(X \neq 0\).
  • Positive semi-definiteness: \(X^T K X \geq 0\) for all \(X\).

Note: \(M\) is always invertible, but \(K\) may not be.

The mass and stiffness matrices define two bilinear forms which are:

  • The mass inner product \[ \langle X, Y \rangle_M = X^T M Y \] for all \(X\) and \(Y\) \(n\)-dimensional column vectors.
  • The stiffness inner product \[ \langle X, Y \rangle_K = X^T K Y \] for all \(X\) and \(Y\) \(n\)-dimensional column vectors.

By symmetry of \(M\) and \(K\), these bilinear forms are symmetrical \(\langle X, Y \rangle_A = \langle Y, X \rangle_A\) with \(A=M,K\). The mass inner product is positive-definite but the stiffness inner product is only positive semi-definite.

Homogeneous, inhomogeneous differential equation

The homogeneous differential equation is the governing equation with a vanishing right-hand side (\(F(t) = 0\)).

\[ M \ddot{X}(t) + C \dot{X}(t) + K X(t) = 0 \quad \text{for} \: t\in\mathbb{R} \]

The set of solutions of the homogeneous differential equation form a 2n-dimensional linear space.

The inhomogeneous differential equation is the governing equation with a an external force (\(F(t) \neq 0\)).

\[ M \ddot{X}(t) + C \dot{X}(t) + K X(t) = F(t) \quad \text{for} \: t\in\mathbb{R} \]

The set of solutions of the inhomogeneous differential equation forms an affine space whose direction is the 2n-dimensional linear space of solutions of the homogeneous differential equation.

Cauchy's Problem

The initial value problem is defined by the inhomogeneous differential equation with initial conditions:

\[ \left\{\begin{array}{ll} M \ddot{X}(t) + C \dot{X}(t) + K X(t) = F(t) & \text{for} \: t\geq 0\\ X(0) = X_0, \quad \dot{X}(0) = V_0 \end{array}\right. \]

The Cauchy-Lipschitz theorem implies that it exists a unique solution when \(F(t)\) is continuously differentiable.

Reduction to order one

We set the \(2n \times 1\) state vectors \(Y(t) = \begin{pmatrix} X(t) \\ \dot X(t) \end{pmatrix}\), \(Y_0 = \begin{pmatrix} X_0 \\ V_0 \end{pmatrix}\), the \(2n \times 2n\) matrix \(A = \begin{pmatrix} 0 & I \\ -M^{-1}K & -M^{-1}C \end{pmatrix}\), and the \(2n\times 1\) column vector \(B(t) = \begin{pmatrix} 0 \\ -M^{-1}F(t) \end{pmatrix}\). This makes sense since \(M\) is invertible. Cauchy's problem then reduces to

\[ \left\{\begin{array}{ll} \dot {Y}(t) = A Y(t) + B(t) & \text{for} \: t\geq 0\\ Y(0) = Y_0 \end{array}\right. \]

The Cauchy-Lipschitz theorem implies that it exists a unique solution which is:.

\[ Y(t) = \int_0^t e^{(t-\tau)A} B(\tau) \, d\tau + e^{tA} Y_0 \]

This solution requires the computation of the exponential of a matrix that requires diagonalization of the matrix \(A\).

Projection on the Modal Basis

We form the square matrix \(\Psi = \left( \psi_1 \cdots \psi_n \right) \) by assembling the eigenvectors in column. Since the family of eigenvectors constitute a basis, the matrix \(\Psi\) is invertible. Let \(Q=\Psi^{-1}X\) the column vector of modal coordinates. Then

\[ \Psi^{T} M \Psi\ddot Q + \Psi^{T} C \Psi \dot Q + \Psi^{T} K \Psi Q = \Psi^{T} F \]

We set

  • \( M' = \Psi^{T} M \Psi \) diagonal matrix since \(\langle \psi_k, \psi_l \rangle_M=0\) when \(k\neq l\)
  • \( K' = \Psi^{T} K \Psi \) diagonal matrix since \(\langle \psi_k, \psi_l \rangle_K=0\) when \(k\neq l\)
  • \( C' = \Psi^{T} C \Psi \) not diagonal matrix in general!

Hence

\[ M' \ddot Q + C' \dot Q + K' Q = \Psi^{T} F \]

This forms a set of coupled linear differential equations. When \(C'\) is diagonal, the set is uncoupled.

Rayleigh Damping

If the damping matrix \(C\) is a linear combination of \(M\) and \(K\):

\[ C = \alpha M + \beta K \]

then \(C\) is called a Rayleigh damping matrix. In the modal basis, \(C' = {}^t \Psi C \Psi\) is diagonal. We write it in the form:

\[ C' = \begin{pmatrix} 2 m_1 \xi_1 \omega_1 & & \\ & \ddots & \\ & & 2 m_n \xi_n \omega_n \end{pmatrix} \]

where \(\xi_k\) is the modal damping ratio for mode \(k\).

