Basic Theory of TMD(1)

Brief Introduction

Tuned mass dampers (TMD) are designed to suppress the vibration of structures by means of resonance. They can be classified as two types: passive TMD and active TMD, the former ones control force arises mechanically from the interaction of the device with the subject structure without feedback of response whereas the latter one is often electro-mechanical and have operation and stability issues.

Basic Theory

Here I just want to remind you that the theory of TMD is based on a two degree of freedom system that treats the structure and the damper as single degrees of freedom respectively.

The basic theory is always easy to understand, which is boring, but what’s really interesting is how do you find tricks from simple theory.😊😊😊

Consider a SDOF structure install with a TMD(don`t worry, the conclusion is also applicable to MDOF).

• x(t)=disp. of structure, which is the target we want to control

• u(t)=disp. of TMD relative to structure

Still use Newton`s second law on the structure`s mass and TMD mass gives respectively:
$$
m\ddot x+kx=c_d\dot u+k_du+F
\\ m_d\ddot u+c_d\dot u+k_du = -m_d\ddot x
$$
Have you noticed something weird about the above equations? We did not take into account the damping of the structure, I mean, we seem to have lost a term \(c\dot x\) in the first equation. Is it acceptable? Sure it does, since the damping of the structure itself is much smaller than TMD, such neglect can greatly simplify the calculation.

Rearranging and writing in terms of freq. & damping ratios:
$$
\ddot x+w_1^2x-\mu (2\zeta_dw_d\dot u+w_d^2u)=F/m…(1)
\\ \ddot u+2\zeta_dw_d\dot u+w_d^2u=-\ddot x…(2)
$$
Subs. (2) into (1) gives:
$$
(1+\mu)\ddot x+w_1^2x+\mu\ddot u=(F_0/m)e^{iwt}…(3)
$$
Where,

Structure freq.\(w_1=\sqrt{k/m}\)

TMD freq. \(w_d = \sqrt{k_d/m_d}\)

TMD mass ratio \(\mu = m_d/m\) often1~5%

TMD damping ratio \(\zeta_d = c_d/2w_d\)

For insight and design purpose one would like to obtain an analytical expression for the steady state amplitude. This is not easy when working in the time-domain, e.g., eigenvalue problem & modal superposition leads to lengthy expressions. It turns out to be much easier when approaching using complex numbers.

Assume the complex-valued harmonic force:
$$
F(t) = F_0e^{iwt}
$$
Assume the following steady-state solutions:
$$
x(t) = Xe^{iwt};\ \ddot x=-w^2Xe^{iwt}
\\ u(t) = Ue^{iwt};\ \dot u(t)=iwUe^{iwt};\ \ddot u=-w^2Ue^{iwt}
$$
Subs. into (2):
$$
(w_d^2-w^2+2\zeta_dw_dwi)Ue^{iwt}=w^2Xe^{iwt}
\\ \Rightarrow U = \frac{w^2X}{w^2_d-w^2+2\zeta_dw_dwi}=\frac{\beta^2X}{\alpha^2-\beta^2+2\zeta_d\alpha\beta i}
\\ \alpha=w_d/w_1,\beta=w/w_1
$$
Now we have gotten the expression of U in terms of X from above. Subs. into (3) to solve for X and simplify:
$$
X = \frac{F_0}{mw_1^2}×\frac{P}{Q}
\\ P = \alpha^2-\beta^2+2\zeta_d\alpha\beta i
\\ Q = [1-(1+\mu)\beta^2](\alpha^2-\beta^2+2\zeta_d\alpha\beta i)-\mu\beta^4
$$
Where,

TMD freq. ratio \(\alpha=w_d/w_1\)

Excitation freq. ratio \(\beta = w/w_1\)

The steady-state amplitude of x(t) is equal to |X|:
$$
|X| = \underbrace{\frac{F_0}{mw_1^2}} _ {static} × \underbrace{\frac{|P|}{|Q|}} _ {Dyn.Amp\ D(w)}
$$
Hence, from the equation above we can say that the steady-state amplitude of x(t) is controlled by \(\alpha,\beta,\zeta_d,\mu\). In other words, now we have a clue about how to control structural response with TMD.

Except for \(\beta\), the parameters \(\alpha,\zeta_d,\mu\) can be controlled by design.

So, how to choose these parameters?