Impulse Response Matrix

The matrix of impulse responses is a square matrix

\[ h(t) = \left( \begin{array}{ccc} h_{11}(t) & \cdots & h_{1n}(t) \\ \vdots & & \vdots \\ h_{n1}(t) & \cdots & h_{nn}(t) \end{array} \right) \]

whose entry \(h_{ij}(t)\) is the reponse of oscillator \(i\) to an impulse at time \(0\) to oscillator \(j\).

The impulse response matrix \(h(t)\) is the solution to:

\[ M \ddot{h}(t) + C \dot{h}(t) + K h(t) = I \delta(t), \quad h(t) = 0 \text{ for } t < 0 \]

where \(I\) is the identity matrix.

Properties:

  • \(h(t) = \dot{h}(t) = 0\) for \(t < 0\) (causality).
  • \(h(t)\) is continuous at \(t = 0\).
  • \(\dot{h}(t)\) is discontinuous at \(t = 0\) with \(\dot{h}(0^+) = M^{-1}\).

The impulse response can be expressed in terms of the modal basis:

\[ h_{ij}(t) = \sum_{k=1}^n h_k(t) \Psi_{ik} \Psi_{jk} \]

where \(h_k(t)\) is the impulse response of the 1-DOF system with modal parameters \(m_k\), \(\xi_k\), and \(\omega_k\).

Solution to Cauchy's Problem

The unique solution to Cauchy's problem is

\[ X(t) = \int_0^t h(t-\tau) F(\tau) \, d\tau + Mh(t) V_0 + C h(t) X_0 + M \dot h(t) X_0 \]

The first term in the right-hand side (integral) is the steady state response and the other terms are the transient response. Caution: the order of terms in products is important since the matrix multiplication is not commutative. In the limit \(t \to \infty\)

\[ X(t) = \int_{-\infty}^\infty h(t-\tau) F(\tau) \, d\tau \]

Frequency Response Matrix

Consider the inhomogeneous differential equation with a harmonic force applied to oscillator \(j\)

\[ M \ddot X(t) + C \dot X(t) + K X(t) = \left( \begin{array}{c} 0 \\ \vdots \\ 0 \\ e^{i\omega t} \\ 0 \\ \vdots \\ 0 \end{array} \right) \]

where the right-hand side is a colum vector whose only non-nul entry is \(e^{i\omega t}\) at row \(j\). Then it exists a unique solution of the form \[ X(t) = \left( \begin{array}{c} H_{1j}(\omega) \\ \vdots \\ H_{nj}(\omega) \end{array} \right) e^{i\omega t} \] The entries \(H_{ij}(\omega)\) are then assembled to form the frequency response matrix.

The frequency response matrix \(H(\omega)\) is the solution to:

\[ [-\omega^2 M + i \omega C + K] H(\omega) = I \]

or equivalently:

\[ H(\omega) = [-\omega^2 M + i \omega C + K]^{-1} \]

If \(C\) is diagonal in the modal basis, then:

\[ H_{ij}(\omega) = \sum_{k=1}^n H_k(\omega) \Psi_{ik} \Psi_{jk} \]

where \(H_k(\omega)\) is the FRF of the 1-DOF system with modal parameters \(m_k\), \(\xi_k\), and \(\omega_k\).

The impulse response matrix and the FRF matrix form a Fourier pair:

\[ H(\omega) = \int_{-\infty}^{\infty} h(t) e^{-i \omega t} \, dt, \quad h(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} H(\omega) e^{i \omega t} \, d\omega \]

3. Continuous Systems

Continuous systems (e.g., strings, beams, membranes, plates, shells, elastodynamic solids or gases) are described by partial differential equations (PDEs).

u(x,t) f(x,t)

Figure 2: Schema of beam in transverse vibration.

The governing equation for a continuous system is:

\[ m(r) \ddot{u}(r, t) + c(r) \dot{u}(r, t) + \mathcal{K} u(r, t) = f(r, t) \]

where \(r\) is the spatial coordinate vector, \(t\) the time, \(u(r, t)\) the displacement field, and \(\mathcal{K}\) is a spatial differential operator.

Examples of continuous systems:

System Governing Equation Operator \(\mathcal{K}\)
String \(m \frac{\partial^2 u}{\partial t^2} - T \frac{\partial^2 u}{\partial x^2} = f(x, t)\) \(\mathcal{K} = -T \frac{d^2}{dx^2}\)
Beam \(m \frac{\partial^2 u}{\partial t^2} + EI \frac{\partial^4 u}{\partial x^4} = f(x, t)\) \(\mathcal{K} = EI \frac{d^4}{dx^4}\)
Membrane \(m \frac{\partial^2 u}{\partial t^2} - T \Delta u = f(x, y, t)\) \(\mathcal{K} = -T \Delta\)
Plate \(m \frac{\partial^2 u}{\partial t^2} + D \Delta^2 u = f(x, y, t)\) \(\mathcal{K} = D \Delta^2\)
Acoustics \(\frac{1}{c^2} \frac{\partial^2 p}{\partial t^2} - \Delta p = f(x, y, z, t)\) \(\mathcal{K} = -\Delta\)

Notation:

  • \(m\): mass per unit length/area.
  • \(T\): tension.
  • \(E\): Young's modulus.
  • \(I\): moment of inertia.
  • \(D\): bending stiffness.
  • \(\Delta\): Laplacian operator.
  • \(c\): speed of sound.

Mass and Stiffness Operators

The mass operator is defined as:

\[ \mathcal{M}: u(r) \mapsto m(r) u(r) \]

\(\mathcal{M}\) is invertible (since \(m(r) \neq 0\) for all \(r\)). The stiffness operator is:

\[ \mathcal{K}: u(r) \mapsto \mathcal{K} u(r) \]

\(\mathcal{H}\) may not be invertible.

The damping operator is:

\[ \mathcal{C}: u(r) \mapsto c(r) u(r) = \alpha m(r) u(r) \]

For proportional damping \(c(r) = \alpha m(r)\), where \(\alpha\) does not depend on the position \(r\).

Initial Value Problem, Boundary Value Problem

\[ \left\{\begin{array}{ll} m(r) \ddot{u}(r, t) + c(r) \dot{u}(r, t) + \mathcal{K} u(r, t) = f(r, t) & \text{for} \: r\in\Omega, \: t>0\\ u(r,0) = u_0(r), \: \dot u(r,0)=v_0(r) & \text{for} \: r\in\Omega\\ \text{boundary condition} & \text{for} \: r\in\partial\Omega, \: t>0 \end{array}\right. \]

Impulse Response

The impulse response \(h(r, s, t)\) is the displacement at position \(r\) due to an impulse applied at position \(s\) and time \(0\):

\[ m(r) \ddot{h}(r, s, t) + c(r) \dot{h}(r, s, t) + \mathcal{K} h(r, s, t) = \delta(t) \delta(r - s) \]

with \(h(r, s, t) = 0\) for \(t < 0\) and homogeneous boundary conditions.

The impulse response can be expressed in terms of the modal basis:

\[ h(r, s, t) = \sum_{k \geq 1} h_k(t) \psi_k(r) \psi_k(s) \]

where \(h_k(t)\) is the impulse response of the 1-DOF system with modal parameters \(m_k\), \(\xi_k\), and \(\omega_k\).

The solution to Cauchy's problem is:

\[ u(r, t) = \int_{\Omega} \int_0^t h(r, s, t - \tau) f(s, \tau) \, d\tau \, d\Omega_s + \int_\Omega \left[ m v_0(s) + c u_0(s) \right] h(r,s,t) + m u_0(s) \dot h(r,s,t) \, d\Omega_s \]

In the limit \(t\to\infty\), the steazdy state response is:

\[ u(r, t) = \int_{\Omega} \int_{-\infty}^\infty h(r, s, t - \tau) f(s, \tau) \, d\tau d\Omega_s \]

Frequency Response Function

Consider a harmonic load of frequency \(\omega\) applied at position \(s\):

\[ m(r) \ddot{u}(r, s, t) + c(r) \dot{u}(r, s, t) + \mathcal{K} u(r, s, t) = e^{i\omega t} \delta(r - s) \]

and homogeneous boundary conditions. The frequency response function is the function \(H(r, s, \omega)\) such that \(H(r, s, \omega) e^{i\omega t}\) is the unique complex solution of this type to the above problem.

When \(u\) is a displacement and the load is a force, the frequency response function is called receptance, between a velocity and a force, the frequency response function is a mobility, between an acceleration and a force, we have an accelerance, and between two forces, a transmissiblity.

In case of proportional damping, the frequency response function can be expressed in terms of the modal basis:

\[ H(r, s, \omega) = \sum_{k \geq 1} H_k(\omega) \psi_k(r) \psi_k(s) \]

where \(H_k(\omega)\) is the frequency response function of the 1-DOF system with modal parameters \(m_k\), \(\xi_k\), and \(\omega_k\).

The impulse response function and the frequency response function form a Fourier pair:

\[ H(r, s, \omega) = \int_{-\infty}^{\infty} h(r, s, t) e^{-i \omega t} \, dt, \quad h(r, s, t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} H(r, s, \omega) e^{i \omega t} \, d\omega \]